通訊原理第六章：解調性能分析

教育部資通訊科技人才培育先導型計畫

通訊原理

第六章解調性能分析

1第六章解調性能分析


大綱線性調變接收性能 DSBAMSSB

訊雜比基帶系統線性雙邊帶調變 (DSB) 的解調性能單邊帶調制系統 (SSB) 的解調性能振幅調變 (AM) 的解調性能

角度調制中的雜訊在雜訊中解調輸出雜訊頻譜輸出雜訊比利用去強提高性能



信號與雜訊假設信號功率是有限的加入的雜訊其頻寬 B gtW 且其雙邊功率頻譜密度為

TP

(WHz) 2

0N

信號

雜訊

0-W W-B B

0N2

1

f



訊雜比 - 基帶系統為了有一個比較系統性能的基礎我們將所得之信號通過一個低頻濾波器其頻寬為 W 以過濾多餘的雜訊此一系統稱為基頻帶系統

低通濾波器頻寬=W

訊息信號=m(t)

訊息頻寬=W)t(yD

雜訊頻寬=B



訊雜比 - 基帶系統

在頻寬 B內的雜訊總功率為

如此可得濾波器輸入訊雜比 (SNR) 為

0 0-

1 =

2

B

BN df N B

i0

(SNR) = TP

N B

信號

雜訊

0-W W-B B

0N2

1

f



訊雜比 - 基帶系統假設訊息信號 m(t) 是有限頻寬則一個簡單的低通濾波器可以用來加強訊雜比如圖所示

則通過濾波器的雜訊功率為

0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f




故濾波器輸出的訊雜比為

濾波器將訊雜比加大 O 0

i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f




由於加了濾波器所以頻帶以外的雜訊全被濾波器濾除因此濾波器輸出的訊雜比可作為系統性能的合理標準此訊雜比的參考標準在以下的討論中將廣被使用

WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD



雙邊帶調制系統 (DSB-SC)的解調性能我們現在來計算同調雙邊帶 (DSB) 解調器的雜訊性能如下圖圖中前面加一預檢濾波器的同調解調器這就如前面章節所討論的中頻 (IF) 濾波器濾波器的輸入是已調制信號與雙邊功率頻譜密度的白色高斯雜訊

( )cx t(WHz)

20N

預檢( I F)濾波器

後檢低通濾波器

)tcos(2 c

)t(n)t(x)t(x cr )t(e2 )t(e3)t(yD



雙邊帶調制系統 (DSB-SC)的解調性能

其中功率雜訊 WNWN

00 222

2

0N2

1

)f(S0N

Wfc Wfc Wfc Wfc f ff



雙邊帶調制系統 (DSB-SC)的解調性能由於發射信號假定為雙邊帶調制信號接收信號可寫成

其中 m(t) 為訊息而用來標示載波相位 ( 或時間原點 ) 的不確定性

若預檢濾波器的頻寬為 2W且雙邊帶調制信號全部通過則在預檢濾波器的輸出可展開成同相和正交成分

其中雜訊之功率

rx (t)= ( )cos( ) ( )c cA m t t n t

2 ( ) ( ) cos( ) ( ) cos( ) - ( )sin( )c c c c s ce t A m t t n t t n t t

WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn




其中信號功率為雜訊功率為

在乘法器輸入端的預檢訊雜比為2 2

T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c




後驗訊雜比須先計算

由於倍頻項被後驗濾波器移除故解調輸出為

後驗信號功率為

後驗雜訊功率為

雙邊帶調制系統 (DSB-SC)的解調性能3( )e t

2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S

e t A m t A m t t n t

n t t n t t

WW

0N

)f(S0N

f



雙邊帶調制系統 (DSB-SC)的解調性能在解調器輸入處的加成性雜訊導致在解調器輸出處的加成性雜訊這是因為線性性質

故後驗訊雜比為

因上式可改寫成

這等同於理想的基頻帶系統

0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT



雙邊帶調制系統 (DSB-SC)的解調性能與的比值稱為檢波增益(detection gain) 常用來作為解調器優劣的指標而對同調雙邊帶解調器而言檢波增益為

D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1

考慮一信號經由DSB-SC 調變器調變後在一帶有雜訊的通道上被傳送雜訊的功率頻譜如圖所示其中

信號頻寬載波頻率

請計算預檢訊雜比後驗訊雜比及檢波增益

第六章解調性能分析 16

cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1

因此預檢訊雜比為

解調器輸出訊雜比為

檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W


單邊帶調制系統 (SSB)的解調性能對於單邊帶調制系統預檢濾波器的輸入可寫成

其中代表的希爾伯 (Hilbert) 轉換而正號代表下邊帶的單邊帶調制負號代表上邊帶的單邊帶調制

對單邊帶調制而言預檢帶通濾波器的最小頻寬為 W

ˆ( ) [ ( ) cos( ) ( )sin( )] ( )r c c cx t A m t t m t t n t

ˆ ( )m t ( )m t



)tcos(2 c




將雜訊對中心頻率展開則預檢濾波器的輸出可寫成

即

所以

單邊帶調制系統 (SSB)的解調性能0( ) ( )cx t n t

2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t

2 ˆ( ) [ ( ) ( )]cos( ) [ ( ) ( )]sin( )c c c c s ce t A m t n t t A m t n t t



)tcos(2 c




單邊帶調制系統 (SSB)的解調性能

由圖可知雜訊功率為 2 2 20 0T c sN n n n N W


0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+



單邊帶調制系統 (SSB)的解調性能因為解調是將乘上解調載波再經低通濾波來達成所以我們可用雙邊帶解調器來解決單邊帶解調的問題結果可得

2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c




單邊帶調制系統 (SSB)的解調性能因此我們對下邊帶單邊調制作解調則分別可算出後驗濾波器的雜訊功率為

後驗濾波器的信號功率為


2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W



單邊帶調制系統 (SSB)的解調性能預檢濾波器信號功率為

經由前面章節得到一函數與希爾伯轉換為正交若則

所以預檢濾波器信號功率可表示成

且函數及希爾伯轉換有相同功率所以

( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m

2T ˆ [ ( ) cos( ) - ( )sin( )]c c cS A m t t m t t



單邊帶調制系統 (SSB)的解調性能同理預檢和後驗雜訊有相同的功率

預檢訊雜比為

所以檢波增益為

2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m

因此雙邊帶調制和單邊帶調制的同調解調和基帶系統有相同的性能不過在此假設有完美的同調解調當有相位誤差情況下則單邊帶解調的效應會比雙邊帶解調嚴重許多

2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2

考慮一信號經由SSB-SC 調變器調變後在一帶有雜訊的通道上被傳送雜訊的功率頻譜如圖所示其中




cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為

如同 DSB-SC 在預檢情況下其信號功率和雜訊功率與輸出時相同因此為



2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W


振幅調制系統 (AM) 同調檢波振幅調制信號

其中 a為調制指數而

利用和雙邊帶調制系統一樣的推導得到有雜訊時解調輸出為

( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn



振幅調制系統 (AM) 同調檢波

中不會出現直流項理由有二

1 因為直流項中不包含來源訊息

2 大部份實用的振幅解調器並非直流耦合故實際系統輸出並不含直流部份

( ) [1 ( )]cos( )c c n cx t A am t t ( ) ( ) ( )D c n cy t A am t n t

)(tyD



)tcos(2 c




振幅調制系統 (AM) 同調檢波由輸出中可得到信號功率為

而雜訊功率為

所以後驗訊雜比為

( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W




在預檢的情況下其信號功率為

而雜訊為

所以預檢訊雜比為

2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W



振幅調制系統 (AM) 同調檢波因此檢波增益為

振幅解調傳輸系統的效率定義可表示成

若效率能達到 100 則振幅調制和理想雙邊帶調制及單邊帶調制系統具有相同的後驗雜訊比但實際上卻不然

2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W

邊帶功率發射信號總功率

2 2

2 21n

ff

n

a mE

a m



範例 6-3

考慮一信號經由AM 調變器調變後在一帶有雜訊的通道上被傳送雜訊的功率頻譜如圖所示其中




cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3


輸出的信號功率與雜訊功率分別為

在預檢的情況下信號功率與雜訊功率分別為2 2 2 2 10

1 1=10 2 4 10

2 2T c c n TS A A a m N N W


2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3


後驗訊雜比比為

檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR


範例 6-4在調變指數為 05 且訊號功率為 01 下運作的振幅調變系統其效率為

而後驗訊雜比為

檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4

低於需要相同頻寬的理想系統 16dB 以上不過使用振幅調制的動機並非是雜訊性能而是可使用簡單的包跡檢波解調效率較差的原因當然是總發射功率中一大部分用在載波分量上而此分量不是訊息信號的函數因而不帶任何訊息

( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W



振幅調制系統 (AM) 包跡檢波假設接收信號為加上窄頻帶雜訊因此

並且可依下方向量圖求得

( )cx t

( ) [1 ( )]cos( ) ( ) cos( )

- ( )sin( )r c n c c c

s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc

)t( )t(cos)t(tcos)t(r)t(x ncr



振幅調制系統 (AM) 包跡檢波

其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns



振幅調制系統 (AM) 包跡檢波假設包跡檢波器為交流偶合使得

其中為包跡振幅的平均值

首先考慮訊雜比很大時則

因此

所以由上式可知當訊雜比很大時則包跡檢波器的輸出和同調檢波器的輸出相同

( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t




當訊雜比很小時會因為包跡檢波器的非線性造成嚴重損失此種效應稱為門限效應 (threshold effect) 在屬於線性的同調檢波器中如果信號和雜訊在檢波器輸入是相加的則在輸出也是相加的也就是說訊雜比低時信號仍維持不變基於這點當訊號大時包跡檢波通常比較適合



振幅調制系統 (AM) 平方律檢波器平方律檢波器可用平方元件後加低通濾波器來實現如下圖所示此檢波器對振幅調制信號的響應為

令並完成平方運算可得

2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t

A n t A n t t n t n t

平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD



振幅調制系統 (AM) 平方律檢波器

假設檢波器輸出為交流耦合故輸出信號和雜訊成份分別為

和

信號成份功率為

雜訊功率為

2( ) 2 cosD C my t A a t

2 2( ) 2 ( ) 2 ( )cos ( ) ( )D C C C C m C Sn t A n t A an t t n t n t

4 22D CS A a

2 2 2 2 2 4 24 ( ) 2 ( ) 2 ( ) - ( )D C C C C C CN A n t A a n t n t n t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t




振幅調制系統 (AM) 平方律檢波器因此總預檢雜訊功率為

平方律檢波器輸出的訊雜比為

2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A




將上式與基帶系統比較弦波調制之振幅系統總發射功率為

並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A




此式很清楚的說明了門限效應當很大時

而對於小的值0TP N W22

20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A










教育部資通訊科技人才培育先導型計畫通訊原理第六章 50

前言

前面已討論過線性調變系統中雜訊的效應現在將注意力轉移到角度調變我們會發現在有考慮雜訊效應時線性調變和角度調變間有極大的差異甚至在 PM 和 FM 間也有顯著的差異

最後還會瞭解在雜訊背景下 FM 系統可提供比線性調制和 PM 系統更大的改善性能


在雜訊中解調

考慮系統

預檢濾波器的頻寬可由卡森準則決定

換言之

其中W 為訊息信號的頻寬D 為偏差比即尖峰頻率偏移除以 W

TB

預檢濾波器鑑別器後驗

濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調

預檢濾波器的輸入假設為經調制的載波

加上雙邊功率頻譜密度為的白色雜訊 N(t)即

WHz2

10N

( ) cos[ ( )]c ccx t A t t


濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx


在雜訊中解調 - PM 調變訊號

對 PM 調變訊號

其中為相位偏移常數單位為弳度每單位而為經正規化使得的尖峰值為 1 的訊息信號



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm


在雜訊中解調 - FM 調變訊號

對 FM 的情況

其中為偏移常數單位為 Hz 每單位

若的最大值不為 1 ( 一般是如此 ) 則由定義所得的比例常數 K 包含在或中

PM 和 FM 二種情況我們會一起分析之後再以適當函數代入取得各自的結果



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調

預檢濾波器的輸出可寫成

其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成

其中為雷萊分布雜訊包跡為均勻分布相位



濾波器)(txr

)(1 te)(tyD

1( ) cos[ ( )] ( )cos( ) ( )sin( ) (a)c c c c s ce t A t t n t t n t t

1( ) cos[ ( )] ( )cos[ ( )] (b)c c n c ne t A t t r t t t

)(trn )(tn


在雜訊中解調

以取代則 (b) 式可寫成

即



濾波器)(txr

)(1 te)(tyD


)()()( tttt nc )(tt nc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成

其中為因雜訊產生的相位偏移可表示成



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t

)]()(cos[)()(1 ttttRte ec

)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne


在雜訊中解調因為加至傳送訊息信號的中故為所要考慮的雜訊成份當時預檢濾波器輸出的相量如圖所示若表成

)(te )(t)(trA nc )(1 te

1 cos ( ) (( ) ( ) ( )cos[ ( )] (d)) cc ee t R t Rt t t t tt

)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調

則鑑別器輸入的相位偏移為

若預檢訊雜大則在大部分時間上對此情況 (c) 式變成

(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調

鑑別器輸入的相位偏移為

若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(


在雜訊中解調 - 若預檢訊雜比大

若預檢訊雜大

值得注意的是若發射信號的振幅增大則雜訊的效應變小因此即使在門限效應之上操作輸出雜訊仍受發射信號之振幅的影響



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號


在雜訊中解調 - 若預檢訊雜比小

如果預檢 SNR 小則在大部分時間上此時的相量圖如下

)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調

如果預檢 SNR 小則在大部分時間上則

即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調

信號包含在中而僅含雜訊

若預檢訊雜 SNR 大

預檢濾波器輸出的相位為二項的和第一項僅含信號第二項包含信號和雜訊分量



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調


若預檢訊雜 SNR 小時預檢濾波器輸出的相位

並沒有僅含信號的項

因此 FM 解調器具有和 AM 包跡檢波器類似的門限效應此效應在往後會詳細說明



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調

假設鑑別器含有一限制器因而對於振幅的變化並不敏感因此 (d) 式中的可假設為一常數

在 PM 的情況下後驗濾波器輸出為

而對 FM 情況則為

其中為鑑別器常數

)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK


在雜訊中解調 - PM

當預檢訊雜 SNR 大

則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP


在雜訊中解調 - FM


則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc

K d t K d r tn t t t

dt dt A


在雜訊中解調PM 解調輸出可用訊息信號表成

PM 輸出信號的功率為

FM 解調輸出可用訊息信號表成

FM 輸出信號的功率為

在計算輸出訊雜比前須先求輸出雜訊的功率頻譜密度

)()()( tntmkKty PnpDDP

)()()( tntmfKty FndDDF

2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜

對於 SNR 大情形 PM 與 FM 之鑑別器輸出雜訊的功率頻譜密度可分別由 (h) 式和 (i) 式來計算

分析的第一個步驟是令使得和僅為雜訊和的函數而不含信號的成分

此一假設雖能大大簡化分析程序但似乎有太多的自由

( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜

對於 SNR 大情形在不假設為零下仍可進行分析推導過程會複雜許多且可發現的效應是當時產生解調輸出中的頻率成分 W 為訊息信號的頻寬

但此一效應可利用與鑑別器輸出串聯的低通濾波器除去

)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜 - PM

當且其它地方為零時 ( 如圖所示 ) 則 PM 情況下輸出雜訊的功率頻譜密度為

TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t



因此鑑別器輸出雜訊功率為

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為

因為預檢頻寬比大所以在鑑別器後加一頻寬的低通濾波器可改善輸出的 SNR 此濾波器不影響信號但可使輸出雜訊功率降低成

W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS


輸出訊雜比 - PM

現在可輕易求出鑑別器輸出的訊雜比對 PM 系統

鑑別器輸出 SNR 為定義信號功率除雜訊功率即

2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA


輸出雜訊頻譜 - FM

因微分的關係故在 FM 的情況下輸出雜訊功率的計算稍微困難些

若則將此結果應用在上式則在時可得

而其它地方為零

dtdxty )( )()2()( 2 fSffS xy

TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜

此頻譜如下圖所示拋物線形的雜訊頻譜是因 FM 鑑別器的微分效應所造成對含雜訊之 FM 系統的性能有深遠的影響

下圖中顯示低頻訊息信號成分比高頻成分的雜訊位準低

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B



再次假設低通濾波器僅有足夠的頻寬讓鑑別器的訊息信號通過則輸出雜訊功率為

23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B


輸出訊雜比 - FM

現在可輕易求出鑑別器輸出的訊雜比對 FM 系統


2

2c

TPA

)()()( tntmfKty FndDDF 2

302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W


PM 與 FM 之雜訊功率比較

將 PM 與 FM 式寫成的形式很有用因為對 PM 與 FM 故有

因此對於 PM

而對於 FM 鑑別器輸出的雜訊功率為

注意到對 PM 與 FM 在鑑別器輸出端的雜訊功率都與成反比

WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0


PM 與線性調制比較

上式顯示了 PM 對線性調制的改進和相位偏移常數與調制信號的功率有關

當 PM 信號的相位偏移超過徑解調的唯一性無法達成所以的尖峰值為而最大值為

這比基帶改良最大達 10 dB 左右實際上由於遠低於最大值所以實質的改進程度低很多

若加上「相位解調器的輸出為連續」的限制則有可能超過而解調雜訊比也隨之增加

2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W


FM 與 PM 調制比較

對 FM 在鑑別器輸出端的訊雜比為信號功率除以 ( 雜訊功率即

由於尖峰偏移與 W 的比值為偏移比 D 故輸出的 SNR 可表示成

22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df



的最大值為 1 須注意由的最大值連同和

決定

對 FM 的情況並沒有 PM 中輸出 SNR 大小受限的情況事實上 D 似乎可無限制的增加因而使輸出 SNR 增加到任意大的值

WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn



增大 SNR 所付出的一個代價是過高的傳輸頻寬當時所需頻寬大約為

如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W



此式說明存在於頻寬與輸出 SNR 之間的權衡取捨不過僅在鑑別器輸入的 SNR 大到足以在門限之上運作時才有效

因此只增加偏差比和傳輸頻寬並不能將輸出 SNR 增加到任意值

22

0

3(SNR)

4T T

DF n

B Pm

W N W


範例 6 - 5考慮下圖的系統信號 x(t ) 定義

低通濾波器在通帶上的增益為 1 且頻寬為 W 其中

n(t) 是雙邊功率頻譜密度為的白色雜訊 y(t) 的信號成

分定義成在頻率的成分試求 y(t) 的訊雜比

通訊原理第六章 89

cos(2) )( cx t A f t

cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc

d x t d n ty t A f f t n t

dt dt

N N fS f j f t f

W

N N f NE n t f df f W

A f A fSNR

NW

502N W


利用去強提高性能去強濾波器通常是直接置於鑑別器輸出的一階低通 RC 濾波器

調制之前信號先通過預強濾波器此濾波器和去強濾波器二者轉移函數的組合對訊息信號無淨效應

去強濾波器之後接一頻寬 W 的理想低通濾波器以濾去頻帶外雜訊

設去強濾波器有振幅響應

其中為 3 dB 頻率 Hz 2

3

1( )

1 ( )DEH f

f f

3f )2(1 RC


利用去強提高性能

則含去強的雜訊總輸出功率為

代入得

2( ) ( )

F

W

DF DEW nN H f S f df

2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f



在一般情況下所以通常比小

對此情況

而輸出的 SNR 變成

以上顯示當時使用預強和去強可獲得顯著的改良

Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f


範例 6 - 6商用 FM 中

而的標準值為 21 kHz 假設

則 FM 在無預強與去強時

若有預強與去強則

75 Hz W=15 Hz 5 ddf k k D

f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6

用所選的值不含去強時 FM 優於基帶 875 dB 而含去強的 FM 則好 21 dB

這 1225 dB 的差約相當於 16 倍這意味著有用預強與去強時發射機功率可減少 16 倍

此種顯著的改進使預強與去強的使用物超所值

2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0

(SNR) 12755d T TDF pe n

f P Pm

f N W N W








TP

(WHz) 2

0N

信號

雜訊

0-W W-B B

0N2

1

f





訊息信號=m(t)

訊息頻寬=W)t(yD

雜訊頻寬=B






0 0-

1 =

2

B

BN df N B

i0

(SNR) = TP

N B

信號

雜訊

0-W W-B B

0N2

1

f





0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f






i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



TP

(WHz) 2

0N

信號

雜訊

0-W W-B B

0N2

1

f





訊息信號=m(t)

訊息頻寬=W)t(yD

雜訊頻寬=B






0 0-

1 =

2

B

BN df N B

i0

(SNR) = TP

N B

信號

雜訊

0-W W-B B

0N2

1

f





0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f






i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




訊息信號=m(t)

訊息頻寬=W)t(yD

雜訊頻寬=B






0 0-

1 =

2

B

BN df N B

i0

(SNR) = TP

N B

信號

雜訊

0-W W-B B

0N2

1

f





0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f






i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





0 0-

1 =

2

B

BN df N B

i0

(SNR) = TP

N B

信號

雜訊

0-W W-B B

0N2

1

f





0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f






i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




0 0-

1

2

W

WN df N W

信號

雜訊

0-W W

0N2

1

f






i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





i 0

(SNR)= =

(SNR)T

T

N BP B

N W P W

WN

PT

0oSNR

信號

雜訊

0-W W

0N2

1

f





WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




WN

PT

0oSNR

i0

(SNR) = TP

N B


雜訊

訊息信號=m(t)

訊息頻寬=W)t(yD




( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



( )cx t(WHz)

20N



)tcos(2 c






00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




00 222

2

0N2

1

)f(S0N










WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W








WNtntntn sc 0222

0 2)()()( )IF()(0 n(t)tn






T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





T0

(SNR)4

cA m

WN

2 212 cA m 02WN




)tcos(2 c









2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







2 2CA m

202Cn WN

( ) ( ) ( )D C Cy t A m t n t

3( ) ( ) ( )cos( ) ( )

( )cos( ) ( )sin( )C C C

C C C

C

S


n t t n t t

WW

0N

)f(S0N

f







0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W






0

( ) TD

PSNR

N W

WN

mACD

0

22

2SNR

22

2

1mAP CT




D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



D(SNR) T(SNR)

2 20D

2 2T 0

4(SNR)2

(SNR) 2c

c

A m N W

N W A m

2 2

T0

(SNR)4

cA m

WN

WN

mACD

0

22

2SNR



)tcos(2 c




範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-1





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-1




)tcos(2 c



2 2 10=2 2 4 10T D c T DS S A m N N N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-1



檢波增益為


52

25D

T

SNR

SNR

2 2

0

254

cT

A mSNR

N W

2 2

0

52

cD

A mSNR

N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W






ˆ ( )m t ( )m t



)tcos(2 c





即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



即

所以


2 ˆ( ) [ ( ) cos( ) ( )sin( )]

( ) cos( ) ( )sin( )c c c

c c s c

e t A m t t m t t

n t t n t t




)tcos(2 c







0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





0N2

1

)f(S0N

Wfc Wfc f ff

- +

0N2

1)f(S

0N

W Wf

-+




2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



2 ( )e t 2cos( )ct

Dy ( ) ( ) ( )c ct A m t n t



)tcos(2 c







2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





2D 0CN n N W

2 2D CS A m

2 2

0

SNR D

D

CD

S

N

A m

N W







( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W






( ) 0m t

ˆ ˆ( ) ( ) ( ) ( ) =0m t m t m t m t

2 2 21 1ˆ( ) ( )

2 2T cS A m t m t

2 2T cS A m





預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



預檢訊雜比為


2 20

1

2D T c sN N n n N W

2 20D

2 2T 0

(SNR)1

(SNR)c

c

A m N W

N W A m


2 2

0

( ) T CT

T

SNRS A m

N N W



範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-2





cos mm t t

60

110 A

22cN

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N



範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-2

輸入信號為




2 2 2 1D 02 2 10T c cS A m N n N W



)tcos(2 c



範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-2


後驗雜訊比為

檢波增益為


101

10D

T

SNR

SNR

2 2

0

10cT

A mSNR

N W

2 2

0

10cD

A mSNR

N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





( ) [1 ( )]cos( )c c n cx t A am t t

( ) ( ) ( )D c n cy t A am t n t

)( min

)()(

tm

tmtmn








)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







)(tyD



)tcos(2 c





而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



而雜訊功率為


( )Dy t

2 2 2D c nS A a m

202D CN n N W

2 2 2

0

( )2

D C nD

D

S ASNR

a m

N N W





而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




而雜訊為


2 2 2 21 1

2 2T c c nS A A a m

02TN N W

2 2 2 2 2 2 2

0 0

1 11

2 2( )2 4

C C n C nT

TT

A A a m A a mS

N N WSNR

N W






2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





2 2 2 2 2

0

2 22 2 2 20

( ) 2 2

( ) 14

D c n n

T nc c n

SNR A a m N W a m

SNR a mA A a m N W


2 2

2 21n

ff

n

a mE

a m



範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-3





cos mm t t

60

110

2N

4W kHz 200cf kHz

TSNR DSNR

0N2

1

)f(S0N


05a 4cA


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-3




1 1=10 2 4 10



2 2 2 2 10=4 2 4 10D c n D cS A a m N n N W

( )Dy t



)tcos(2 c



範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-3



檢波增益為


2 2 2

0

102c n

D

A a mSNR

N W

2 2 2 2

0

254

c c nT

A A a mSNR

N W

1004

25D

T

SNR

SNR




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




檢波增益為

2

2

05 01=00244

1 05 01ffE

0

( ) 00244 TD

PSNR

N W

( )2 =00488

( )D

ffT

SNRE

SNR

)(tmn



範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-4


( )2 =00488

( )D

ffT

SNRE

SNR

0

( ) 00244 TD

PSNR

N W





( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




( )cx t

( ) [1 ( )]cos( ) ( ) cos( )


s c

x t A am t t n t t

n t t

( ) ( ) cos[ ( )]r cx t r t t t

)t(am1A nc )t(nc

)t(r

)t(

)t(ns

)t(am1A nc





其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



其中

且

2 2( ) [1 ( )] ( ) ( )c n c Sr t A am t n t n t

-1 ( )( ) tan

[1 ( )] ( )s

c n c

n tt

A am t n t

)t(am1A nc


)t(am1A nc )t(nc

)t(r

)t(

)t(ns






因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





因此


( ) [1 ( )] ( )c n cr t A am t n t

( ) ( ) ( )D c n cy t A am t n t

( ) ( ) - ( )Dy t r t r t

( )r t









2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W








2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




2 2 2( )= [1 ( )] ( ) ( )C SC nr t A am t n t n t

( ) cosn mm t t

2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t


平方律檢波器

低通濾波器

)t(r )t(r2 )t(yD





和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




和


雜訊功率為



4 22D CS A a


2 2 2 2 2 2 2C C C C

2 2

1 1( )= +2 a cos + a + a cos2

2 2

+2 ( )+2 a ( )cos + ( )+ ( )

m m

C C C C m C S

r t A A t A A t






2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




2 2 20 02 ( ) ( )n T c sN B N W n t n t

2 2 2

2 2 2( )

(2 ) 2( )c n

Dn c

A aSNR

a A





並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




並以取代可得

2 21 11

2 2T cP A a 2

n02N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A






20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





20

( ) 22

TD

PaSNR

a N W

0

TP

N W2

20

( ) 22

TD

PaSNR

a N W

2 2 2 2

2 2 2 2( ) 2

2 (2 ) 2( )c n

Dn c

A aaSNR

a a A











前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W










前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


前言




在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調

考慮系統


換言之


TB


濾波器)(txr

)(1 te)(tyD

2( 1) HzT DB W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調



WHz2

10N



濾波器)(txr

)(1 te)(tyD

( ) ( ) ( )r ct x t N tx



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



對 PM 調變訊號




濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

)()( tmkt nP

Pk)(tmn )( tm



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



對 FM 的情況






濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

( ) 2 ( ) t

d nt f m d

df

)( tm )()( tKmtm n

Pk df

)(t


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調


其中



濾波器)(txr

)(1 te)(tyD

( ) cos[ ( )] ( )r ccx t A t t N t

1( ) cos[ ( )]

( )cos( ) ( )sin( ) (a)c c

c c s c

e t A t t

n t t n t t

Tsc BNnn 022


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調

(a) 式可寫成




濾波器)(txr

)(1 te)(tyD



)(trn )(tn


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調


即



濾波器)(txr

)(1 te)(tyD



)](sin[)]()(sin[)(

)](cos[)]()(cos[)(

)](cos[)(1

tttttr

tttttr

ttAte

cnn

cnn

cc

)](sin[)]()(sin[)(

)](cos[)]()(cos[)()(1

tttttr

tttttrAte

cnn

cnnc


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調

上式可寫成




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( ) ( )]cos[ ( )]

( )sin[ ( ) ( )]sin[ ( )]c n n c

n n c

e t A r t t t t t

r t t t t t


)(te

(c) )]()(cos[)(

)]()(sin[)(tan)( 1

tttrA

tttrt

nnc

nne





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





)(tn

)(trn

)(t)(tR

)(t )(t

)(te

)(sin)( ttrn

)(cos)( ttrn cA


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調



(e) )()()( ttt e

T)SNR( )(trA nc


濾波器)(txr

)(1 te)(tyD


1 1( )sin[ ( ) ( ( )ta

)]n sin ( ) ( )( ) tan

( )cos[ ( ) ( )]n n

ec n

nc

n c

r t t rtt

A r

t

t tt t

At


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調


-02 -01 0 01 02 03

-02

-01

0

01

02

y=xy=tan(x)

y=tan-1(x)

1 1 ( )tan

( )sin[ ( ) ( )]( ) tan

( )cos[ ( ) (sin ( ) ( )

( ) sin )

)]

( ( )

nc

c

nc

c

n ne

c n n

rr t t tt

A r t t

tt t

A

r tt t

A

t


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調


若預檢訊雜大

使得為



濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

)(t

(e) )()()( ttt e

( )sin( ) ( ) ( )e

nc

c

tr t

t tA

T)SNR(



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



若預檢訊雜大




濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

T)SNR(

(f) )]()(sin[)(

)()( ttA

trtt n

c

n

cA )(trn

PM 或 FM訊號




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




)(trA nc

cA

)(tn

)(te

)(t

)(t

)(t

)(trn

cA

)(cos tAc )(sin tAc

)(tR


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調


即


)(trA nc

1 1( )sin[ ( ) ( )] (tan

)sin[ ( ) ( )]( ) tan

( )cos[ ( ) ( )] ( )cos[ ( ) ( )

]

( ) ( )

n n n ne

c n n n n

n

r t t t r t t tt

A r t t t r t t

t t

t

( ) () ) ( )( e nt tt t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調






濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )]n

nc

r tt t t t

A

)(tm )(t )(trn

)(t


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調







濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

)(tm )(t )(trn

)(t

( ) ( ) ( ) ( ) ( ) (( )) e n nt t t tt t t


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


在雜訊中解調





)(tR

)(tyD

( ) ( )DP Dy t K t

1 ( )( )

2DF D

d ty t K

dt


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

DK




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




則

其中

( ) ( )DP Dy t K t


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

)()()( tntKty PDDP

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




則

其中


濾波器)(txr

)(1 te)(tyD

1( ) ( )cos[ ( )] (d)ce t R t t t

( )( ) ( ) sin[ ( ) ( )] n

nc

r tt t t t

A

1 ( )( )

2DF D

d ty t K

dt

1( ) ( )

( )

2DF D F

dy t K n t

dt

t

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W









2 2 2DP p nDS K k m

2 2 2DF D nDS K f m


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


輸出雜訊頻譜




( ) 0t )(tnP )(tnF

(h) )]()(sin[)(

)()( ttA

trKtKtn n

c

nDeDP

( ) ( )( ) sin[ ( ) ( )] (i)

2 2D e D n

F nc


dt dt A

( )nr t ( )n t


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


輸出雜訊頻譜



)(t)(t

Wf


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


輸出雜訊頻譜

當

其中因此

( )( ) sin[ ( ) ( )] ( )sin ( ) Dn

n nc

P D nc

Kr t t

A

r tn t K t t

A

( )( ) sin[ ( ) ( ) = ( )

2] sin[ ( )]

2D n

F n nD

ccn

K d Kt

A

r t dn t t t r t

dt A dt

0)( t

( )si )n[ ( ) (]n n sr t t n t

n )( () DP s

C

nK

At t

( )( )

2D s

Fc

K dn tn t

A dt




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




TBf2

1

2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n ( )( ) PD

sC

Kt

An t




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




2 2

02 2( ) ( )

P sn nD D

c c

K KS f S f N

A A

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS

)( fSnP

WWTB2

1

TB2

1

02

2

NA

K

c

D

f

n )( () DP s

C

nK

At t



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W



信號頻率為


W

TB W2W

2

02 ( ) 2 (n)P

DW

cnDP W

KN N WS df

Af

2

2

022( )

T

PT

BD

DP TBc

n

KN Nf df B

AS





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





2 2 22 2

20

10

(SNR)( )

DP D P n TP n

DP DDP

T

K k mk m

K

S P

N P N W N W


2 2 2DP p nDS K k m

2

02( ) 2P

DDP

c

W

nWS f df

KN N W

A

2

2c

TPA







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







TBf2

1

2 22 2

22 20 0( ) (2 ) (o)(2 )F

D Dn

c c

K KS f f N N f

A A

( )( )

2D s

Fc

K dn tn t

A dt


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


輸出雜訊頻譜



)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




23

02

2

3( )

F

WD

WD nFc

S f dK

N f N WA

)( fSnF

WWTB2

1 TB2

10

202

2

fNA

K

c

D

f

2

22

0 | |2

( ) F

D Tn

c

KS f N f f

A

B





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





2

2c

TPA


302

2

3( )

F

WD

WD nFc

S f dK

N f N WA

2 2 2DF D nDS K f m

22

0

( ) 3 d TDF n

f PSNR m

W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




因此對於 PM



WNPT 0

22cT AP

WN

A

WN

P cT

0

2

0 2

1

2

0

TDP D

PN K

N W

1

2 2

0

1

3T

DF D

PN K W

N W

WNPT 0







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







2

2

)(tmk nP

)(tmk nP

22nP mk

22nP mk

2 2

0

(SNR) DP TP n

DPDP

Sm

Nk

P

N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





22 2 22

102

30

( ) 313

D d n d TDF n

TD

K f m f PSNR m

W N WPK

N W

WN

PmD T

nDF0

223)SNR(

df




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




決定


WN

PmD T

nDF0

223)SNR(

)(tmdf WD)(tmn




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




如此得

1D

TB DW2

WN

PmD T

nDF0

223)SNR(

22

0

3(SNR)

4T T

DF n

B Pm

W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W





22

0

3(SNR)

4T T

DF n

B Pm

W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W








cf W

0

1

2N

cf

低通濾波器dt

d

dt

d)(tx

)(tn


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6 - 5

0

2 22

2 2

4 40 0

5 42 4 4 4 50 0 0

0

2 42 4

450

( ) ( ) ( ) LPF (2 ) cos(2 ) ( )

( ) LPF 2 (2 ) ( )2 2 2

(2 ) ( ) (2 ) (2 ) 2

2 2 5 5

1(2 ) 52

(2 )5

c c o

N c

WW

o W

cc


dt dt

N N fS f j f t f

W


A f A fSNR

NW

502N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







3

1( )

1 ( )DEH f

f f

3f )2(1 RC




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




代入得

2( ) ( )

F

W


2 2

20 32 2 2

3

23 1

0 323 3

2 tan

WD

DF Wc

D

c

K fN N f df

A f f

K W WN f

A f f

2

22

0 |( ) |F

Dn

c

KS f N f

Af W

23

1( )

1 ( )DEH f

f f




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W




對此情況



Wf 3

2

1)(tan 3

1 fW 3 fW

WfNA

KN

c

DDF

2302

2

2

2

2

3 0

(SNR) d TDF pe n

f Pm

f N W

Wf 3

23 1

0 323 3

2 tanDDF

c

K W WN N f

A f f







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W







f

W

102 nm3f

2 2 2

0 0 0

(SNR) 3 3 5 0 51 7T T TDF n

P P PD m

N W N W N W

2

2

3 0 0

2

0

(SNR) 1275

01 7 52

51

d T T TDF pe n

kf P P Pm

f N W Nk W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W


範例 6-6




2 2

0 0

(SNR) 3 75T TDF n

P PD m

N W N W

2

2

3 0 0


f P Pm

f N W N W

Documents

通訊原理 第六章： 解調性能分析

通訊原理第六章：解調性能分析