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NGN HNG THIMn: L THUY

T THNG TIN

S tn ch : 4SDNG CHO NGNH IN T- VIN THNG V CNG NGH THNG TIN

HI HC TXA

CHNG I: NHNG VN CHUNG V KHI NIM CBN

1/ Chn cu ng sau :a Tin lun c biu din di dng sb Tm nh, bn nhc, bc th . . . khng phi l cc tinc Tin l mt nh x lin tc n ngi nhnd Tin l dng vt cht biu din hoc th hin thng tin.

2/ Chn pht biu ng trong nhng cu sau :a Thng tin l nhng tnh cht xc nh ca vt cht m con ngi nhn c t th gii vt cht

bn ngoib Thng tin khng th xut hin di dng hnh nhc Thng tin tn ti mt cch ch quan, ph thuc vo h th cm.d Thng tin khng th xut hin di dng m thanh

3/ Mn l thuyt thng tin bao gm vic nghin cu:a Vai tr ca thng tin trong k thutb Cc qu trnh truyn tin v L thuyt m ha.c L thuyt ton xc sut ng dng trong truyn tin.d Cch chng nhiu phi tuyn trong v tuyn in

4/ Chn cu ng nht v ngun tina Ngun tin l ni sn ra tinb Ngun tin l tp hp cc tin c xc sut v k hiu nh nhauc Ngun tin lin tc sinh ra tp tin ri rc.

d Ngun tin ri rc sinh ra tp tin lin tc.

5/ Chn cu ng nht vng truyn tina L mi trng Vt l, trong tn hiu truyn i t my pht sang my thub L mi trng Vt l m bo an ton thng tinc L mi trng Vt l trong tn hiu truyn i t my pht sang my thu khng lm mt

thng tin ca tn hiu.d ng truyn tin chnh l knh truyn tin.

6/ bin i mt tn hiu lin tc theo bin v theo thi gian thnh tn hiu s, chng ta cnthc hin qu trnh no sau y:

a Ri rc ha theo trc thi gian v lng t ha theo trc bin b Gii m d liuc M ha d liu.d Lng t ha theo trc thi gian v ri rc ha theo trc bin

HC VIN CNG NGH BU CHNH VIN THNGKm10ng Nguyn Tri, Hng-H Ty

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CHNG II: TN HIU V NHIU

1/ Chn cu ng v tn hiu:a Tn hiu l mt nh x lin tc n ngi nhnb Tm nh, bn nhc, bc th . . . khng phi l cc tinc Tin lun c biu din di dng sd Tn hiu l qu trnh ngu nhin .

2/ Chn pht biu ng nht vc trng thng k :a c trng cho cc qu trnh ngu nhin chnh l cc quy lut thng k v cc c trng thng

kb K vng, phng sai, hm t tng quan, hm tng quan l cc quy lut thng kc Cc hm phn b v mt phn b l nhng c trng thng kd Tn hiu v nhiu khng phi l qu trnh ngu nhin theo quan im thng k

3/ Chn cu ng nht v hm t tng quan :a Hm t tng quan l quy lut thng k th hin ca qu trnh ngu nhin.

b Hm t tng quan x 1 2R (t , t ) lun c tnh bng biu thc sau[ ]{ }21 2( , ) ( ) ( )x xR t t M X t m t=

c Hm t tng quan x 1 2R (t , t ) c trng cho s ph thuc thng k gia hai gi trhai thiim thuc cng mt

d Hm t tng quan x 1 2R (t , t ) lun bng phng sai ( )xD t vi mi t

4/ Vic biu din mt tn hiu gii hp thnh tng ca hai tn hiu iu bin bin thin chm s lmcho vic phn tch mch v tuyn in di tc ng ca n trnn phc tp

a ngb Sai

5/ Ngi ta gi tn hiu gii rng nu b rng ph ca n tho mn bt ng thc sau:

0

1

. Cc tn hiu iu tn, iu xung, iu xung ct, manip tn s, manip pha, l cc tn hiu

gii rng.a Saib ng

6/ Chn cu ng v cng thc xc nh mt ph cng sut

a

22

TT T1

E x (t)dt S ( ) d2

= =

l cng thc xc nh mt ph cng sut ca cc qutrnh ngu nhin.

b

2

T

T

S ( )

G ( )T

= l cng thc xc nh mt ph cng sut ca cc qu trnh ngu nhin.

c { }

2

T

xT

S ( )

G( ) M G ( ) M lim T

= = l cng thc xc nh mt ph cng sut ca cc qutrnh ngu nhin.

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d

2

T

x TT T

S ( )

G ( ) lim G ( ) limT

= =l cng thc xc nh mt ph cng sut ca cc qu

trnh ngu nhin.

7/ Chn cu ng v cng thc quan h gia mt ph cng sut v hm t tng quan

a

2T

x TT T

S ( )

G ( ) lim G ( ) limT

= =

b{ }

2

T

xT

S ( )

G( ) M G ( ) M limT

= =

c

22

TT T1

E x (t)dt S ( ) d2

= =

d

jG( ) R( )e d

=

8/ Trong trng hp h thng tuyn tnh thng c suy gim th nhng thi im t >> t 0 = 0 (thiim t tc ng vo), qu trnh ngu nhin u ra sc coi l dng. Khi hm t tng quanv mt ph cng sut ca qu trnh ngu nhin u ra s lin h vi nhau theo biu thc sau :

jra ra

1R ( ) G ( )e d

2

=

a Saib ng

9/ ( )*aS t l hm lin hp phc ca ( )aS t : ( ) ( ) ( )

= +aS t x t jx t l tn hiu gii tch.

ng bao ca tn hiu gii tch c th biu din bng cng thc sau: ( ) ( ) ( )= *a aA t S t .S t

a Saib ng

10/ Mt mch v tuyn in tuyn tnh c tham s khng i v c tnh truyn t dng ch nht(hnh di) chu tc ng ca tp m trng dng. Tm hm t tng quan ca tp m ra theo cng

thc

2

0ra

0

GR ( ) K( ) cos d

2

=

ta c kt qu no ?

GV()

2N0

a.

0 1 0 2

| K() |

b.

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a

2ra ra 0

sin2R ( ) cos2

=

b

22

ra ra ra v1 1

R (0) G ( )d P K( ) G ( )d2 2

= = = =

c

jra ra

1R ( ) G ( )e d2

=

d 0=raR

11/ Cho qu trnh ngu nhin dng c biu thc sau: ( ) ( )= + 0X t A cos 2 f t

Trong A = const, 0f = const, l i lng ngu nhin c phn bu trong khong ( ) , .

Tnh k vng { }( )M X t theo cng thc { }( ) ( ) ( )M X t X t w d

= ta c gi tr no di y

a { }( ) 0M x t =

b { }( ) 2M x t =

c { }( ) 1M x t =

d { }( ) 1M x t =

12/ Cho qu trnh ngu nhin dng c biu thc sau: ( ) ( )= + 0X t A cos 2 f t Trong A =

const, 0f = const, l i lng ngu nhin c phn bu trong khong ( ) , . Tnh hm ttng quan 1 2( , )R t t theo biu thc { }1 2( , ) ( ). ( )R t t M X t X t = + ta c gi tr no di y:

a 21 2 0( , ) cos 2R t t A f =

b 1 2( , ) 0R t t =

c 21 2 01

( , ) cos 22

R t t A f =

d 21 2 0( , ) cos 2R t t A f =

13/ Tn hiu in bo ngu nhin X(t) nhn cc gi tr + a; - a vi xc sut nh nhau v bng 1/2. Cn

xc sut trong khong c N bc nhy l: ( ) ( ) = >N

P N, e 0N!

(theo phn b Poisson). T cc gi thit trn tnh c hm t tng quan 2 2( )xR a e

= . Khi

mt ph cng sut ( )xG ca X(t) c tnh theo cng thc0

( ) 2 ( )cosx xG R d

= ta c gi

tr no sau y:

a2

2 2

4( )

4xa

G

=+

b2

2 2

4( )

4x

aG

+=

+

c ( ) 0xG =

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d2

2 2

4( )

4xa

G

=

14/ Mt qu trnh ngu nhin dng c hm t tng quan: 2 0( ) cosxR e

=

Khi mt ph cng sut ca cc qu trnh ngu nhin trn l

a2

2 20 0

2 2

( ) ( ) ( )xG

= + + + +

b 22 2

0 0

2 2( )

( ) ( )xG

= + + +

c 22 2

0 0

1 1( )

( ) ( )xG

= + + +

d 22 2

0 0

1 1( )

( ) ( )xG

= + + + +

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CHNG 3: CSL THUYT THNG TIN THNG K

1/ Khi nim lng tin c nh ngha da trn:a Nng lng ca tn hiu mang tinb bt nh ca tinc ngha ca tin

d Nng lng ca tn hiu mang tin v ngha ca tin

2/ Chn pht biu ng nht v Entropy ca ngun tin, H(X):a L i lng c trng cho bt nh trung bnh ca ngun tinb c tnh theo cng thc ( ) ( ) log ( )

x X

H X P X P x

=

c t cc tiu khi ngun l ng xc sutd t cc i khi ngun l ng xc sut

3/ Chn pht biu sai v bt nha bt nh ca php chn t l nghch vi xc sut chn mt phn t

b bt nh gn lin vi bn cht ngu nhin ca php chnc bt nh ca mt phn t c gi tr 1 bit khi xc sut chn phn t l 1d bt nh cn c gi l lng thng tin ring ca bin c tin

4/ Entropy ca ngun ri rc nh phn ( ) ( )H(A) p log p 1 p log 1 p=

Khi p=1/2 th H(A) t max Chn cu ng v ax( )mH A

a ax( )mH A =3/2 bt;

b ax( )mH A =1/2 bt;

c ax( )mH A =1 bt ;

d ax( )mH A =0

5/ Trong mt trn thi u bng Quc t, i tuyn Vit Nam thng i tuyn Brazin, thng tin nyc bt nh l

a bng 0 ;b V cng lnc nh hn 0;d ln hn 0

6/ Hc sinh A c thnh tch 12 nm lin t danh hiu hc sinh gii, hc sinh B lc hc km Thi ttnghip ph thng trung hc, hc sinh A trt, hc sinh B th khoa Thng tin v hc sinh B thkhoa, hc sinh A trt c bt nh l:

a bng 0 ;b V cng lnc nh hn 0;d ln hn 0

7/ Chn ngu nhin mt trong cc st 0 n 7 c xc sut nhnhau Khi xc sut ca scchn ngu nhin l:

a 7b 1/8

c 8d -7

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8/Mt thit b v tuyn in gm 16 khi c tin cy nhnhau v c mc ni tip Gi sc mtkhi no b hng, khi xc sut ca mt khi hng l:

a 1/16b 16c -16d -1/16

9/B t lkh52 qun (khng kfng teo), A rt ra mt qun bi bt k Xc sut v qun bi m A rt l:

a Bng 1/52b Nh hn 5c Ln hn 5 nh hn 6d Bng 1/52

10/ Chn cu sai trong cc cu sau :a bt nh s trthnh thng tin khi n b th tiub bt nh chnh l thng tinc Lng thng tin = bt nh tin nghim + bt nh hu nghim

d Lng thng tin = bt nh tin nghim - bt nh hu nghim

11/ Chn cu ng sau :a Lng thng tin = thng tin tin nghim - thng tin hu nghimb Thng tin hu nghim chnh l thng tin ringc Lng thng tin = thng tin hu nghim - thng tin tin nghimd Lng thng tin = thng tin tin nghim + thng tin hu nghim

12/ Chn cu sai trong cc cu sau :a Thng tin tin nghim (k hiu I(

kx )) c xc nh theo cng thc sau:I(

kx ) = log P(

k);

b Thng tin tin nghim cn gi l lng thng tin ring;c Thng tin tin nghim (k hiu I(k

x )) c xc nh theo cng thc sau:I(k

x ) =- log P(k

x );

d Thng tin tin nghim cn gi l lng thng tin ring c xc nh theo cng thc sau :

I(k) =

1log

( )k

P x;

13/ ( , )k l

I x y l lng thng tin cho vk

x dol

y mang li c tnh bng cng thc no sau y :

a1 1

log log( ) ( / )k k lP x P x y

b ( ) ( / )j j kI y I y x c

1log log ( / )

( ) k lkP x y

P x

d( / )

log( )

l k

l

P y x

P y

14/ ( / ) log ( / )k l k l I x y p x y= l thng tin hu nghim v k ( / ) 1k lp x y = khi vic truyn tin khng

b nhiu. Chn cu sai trong nhng cu v ( / )k lI x y di y:

a ( / )k lI x y =0 khi knh khng c nhiu

b ( / )k lI x y l lng tin b mt i do nhiuc ( / )k lI x y l lng tin c iu kin

d ( / )k lI x y =1/2 khi knh khng c nhiu

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15/ ( / )k lI x y l thng tin hu nghim v kx ( / ) 1k lp x y = khi vic truyn tin khng b nhiu Chn cu

sai trong nhng cu v ( / )k lI x y di y:

a ( / )k lI x y l lng tin b tn hao do nhiu

b ( / )k lI x y = 0 khi knh khng c nhiu

c ( / )k l

I x y >1/2 khi knh khng c nhiu

d ( / )k lI x y l lng thng tin v kx khi bit ly

16/ ( / )k lI x y l lng thng tin ring ca kx khi bit ly v ( / )k lI x y = 0 khi khng c nhiu Cu

ny ng hay sai ?a ngb Sai

17/ Chn cu sai trong nhng cu sau:a Lng tin cn li ca k sau khi nhn c ly k hiu l ( / )k lI x y

b ( , )k lI x y l lng tin ring ca kv ly

c Lng tin ( )kI x l lng tin ban u ca kd Lng tin ( )kI x l lng tin ban u ca kx , Lng tin cn li ca kx sau khi nhn c

ly k hiu l ( / )k lI x y

18/ Cho tini

x c xc sut l ( ) 0,5i

P x = , lng tin ring ( )i

I x ca tin ny bng cc i lng no

di y :a 4 btb 1 btc 1/4 btd 2 bt

19/ Cho tinic xc sut l ( ) 1/ 4

iP x = , lng tin ring ( )

iI x ca tin ny bng cc i lng no

di y :a 2 btb 4 btc 3 btd 1/2 bte 1/4 bt

20/ Cho tini

x c xc sut l ( ) 1/ 8i



di y :a 5 btb 3 btc 4 btd 1/4 bt

21/ Cho tini

x c xc sut l ( ) 1/16i



di y:a 1/4 btb 2 bt

c 3 btd 4 bt

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22/ Cho tini

x c xc sut l ( ) 1/ 27i



di y:a 2log7 btb Log1/9 bt;c Log27 bt;d Log1/27 bt;

23/ Cho tin i c xc sut l ( ) 1/ 9iP x = , lng tin ring ( )iI x ca tin ny bng cc i lng nodi y :

a Log9 btb Log1/3 btc 2log3 btd Log1/9 bt

24/ Cho tini

x c xc sut l ( ) 1/ 25i



di y :a Log2/5 bt

b 2log5 btc Log1/25 btd - Log25 bt

25/ Tm cu sai trong nhng cu di ya bt ngca tin

ix trong ngun tin

NX c tnh bng entropy ca lp tin

itrong ngun

tinN

X

b bt nh ca tin v lng tin v ngha tri ngc nhau nhng v gi tr li bng nhauc bt nh ca tin v lng tin c ngha nh nhau nhng gi tr khc nhaud Lng tin trung bnh c hiu l lng tin trung bnh trong mt tin bt k ca ngun tin

cho

26/Lng thng tin ring ( btnh) ca mt bin ngu nhin kx l ( )kI Chn biu thc sai trong cc biu thc di y

a ( ) ( )k kI ln px x= n vo l bit;b ( ) ( )k kI lgp x= n vo l hart;c ( ) ( )k kI ln p x= n vo l nat;d ( ) ( )k 2 kI log p x= n vo l bt

27/ Lng thng tin ring ( bt nh) ca mt bin ngu nhin kl ( )kI c tnh bng biu thcno di y :

a ( ) ( )k kI k ln px x= ;b ( ) ( )k kI ln px x= n vo l bit;c ( ) ( )k 2 kI log px x= n vo l nat;d ( ) ( )k kI lg px x= n vo l hart;

28/ Lng thng tin ring ( btnh) ca mt bin ngu nhin k l ( )kI x c tnh nh sau( ) ( )k kI k ln px x= , trong k l h s t l Tm cu sai v cch chn k trong cc cu di y :a Chn k = 1 ta c ( ) ( )k kI ln p x=

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b Chn k =-1 ta c ( ) ( )k kI ln p x= ;

c Chn1

kln10

= ta c ( ) ( )k kI lg px x= ;

d Chn1

kln 2

= ta c ( ) ( )k 2 kI log px x=

29/ Entropy ca ngun tin ri rc A l trung bnh thng k ca lng thng tin ring ca cc tin thucA K hiu:

( )1H A ; ( ) ( )1 iH A M I a= vi ( ) ( ) ( )

1 2 s

1 2 s

a a ... aA

p a p a ... p a

=

( )i0 p a 1 ;

( )s

ii 1

p a 1=

= ; ( )1H A c tnh bng biu thc no di y:

a ( ) ( ) ( )s

1 i i

i 1

H A p a log p a

=

= (bt) ;

b ( ) ( ) ( )s 1

1 i ii 1

H A p a log p a

=

= (bt) ;

c ( ) ( ) ( )s

1 i ii 0

H A p a log p a=

= (bt) ;

d ( ) ( ) ( )s

1 i ii 1

H A p a log p a=

= (bt) ;

30/ Entropy ca ngun tin ri rc A l trung bnh thng k ca lng thng tin ring ca cc tin thucA

K hiu: ( )1H A ; ( ) ( )1 iH A M I a= vi ( ) ( ) ( )

1 2 s

1 2 s

a a ... aA

p a p a ... p a

=

( )i0 p a 1 ; ( )s

ii 1

p a 1=

= ( )1H A c tnh bng biu thc no :

a ( ) ( ) ( )s

1 i ii 1

H A p a log p a=

= (bt) ;

b ( ) ( ) ( )s

1 i ii 0

H A p a log p a=

= (bt)

c ( ) ( )( )

s

1 ii 1 i

1H A p a log

p a== (bt)

d ( ) ( ) ( )s 1

1 i ii 1

H A p a logp a+

=

= (bt) ;

31/ Entropy ca ngun tin ri rc A l trung bnh thng k ca lng thng tin ring ca cc tin thucA

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K hiu: ( )1H A ; ( ) ( )1 iH A M I a= vi ( ) ( ) ( )

0 1 s 1

0 1 s 1

a a ... aA

p a p a ... p a

=

( )i0 p a 1 ; ( )s 1

ii 0

p a 1

=

= ( )1H A c tnh nh sau:

a ( ) ( ) ( )

s

1 i ii 0H A p a log p a== (bt)

b ( ) ( ) ( )s 1

1 i ii 1

H A p a log p a

=

= (bt)

c ( ) ( ) ( )s

1 i ii 1

H A p a log p a=

= (bt) ;

d ( ) ( ) ( )s 1

1 i ii 0

H A p a logp a

=

= (bt);

32/ A v B l hai trng bin c bt k, Entropy ca 2 trng bin cng thi C=AB l H(AB)Trong cc tnh cht ca H(AB) di y, tnh cht no sai:

a ( ) ( ) ( )H AB H A H B A= + ;b ( ) ( ) ( )H AB H B H A B= + ;

c ( ) ( ) ( )s

1 i ii 1

H A p a log p a=

= (bt) ;

d ( ) ( ) ( )s 1

1 i ii 0

H A p a logp a

=

= (bt);

33/ Entropy c iu kin v 1 trng tin A khi r trng tin B l H(A/B)Trong cc tnh cht ca H(A/B) di y, tnh cht no sai

a ( ) ( )H A B H B/ A ;b ( )0 H A B ;c ( ) ( )H A H A B d ( ) ( )H A B H A ;

34/ Entropy c iu kin v 1 trng tin B khi r trng tin A l ( )H B/ A , Tnh cht no ca( )H B/ A di y l nga ( )0 H B/ A ;b ( )0 H B/ A ;c ( ) ( )H A B H B/ A ;d ( ) ( )H B/ A H A ;

35/ Entropy ca trng bin cng thi H(AB) c tnh bng cng thc no sau y

a H(A) + H(A/B);b H(A) + H(B)c H(A) + H(B) - H(A/B) - H(B/A);d H(B) + H(A/B);

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36/ Lng thng tin cho trung bnh (k hiu l I(A,B) c cc tnh cht no sau y

a ( ) ( )I A,B H A= khi knh c nhiu;b ( ) ( )0 I A,B H A ;c ( ) ( )H A I A,B 0 ;

d ( ) ( )I A,B H A= khi knh c nhiu;

37/ Lng thng tin cho trung bnh (k hiu l I(A,B) Trong cc tnh cht di y, tnh cht nosai:

a ( ) ( )I A,B H A ;b ( ) ( )I A,B H A= khi knh c nhiu;c ( )0 I A,B ;d ( ) ( )I A,B H A= khi knh khng c nhiu;

38/ Lng thng tin cho trung bnh (k hiu l I(A,B) c cc tnh cht no sau ya ( )I A,B 1= khi knh bt;b ( )0 I A,B v ( )I A,B 0= khi knh bt;c ( )0 I A,B d ( ) ( )I A,B H A= khi knh c nhiu;

39/ Lng thng tin cho trung bnh (k hiu l I(A,B) ), trong cc tnh cht di y ca I(A,B), tnhcht no sai

a ( ) ( )I A,B H A= khi knh khng c nhiu;b ( )I A,B 0= khi knh bt;c ( ) ( )H A I A,B ;d ( )I A,B 1 khi knh bt;

40/ Lng thng tin cho trung bnh (k hiu l I(A,B)) , tm biu thc sai trong cc biu thc diy

a I(A,B) = H(A) - H(A/B);

b I(A,B) = ( ) ( )( )

s t

i ji ji 1 j 1 i

p a bp a b log p a= =

c I(A,B) = H(A) - H(B/A);d I(A,B) = H(B) - H(B/A) ;

41/ Mnh no sau y saia H(A/B) H(A) ;b H(A,B) H(A) + H(B)c I(A,B) = H(A) + H(B) + H(AB);d I(A,B) = H(A) + H(B) - H(AB);

42/ Chn ngu nhin mt trong cc s t 0 n 7 c xc sut nhnhau btnh ca scchn ngu nhin l

a 1/8 bt;

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b -3 bt;c 3 bt;d 8 bt;

43/ Mt thit b v tuyn in gm 16 khi c tin cy nh nhau v c mc ni tip Gi s cmt khi no b hng, bt nh ca khi hng l:

a 1/16 bt;

b 16 bt;c 1/4 bt;d 4 bt;

44/ B t lkh52 qun (khng k fng teo), A rt ra mt qun bi bt k bt nh v qun bim A rt l:

a Nh hn 5 bt;b Ln hn 5 nh hn 6 bt;c Bng 6 btd Ln hn 6 bt ;

45/ Mt hp c 8 ng tin kim loi , trong c 02 ng tin 500 ng; 02 ng tin 1000 ng, 2ng tin 2000 v 2 ng tin 5000 ng Chn ngu nhin 1 trong 8 ng tin Khi xc sut cang tin c chn ngu nhin l:

a 8 bt ;b -1/2 bt;c 1/8 bt;d 1/4 bt ;

46/ Mt hp c 8 ng tin kim loi , trong c 02 ng tin 500; 02 ng tin 1000, 2 ng tin2000 v 2 ng tin 5000 Chn ngu nhin 1 trong 8 ng tin Khi bt nh ca ng tinc chn ngu nhin l:

a 1/4 bt;b 8 bt ;c -1/2 bt;d 2 bt;

47/ Cho ngun tin X = {x1, x2, x3} vi cc xc sut ln lt l {1/2, 1/4, 1/4}, Entropy ca ngun tinH(X) c tnh l:

a 1 1 1log2 + log4 + log82 4 4

b 1 1 1log2 + log4 + log42 4 2

c 1 1 1log2 + log4 + log42 4 4

;

d 1 1 1log2 + log4 + log42 2 4

48/ Cho ngun tin X = {x1, x2, x3, x4, x5, x6, x7, x8, x9} vi cc xc sut ln lt l {1/4, 1/8, 1/8,1/8, 1/16, 1/16, 1/8, 1/16, 1/16}. Trong cc kt qu tnh Entropy di y, kt qu no sai:

a 1 1 1log4 + 4 log8 + 4 log164 8 8

b 1 3log4 + log2 + log22 2

c 1 1 1log4 + 4 log8 + 4 log164 8 16

d 1 3log4 + log2 + log24 2

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49/Entropy ( )1H A ca ngun ri rc

( ) ( ) ( )1 2 s

1 2 s

a a ... aA

p a p a ... p a

=

vi ( )i0 p a 1 ; ( )

s

ii 1

p a 1=

= Trong cc tnh cht ca H(A) di y tnh cht no l sai

a Khi ( )kp a 1= , ( )ip a 0= vi i k th ( ) ( )1 1 minH A H A 1= = b Ngun tin ri rc A c s du ng xc sut cho entropy cc i ( )1 maxH A logs=

c Khi ( )kp a 1= , ( )ip a 0= vi i k th ( ) ( )1 1 minH A H A 0= =

d Entropy ca ngun ri rc A l mt i lng gii ni ( )10 H a logs

50/ Cho ngun ri rc A

( ) ( ) ( )1 2 s

1 2 s

a a ... aA

p a p a ... p a

=

vi ( )i0 p a 1 ; ( )

s

ii 1

p a 1=

=

Gi entropy ca ngun A l ( )1H A , trong cc biu thc tnh ( )1 maxH A logs di y, biuthc no sai:

a ( )1 maxH A logs = ( )s

ii 1 i

1p a log logs

p(a )=+

b ( )1 maxH A logs = ( )s

ii 1 i

1p a log logs

p(a )=

;

c ( )1 maxH A logs = ( )s s

i i

i 1 i 1i

1p a log p(a )logs

p(a )= = ;

d ( )1 maxH A logs = ( )s

ii 1 i

1p a log logs

p(a )= ;

51/ Cho ngun ri rc A

( ) ( ) ( )1 2 s

1 2 s

a a ... aA

p a p a ... p a

=

vi ( )i0 p a 1 ; ( )

s

ii 1

p a 1=

= Nu ngun A c s du ng xc sut , khi biu thc no di y l sai

a ( )

s s

ii 1 i 1

1p a 0s= =+ =

bi

1p(a )

s=

1 i s ;

c ( )s s

ii 1 i 1

1p a 0

s= = = ;

d 1 1H (A) logs 0 H (A) logs

52/ Kh nng thng qua ca knh ri rc C l gi tr cc i ca lng thng tin cho trung bnh

truyn qua knh trong mt n v thi gian ly theo mi kh nng c th c ca ngun tin A

( ) ( )' ' kA A

C max I A,B v max I A,B= = (bps);

KA

C' .C v m I(A,B)i C = ax=

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K biu th s du m knh truyn c (c truyn qua knh) trong mt n v thi gianI(A,B) l lng thng tin truyn qua knh trong mt n v thi gian C c gi l kh nngthng qua ca knh i vi mi du C c cc tnh cht no di y :

a C 0, C = 0 khi A v B c lp ; C Klogs , ' kC v logs= khi knh khng nhiub C = 0 khi v ch khi A v B c nhiu

c

'

kC v logs= khi knh c nhiud ' kC v logs= khi cc knh c lp

53/ I(A,B) l lng thng tin trung bnh c truyn qua knh ri rc c tnh cht : I(A,B) H(A)V mt snh ngha :

( ) ( )' ' kA A

C max I A,B v max I A,B= = ,

KA

C' .C v m I(A,B)i C = ax=

K biu th s du m knh truyn c trong mt n v thi gian. T cc tnh cht v nhngha trn cho bit cc biu thc di y, biu thc no sai

a K I(A,B) K H(A);

b K I(A,B) K H(A);

c K maxI(A,B) K maxH(A)

d max( K I(A,B)) max( K H(A));

54/ Cho ngun ri rc ch c hai du:1 2

1 2

a aA

p(a ) p(a )

=

Ngun ri rc nh phn l ngun A trn tho mn iu kin sau

1

2

a "0" v ) p

a "1" v ) 1 p1

2

i xc sut p(a

i xc sut p(a

=

= Khi ngun ri rc nh phn A c th vit biu thc no

a1 2

p 1 pA

a a

=

b1

2

a pA

a 1 p

=

c 1 0Ap 1 p

=

d1 2a aA

p 1 p

=

55/ 1 2a a

Ap 1 p

=

l ngun ri rc nh phn Tho mn iu kin

1

2

a "0" v ) p

a "1" v ) 1 p1

2

i xc sut p(a

i xc sut p(a

=

= Khi Entropy ( )1H A c tnh bng cng thc no sau y

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a ( ) ( )p log p 1 p log 1 p

b ( ) ( )p log p 1 p log 1 p ;

c ( ) ( )p 1 p log 1 p p log ;

d ( ) ( )p log p 1 p log 1 p

56/ ( )lH A / b l lng thng tin tn hao trung bnh ca mi tin u pht khi u thu thu cjb ( ) ( ) ( )

s

l i l i li 1

H A / b p a / b log p a b=

= ; iH(B / a ) l lng thng tin ring trung bnh cha trong mi tin u thu khi u pht pht i mt tin

ia c tnh theo cng thct

i i ij 1

H(B/ a ) / a ) log p / a )j jp(b (b=

=

Trong trng hp knh bt (b nhiu tuyt i) ta c biu thc no di y l sai :

a ( ) ( )jH A b H A H(B)= + b ( ) ( )iH B a H B= ;c ( ) ( )H B A H B=

d ( ) ( )jH A b H A= = ( )H A B

57/ ( )lH A / b l lng thng tin tn hao trung bnh ca mi tin u pht khi u thu thu c

jb

( ) ( ) ( )s

l i l i li 1

H A / b p a / b logp a b=

= ; iH(B / a ) l lng thng tin ring trung bnh chatrong mi tin u thu khi u pht pht i mt tin c tnh theo cng thc:

t

i i ij 1

H(B/ a ) / a ) logp / a )j jp(b (b=

=

Trong trng hp knh khng nhiu biu thc no di y l ng :

a ( )kH A b H(A / A) 0= = ;b ( )kH A b H(B/ A) 0= = c ( )kH A b H(B/ B) 0= = ;d ( )kH A b H(A / B) 0= = ;

58/ ( )lH A / b l lng thng tin tn hao trung bnh ca mi tin u pht khi u thu thu c

bj . ( ) ( ) ( )s

l i l i li 1

H A / b p a / b logp a b=

= ; iH(B / a ) l lng thng tin ring trung bnh chatrong mi tin u thu khi u pht pht i mt tin aic tnh theo cng thc sau:

t

i i ij 1

H(B/ a ) / a )log p / a )j jp(b (b=

=

Trong trng hp b nhiu tuyt i, A v B l c lp nhau suy ra :

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i j ip(a / b ) p(a )= ; j i jp(b / a ) p(b )= i j i jp(a b ) p(a )p(b ) = , khi ta c biu thc no sau

y l ng:

a ( ) ( ) ( )s

j i ii 1

H A / b p a log p a=

= v ( ) ( ) ( )t

i j jj 1

H B/ a p b logp b=

=

b ( ) ( ) ( )

t

i j jj 1H B/ a p b log p a== c ( ) ( ) ( )

s

j i ii 1

H A / b p a log p b=

= v ( ) ( ) ( )t

i j jj 1

H B/a p b log p a=

=

d ( ) ( ) ( )s

j i ii 1

H A / b p a log p b=

=

59/ Entropy c iu kin v 1 trng tin A khi r trng tin B l ( )H A B , c xc nh theocng thc sau:

( ) ( ) ( )s t

i j i ji 1 j 1

H A B p a b log p a b= =

= Trong trng hp b nhiu tuyt i, A v B l c lp nhau, suy ra :

i j ip(a / b ) p(a )= ; j i jp(b / a ) p(b )= i j i jp(a b ) p(a )p(b ) = , khi biu thc no sau y l

ng:

a ( ) ( ) ( )s t

i j i ji 1 j 1


=

b ( ) ( ) ( ) ( )t s

j i ij 1 i 1H A B p b p a log p a= ==

c ( ) ( ) ( )s t

i j i ji 1 j 1

H A B p a / b log p a b= =

=

d ( ) ( ) ( ) ( )t s

j i ij 1 i 1

H A B p b p a log p a= =

=

60/ Entropy c iu kin v 1 trng tin B khi r trng tin A l ( )H B A , c xc nh theo

cng thc sau: ( ) ( ) ( )s t

j i j ii 1 j 1

H B/ A p b a log p b a= =


i j ip(a / b ) p(a )= ; j i jp(b / a ) p(b )= i j i jp(a b ) p(a )p(b ) = Khi biu thc no sau y l ng

a ( ) ( ) ( ) ( )s t

i j ji 1 j 1

H B/ A p a p b logp b= =

= ;

b( ) ( ) ( ) ( )

t s

j i ij 1 i 1

H B/ A p b p a logp a= =

=

;

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c ( ) ( ) ( )s t

i j i ji 1 j 1

H B/ A p a b log p a b= =

= ;

d ( ) ( ) ( )s t

i j i ji 1 j 1

H B/ A p a / b log p a b= =

=

61/ Entropy c iu kin v 1 trng tin A khi r trng tin B l ( )H A B , c xc nh theo

cng thc sau: ( ) ( ) ( )s t

i j i ji 1 j 1




ng:

a ( ) ( ) ( ) ( )t s

j i ij 1 i 1

H A B p b p a logp a H(A)= =

= =

b( ) ( ) ( )

s t

i j i ji 1 j 1

H A B p a b log p a b H(A)= =

= = ;

c ( ) ( ) ( )s t

i j i ji 1 j 1

H A B p a / b log p a b H(A)= =

= =

d ( ) ( ) ( ) ( )t s

j i ij 1 i 1

H A B p b p a log p a H(A)= =

= =

62/ Entropy c iu kin v 1 trng tin B khi r trng tin A l ( )H B A , c xc nh theocng thc sau:

( ) ( ) ( )s t

j i j ii 1 j 1

H B/ A p b a log p b a= =



ng:

a ( ) ( ) ( )s t

i j i ji 1 j 1

H B/ A p a / b log p a b H(B)= =

= =

b ( ) ( ) ( ) ( )t s

j i ij 1 i 1

H B/ A p b p a log p a H(B)= =

= =

c ( ) ( ) ( ) ( )s t

i j ji 1 j 1

H B/ A p a p b log p b H(B)= =

= =

d ( ) ( ) ( )s t

i j i ji 1 j 1

H B/ A p a b logp a b H(B)= =

= =

63/ Entropy c iu kin v 1 trng tin A khi r trng tin B l ( )H A B , c xc nh theocng thc sau:

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( ) ( ) ( )s t

i j i ji 1 j 1


= vi i ji jj

p(a b )p(a / b )

p(b )=

T cng thc ny c th khai trin ( )H A B thnh cng thc no sau y:

a ( ) ( )s t

i j i ji 1 j 1

p a b logp b a= =

b ( ) ( )s t

i j i ji 1 j 1

p a / b logp a ,b= =

c ( ) ( )t s

j i j i jj 1 i 1

p(b ) p a / b logp a b= =

;

d ( ) ( )s t

i j i ji 1 j 1

p a / b logp a b= =

;

64/ Lng thng tin cho trung bnh (k hiu l I(A,B)) :

i jI(A,B) M I(a ,b )

= vi ( )( )

( )i j

i ji

p a bI a ,b log

p a= Xc sut c thng tin i j

I(a ,b )l

i jp(a b ) Do c th vit I(A,B)) bng cng thc no sau y:

a ( )( )( )

s ti j

i ji 1 j 1

i

p a bp a b log

p a= = ;

b ( )( )

( )

s ti j

i ji 1 j 1 i

p a bp a b log

p a= = ;

c ( )( )

( )

s ti j

i ji 1 j 1 i

p a bp a / b log

p a= = ;

d ( )( )( )

s ti j

i ji 1 j 1 i j

p a bp a b log

p a b= =

65/ Lng thng tin cho trung bnh (k hiu l I(A,B)) c vit thnh :

( ) ( )( )

( )

s ti j

i ji 1 j 1 i

p a bI A,B p a b log

p a= == Khi c th khai trin I(A,B) thnh

I(A,B)= ( ) ( ) ( )s t

i j i j ii 1 j 1

p a b logp a b logp a= =

a Saib ng


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( ) ( )( )

( )

s ti j

i ji 1 j 1 i



I(A,B)= ( ) ( ) ( ) ( )s t s t

i j i j i j ii 1 j 1 i 1 j 1

p a b logp a b p a b logp a= = = =

a ngb Sai


( ) ( )( )

( )

s ti j

i ji 1 j 1 i



I(A,B)= ( ) ( ) ( )s t

i j i j ii 1 j 1

p a b logp a b p b= =

a Sai

b ng


( ) ( )( )

( )

s ti j

i ji 1 j 1 i


p a= == Trong cc biu thc khai trin I(A,B) di y, biu thc

no sai

a I(A,B) = ( ) ( ) ( )s t

i j i i ji 1 j 1

p a b logp a logp a b= =

;

b I(A,B) = ( ) ( ) ( )s t

i j i j ii 1 j 1

p a b logp a b logp b= =

+ ;

c I(A,B) = ( ) ( ) ( ) ( )s t s t

i j i j i j ii 1 j 1 i 1 j 1

p a b logp a b p a b logp a= = = =

;

d I(A,B) = ( ) ( ) ( )s t

i j i j ii 1 j 1

p a b logp a b logp a= =

;


( ) ( ) ( )( )

s t i ji j

i 1 j 1 i

p a bI A,B p a b logp a= =

=

Entropy c iu kin v 1 trng tin A khi r trng tin B l ( )H A B , c xc nh theo cngthc sau:

( ) ( ) ( )s t

i j i ji 1 j 1

H A B p a b logp a b= =

= Khi trong cc kt qu tnh I(A,B), kt qu no sai

a I(A,B) = H(A) - H(A/B) ;

b I(A,B) = H(A)+ H(A/B) ;c I(A,B) = H(A) + H(B) - H(AB);d I(A,B) = H(B) - H(B/A);

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70/ Xt 2 trng s kin A v B sau :( ) ( )

ji

i j

baA i 1,s ;B j 1, t

p a p b

= = = =

Khi , trng s kin ng thi C A.B= Nu A v B l c lp th C c th vit thnh biu thcno di y:

a ( ) ( )i j

i j

a / b

C p a / p b

=

;

b( ) ( )

i j

i j

a bC

p a p b

+ =

+

cji

ji

baC

p(b )p(a )

=

;

d ( ) ( )

i j

i j

a b

C p a p b

= ;

71/ Xt 2 trng s kin A v B sau :( ) ( )

ji

i j

baA i 1,s ;B j 1, t

p a p b

= = = =

Trng s kin ng thi C A.B= c entropy H(C) c tnh bng cng thc no di y:

a ( ) ( ) ( )s t

i j i ji 1 j 1

H C p a b log p a b= =

=

b ( ) ( ) ( )s t

i j i ji 0 j 0


=

c ( ) ( ) ( )s 1 t 1

i j i ji 1 j 1

H C p a b log p a b

= =

= ;

d ( ) ( ) ( )s t

i j i ji 1 j 1


= ;

72/ Chn cu sai :

a Xc sut xut hin cng ln, lng tin thu c cng lnb Mt tin x c xc sut xut hin l p(x), nu p(x) cng nh th lng tin khi nhn c tin nycng s cng ln

c Nu p(x) cng ln th 1/p(x) cng nhd Mt tin x c xc sut xut hin l p(x), nu p(x) cng ln th lng tin khi nhn c tin ny

cng s cng nh

73/ Chn cu sai sau :a Xc sut xut p(x) cng ln th lng tin khi nhn c tin ny cng s cng lnb Xc sut xut hin ca mt tin t l nghch vi bt ngkhi nhn c mt tinc Xc sut xut p(x) cng ln th lng tin khi nhn c tin ny cng s cng nh

d Lng tin ca mt tin t l thun vi s kh nng ca mt tin v t l nghch vi xc sutxut hin ca tin

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74/ Lng tin c iu kin hu nghim v Kx ( thng tin ring v Kx sau khi c y ) c nh nghal ( / ) log ( / )

k l k l I x y p x y=

Chn cu sai trong cc cu sau :a Xc sut ( / ) 1

k lP x y = ch xy ra khi knh truyn khng c nhiu

b Khi ( / ) 1k l

P x y th ( / ) 1k lI x y v ngc li

c Xc sut ( / ) 1/ 2k lP x y = ( / ) 1k lI x y = bt ,d Khi ( / ) 1k l

P x y th ( / ) 0k l

I x y v ngc li

75/ Lng thng tin v Kx khi r tin y l( / )

( , ) log( )k l

k l

k

p x yI x y

p x=

Chn cu sai sau :

a ( / ) 1k l

p x y = tc l l lng tin ca Kx c truyn nguyn vn

b Nu ( / ) 0k l

p x y = suy ra1

( , ) log( )k l

k

I x yp x

=

c Nu ( / ) 1k l

p x y = suy ra 1( , ) log( )k l

k

I x yp x

=

d Nu ( / ) 1k l

p x y = , c ngha l khi y nhn c tin th chc chn Kx pht

76/ Lng thng tin hu nghim v Kx ( thng tin ring v Kx sau khi c y ) c vit l :1

( / ) log( / )k l

k l

I x yp x y

= . Lng thng tin ring v Kx l1

( ) log( )k

k

I xp x

=

Lng thng tin cho v Kx do y mang li l :1 1

( , ) log log

( ) ( / )k l

k k l

I x y

p x p x y

=

Tm cu sai sau :a Lng thng tin ring bng tng lng thng tin cho v lng thng tin hu nghimb Lng thng tin ring c th mc Tng lng thng tin cho v lng thng tin hu nghim bng lng thng tin ringd Lng thng tin ring lun dng

77/ Cho 2ngun tin A v B c cc xc sut tng ng l :

1 2 3 4a a a aA1/ 2 1/ 4 1/8 1/8

=

;

1 2 3 4b b b bB0,5 0,25 0,125 0,125

=

Entropy ca ngun A ( k hiu l H(A)), entropy ca ngun B ( k hiu l H(B)) c quan h theo cch thc no di y

a H(A)=H(B);b H(A) > H(B);c H(B)>H(A);d H(A)=2H(B)

78/ Cho ngun tin A c cc xc sut tng ng l :

1 2 3 4a a a aA1/ 2 1/ 4 1/8 1/8

=

; Khi Entropy ca ngun A ( k hiu l H(A)) bng cc i lngno di ya H(A) = 1,85 bt;b H(A) = 1,75 bt ;

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c H(A) = 1,7 bt;d H(A) = 1,65 bt ;


1 2 3 4 5a a a a aA0.45 0.2 0.15 0.1 0.1

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) gn bng cc i lng no di ya H(A) = 2,75 bt;b H(A) = 2,06 bt;c H(A) = 2,7 bt;d H(A) = 2,85 bt;


1 2 3 4 5a a a a aA0.4 0.25 0.15 0.1 0.1

=

;



1 2 3 4 5a a a a aA0.4 0.25 0.15 0.15 0.05

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) gn bng cc i lng no di y

a H(A) = 2,85 bt;b H(A) = 2,7 bt;c H(A) = 2,06 bt;d H(A) = 2,07 bt;


1 2 3 4 5a a a a aA0.4 0.25 0.2 0.1 0.05

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) gn bng cc i lng no di ya H(A) = 2,06 bt;

b H(A) = 2,85 bt;c H(A) = 2,04 bt;d H(A) = 2,07 bt;


1 2 3 4 5a a a a aA0.4 0.3 0.15 0.1 0.05

=

;


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84/ Gi s ngun tin A c cc xc sut tng ng l :

1 2 3 4 5a a a a aA0.4 0.3 0.2 0.05 0.05

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) gn bng cc i lng no di ya H(A) = 1,95 bt;b H(A) = 2,07 bt;

c H(A) = 2,01 bt;d H(A) = 2,04 bt;


1 2 3 4 5a a a a aA0.35 0.35 0.2 0.05 0.05

=

;

Khi Entropy ca ngun A (k hiu l H(A)) gn bng cc i lng no di ya H(A) = 2,04 bt;b H(A) = 2,01 bt;c H(A) = 2,07 bt;d H(A) = 1,96 bt;


1 2 3 4 5a a a a aA0.35 0.3 0.25 0.05 0.05

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) gn bng cc i lng no di ya H(A) = 2,01 bt;b H(A) = 1,9 bt;c H(A) = 1,98 bt;

d H(A) = 2,04 bt;


1 2 3 4 5a a a a aA0.35 0.3 0.2 0.1 0.05

=

;



1 2 3 4 5a a a a aA0.3 0.3 0.2 0.1 0.1

=

;


89/ Gi s ngun tin A v B c cc xc sut tng ng l :

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1 2 3 4 5 6 7 8a a a a a a a aA1/ 4 1/ 8 1/16 1/16 1/ 4 1/ 8 1/16 1/16

=

;

1 2 3 4 5 6 7 8b b b b b b b bB0, 25 0,125 0, 0625 0, 0625 0, 25 0,125 0, 0625 0, 0625

=

Entropy ca ngun A ( k hiu l H(A)), entropy ca ngun B ( k hiu l H(B)) c quan h theo cch thc no di y

a H(A) = 2H(B)b H(A)=H(B) ;c H(B)>H(A);d H(A) > H(B);


1 2 3 4 5 6 7 8a a a a a a a aA1/ 4 1/8 1/16 1/16 1/ 4 1/8 1/16 1/16

=

;

Khi Entropy ca ngun A ( k hiu l H(A)) bng cc i lng no di y

a H(A) = 2,7 bt;b H(A) = 2,75 bt;c H(A) = 2,85 bt;d H(A) = 2,80 bt;

91/ Gi s ngun tin B c cc xc sut tng ng l :

1 2 3 4 5 6 7 8b b b b b b b bB0,25 0,125 0,0625 0,0625 0,25 0,125 0,0625 0,0625

=

;Khi Entropy ca ngun B ( k hiu l H(B)) bng cc i lng no di y

a H(B) = 2,80 bt;

b H(B) = 2,7 bt;c H(B) = 2,75 bt;d H(B) = 2,85 bt;

92/ Cho mt knh nh phn nh hnh bn:

p(b1/a1) = pda1 b1

a2 b2p(b2/a2) = pd

p(b1/a2)= ps

Trong :Phn b xc sut ca tin u ra p( 1b )c tnh theo cng thc sau :

2

1 11

( ) ( ) ( / )i i

i

p b p a p b a=

=

T cng thc ny, c th khai trin p( 1b ) thnh cng thc no di y:

a ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 2p b p a .p b a p a .p b a= + ;

b ( ) ( ) ( ) ( ) ( )1 2 1 1 2 1 2p b p a .p b a p a .p b a= + ;c ( ) ( ) ( ) ( ) ( )1 1 1 1 2 1 2p b p a .p b a p a .p b a= + ;

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95/ Cho knh i xng nh phn nh hnh bn:

p(b1/a1) = pda1 b1

a2 b2p(b2/a2) = pd

p(b1/a2)= ps

Bit

( )1p a p= ; ( )2 1p a p= ( ) ( )1 2 2 1 1s dp b a p b a p p= = =

p(b1/a

1) = p(b

2/a

2) = p

d

( ) ( ) ( )s s s sH B A p log p 1 p log 1 p = +

T truyn tn hiu cho knh 1kv T= v ( ) ( ) ( )'

A A

1 1C max I A,B max H B H B AT T= =

Khi '

'max

C

Cc tnh theo biu thc no di y

a ( ) ( )'

s s s s'max

C1 p log p 1 p log 1 p

C= +

b ( ) ( )'

s s s s'max


C= +

c ( ) ( )'

s s s s'max

C 1 p log p 1 p log 1 pC

= + +

d ( ) ( )'

d d d d'max


C= + +

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CHNG 4: CSL THUYT M

1/ Chn cu ng v m haa M ha l php bin i mi dng tn hiu thnh tn hiu lin tcb M ha l nh x che du thng tin

c M ha l mt nh x 1- 1 t tp cc tin ri rc ia ln tp cc t min

i ;in

i i: af d M ha l php bin i dng s nh phn thnh dng tn hiu ban u

2/ Chn cu ng v m :a M l mt tp cc t m c lp nn mt cch ngu nhinb M l qu trnh phc hi tin tcc M l nh x che du thng tind M (hay b m) l sn phm ca php m ha

3/ Chn cu ng v m :a M l qu trnh phc hi tin tc

b M l mt tp cc t m c lp nn theo mt lut nhc M (hay b m) l sn phm ca php bin i tn hiu ri rcd M (hay b m) l sn phm ca php bin i tn hiu lin tc

4/ di t m in l s cc du m cn thit dng m ha cho tin ia Chn cu ng v di tm

a di t m in l mt s phcb Nu in const= vi mi i th mi t m u c cng di B m tng ng c gi l b

m u

c di t m in cng ln th php m ha cng ti ud di t m in l mt s nguyn c th m hoc dng

5/ di t m in l s cc du m cn thit dng m ha cho tin ia Chn cu sai v di t m

a Nu in const= vi mi i , khi b m tng ng c gi l b m ub di t m in l mt s nguyn lun ln hn hoc bng 1c Nu in const= i c ngha l tt c cc t m u c cng di

d di t minl mt s nguyn c th m

6/ di t m in l s cc du m cn thit dng m ha cho tin ia Chn cu ng v di t m

a Nu i jn n th b m tng ng c gi l b m khng u

b di t m in cng ln th php m ha cng ti uc Nu in = 1 th b m tng ng c gi l b m khng ud Nu in const= i , c ngha l php m ha l ti u

7/ S cc du m khc nhau (v gi tr) c s dng trong b m c gi l cs m Ta k hiugi tr ny l m Chn cu sai vcc du m m:

a Nu m = 2 th b m tng ng c gi l m nh phn

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b Nu m = 3 th b m tng ng c gi l m tam phnc Nu m = 0 th b m tng ng c gi l m ud Nu m = p th b m tng ng c gi l m p phn

8/ Gi s c t m 7i 0 1 1 0 1 0 1 = Chn cu ng nht v t m7i

a T m 7i c di bng 4

b T m 7i trong b m nh phn c m=2 ( tc l c 2 du m l 0 v 1) v c di l 7

c T m 7i c 7 du m

d T m 7i c 3 du m

9/ Gi s c t m 7i 0 1 1 0 1 0 1 = Chn cu ng v t m7i

a T m 7i c 7 du m

b T m 7

i trong b m nh phn c m=3, tc l c 3 du m l 0

c T m 7i c di bng 7

d T m 7i c di bng 3

10/ Chn cu ng v di t m

a di trung bnh ca t m n l k vng ca i lng ngu nhin in c xc nh nh

sau : ( )s

i ii 1

n p a n=

=

b di trung bnh ca t m n l k vng ca i lng ngu nhin in c xc nh nh

sau : ( )s

i ii 1

n p a n=

= c di t m in l s cc du m cn thit dng gii m cho tin ia d di t m in l S cc du m khc nhau (v gi tr) c s dng trong gii m

11/ Ni dung ca nh l m ho th nht ca Shannon (i vi m nh phn) c pht biu nh sau:Lun lun c thxy dngc mt php m ho cc tin ri rc c hiu qu m di trung bnhca tm c thnh tu , nhng khng nh hn entropie xc nh bi cc c tnh thng k ca

ngun . Chn cu ng v di t m :a S t m nh nhtb S t m khng ic Chiu di trung bnh ca cc t m nh hn hoc bng entropy ca ngund Chiu di trung bnh cc t m nh nht trong tt c cc cch m ha

12/ Ni dung ca nh l m ho th nht ca Shannon (i vi m nh phn) c pht biu nh sau:Lun lun c th xy dng c mt php m ho cc tin ri rc c hiu qu m di trung bnhca t m c th nh tu , nhng khng nh hn entropie xc nh bi cc c tnh thng k cangun. Chn cu sai v di t m :

a Chiu di trung bnh cc t m tho mn h thc ( ) ( )

s

i i 1i 1n p a n H A==

b Chiu di trung bnh cc t m nh nht trong tt c cc cch m hac Chiu di trung bnh ca cc t m ln hn hoc bng entropy ca ngun

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d Chiu di trung bnh cc t m tho mn h thc ( ) ( )s

i i 1i 0

n p a n H A=

= =

13/ Khong cch gia hai t m bt k ni vnj l s cc du m khc nhau tnh theo cng mt v

tr gia hai t m ny, k hiu ( )n ni jd , Gi s 7i 0 1 1 0 1 0 1 = ; 7j 1 0 0 1 1 1 0 = Khong cch gia 2 t m ni v

nj l ( )7 7i jd , bng cc i lng no di y

a 1;b 6 ;c 12;d 14;


tr gia hai t m ny, k hiu

( )n n

i j

d , . Tm biu thc sai v khong cch m d trong cc biu

thc sau:

a ( )n ni jn d , ;b ( )n ni jd , 0 c ( ) ( )n n n ni j j id , d , ;d ( ) ( )n n n ni j j id , d , =


tr gia hai t m ny, k hiu ( )n ni jd , Tm biu thc sai v khong cch m d trong cc biuthc sau:

a ( ) ( ) ( )n n n n n ni j j k i k d , d , d , + ;b ( )n ni jd , 0 ;c

( )n n

i jd , 0

d ( )n ni jn d ,


tr gia hai t m ny, k hiu ( )n ni jd , . Chn cu ng sau :a ( )n ni j0 d , 1 ;

b ( ) ( ) ( )n n n n n ni j j k i k d , d , d , + =

c ( )n ni j1 d , ;

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d ( )n ni jd , 0 ; ( )n ni jd , 0 = khi n ni j ;

17/ Trng s ca mt t m ( )niW l s cc du m khc khng trong t m V d:7i 0 1 1 0 1 0 1 = th ( )7iW 4 = . Chn cu ng v cc tnh cht ca trng s ( )niW :a ( ) ( )n n n ni j i jd , w + b ( )niW 0 c ( ) ( )n n n ni j i jd , W + ;d ( ) ( )n n n ni j i jd , W = + v ( )ni0 W 1 ;

18/ Trng s ca mt t m

( )n

iW l s cc du m khc khng trong t m Gi s

7i 0 1 1 0 1 0 1 = , th trng s ( )7iW bng s no di ya 4 ;b 3;c 5 ;d 7 ;

19/ Coi mi t m ni l mt vctn chiu trong mt khng gian tuyn tnh n chiu nV , khi phpcng c thc hin gia hai t m tng t nh php cng gia hai vcttng ng c thc hin

trn trng nh phn GF(2) Php cng theo modulo 2 ny c m t nh sau:

Cho ( )7i 0 1 1 0 1 0 1 0, 1, 1, 0, 1, 0, 1 =

( )7j 1 0 0 1 1 1 0 1, 0, 0, 1, 1, 1, 0 =

Khi 7 7 7k i j = + bng gi tr no di ya (0 1, 1, 1, 0, 1, 1);b (1 1, 1, 1, 0, 1, 1);c (1 1, 1, 1, 1, 1, 1);d (1 1, 1, 1, 0, 1, 0);

20/ Coi mi t m ni l mt vctn chiu trong mt khng gian tuyn tnh n chiu nV , khi phpcng c thc hin gia hai t m tng t nh php cng gia hai vcttng ng c thc hintrn trng nh phn GF(2) Php cng theo modulo 2 ny c m t nh sau:

Cho ( )8i 1 1 1 0 1 0 1 0 1, 1, 1, 0, 1, 0, 1, 0 =

( )8j 1 0 0 1 1 1 0 1 1, 0, 0, 1, 1, 1, 0, 1 =

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b n =1,51;

c n =1,35;

d n =1,81;

27/ Sau khi thc hin m ha ngun ri rc A ( c entropy l H(A)=1,4 ) c th tnh c di

trung bnh n Vi mi cch m ha khc nhau s tnh c n khc nhau Trong cc kt qu m ha,

gi tr n no di y c gi l ti u nht:

a n =1,51;

b n =1,45;

c n =1,55;

d n =1,75;

28/ Sau khi thc hin m ha ngun ri rc A (c entropy l H(A)=1,9) c th tnh c di trung

bnh n Vi mi cch m ha khc nhau s tnh c n khc nhau Trong cc kt qu m ha, gi tr

n no di y c gi l ti u nht:

a n =1,975;

b n =1,91;

c n =1,905;

d n =1,95;

29/ Sau khi thc hin m ha ngun ri rc A ( c entropy l H(A)=1,905 ) c th tnh c di


gi tr n no di y c gi l ti u nht:a n =1,906;

b n =1,907;

c n =1,91;

d n =1,95;

30/ Sau khi thc hin m ha ngun ri rc A (c entropy l H(A)=2,01 bt), c th tnh c di


gi tr n no di y c gi l ti u:a n =2,07;

b n =2,11;

c n =2,09;

d n =202;

31/ Cho m Cyclic C(n, k) = C(7,4) c a thc sinh lg(x) tha g(x) c bc 4 ;b g(x) l mt c ca x 7 4+ + 1

c g(x) c bc 5 ;d T g(x) c th xc nh c ma trn sinh h thng G v ma trn kim tra H cho b m

32/ Cho m Cyclic C(n, k) = C(7,4), c a thc sinh g(x) l

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a 4( ) 1g x x x= + + ;

b 2 4( ) 1g x x x x= + + +

c 2 3( ) 1g x x x= + + ;

d 2 3 4( ) 1g x x x x= + + +


a 2 3( ) 1g x x x x= + + + ;b 3( ) 1g x x x= + + ;

c 2 4( ) 1g x x x= + +

d 3 4( ) 1g x x x= + +

34/ Cho m Cyclic C(n, k) = C(7,3) , tm cu sai vg(x)

a g(x) khng l c ca7x + 1 ;

b T g(x) c th xc nh c ma trn sinh h thng G v ma trn kim tra H cho b mc g(x) c bc 4 ;

d g(x) l mt c ca 7x + 1 ;


a4( ) 1g x x x= + +

b2 4( ) 1g x x x x= + + + ;

c2 3 5( ) 1g x x x x= + + +

d2 3( ) 1g x x x= + + ;

36/ Cho m Cyclic C(n, k) = C(7,3), c a thc sinh g(x) la

5( ) 1g x x x= + +

b2 3 5( ) 1g x x x x= + + +

c4( ) 1g x x x= + + ;

d2 4( ) 1g x x x x= + + + ;

37/ Pht biu sau ng hay sai : Cho m Cyclic C(n, k) = C(7,3) ,g(x) l a thc bc 4a Saib ng

38/ Cho m tuyn tnh (7,4) S nhcn thit nhb m la 7;b 28;c 4;d 112;

39/ Cho m Xyclic (7,4) S cc nhcn thit nhb m la 7;b 112c 28;

d 8

40/ Cho m tuyn tnh (7,3) S nhcn thit nhb m la 7;

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b 21;c 3;d 56;

41/ Cho m Xyclic (7,3) S cc nhcn thit nhb m la 3b 56

c 21;d 7;

42/ Cho m tuyn tnh (n,k) S nhcn thit nhb m l

akn.2

b kn;c n+k;d n;

43/ Cho m Xyclic (n,k) S cc nhcn thit nhb m l

a kn;b n+k;

ckn.2

d n7;

44/ Chn cu ng sau:a Cc dng tuyn tnh ca k bin c lp 1 2 k, , ,x x x l cc biu thc c dng:

( )k

1 k i ii 1

, ,f x x a x=

= Trong : i Fa , F l mt trng

b Cc dng tuyn tnh ca k bin c lp 1 2 k, , ,x x x l cc biu thc c dng:

( )k

1 k i ii 1

, , )f x x (a + x=


c Cc dng tuyn tnh ca k bin c lp 1 2 k, , ,x x x l cc biu thc c dng:

( )k

1 k i ii 1

, ,f x x a - x=


d Cc dng tuyn tnh ca k bin c lp 1 2 k, , ,x x x l cc biu thc c dng:

( )k

1 k ii 1

, , xif x x a=


45/ Chn cu sai v m tuyn tnh :

a k2 cc vtkhc nhau l tt c cc t hp tuyn tnh c th c ca k vcthng ny

G.H 0= Trong : r n k= b Trong i s tuyn tnh ta bit rng vi mi G s tn ti ma trn r nH tha mn:

c Trong i s tuyn tnh ta bit rng vi mi G s tn ti ma trn r nH tha mn:

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d m t m tuyn tnh, c th s dng ma trn sinh k,nG , ma trn ny cha k vcthng

c lp tuyn tnh to nn khng gian m ( )n,kV TG.H 0= Trong : r n k=

46/ m t m tuyn tnh, c th s dng ma trn sinh k,nG Trong i s tuyn tnh ta bit rng

vi mik,n

G s tn ti ma trnr n

H

tha mn: TG.H 0= Chn cu sai sau :

a k,nG l ma trn k hng, n ct

b TH c gi l ma trn k hng, n ct ;c r nH l ma trn r hng, n ct ;

d TH c gi l ma trn chuyn v ca H

47/ Khi xy dng mt m tuyn tnh ( )0n,k,d ngi ta phi tm c cc m c tha nh nhngli c kh nng khng ch sai ln. Ngi ta thng xy dng m ny da trn cc bi ton ti u Tmcu sai trong cc cu di y:

a Vi k v 0d xc nh, ta phi tm c m c di vi t m l ln nhtTng ng vi biton ny ta c gii hn n = k

b Vi n v k xc nh, ta phi tm c m c khong cch 0d l ln nht Tng ng vi bi

ton ny ta c gii hn Plotkin sau:k 1

0i

i 0

dn

2

=

c Vi k v 0d xc nh, ta phi tm c m c di vi t m l nh nhtTng ng vi bi

ton ny ta c gii hn Griesmer sau:k 1

0 k

n.2d

2 1

d Vi n v s sai khi sa t xc nh, ta phi tm c m c s du thng tin k l ln nht (hay sdu tha l nh nht)Tng ng vi bi ton ny ta c gii hn Hamming

sau:t

n k in

i 0

2 C

=

48/ Chn nh ngha sai v m xyclic trong cc nh ngha sau

a M xyclic (n, k) l Ideal ( )I g X= ca vnh a thc [ ] n2Z x X 1+ b M xyclic (n, k) l mt b m m a thc sinh c bc r = n+k

c M xyclic l mt b m tuyn tnhd M xyclic l mt b m , m nu ( )a X l mt t m th dch vng ca ( )a X cng l mt t

m thuc b m ny

49/ Chn cu ng ca nh l v kh nng sa saia M u nh phn c tha (D > 0) vi khong cch Hamming 0d 4= c kh nng sa c

t sai tho mn iu kin: 0d 1

t2

+

b M u nh phn c tha (D > 0) vi khong cch Hamming 0d 1> c kh nng pht hint sai tho mn iu kin 0t d 1

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c M u nh phn c tha (D > 0) vi khong cch Hamming 0d 3 c kh nng sa c

t sai tho mn iu kin: 0d 1

t2

v c kh nng pht hin t sai tho mn iu kin

0t d 1 d M u nh phn c tha (D > 0) vi khong cch Hamming 0d 1 c kh nng sa c

t sai tho mn iu kin: 0d 1t2

50/ Tnh l m ho th 1 ca Shannon i vi m nh phn ta c : ( ) ( )s

i i 1i 1

n p a n H A=

= T biu thc trn tm cu ng nht trong cc biu thc sau :

a ( ) ( )s

i ii 1

n p a log p a=

= ;

b ( ) ( ) ( )

s s

i i i ii 1 i 1

p a n p a logp a= == ;

c ( ) ( ) ( )s s

i i i ii 1 i 1

p a n p a logp a= =

;

d ( ) ( ) ( )i i i ip a n p a logp a=

51/ Theo nh l m ho th 1 ca Shannon i vi m nh phn ta c:

( ) ( ) ( ) ( )s s

i i 1 i ii 1 i 1

n p a n H A p a log p a= =

= =

T biu thc ny suy ra di t m in v xc sut ( )ip a lin h vi nhau: ( )i i1n log

p a (*)

T (*) tm cu ng sau v nguyn tc lp m tit kim:a Cc t m c di ln sc dng m ha cho cc tin c xc sut lnb Cc t m c di n t l thun vi xc sut Pc Cc tin c xc sut xut hin ln c m ha bng cc t m c di nh v ngc li cc

tin c xc sut xut hin nhc m ha bng cc t m c di lnd Cc t m c di nh sc dng m ha cho cc tin c xc sut nh

52/ Cho ngun tin{ }1 2 3 4 5

, , , ,X x x x x x= vi cc xc sut ln lt l {1/2, 1/4, 1/8, 1/16, 1/16}

Bit 1 c m ha thnh 0, 2x c m ha thnh 10, 3x c m ha thnh 110, 4x c m ha

thnh 1110, 5x c m ha thnh 1111 B m ti u cho ngun trn c chiu di trung bnh tnh

theo cng thc : ( )5

i ii 1

n p x n=

= l :a 1,88b 1,90c 1,875d 1,925

53/ Yu cu ca php m ha: nhng t m c di nh hn khng trng vi phn u ca t m c di ln hn Cc tin c xc sut xut hin ln hn c m ha bng cc t m c di nh vngc li.

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Cho ngun tin { }1 2 3 4 5, , , ,X x x x x x= vi cc xc sut ln lt l {1/2, 1/4, 1/8, 1/16, 1/16}

Bit 1x c m ha thnh 0, 2x c m ha thnh 10 Chn cu ng di y m ha cho 3x

a 101b 011c 110d 100

54/ Yu cu ca php m ha: nhng t m c di nh hn khng trng vi phn u ca t m c di ln hn Cc tin c xc sut xut hin ln hn c m ha bng cc t m c di nh vngc li.Cho ngun tin { }1 2 3 4 5, , , ,X x x x x x= vi cc xc sut ln lt l {1/2, 1/4, 1/8, 1/16, 1/16}

Bit 1x c m ha thnh 0, 2 c m ha thnh 11, 3x c m ha thnh 100

Chn cu ng di y m ha cho 4x

a 000b 110c 001d 1010

55/ Yu cu ca php m ha: nhng t m c di nh hn khng trng vi phn u ca t m c di ln hn Cc tin c xc sut xut hin ln hn c m ha bng cc t m c di nh vngc li Cho ngun tin { }1 2 3 4 5, , , ,X x x x x x= vi cc xc sut ln lt l {1/2, 1/4, 1/8, 1/16, 1/16}

Bit 1 c m ha thnh 0, 2x c m ha thnh 10, 3x c m ha thnh 110, 4x c m ha

thnh 1110. Chn cu ng di y m ha cho 5x

a 1010b 1111c 101d 110

56/ Cho m Cyclic C(7, 4) c a thc sinh l 3( ) 1g x x x= + + v a thc sinh G sau :3

2 4

2 3 5

3 4 6

x x

x x x

x x x

x x x

+ +

+ + + + + +

1

G =

Ma trn no sau y l mt ma trn sinh G ng vi m Cyclic C(7,4) trn

a

1101000

0110100

0011010

0001101

b

1010100

0101100

0010110

0001011

c

1000100

01000000011001

0001011

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d

1010100

0101010

0010101

0001101

57/ Cho m Cyclic C(7, 4) c a thc sinh l 2 3( ) 1g x x x= + + v a thc sinh G sau2 3

3 4

2 4 5

3 5 6

x x

x x x

x x x

x x x

+ +

+ + + + + +

1

G =

Ma trn no sau y l mt ma trn sinh G ng vi m Cyclic C(7,4) trn

a

1010100

0101010

0010101

0001101

b

1011000

0101100

0010110

0001011

c

1010100

0101100

00101100001011

d

1000100

0100000

0011001

0001011

58/ Cho m Cyclic C(7, 4) c a thc sinh l : 3( ) 1g x x x= + + v ma trn sinh G. T ma trn sinh Gtnh c ma trn kim tra H bn:

2 3 4

3 4 5

2 4 5 6

1 x x xH x x x x

x x x x

+ + +

= + + + + + +

Chuyn ma trn kim tra H sang dng khng gian tuyn tnh. Ma trn no sau y l mt ma trn kimtra H ( dng khng gian tuyn tnh)

a

1011100

0101110

0010111

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Tnh di m trung bnh theo biu thc( )

5

i ii 1

n n p a=

=

ta c gi tr no :

a 4;b 3;c 2,1;d 1,2;

67/ Gi s sau khi thc hin m ha, cc tini

a vi xc sut tng ng ( )i

P a c m ha thnh cc

m nh phn c di mi

n tng ng nh bng sau:

Tnh di m trung bnh theo biu thc ( )5

i ii 1

n n p a=

= ta c gi tr no:a 1,8;b 4;c 2,4;d 3;



P a c m ha thnh cc

m nh phn c di mi

n

tng ng nh bng sau :


i ii 1

n n p a=

= ta c gi tr no :a 1,8;b 2,1;c 4;d 2,75;



P a c m ha thnh cc

m nh phn c di mi

n

tng ng nh bng sau :

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Tnh di m trung bnh theo biu thc

( )

5

i ii 1

n n p a=

=

ta c gi tr no :

a 2,1 du m;b 1,8 du m ;c 2,75 du m;d 2,45 du m;



P a c m ha thnh cc

m nh phn c di mi

n tng ng nh bng sau :


i ii 1

n n p a=

= ta c gi tr no :a 2,5;b 2,45;c 1,8;d 2,75;

71/ Mt dy tin { }1 2, ,..., nX x x x= vi ; 1ix X i n = Lng tin I(X) cha trong dy tin X s l:

1 2

1 1 1( ) log log .... log

( ) ( ) ( )n

I XP x P x P x

= + + + Gi s cho ngun { }1 2 3 4 5, , , ,X x x x x x= vi cc xc

sut ln lt l {1/2, 1/4, 1/8, 1/16, 1/16}. Lng tin I(X) cha trong dy tin X={ }1 2 1 1 3 4 1 1 5x x x x x x x x x l:

a 17 bt ;b 15 bt;c 18 bt;

d 16 bt;

72/ Khi xy dng mt m tuyn tnh ( )0n,k,d ngi ta phi tm c cc m c tha nhnhng li c kh nng khng ch sai ln Ngi ta thng xy dng m ny da trn cc bi ton tiu Tm cu sai trong cc cu di y:

a Vi k v 0d xc nh, ta phi tm c m c di vi t m l ln nht. Tng ng vi biton ny ta c gii hn n = k

b Vi k v 0d xc nh, ta phi tm c m c di vi t m l nh nht. Tng ng vi biton ny ta c gii hn Griesmer sau:

k 1

0ii 0

dn 2

=

c Vi n v s sai khi sa t xc nh, ta phi tm c m c s du thng tin k l ln nht (hay sdu tha l nh nht) Tng ng

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d 1 2 3 4 51111 1110 110 10 0

a a a a a

76/ Thc hin m ha Shannon - Fano ngun ri rc A sau :

1 2 3 4 5

( ) 1/ 4 1/2 1/ 8 1/16 1/16i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 510 0 110 1110 1111

a a a a a

b 1 2 3 4 50 10 111 1110 1111

a a a a a

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

d 1 2 3 4 50 10 110 1110 1111

a a a a a

;


1 2 3 4 5

( ) 1/ 8 1/4 1/ 2 1/16 1/16i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 110 1110 1111

a a a a a

;

b 1 2 3 4 5110 10 0 1110 1111

a a a a a

c1 2 3 4 5

0 11 110 1110 1111

a a a a a ;

d 1 2 3 4 50 01 11 111 1111

a a a a a

;


1 2 3 4 5

( ) 1/16 1/4 1/ 8 1/ 2 1/16i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 5

0 10 110 1110 1111

a a a a a

;

b 1 2 3 4 51110 10 110 0 1111

a a a a a

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

d 1 2 3 4 50 11 110 1110 1111

a a a a a

;


1 2 3 4 5

( ) 1/ 2 1/4 1/16 1/ 8 1/16i

i

a a a a a aA

p a

= = sc kt qu no sau y:

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a 1 2 3 4 50 10 1110 110 1111

a a a a a

;

b 1 2 3 4 50 01 11 111 1111

a a a a a

;

c 1 2 3 4 5

0 11 110 1110 1111

a a a a a

;

d 1 2 3 4 50 10 111 1110 1111

a a a a a


1 2 3 4 5

( ) 1/ 2 1/8 1/ 4 1/16 1/16i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 110 1110 1111

a a a a a

;

b 1 2 3 4 50 11 01 111 1111a a a a a

;

c 1 2 3 4 50 10 111 1110 1111

a a a a a

d 1 2 3 4 50 11 110 1110 1111

a a a a a


1 2 3 4 5

( ) 1/ 2 1/16 1/ 8 1/ 4 1/16i

i

a a a a a a

A p a

= = sc kt qu no sau y:

a 1 2 3 4 50 01 11 111 1111

a a a a a

;

b 1 2 3 4 50 10 111 1110 1111

a a a a a

c 1 2 3 4 50 1110 110 10 1111

a a a a a

;

d

1 2 3 4 5

0 10 110 1110 1111

a a a a a

;


1 2 3 4 5

( ) 1/ 2 1/4 1/16 1/16 1/ 8i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 1111 1110 110

a a a a a

b 1 2 3 4 50 10 110 1110 1111

a a a a a

;

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

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d 1 2 3 4 50 11 110 1110 1111

a a a a a

;


1 2 3 4 5

( ) 1/16 1/4 1/ 8 1/16 1/ 2i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 110 1110 1111

a a a a a

;

b 1 2 3 4 50 11 110 1110 1111

a a a a a

;

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

d 1 2 3 4 51111 10 110 1110 0

a a a a a

84/ Gi s sau khi thc hin m ha Shannon - Fano ngun ri rc A :

1 4 5 2 3

( ) 1/ 2 1/4 1/ 8 1/16 1/16i

i

a a a a a aA

p a

= =

Ta c kt qu m ho sau : 1 4 5 2 30 10 110 1110 1111

a a a a a

di t m trung bnh n v entropy H(A)

c tnh theo biu thc

( )5

i ii 1

n p a n=

= v ( ) ( ) ( )5

i ii 1

H A p a logp a=

= s c kt qu no sau y :

a n =H(A)=2,875;

b n =H(A)=1,875;

c n =1,875 v H(A)=1,95;

d n =1,95 v H(A)=1,875


1 2 3 4 5

( ) 0.5 0.25 0.125 0.0625 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 01 11 111 1111a a a a a

;

b 1 2 3 4 50 10 110 1110 1111

a a a a a

c 1 2 3 4 50 10 111 1110 1111

a a a a a

d 1 2 3 4 50 11 110 1110 1111

a a a a a

;


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1 2 3 4 5

( ) 0.25 0.5 0.125 0.0625 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 11 110 1110 1111

a a a a a

;

b 1 2 3 4 5

10 0 110 1110 1111

a a a a a

;

c 1 2 3 4 50 10 110 1110 1111

a a a a a

d 1 2 3 4 50 10 111 1110 1111

a a a a a


1 2 3 4 5

( ) 0.125 0.25 0.5 0.0625 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 110 1110 1111

a a a a a

b 1 2 3 4 5110 10 0 1110 1111

a a a a a

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

d 1 2 3 4 50 10 111 1110 1111

a a a a a


1 2 3 4 5

( ) 0.0625 0.25 0.125 0.5 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 51110 10 110 0 1111

a a a a a

b 1 2 3 4 50 01 11 111 1111

a a a a a

;

c1 2 3 4 5

0 11 110 1110 1111

a a a a a

;

d 1 2 3 4 50 10 110 1110 1111

a a a a a


1 2 3 4 5

( ) 0.0625 0.25 0.125 0.0625 0i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 5

0 11 110 1110 1111

a a a a a

;

b 1 2 3 4 50 10 110 1110 1111

a a a a a

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c 1 2 3 4 51111 10 110 1110 0

a a a a a

d 1 2 3 4 50 01 11 111 1111

a a a a a

;


1 2 3 4 5

( ) 0.5 0.125 0.25 0.0625 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 01 11 111 1111

a a a a a

;

b 1 2 3 4 50 10 111 1110 1111

a a a a a

c 1 2 3 4 50 11 110 1110 1111

a a a a a

;

d 1 2 3 4 50 110 10 1110 1111a a a a a


1 2 3 4 5

( ) 0.5 0.0625 0.125 0.25 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 11 110 1110 1111

a a a a a

;

b1 2 3 4 5

1 00 010 0111 0101

a a a a a

c 1 2 3 4 50 10 111 1110 1111

a a a a a

d 1 2 3 4 50 1110 110 10 1111

a a a a a


1 2 3 4 5

( ) 0.5 0.0625 0.125 0.0625 0.25

i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 10 110 1110 1111

a a a a a

b 1 2 3 4 50 10 111 1110 1111

a a a a a

c 1 2 3 4 50 01 11 111 1111

a a a a a

;

d 1 2 3 4 50 1111 110 1110 10

a a a a a

;


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1 2 3 4 5

( ) 0.5 0.25 0.0625 0.125 0.0625i

i

a a a a a aA

p a

= =

sc kt qu no sau y:

a 1 2 3 4 50 01 11 111 1111

a a a a a

;

b 1 2 3 4 5

0 10 110 1110 1111

a a a a a

c 1 2 3 4 50 10 1110 110 1111

a a a a a

d 1 2 3 4 50 11 110 1110 1111

a a a a a

;


1 4 5 2 3

( ) 0.5 0.25 0.125 0.0625 0.0625i

i

a a a a a aA

p a

= =

Ta c kt qu m ho sau : 1 4 5 2 30 10 110 1111 1110

a a a a a


c tnh theo biu thc ( )5

i ii 1

n p a n=

= v ( ) ( ) ( )5

i ii 1

H A p a log p a=

= s c kt qu no sauy :

a n =H(A)=1,875;

b n =H(A)=2,875;

c n =1,95 v H(A)=1,875;

d n =1,875 v H(A)=1,95;


1 2 3 4 5 6 7 8

( ) 0.25 0.125 0.0625 0.0625 0.25 0.125 0.0625 0.0625i

i

a a a a a a a a aA

p a

= =

Ta c kt qu m ho sau : 1 5 2 6 3 4 7 800 01 100 101 1100 1101 1110 1111

a a a a a a a a

di t m trung bnh n v entropy H(A) c tnh theo biu thc ( )8

i i

i 1

n p a n

=

= v

( ) ( ) ( )8

i ii 1

H A p a log p a=

= s c kt qu no sau y :

a n =1,875 v H(A)=1,95;

b n =1,95 v H(A)=1,875 ;

c n =H(A)=1,875;

d n =H(A)=2,75;


1 2 3 4 5 6 7 8

( ) 1/ 4 1/ 8 1/16 1/16 1/ 4 1/ 8 1/16 1/16i

i

a a a a a a a a aA

p a

= =

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Ta c kt qu m ho sau : 1 5 2 6 3 4 7 800 01 100 101 1100 1101 1110 1111

a a a a a a a a

di t m trung bnh n v entropy H(A) c tnh theo biu thc ( )8

i ii 1

n p a n=

= v

( ) ( ) ( )8

i ii 1

H A p a log p a=

=

s c kt qu no sau y :

a n =1,95 v H(A)=1,875 ;

b n =H(A)=2,75;

c n =H(A)=1,875;

d n =1,875 v H(A)=1,95;

97/ Gi s sau khi thc hin m ha ngun ri rc A :

1 4 5 2 3

( ) 0.5 0.25 0.125 0.0625 0.0625i

i

a a a a a aA

p a

= =

Ta c kt qu m ho sau : 1 4 5 2 3

0 10 110 1111 11100

a a a a a


c tnh theo biu thc: ( )5

i ii 1

n p a n=

= v ( ) ( ) ( )5

i ii 1

H A p a logp a=


a n =H(A)=2,875;

b n =H(A)=1,875;

cn

=1,875 v H(A)=1,95;d n =1,9375 v H(A)=1,875;


1 4 5 2 3

( ) 1/ 2 1/4 1/ 8 1/16 1/16i

i

a a a a a aA

p a

= =

Ta c kt qu m ho sau : 1 4 5 2 30 10 110 1111 11100

a a a a a


c tnh theo biu thc

( )

5

i ii 1

n p a n=

=

v

( ) ( ) ( )

5

i ii 1

H A p a log p a=

=

s c kt qu no sau

y :

a n =1,875 v H(A)=1,95;

b n =H(A)=1,875;

c n =1,9375 v H(A)=1,875;

d n =H(A)=2,875;


1 4 5 2 3( ) 1/ 2 1/4 1/ 8 1/16 1/16

i

i

a a a a a aA

p a

= =

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Ta c kt qu m ho sau : 1 4 5 2 300 10 110 1111 1110

a a a a a



i ii 1

n p a n=

= v ( ) ( ) ( )5

i ii 1

H A p a log p a=


a n =2,5 v H(A)=1,85;b n =1,9375 v H(A)=1,87;

c n =2,375 v H(A)=1,875 ;

d n =2 v H(A)=1,9;


1 4 5 2 3

( ) 0.5 0.25 0.125 0.0625 0.0625i

i

a a a a a aA

p a

= =

Ta c kt qu m ho sau :1 4 5 2 3

00 10 1100 1111 1110

a a a a a

di t m trung bnhn

v entropy H(A)


i ii 1

n p a n=

= v ( ) ( ) ( )5

i ii 1

H A p a log p a=


a n =2,5 v H(A)=1,875;

b n =2,375 v H(A)=1,975

c n =2,55 v H(A)=1,775;

d n =2 v H(A)=1,675;

104/ Cho m Cyclic C(7,4) c a thc sinh l 3( ) 1g x x x= + + tng ng a thc thng tin

( ) 3a x = x + x S dng thut ton 4 bc thit lp t m h thng ta c kt qu no di y :a 1100101b 0101010;c 1000110;d 0110111;


( )2

a x = 1+ x S dng thut ton 4 bc thit lp t m h thng ta c kt qu no di y:a 0101010;b 1000110;c 0011010;d 0110111;


( ) 2a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 0101010;b 1110010;c 1000110;d 0110111;

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( ) 3a x = 1+ x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 0101010;b 1000110;c 1110010;d 0111001;


( ) 2 3a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy:

a 0111001;b 1000110;c 1110010;d 0100011;


( ) 2 3a x = 1+ x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy :

a 0100011;b 0111001;c 1110010;d 1001011;


( )a x = 1+ x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :

a 1001011;b 0100011;c 1011100;d 1110010;


( ) 2a x = 1+ x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy :

a 1001011;b 0101110;

c 1011100;d 1110010;


( ) 3x+2a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy :

a 1011100;b 0101110;c 1001011;d 0010111;


( ) 3xa x = S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :

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a 0101110;b 1001011;c 1010001d 1011100;


( )a x = x

S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 0010111;b 0110100;c 0101110;d 1011100;


( ) 2a x = x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 1110010;b 0101110;

c 0110100;d 0010111;

116/ Cho m Cyclic C(7,4) c a thc sinh l 2 3( ) 1g x x x= + + tng ng a thc thng tin

( ) 3a x = x + x S dng thut ton 4 bc thit lp t m h thng ta c kt qu no di y:a 0101110;b 0110111;c 1000101;d 0101010;


( ) 2a x = 1+ x S dng thut ton 4 bc thit lp t m h thng ta c kt qu no di y:a 0011010;b 1000110;c 0101010;d 0111010;


( ) 2a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :

a 1110010;b 0110111;c 1000110;d 0010110;


( ) 3a x = 1+ x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 1101001;b 0101010;c 0111001;

d 1000110;

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( ) 2 3a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy:

a 1010011;b 0111001;c 1000110;

d 0100011;


( ) 2 3a x = 1+ x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy :

a 0001011;b 0100011;c 0111001;d 1110010;

122/ Cho m Cyclic C(7,4) c a thc sinh l 2 3( ) 1g x x x= + + tng ng a thc thng tin( )a x = 1+ x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 1110010;b 1001011;c 0101100;d 0100011;


( ) 2a x = 1+ x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di

y :a 1001011;b 1001110;c 1110010;d 1011100;


( ) 3x+2a x = x + x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no diy :

a 1001011;

b 0100111;c 1011100;d 0101110;


( ) 3xa x = S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 1001011;b 0111011c 1011100;d 0101110;


( )a x = x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :

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a 1110100;b 1011100;c 0010111;d 0101110;


( )2

a x = x S dng thut ton 4 bc thit lp t m h thng, sc kt qu no di y :a 0101110;

b 0110100;c 1110110;d 0010111;

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CHNG 5 : L THUYT THU TI U

1/ Thu tn hiu khi c nhiu l mt bi ton thng k:a ngb Sai

2/ Nhim v ca my thu l phi chn li gii, do my thu cn c gi l s gii Yu cu cas gii l phi tm li gii ng (pht i ta phi tm c i ) Trong thc t c rt nhiu sgii Trong tt c cc s gii c th c th ti mt s bo m xc sut nhn ln phi ng l lnnht (xc sut gii sai l b nht) S ny c gi l s gii ti u. Khi ngi ta gi Mythuxy dng theo s gii ti u c gi l my thu ti u Kt lun ny ng hay sai ?

a Saib ng

3/ ( )( )

( )

i/i

pu

p i l

l

l

>

i = 1,mVi l quy tc gii ti u vit di dng hm hp l. Chn cu

ng trong cc cu sau:

a Nu mi tn hiu gi i u ng xc sut th ( )/i u 0 il l = Vi b Nu mi tn hiu gi i u ng xc sut th ( )/i u 1 il l > Vi c Nu mi tn hiu gi i u ng xc sut th ( )/i u 1 il l < Vi d Nu mi tn hiu gi i u ng xc sut th ( )/i u 1 il l = Vi

4/ nh ngha b lc phi hp tuyn tnh thng : i vi mt tn hiu xc nh, mt mch tuyn

tnh thngm bo t s rara

N

S = cc i mt thi im quan st no y sc gi l

mch lc phi hp tuyn tnh thng ca tn hiu Trong ra l t sgia cng sut trungbnh ca nhiu u ra b lc y v cng sutnh ca tn hiunh ngha ny ng hay sai ?

a Saib ng

5/ Qu trnh x l tn hiu trong my thu ti u c gi l x l ti u tn hiu X l nhn li giic xc sut sai b nht Chn cu ng nht v x l ti u cc tn hiu:

a Da vo cc tiu chun ti u, bng cng c thng k ton hc ngi ta xc nh c quy tc

gii ti u, do ngy nay l thuyt truyn tin cho php bng ton hc tng hp c mythu ti u.b Vic tng hp cc my thu (xy dng s gii) ch cn c vo cc tiu chun cht lng

mang tnh cht chc nng m khng mang tnh cht thng k.c nh hng ca nhiu ln cht lng ca my thu ti u chc tnh theo t s tn/tpd Vic tng hp my thu ti u l da vo trc gic, kinh nghim, th nghim

6/ Gi s i l tn hiu gi i, c xc sut ( )ip - c gi l xc sut tin nghim my thu tanhn c ( )u t , t ( )u t qua s gii ta s c li gii l no Nh vy l c gi i vimt xc sut ( )p / ul - c gi l xc sut hu nghim. Do xc sut gii sai s l:

( ) ( )p sai / u, 1 p / ul l = Xt hai s gii:- T ( )u t cho ta 1 - gi l s (1)

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- T ( )u t cho ta 2 - gi l s (2)Nu ( ) ( )1 2p sai / u, p sai / u, < ta rt ra kt lun s (2) ti u hn (1) Khng nh ny nghay sai ?

a Saib ng

7/ Tn hiu tng qut c dng: ( ) ( ) ( )( )i 0i 0C t C t cos t t= + + Chn cu sai trong cc cu sau:

a Vic x l ti u tn hiu khng ph thuc ng bao ( )0iC t v tn s tc thi

( )( )

id t

tdt

= +

b Khi x l ti u tn hiu ta cn bit ng bao ( )0iC t v tn s tc thi ( )( )

id t

tdt

= +

c Nu vic thu ( )iC t cn bit 0 (iu chnh h thng thu) th c gi l thu kt hp Nu

vic thu ( )iC t khng cn bit 0 (iu chnh h thng thu) th c gi l thu khng kthp

d Thc t khi thay i s lm cho 0 thay i ch bin thin t nhng cng lm cho 0 thay i rt mnh Khi ta phi chuyn sang thu khng kt hp.

8/ Cho u vo mch tuyn tnh thng mt dao ng c dng: ( ) ( ) ( )iy t C t n t= + Trong

( )iC t l th hin ca tn hiu pht i (cn c gi l tn hiu ti) ( )n t l nhiu cng, trng, chun

Bi ton tng hp mch l tm biu thc gii tch ca hm truyn phc ( )iK ca mch tuyn tnh

thng sao cho mt thi im quan st (dao ng nhn c) no ra t max, p dng

cng thc bin i ngc Fourier tnh c ( )2

ra iv0

1S 2 d

Nf f

Trong ( )ivS l mt

ph (bin) phc ca th hin tn hiu u vo mch tuyn tnh

Theo nh l Parseval, ta c: iramax0

E

N = (*)

trong ( )2

i ivE S 2 df f

= l nng lng ca tn hiu ti

Chn cu sai trong nhng nhn xt sau :

a T (*) chng t t s rara

S

N =

ch ph thuc vo nng lng ca tn hiu v bi ton pht

hin dng ca tn hiu l khng quan trngb T (*) chng t : T s gia cng sut nh ca tn hiu v cng sut trung bnh ca nhiu

u ra b lc y ch ph thuc vo nng lng

c T (*) chng t t s rara

S

N =

ch ph thuc vo nng lng ca tn hiu m hon ton

khng ph thuc vo dng ca n

d T (*) chng t t s rara

SN =

hon ton ph thuc vo dng ca n

ca tn hiu m hon ton khng ph thuc vo dng ca n.

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9/ Cho knh nh phn, i xng, khng nhc nhiu cng, trng, chun theo m hnh sau:

1

2 2

1 ( )1 1p /

( )2 2p /

( )1 2p / ( )2 1

p b /

( )11

p2

=

( )21

p2

=

Xc sut sai ton phn sp ( xc sut sai khng iu kin) l

( ) ( ) ( ) ( )= + s 1 2 1 2 1 2p p .p / p .p /

Gi s tnh c ( )

= =

0

P T

4N 2

2 1

0

P T1p / exp d

2 2G2

khi sp s bng biu thc no di y:

a

= +

s0

P Tp 1

2G

b

= +

s0

P T1p

2 2G

c

= +

s0

P T1 1p

2 2 2G

d

=

s0

P Tp 1

2G

10/ Ti u vo b lc tuyn tnh tc ng tn hiu: x(t) = s(t) + n(t)Trong n(t) l tp m trng, chun, dng Cn s(t) l xung th tn c lp vi n(t) v c dng:

( )( ) =

>

A t TA.e t T

s t0 t T

Hm truyn = * j T0K (j ) kS (j )e

Hm truyn ca b lc ( = * j T

0K (j ) kS (j )e ) sao cho t s tn trn tp u ra ca b lc t cci s l biu thc no di y

a = + 0kA

K (j )A j

b =+ 0kA

K (j )A j

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