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23/4/19 1Zhongguo Liu_Biomedical Engineering_Shandong U

niv.

Biomedical Signal processing

Chapter 4 Sampling of Continuous-Time Signals

Zhongguo Liu

Biomedical Engineering

School of Control Science and Engineering, Shandong University

山东省精品课程山东省精品课程《《生物医学信号处理生物医学信号处理 ((双语双语 )) 》》http://course.sdu.edu.cn/bdsp.htmlhttp://course.sdu.edu.cn/bdsp.html

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Chapter 4: Sampling of Continuous-Time Signals

• 4.0 Introduction

• 4.1 Periodic Sampling

• 4.2 Frequency-Domain Representation of Sampling

• 4.3 Reconstruction of a Bandlimited Signal from its Samples

• 4.4 Discrete-Time Processing of Continuous-Time signals

3

4.0 Introduction• Continuous-time signal processing can be

implemented through a process of sampling, discrete-time processing, and the subsequent reconstruction of a continuous-time signal.

f=1/T: sampling frequency ,cx n x nTn

sradTs /,2

T: sampling period

4

( )n

t nT

4.1 Periodic

Sampling

Continuous-time signal

T: sampling period

[ ] ( )cx n x nT

s cn

x t x nT t nT

impulse train sampling

Sampling sequence

Unit impulse

train

5

T ： sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate

n

s t t nT

2s

k

S j kT

2s

k

S j kT

sjk tk

k

a e

s(t)为冲激串序列，周期为 T，可展开傅立叶级数

1sjk t

k

eT

2 ( )sjk t Fse k

-T

1

tT0

( )s t

…… ……/2

/2

1 1( ) s

T jk tk T

a t e dtT T

0

( )S j

…… ……2

T

2

T

2

T

冲激串的傅立叶变换：

6

4.2 Frequency-Domain Representation of Sampling

[ ] ( ) | ( )c t nT cx n x t x nT

T ： sample period; fs=1/T:sample rateΩs=2π/T: sample rate

n

nTtts

n

s c cn

c x t t nTx t x t s t x nT t nT

2s

k

S j kT

1c s

k

X j kT

1( )

21 2 1

1*

2

( ) ( )2

c

s c sk

s c

ck

S j X j d

k X j

X j X j S j

d k X j dT T

Representation of

in terms of jweX

jX s

7

s c c cn n

x t x t s t x t t nT x nT t nT

cn

j Tnx nT e

( ) j ts c

n

X j x nT t nT e dt

[ ] ( )cx n x nT

DTFT

Representation of in terms of , jweX jX s cX j

T 数字角频率 ω，圆频率， rad模拟角频率 Ω,

rad/s

2s T

1s c s

k

X j X j kT

采样角频率 , rad/s

( )j TX e ( )jc

n

j nx nT X ee

8

( ) ( )j j TX e X e

1 2( ) c

k

j kX X j

T T Te

0,cif X jT

1( ) c

jthen X X jT T

e

1s c s

k

X j X j kT

DTFT

Representation of in terms of , jweX jX s cX j

Continuous FT/ T

T

T

9

Nyquist Sampling Theorem

• Let be a bandlimited signal with

. Then is

uniquely determined by its samples

, if

• The frequency is commonly referred as the Nyquist frequency.

• The frequency is called the Nyquist rate.

tX c

Nc forjX 0 tX c

,2,1,0, nnTxnx c Ns T 2

2

N

N2

10

NNs

NNs aliasing

frequencyT2

s

2

No aliasing

aliasing

k

scs kjXT

jX ))((1

)(

( ) ( ) |

1 ( 2 )( )

js

T

ck

X e X j

j kX

T T

frequency spectrum of ideal sample signal

2s

k

S j kT

Compare the continuous-time and discrete-time

FTs for sampled signal

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Example 4.1: Sampling and Reconstruction of a sinusoidal signal

cos 4000 ,cx t t1 6000.if sampling period T

02cos 4000 cos cos3cx n x nT Tn n w n

0

212000 2ssampling frequency

T

0: 4000The highest frequency of the signal

Solution:

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Example 4.1: Sampling and Reconstruction of a sinusoidal signal

4000 4000cX j

40004000 0

( ) ( )j j TX e X e 1s c s

k

X j X j kT

ttxc 4000cos

02cos cos3x n n w n

continuous-time FT of

discrete-time FT of

( ) ( )j j TX e X e 1 2

ck

kX j

TT T

1s c s

k

X j X j kT

从积分 ( 相同的面积 ) 或冲击函数的定义可证 TT

2 cos 3same x n n ttxc 4000cos

ttxc 000,16cos

1 6000T 4000 4000

2 3

T T

2

3

Compare the continuous-time and discrete-time

FTs for sampled signal

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Example 4.2: Aliasing in the Reconstruction of an Undersampled sinusoidal signal

cos 16,000cx t t

60001Tperiodsamplingif

cos 16,000 cos 16,000 / 6000

2cos 2 2 / 3 cos 3

cx n x nT Tn n

n n n

0

212000 2ssampling frequency

T

0 16,000:The highest frequency of the signal

Solution:

15

rn

n nx h Tt

Tt

Ttthr

sin

sin

n

nT Tx

T

tn

t nT

Tjrr eXjHjX

Gain: T

sn

x t x n t nT

*r s rx t x t h t

rn

x n nT h t d

4.3 Reconstruction of a Bandlimited Signal from its Samples

rn

x n n dT h t

16

4.4 Discrete-Time Processing of Continuous-Time signals

nTxnx c

1 2

jw

ck

X e

w kX j

T T T

n

r TnTt

TnTtnyty

sin

,

0,

j TTY eT

otherwise

jwH e

j Tr rY j H j Y e

,

0,r

TH j Totherwise

:T

a half of the sampling frequency

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C/D Converter

• Output of C/D Converter

nTxnx c

kc

jw

T

k

T

wjX

TeX

21

18

D/C Converter

• Output of D/C Converter

n

r TnTt

TnTtnyty

sin

otherwiseT

eTY

eYjHjY

Tj

Tjrr

,0

,

,

0,r

TH j Totherwise

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,

0,

j TcH e X j

T

T

4.4.1 Linear Time-Invariant Discrete-Time Systems

cX j jweX rY j jweY jweH

= T

,jw jw jwY e H e X e

1 2

j T j Tr r

j Tr c

k

Y j H j H e X e

kH j H e X j

T T

j Tr rY j H j Y e

Is the system Linear Time-Invariant ?

1 2=j T jw

ck

kX e X e X j

T T

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Linear and Time-Invariant

• Linear and time-invariant behavior of the system of Fig.4.11 depends on two factors:

• First, the discrete-time system must be linear and time invariant.

• Second, the input signal must be bandlimited, and the sampling rate must be high enough to satisfy Nyquist Sampling Theorem.( 避免频率混叠 )

21

kc

Tjr

TjTjrr

T

kjX

TeHjH

eXeHjHjY

21

0 ,cIf X j for T

,

0,

j Tc

r

X jH eY Tj

T

effr cH jY j X j

,

0,

j T

eff

H eTH j

T

,

0,r

TH j Totherwise

effective frequency response of the overall LTI continuous-time

system

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4.4.2 Impulse Invariance

Given:

jwH eDesign:

ch n Th nT

. . i e h n ch nT

jX c jweX jYr jweY jwH e

,cH j ch t

,

0,

j T

c eff

H eTH j H j

T

eff cH j H j

impulse-invariant version of the continuous-time system

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4.4.2 Impulse Invariance Two constraints

,jcH e H j T

TjH

thatsuchchosenisT

c ,0

1.

2.

ch n TTh nThe discrete-time system is called an impulse-invariant version of the continuous-time system

,

ch n h nT ( )1

cjH j

TTHe

ch n TTh n ( ) c

jH H jT

e

/C T 截止频率T

= T

, T

24

4.5 Continuous-time Processing of Discrete-Time Signal

jweX jX c jYc jweY

sinc

n

t nT Tx t x n

t nT T

sinc

n

t nT Ty t y n

t nT T

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4.5 Continuous-time Processing of Discrete-Time Signal

,j TcX j TX e T

,c c cY j H j X j T

1 1,

wj Tjw T

c c

w wY e Y j H j TX e w

T T T T

w

T

wjHeH c

jw ,

TeHjH Tjc ,

=w T

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4.5 Continuous-time Processing of Discrete-Time Signal

Figure 4.18 Illustration of moving-average filtering. (a) Input signalx[n] = cos(0.25πn). (b) Corresponding output of six-point moving-average filter.

Errata

27

The Nyquist rate is two times the bandwidth

of a bandlimited signal. The Nyquist frequency is half the sampling

frequency of a discrete signal processing

system.( The Nyquist frequency is one-half

the Nyquist rate)

What is Nyquist rate？What is Nyquist frequency？

Review

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DTFT derived from the equation. impulse train sampling xs(t) and x[n] have the

same frequency component.

Review

What is the physical meaning for the equation: DTFT of a discrete-time signal is equal to the FT of a impulse train sampling .

1s c s

k

X j X j kT

[ ] ( )cx n x nT ( )j

n

j nx n X ee

s c cn

x t x t s t x nT t nT

cn

j Tnx nT e

T

29

How many factors does the linear and time-invariant behavior of the system of Fig.4.11 depends on ?

Review

First, the discrete-time system must be linear and time invariant.

Second, the input signal must be bandlimited, and the sampling rate must be high enough to satisfy Nyquist Sampling Theorem.( 避免频率混叠 )

30

Assume that we are given a desired continuous-time system that we wish to implement in the form of the following figure, how to decide h[n] and H(ejw)?

Review

,

0,

j T

c eff

H eTH j H j

T

ch n Th nT

23/4/19 31 Zhongguo Liu_Biomedical Engineering_Shand

ong Univ.

Chapter 4 HW

•4.5

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