giải bài tập xử lý tín hiệu số và matlab

5/26/2018 Gia i Bi T p X L Tn Hi u S V Matlab

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hS. RN H HC LINH

Gii bi tp

l i l t i n h i u s v Matlab

NH XU BN HNG IN V RUYN HNG


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Chng 1

TNHIU V H THNG RI RC

A. TM TT L THUYT

1.1. nh l av mu

Ta ch rng int tn hiu s c khi phc khitn s ly mu phi ln hn hoc bng hai

ln b rng ph ca tn hiu. > 2B (B mdx)

1.2. Phn loai tn hiu

T N H U

I

T N H I U L I N T C T N H I U R I R C

Bin: lin tuc Bin: ri rac

Bin ; lin ruc hoc ri rac Bin : lin tuc hoc ri rac

i T n h i u t o n g t T n h i u l i m n i t T n h i u ly m u T n h i u s

Bin; lin tuch o

Bin: ri rac Bin: ri rac

Bin d: lin tuc Bicn: licn tuc

Bin d: r rac

Bin : lin tuc Bin d: ri rac

1.3. Cc h thng x l tn hiu

VoH THNG

Ra

-----------------------------------^

Tn hiu tonu

TNG T

Tn hiu tong

VoH THNG S

Ra

---------------------------------- ^

rn hiu s Tn hiu syj(n:

Vo

X a ( t )


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ii b! tp x ly tin hiu s v Matlab

1.4. Tn hiu ri rc

L 4. L B i u d i n tn l ii n )i r c

- Biu din bniz lim s

- Bicu din bni bn

- Biu dicn bniz dy se^

- Biu din bnii ih

Cl ; m t t n h i u b t k x ( n ) d e u d i i c b i u d i n h L i q u a p i m X U I U d n u l n ^ q u t

nh sau:

(n) =

L 4 .2 M ( s (iv c hna) Dy xung coi v:

Trong min n, dy xung on v c dnh nha nh sau:

n - 0

[0

h) Dy nhay clov v:

TYong min n, dy nhy on v uc nh imlia nh sau:

1 n>( )

0 n 7 0

c) D y ch nll:

u( n )

Tronu min n, dy chi nht e dn nuha nhu' sau: rccl^ (n ) -

(i) Dy dc don v:

' ron g min n, dy dc on v c nh ngha nh sau: r (n ) -

e) Dy h :

Trong min n, dy hm m c nh ngha nh sau:

0 < n < N - 1

n cn li

n > 0

n c l li

e (n) =a" n > 0

0 n cn li

L 4 3 , M s n h n gh a

a. Dy tun hon:

Ta ni rng mt dy x(n) l tun hon vi chu k N nu tha mn iu kin sau y:

x ( n ) X (n + N)"= X ( n + f N )

K hiu: x( n ) ,.


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h. /)/v c ci di hu hn:

Mt dy c xc dnh vi s hu n N Iiu la gi l day c chiu di hu hn vi N l

chiu di cua dv.

c. ixrng cu a v:

Nnu lnu cua ml ds' x(n) d'c dnh nuli a nliLi' sau:

d. Cen linl bt bin l xun n v(S(n) thi u ra l dp ng

xLiim h(i). p nu XLum h(n) l dc trim hon ton cho h tlinu luyn tnh ht bin.

.4.6. Php chp:

Dy l php ton quan Irn^ ll troii xu l ln hiu d xc nl u ra v(!) h llng khi

bict u vo x(n) v dp im xLum h( n) .

y( n ) ^ x ( n ) * h ( n ) ^ x ( k ) .h{ n ~ k ) h ( k ) . x ( n - k )k~ - k -

Php chp c tnli cht: uiao hon, phn phi, kt hp.

L4,7. H th ong TT BB nhn (iiy tn hiu h qu

JI thng TTBB dc ui l h tliHii nhn qu khi p iii xunt h(n) cua n tho mn:

h(n ) ^ 0 v i Vn


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ii bi tp x l tn hiu s v Matlab

L4.8. Phung tr nh saiphn tuyn tinh h s hng (P TSPTTHSH)

Qua n h vo ra ca h tliim tuyn tnh bt bin s irc m t xVi ph oi m trnh sai phn

tuyn tnh h s him c dni nh sau:

^ a , ( n ) y ( n - k ) = 2 ] b , . ( n ) x ( n ^ r )k=() r O

Trong : - Xu vo.

- y u ra.

Cc h s ai,, br c triii hon ton cho h thn, c vai tr tong t nlur p ng xun" h(n).

Vic gii PTSPTTHSH tim ra u ra y(n) c hai phu'onz php chnh:

- Phoim php th.

- Phong php tim nghim ring (yp(n)) v nghim thun nht (yo(n)). suy ra nghim tngqut y(n) - yo(n) + yp(n).

T PTSPTTHSH trn ta s c ml s khi nim v;

- H th ng khng quy khi N "= 0. Bn cht ca h thno ny lkhng c thnhphn hi

tip.

- H thn qu ykhi N ^ 0. Bn cht ca h thng ny l c thnh phn hi tip.

- H th ng quy th un tu khi N 0. M 0. H thn ny chi gm duy nhtccthnh

ph n quy.

Lim : Nh vy n y ta c hai cch biu din quan h vo ra h thng r'i rc.

- Biu din theo php chp: v(n) x(n)*l(n)

M N- Biu din theo phng trinh SPTTHSH: y(n) ^ b ^ x ( n ~ r ) - ^a ^ y (n k) (th('mg

r = 0

pha i chu ho Q ^ I)

1,4.9. Thc hit h thng

Cc phn t thc hin h thng bao gm: phn t cng, phn t nhn, nhn vi hng s,

phn t tr D.

Khi thc hin h thng phi da vo P TSPT THSH , lun nhcT phi chun ho h s a,) = 1 M N

C y ( n ) - ^ b j . x ( n - r ) - ^ a | . y ( n k) ri m i v SO' h t h ng . T r n th-c t ng' i t a s d n g

r-o k=!

rc b x l ton hc ALU, cc thanh ghi dch... thc hin h thng x l tn hiu s theo SO' .

L 4 J 0 . T ng quan tn hiu

Php tong quan thng dng nhn bit cc tn hiu, phn bit tn hiu vi nhiu, pht

hin vt th... C hai loi tng quan:

fOO

- T to'ng quan: Tng quan tn hii x(n) vi chnh n: R^ ^( n) = ^ x(m) .x(m - n).

ttl = -CC'

- Tng quan cho: Toig quan tn hiu x(n) vi y(n); R^y(n) = ^ x(m) .y(m - n).


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Ch n g 1: Tn h iu v h thng r i rc

B. BI TP C BN

L l . Cho tn hiu x, ( t ) ^ 3cosl007Tt

a) Xc nh tc lay mu nh nhl cn thit trnh s chng mu.b) Gi s tn hiu 'c ly mu ti tc = 200 Hz. Tn hiu r'i rc no s c 'c sau

ly mu?

c) Gi s tn hiu 'c ly mu ti tc F. - 75 Hz. Tn hiu ri rc no t c sau ly

mu?

d) Tn s F < F^ /2 ca mt hnh sin c cc mu ng nht vi cc mu trong phn c) l bao

nhiu ?

Li gi i:

a) Tn s ca tn hiu ton g t l F = 5 0 Hz. Vi th, tc ly mu ti thiu cn thit tr nh hin tni ch ng mu l - 100 Hz.

b) Ne u tn hiu 'c ly mu ti - 200 Hz th tn hi u ri rc c dng:

x(n) = 3cos(l007/200)n = 3 cos(7 / 2 )n

c) Nu - 75 Hz, tn hiu ri rc c dng

x ( n ) = 3 c o s ( l 007i / 7 5 ) n ^ 3 c o s ( 47t / 3 ) n - 3 c o s ( 2 k - 27 / 3 ) n := 3 c o s ( 2 Tc/3 ) n

d) i vi tc ly mu = 75 Hz, ta c; F = fF =: 75 f

Tn s ca tn hiu sin trong phn c) l f - 1/3 . Do : F = 25 Hz

Tn hiu sin i: (t) - 3cos27iFt - 3cos507tt c ly mu ti R - 7 5 mu/s sinh ra cc

mu ng nht. V th F = 50 Hz l b danh (alias) ca F - 2 5 H z ng vi tc ly mu

*: = 75 h I

1.2 . Xl tn hiu t ong t

Xg (t) = 3cos507t + 10sin3007il "CoslOTit

Hy xc nh tc Nyquist i vi tn hiu ny?

Li g ii: Tn hiu trn c cc tn s thnh phn sau:

F, = 2 5 H z , P . - I S O H z , F 3 - 5 0 H z

Nh vy, =150 Hz v theo nh l ly mu ta c: 300 Hz

Tc Nyq uist l . Do , ==300 Hz.

Nhn xt : Ta nhn thy rng, khi ly mu thnh phn tn hiu 10sin3007Tt vi tc

Nyquist - 300 Hz s to c cc mu lOsinTin c gi tr lun lun bng khng. Ni khc i, ta

ly mu tn hiu hnh sin ti cc im n ct trc honh, v th thnh phn ca tn hiu ny b

mt hon ton. Hin tng ny s khng xut hin, nu dao ng sin c mt s lch pha 0 no .Kh , ta c 10sin (3007 r+ 0) v nu ly mu ti tc Nyquis t ta c:

10sin(Tcn + 0) - io (s in 7m c o s 0+ cosTinsinG) = lOs in 0 cos7n -


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Ncu 0 0 hoc Kthi cc mu S!1!, \ li tc d N\tj ii isl s klic klnii. Tu\ nhicn. bin d

cua cc niu \'n chu a xc nh tlirc nii cch chnli xc \ pha 0 \ii clura hict. Bin plip di

Xn dc trnh hin Urn 2P' 12 k\7.* ^ MIU '> fUil \

Tc Nyquist l:F;, -12kl/ .

b) Vi la chn - 5 k l z, nn lii s up s i: F^/2 2.5 kHz

dv l tn s cc i c ti hiu ly mu th hin t ccli duy Iiht. l'a c:

\ (n) (n r) ^ (n /R ) = 3c os 2n ( /5 )n + 5sin 27(3/5)n -t- l)cos27r(6/5)n

3cos 2Tt( l/5) n -f 5sin 2t( - 2/5 )n f H)cos27( l --i-1/5) n

- 3 cos 2ti(1/5) n + 5 sin 2 (-- 2/5) n -r 0 cos 271 (1/5) n

C u i c n g , l a c : X ( n ) 1 3 c o s 2 t(1 / 5 )n - 5 s i n 2 tt( 2 / 5 ) n

c) Vi chi c cc thnh phn tnn s 1 ki I/ v 2 k l / l hin din Iroim tn hiu cl ly mau. ncn

tn hiu tonu t c tl phc hi clirc l

(t ) 13cos20()()7t - 5sin 4000nt

Dy l kt qu khc ng k st vi n li iu I c x,^ ( t ) . Vic Iio tin hiu lo'imXlioc nh llie

ny l do nh hng cua hin tirng chnn mu Ii niTuvn nhn chnh l v ln s lay mu thp.

1.4. Phn loi cc tn hiu sau theo cac iu ch (1) tn hiu mt chiu hay nhiu chiu, (2) tn hiu

on knh hay a knh, (3) tn hiu lin c hay r'i rc theo thi gian, v (4) tn hiu lng t hay s

(theo bin ). Hv a ra gii thch ngan min.

a) Gi n nu cua cc chim khon trcn th tr'nu chne khon Vit Nam.

b) Mt b phim mu

c) V tr ca bnh li ca mt xe hoi khi chuyn ng i vi vt tham chiu l ihn xe.

d) V tr ca bn h li ca ml xe hoi khi chuyn nu i vi vt Iham chiu l mt t.

e) Cc s o trng l'ng v chiu cao cua mt a tr hng thnu.

Li *ii:

Gi :

10 ii bi p x l tin hiu s v Matlab


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Chng 1: Tn hiu v h hng ri rc 11- L ln hiu mt chiLL a knh, ri l ic theo llii uian. v l tn hu s.

' I . ln hiu da ch iu, do n knh, licn tc theo thi g ian, v l tn hiu u ur nu t.

- , tin hiu int cliiu, on knh, lin lc theo Ihi gian, v l tn hiu t'0ng t.

- . tn hiti mt chiu, on knh, lin lc theo thi gian, v l tin hiu URTng t.- I. ln hiu mt chiu, a knl. ri rc llieo ihi man, v l tn hiu s.

1.5* Hy xc dnh xem liu cc tn hiu sau y c phi l cc tn hiu tui hon khng. Tvoi

tru n u h p l tn hiu tun ho n, hy xc dnli chu k COb an c a tn hiu .

a) X,,( t ) = co s 5 t + J \ / I

V J

b)x ( n ) = 2 co s 5 n---^ I\ ^ '

c) \ ( n ) =: 3exp J - TtV y

d) X(n) = 2cosnTT'' ( n

cos -- ! 1 ' 4\ y V /'

e) X(n) ~ cos' riT ^ . (ITT nn n'] __ "Sin i4-2cos----------

2 ; l 8 ; l 4 3 ,

L i gi i:

- 2 7Tn hiu tun hon vi chu ki r - .

5

_ 5

2H

1 2 t

1'oii^ t la c; cos

tn hiu khn tun hon.

>tn hin khnu tun hon.

/ \nu l tn hiu khng tun hon; cos ^ ] l tn hiu tun hon, suy raV 4 ;

tch ca chng l tn hiu khng lun hon.

^ 7cos

CCS

cos

Tn

8 y

7in 7

l tn hiu tun hon vi chu k Tj,| - 4

l tn hiu tun hon v'i chu k T - 1 6

4 3l tn hu tun hon vi chu k = 8

Do x(n) tun hon vi chu k Tp = 16 ( v 16 l bi s chung nh nht ca 4,8,16).

1.6. Chn minh rng tn s CO'bn Np ca cc tn hiu:

s j n j - k - 0 , 1, 2, .. .


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c dnii Np = N/U SCI .N(k.N)

(U'SCLN(k,N) l c s cliuii ln nht ca k v N)

b) X c nh chu ki c baii cua tp hp ny i vi N =7

c) Cu hi nh b) i vi N = 16

r -. 27rk k L i iai: a) co =------ c naha l t = .

N N

t a =U'SCLN(k,N). c ntha lk = k a . N = N a

Do , f = ~ , c ngha l N = N = ^ (pcm)N a

b) N - 7

k = 0 1 2 3 4 5 6 7SCLN(k,N) = 7 1 1 1 1 1 1 7

Np = 1 7 7 7 7 7 7 1

c) N = 16

k = 0 1 2 3 4 5 6 7 8 9 10 11 1 2 . . . 16

SCLN(k ,N) = 16 ! 2 1 4 1 2 1 8 1 2 14 . . . 16

Np = 1 6 8 1 6 4 16 8 16 2 16 8 1 6 4 . . . 1

1.7. Xt tn hiu toTig t hnh sin nh sau:

(t ) = 3sin(1007:t)

a) V tn Iiiu (t ) vi 0 < t < 30ms .

b) T n hi u (t) 'c ly mu vi tc ly mu = 400 mu/s . Mv xc nh tnsca

tn hiu ri rc x ( n ) = (n' ]') ,T = v chng minh rng x(n) l tun hon.^s

c) Tnh gi tr ca cc mu troni> mt chu k ca x(n). V x(n) trong cng mt hnh v vi

( t ) . Xc nh chu k ca tn hiu ri rc theo ms.

d) C th tim thy mt tc ly mu sao cho ln hiu x(n) t ti gi tr nh ca nl 3?

Gi ti' nh nh t tho mn iu kin l bao nhiu?

L i g i i: a) Hnh 1.1

b) x ( n ) = X3 (nT) ,T = ^

x (n ) = (n / F^) = 3 s in( l ( )07n / = 3s in (7n / 4 )

12 ii bi tp x l tin hiu s v Matlab

f = -L2n

/ \

nv 4 .


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Chng 1: T in h iu v h thng r i rc 13Tin hie hnh sin xa(t)

0.0)5 0.01 0.D16 0,02 -i. Q025 0.03gitn ln tuc t (c) ^ '

c )Hinh 1.2

(n) = 0 , . 0 ,

H n h L I

, T = 8.

o : :. . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . .

i

i

i

:

. . . . . . . . . . . . . . . . . . . . . . . ' ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i

1 : I ' 1

1i

i... _ _ _ _ _ _ _ L . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . < :

i.......... i .......

j . . . . . . . . . . . . . . . . . . . . .

i.........

7>

Thoi gian roi rac rt

H n h 1.2

d) C.

x( l) = 3 = 3sin' i OOtc'

=:> F = 200 mu/s

1.8. Mt tn hiu hinh sin licn tc theo thi gian ( t ) c tn s CO' bn - 1/ 'c ly mu

tc lv mu = 1/ T e to ra mt tn hiu hnh sin ri rc x( n) = ( n T ) .

a) Chng minh rn s x(n) l lun hon nu T / = k / N , (ngha l T / Tp l mt s hiru t)

b) Nu x(n) l tun hon, xc nh chu k c bn Tp ca n theo gi y?

c) Gii thch pht biu sau y; x(n) l tun hon nu chu ki CO' bn Tp (s) ca n b n g m t

s nguyn ln chu ki ca x, ( t ) .

Lii: a) x (n ) = Acos(27tF )n /1 ; + 0) Acos 1271 ( T / T p ) n + 0


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M T / T l mt s hu tv\ suy ra x(n) l iLin hon.' N

b) Neu x(n) l tun hon th f ky'N, tron N l chu k. Do .

14 ii bi tp x ly tn hiu s v Matlab

T, - ~ T = k d

nuha l cn k chu ki Tp ca tn hiu to'ng t to nn mt chu k cua n hiu ri rc.

c) T kT,^ => NT = k T =:> = ^ f l s hu tv => X(n ) tLii! hoxn./ a p p - 1 J - V ./

1.9. Mt tn hiu torm t c cc tn s ln n 20 kllz.

a) Xc nh ph m VI cua ln s ly mu c th khi ph c chi ih xac l in hu ny t cc

mau ca n.

b) Gi thit r ng chn ta ly mu tin hiu ny vcVi tan s lv mau ^ 16k}Iz . H y xc nh

diu g s xv ra i vi tn so Fj = 10kHz

c) Lp li cu b) vi F, ~ 18kHz

Li gi i:

a ) F _ = 2 0 k H z = i > F , > 2 F _ = 4 0 k H z

b) i vi \\ ^16kHz . F.p(mid) F, / 2 = 8 kHz, suy ra10kHz s

c) F = 18 klz s l nh (alias) ca 2 kHz

1.10. Mt tn hiu in tm m cc tn s hu dng ln n lOO Hz.

a) 'rn s Nyquist cho tn hiu ny l bim bao nhiu'.^

b) Gi ihil l la lv mu in hiu ny lc d 200 mu/s. Hv xc dnl tn s ln nll m

tn hiu c th c biu din duy nht ti tc lv mu ny.

Li gid i:

a) =100kHz, l' > = 200kHz => = 200kMz

b) F,,p(,id) = F, /2 = 100Hz

1.11. Mt ln hiu tuxTng t X,, (t ) = 2sin( 240Ttt ) + 3s in (720 i t) uxrc lv mu 600 ln ,'s.

a) Xc iili tc lv mu Nyquis t cho (t ) ?

b) Xc dn i tn s gp (blding tVequency)?

c) Xc nli cc tn s theo radian trong tn hiu ri rc x(n)thuu''c?

d) Nu \( n) 'c cho qua mt b bin i s/ toim t,hyxc nh tin hiu ^^^khi

phc c.

Li i:

a) =360Hz ,F ,= 2F , =720Hz

b) Ppp = F, /2 = 600/2 = 300 Hz


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c) x ( n ) = x,, ( n T ) = x , ( n / i g

= 2s in (24() 7m / 60 0) 3s!n ( 72()7:n / 60 0) = 2 si n (27TI1 / 5) - 3s in (Ttn / 5)

2 t : _ I n

' s " 5

d ) y ^ ( t ) = x( l -' t ) = 2 s i n ( 24 07 X1 ) 3 s m ( 7 2 0 T i t )

I.I2. Mt cLrtrnLi truyen hnii tin s nianu cc m nh phn hicii cin cc mu ca ml tn hiu

vo:

( l) 3cos(6007t) r 2cos(l SOOit)

Diriiu truyn hot dim ti lOOOO bit/s v mi mu u vo 'c liriiu t ho thnh 1024

nic in p khc nhau.

a) 1y xc nh tn so ly mu \ tn s lp.

b) Mv xc nh tn s Nyqu isl i \ i ln hiu (t)

c) l v xc nh cc tn s tronu n hiu ri rc x(n) thu irc.

d) Fv xc nh phn uiai A

Li iii:

S bi l/mu log: {1024) ^ 10

.;, ~ ^ -------- =: 1OOOmai ! slObi / mau

Suy ra: F, , : ip = Fs / 2 = 500 IIz

^0 .... ^

. I ' , , = . 2 l - ; = 1 8 0 0 1 1 / .

Cc tn s trone tn hiu ri rc x(n) thu dc:

. OOt'

Ch ng 1: Tn hiu v h th ng ri rc 15

= 0,3

= 0.92 tc i F

Xct thy f\ = 0,9 > 0,5 = 0.5) => i\ =0,1 (tn s cua aiih_al iasing)

Suy ra: X(n) .lco s"2 n(0. 3)nl + 2cos 2 ;i (0 . l)n

m - 1 1023 1023

1.13. Xt mt i thnu x l ln liiu n liian nliu' trn linh v SUI > . Cc CLI k ly mu cab bin i A/[) v D/A tuoi iu im l 1' = 5 ms v T' " 1 ms. Hy xc n u ra Ya (t) cua h

thiiu, nu du vo l:


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X,, (t ) - 2cos(l 007i t) + 3sin(2507it ) t: tnh theo giy

B ic pha sau (post tllter) lc b bt c thnh phn tn s no ln hn F, /2.

X a ( t )

Hnh 1.3

Li gi i: x (n ) = x ^( nT ) = 2cos|^100Ttn,5.10' j + 3sin2507m.5.10'^ j

x( n) = (nT) = 2cos ^nn ^+ 3sin ^57m^ = 2cos ^ 7rn ^ - 3sin " 37in ''[ 2 J V 4 , V 2 , V 4 J

T ' = - - = > y a { t ) = x f t / T ' 1nnn \ 1000

= 2cos''TclOOOt''

- 3sin37tlOOOl

==2cos(5007t) -3si n(7 50 Tt )

r tiooot ^ - 3sin ^3Tcl000tl 2 y V 4 Jy f t ) c cc tn s f, - - 250Hz,f-> - ~ 375 Hz C 2 tn s ny u nh hoTi

^ 2 t ' 2 7 1

p / 2 - 500Hz nn u ra ca h thng l; (t) = 2cos

1.14. Xc nh nng luxTiig ca chui

x i n ) . ! * ' / - * * '[ 3 n < 0

Li g i i: Theo nh ngha ta c:

co / 1 -1

E = M " ) i ' = ml l = : - O C 1 = - ( ) V / n = - c o1 16 9 ,

_ -------------------------- . \ _ ---------------- ----------------1 :1 J_ u j 15 8

16 9 , 143= + - 1 = --- -< +C0

5 8 120

16

V nng lng E l hiLi hnnn tn hiu ang xt l tn hiu nng lnu.

1.15. Xc nh nng liig ca tnhiu nhy bc on v u (n)? Tn hiu Li(n) c phi l mt tn hiu

cng sut khng?

co 00

Li gi i: The o nh ngha ta c: E - ^ x ( n ) - ^ 11'( n ) - y^ lI 1= - C C i i= - c o 1 1= 0


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Nn lnu ca chui l v hn. Do d, tn hiu nhay bc n v khi t phai l tn hiu nng

l i i i m ,

('nu sut truim binh cua tn liiii l:

p - l i m----------> i r n ^ l i m --------- = l i m------ -----= n - - 2 N + 1; V - ^ 2 N ^ 1 N - - 2 + 1 N 2

Do d. tn hiu nhy bc n v l mt tn hiu cni sul.

1.16.Xc nh Iins lng cua dy:

Tronu d. C0,J v A l cc hnu s.

Dy x(n) c phi l dy cnu sut '!

Li


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18 ii bi tp x l tn hiu s v Ma tlab

'v. x (n ) l tc ng vo, y (n ) l p ng ra ca h.

L i gi i:

Hnh 1.4

1.19. Hy v s khi ca h thng ri rc 'c m t bi phog trnh vo - ra sau:

y (n ) = y ( n - l ) + 3 x(n ) + 4 x ( n - 2 )

Li gi i:

y { n )

H n h 1.5: S khi ca h = v(/7 1) + 3x (; ) + 4a '(a2 - 2 )

1.20. Hy xc nh xem cc h tha g c phoTig trnh vo - ra di y c tuvntnh hay khng:

a ) y ( n ) - n x ( n )

b) y ( n ) - x ( n ^ )

c) y ( n ) - x - ( n )

d) y(n ) = Ax(n ) + B

e) y (n ) = e-


17/279

Ch ng 1: T in h iu va h thng r i rc 19

a,y, (n ) + a , y , {n) = a,nX| ( n) + a, 0X3 ( 0 ) (3)

So snh (2) v (3), ta suy ra h thng l txiyn tnh.

b) ' i rng t nh phn a), ta ti m p ng ca h i vi hai tn hiu ri ng r X, (n) v

X, (n) . Ket qu l:

y , ( n ) = x , ( n - )

y , ( n ) = x , ( n - ) (4)

u ra ca h khi tc ng lin hp tuvn tnh Xj (n ) v (n ) l:

y 3(n) = H[a, x, (n) + a, x, (n) ] = a,X| (n -l + a^x, ( i r ) (5)

Lin hp tuyn tnh ca hai li ra trong (2.2.36) c dng:

a,y, (n) + a, y 2 {n) = a ,x, [n -) + a ,x , n ' ) (6 )

So snh (5) vi (6), ta suy ra h thna l tuyn tnh.

c) u ra ca h l bnh phong ca u vo, (Cc thit b in thng c qui lut nh th v

ai l thit b bc 2). T tho lun trc v, ta thv r rng h l khng nh. By gi ta ch r h

l tuyn tnh hav khng?

p ng ca h i vi hai tii hiu vo ring r l:

y, (n) = x,- (n)

y , ( n ) - x ^ n )

p ng ca h vi lin hp tuyen tnh hai tn hiu l:

y 3 (n ) = H a,X (n ) + a .X ; , (n ) ~ a ,X| ( n ) + a^ x , ( n )

a;^x;^ (n) + -2'd3.^X ( n ) x T ( n ) 4 - a 2 X ^ n )

Ng c li. nu h tuyn tnh, n s lo ra lin h p tuyn tnh t hai tn hiu cho, tc l:

a iy i (n ) + a3 y , ( n ) = a,X| (n ) + a ,X2(n ) (9)

V tn hiu ra ca h nh cho trong ( 8) khng bng (9), nn h l khng tuyn tnh.

d) Gi thit l h thng 'c kch thch ring r bi X| (n) v X, (n) ta c;

y, (n ) = AX|(n) + B

Y2(n) = Ax ,(n ) + B

Lin hp tuyn tnh ca X| (n) v X3 (n) cho tn hiu ra l:

y, (n ) = H 'a,X| (n ) +(n)" = A a,X| (n) + a, X2(n ) " + B

= Aa,X| (n) + a, Ax, (n) + B

Nu h tuyn tnh, th tn hiu ra i vi lin hp tuyn tnh X| ( n ) , X, (n) s l:

a,y, (n) + a. y, (n) = a, AX| (n) + a^Ax, (n )+ B (12)

(7 )

(8)

(10)

(11)


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R rnu ( 11) v ( 12) khc nhau nen h kim lioa nin icu kin l u y n lnh. Trn ihc tc. h

uc ni ta bim plnroi m trinh tuvn lnh, tuy lliin c mt ihani s B d lni cho iu kin tuvn

tnh ca h mt i. p nm ra cua ph lluc ca tc nii \ o v h s B^ ) . V th, nu B 7 0,

h l khn u tri tiu. Nmr c li. nu B - 0 h tnt tiu v thoa mn iu kin tuyn tnh.

e) Ch rim, l i 'c m ta bnu hiu ihc vo ra: v (n ) = l h gim dn. Nu

x ( n ) = 0 , ta c y ( n ) = 1 . iu ny ni ln rim h l khnu uven tn.

1.21. Xc nh xem cc h c m la bim nl nm plLiCim tr inh diri y l nhn qu hav khni :

a) y ( n ) = x ( n ) - x ( n - l ) ;

b) y ( n ) = J x (k ) ;

c) y(n ) = ax( n) ;

d) y( n) = x(n ) + 3x(n + 4) ;

e ) y ( n ) - x n ) ;

0 y ( n ) = x ( 2n ) ;

g ) y ( n ) - x { - n ) ;

Li i: Cc h thuc phan a), b) v c) r rng l nhn qua v u ra ch ph llLic hin ti v qu

kh ca u vo. Ngc li cc h ' phn d). e) v ) l khng nhn qu vi du ra ph thuc c vom Ir tirong lai cua u vo. H g) cni kliim nhn qua vi neu la chn n ^ -1 th y ( l) =x ( l ) .

Nh vy u ra li n - - 1 ph thuc vo u vo ti n = 1 cch n hai on v thi gian vpha

tircrnu lai.

1.22. Hy xc nh u ra y ( n ) i vi li U 1uiani dn, c dp im xunii:

h(n ) ^a " . u( n) | a |< l khi t n h iu vo l chui nhy bc n v

x ( n ) = u ( n )

L i ii: Troi m tririig lp ny c h( n ) v x (i ) l cc chui v hn. Ta dng cng thc chp

20 ii bi tp x l tn hiu s v Ma lab

Cc chui h ( k ) , x ( k ) v x(--k) c dnii nh hnh 1.6. Chui tch v, ,( k) ,V |( k) v

v . ( k ) tLi0 'nu ng x ( - k ) . h ( k ) , x ( l ~ k ) . h ( k ) . \ ( 2 - k ) . h ( k ) irc n i t a h i n h 1.6c,d v e. N h

vv ta c cc ui tr ra;

y ( o ) - i

y(l) = l + ay ( 2 ) = 1+ a + a'

R rni . vi n > 0 , tn hiLi ra l:


19/279

Ch ng 1: Tin hiu v h thn g ri rc 21

y (n ) ~ 1+ a a -f ... -a" ( 1)

Ntuxrc li, vi n < 0 , chui lun bnu 7.ero. V ihc:

y( n ) = -0 n < 0

D th ca tn hiu ra y (n ) irc minh lo O' hnh l. 6f, vi 0 < a < . Ch rn


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22 ii bi tp x tn hiu s v Matlab

1.23. Xc nh p ng xung ca h thng gm 2 h thni tuyn tnh bt bin (TBB) ni tip

nhau, bit p ng xung ca 2 h thng TTB B ny l:

f1Xh , ( n ) = u ( n )V /

Li gi i:

p ng xung ca h thng tng qut:

Ta nh ngha chui tch:

' ( k) = h , ( k ) h , ( n - - k ) =v 2 . V y

Chui ny khc 0 i vi k > 0 v n - k > 0 hay n > k > 0 .Ni c ch khc, n < 0 chng ta c

(k) = 0 i vi mi k, suy ra:

h(n ) = 0 n < 0

i vi n > k > 0, tng ca cc gi tr ca chui tch (k ) i vi mi k l:

k=0

Ii-k

3 ' =3

k=0 v 2 .

1-

v 3 .2

v 3 .,n > 0

1.24. Phn tch s khi ca hnh 1.7 v xc nh mi quan h gia y(n) v x(n)

(a)

mv[n-l]

(b)

(c)


21/279

Chng 1: Tn hiu v h thng ri rc 23vn-:'

l i .

Vx.

K5>-^ v[:i]

(d)

H n h . 7

L i g i i:

a) T hnh 1.7a ta c:

v ( n ) = x ( n ) + a v ( n " l )

y ( n ) - p v ( n - l ) + v(n ~ l ) = ( p + 7 ) v ( n - 1)

Suy ra:

v(n - 1) ^ x(n - 1) + av (n 2 )

y ( n - l ) = (P + y ) v ( n - 2 )

Do , y( n) = ([3 + y) v(n - 1) = (p + y) x( n -1 ) + a( |3 + y) v(n - 2)

= (3 + ) x ( n - l ) + a ( P + y ) - ^ ^ - ^ = ([3 + Y) x ( n -^ l ) + a y ( n - l )

b) T hnh 1.71) ta c:

y(n ) = yx(n - 2) + p x(n l ) + x(n - 3) j + a x( n) + x(n - 4)

c) T hnh 1.7c ta c;

v (n ) = x ( n ) - d j v ( n - l )

y( n) = d,v( n) + v(n - 1)

Do ta c th vit li phoTig trinh th 2 nh sau:

y {n ) = d| " x ( n ) - d | V ( n - l ) + v ( n - l ) = : d | X ( n ) + l - d ; ^ j v ( n - l ) ( 1 )

= d|x (n ) + l -d;^ j "x(n - 1) -d|V(n - 2 ) '

= d |X(n) + l -dj^ jx (n - ) -d , l jv ( n - 2 )

T pho ig t ri nh ( l ) y ( n - l ) = d | X ( n - l ) + ( l - d M v ( n - 2 ) , hay toTig ong vi

d| y(n - l) = d;^x(n - 1) + d| l - jv (n - 2). Do :

y{n) + d | y ( n - l ) = d|X(n) + ^ l - d j x { n - ) - d | l - d , j v ( n - - 2 ) + d f x ( n - l ) +

+ d , ^l - d ^ j v ( n - 2 )

= d,x (n) + x(n - 1) , h ay y ( n) = d| X(n) + x ( n - l ) - d | y ( n - l )


22/279


d ) v ( n ) - x ( n ) - v v ( n ) , w ( n ) dj v ( i - I ) r d . u ( n ) , u ( n ) v ( n - 2 ) + X ( n )

T cc phOTig trnh trn ta c:

w ( n ) = d . x ( n ) + d |X (n - l) + d .x ( n - 2) - d j w ( n - 1 ) - d . \ v ( n - 2)

T hinh 1.7d ta c:

y ( n ) = v ( n - 2 ) + w(n ) ~ x(n - 2 ) + \v(n) - w ( n - 2 ) , siiv ra:

djy (n - 1) = d;x(n - 3) + d|\v (n - l) - dW(n -3)

dTy(n - 2) = d.x (n - 4) + d. w (n - 2) - d. w (n - 4)

Do :

y(n ) + d 5y(n -1 ) + d.y (n ~ 2) = x(n - 2) + diX(n - 3) -f d. x n -4 )

+ w ( n ) + d j w ( n - l ) - t - d . \ v ( n - 2 ) j - i \ v ( n -2 ) + d , \ v ( n - 3 ) + d . w ( n - 4 )

= x( n - 2) + d;,x(n) + d |X(n ~ 1)

Hay tng ong vi:

y( n) = c2x ( n ) + d , x ( n - l ) + x ( n - 2 ) - d ,y(n ~ l ) - d .y (n - -2 )

1.25. Cho tn hiu ri rc xc nh bci:

n

(n) =

1+ - 3 < n < -3

Xc nh cc gi tr v v x(n)

a) V cc tn hiu nu:

u tin ta o x(n) sau cho ln hiu ny tr di 4 o'ii v.

u tin ta cho x(n) tr i 4 crn v sau do li tn hiu .

b) V tn hiu x("i + 4)

c) So snh kt qu phn (b) v (c) v rl ra quy tc c uc tn hiu x(-n + k) t tn

hiii x(n)

d) C th biu din tn hiu x(n) du''i dni tn hiu (n) v Ii(n) ir'c khng?

Lo i i : () x( n) = ,1,1,1,1,0,... 1

th cho trn hnh 1.81/2

-4

F-1 0 1 2 3 4

Hnh 1.8


23/279

Chng 1: Tin hiu v h thng ri rc 25(b) Sau khi ao \(n ) ta c:

x ( - n ) = I ......0 , 1, 1, 1, 1 . . o .

'rhc hin re tn hiu nv i 4 oi v. ta c:

' x(-n + 4 ) - I...... J . , l , - , 0 , 2 2

Mt klic, nu ta thc hin tr x(n) di 4 crn v la ir'c:

x(n - 4 ) = {......0, - . 0 , - - V l . 1,1.1, 0 . . . [

By gi' ta o li tn hiu x(n - 4). irc;

x ( - n - 4 ) = {......0,1,1,1. 1 . .0. - - . . . . 9 7 T

( c) x ( - n + 4 ) = { ...... 0 , l , , l , . o .

1/2

Ilih 1.9

(d) c -c x(-n + k), u tin ta o x(n) s nhn c dy x(-n). Sau a thc hin

dch dy x(-n) i k n v v bn phi nu k>(), hoc k on v v bn ri vi k


24/279

26 ii bi tp x l tn hiu s v Matlab

b)

x ( 4 - n ) =

c) x( n + 2) = . . . ,0 ,1 ,1 ,1 , 1 , - , - ,0 , . . . ^ [ T 3 3

d ) x ( n ) u ( 2 - n ) =

e) x(n- l ) ( n - 3 ) = |...,0,0,0,0,1,0,...

1,1,1,1,0,3 3T

0

x(n-) = {...,0,x(4),x(l),x(0),x(l),x(4),0,...

g)x ( n ) + x ( - n )

1 1x ( - n ) = - ...,0 , - , - , 1, 1,, 1,0 ,...

h)

(5 6 2

x ( n ) - x ( - n ) 'x(n) = -------- ^-------

x ( n ) = [ . . , 0 , ^ i - i 0 , 0 , 0 , 0 l 0,...

[ 6 6 2 ^ 2 6 6

2 6 6

1.27. Hy biu din dy xung on v theo dy nhy on v v ngc li.

Li gi i:

u ( n j = 0

0 n < 0

1 n > 1

0 n < 1

1 n = 0

0 n 0

u ( n - l ) =

Suy ra: 5 (n) = u( n) - u (n - 1)+CO

Mt khc t a t hy r ng: u ( n ) = ^ ( n - k )k=0


25/279

Ch n g 1: Tin h iu v h thng r i rc 27

1.28. Chng minh rng bt c tn hiu no cng c th 'c phn tch thnh mt thnh phn chn

v mt thnh phn l. Vic phn tch ny c phi l duy nht khng? Minh hoa pht biu trn bng

tn hiii: x( n) = |2,3,4, 5,6

Li i:

t:x(n) + x( -n )

x ( n ) - x ( - n )

Vi:

X o ( n ) =I

X e ( - n ) = X e ( n )

Xo(-n) = - x(n)

x(n) = X o ( n ) + x,(n)Do :

Cch phn tch nv l duy nht.

i vi: x( n) = {1,3,4,5,7}

Ta c: x^(n) = {4,4,4,4,4} v x(n) =

1.29. Ch ng minh r ng nng lng ca mt tn hiu nng lng gi tr thc bng tng ca cc nng

lng ca cc thnh phn chn v l ca tn hiu.

Li g i i:

00

u tin ta chng minh rng: ^ x^.(n)xp(n) = 0I l = - c c

X ^ c ( n K ( n ) = x^ (m )x(m)n=-D ni=-co

co co

= ^e(n)Xo(n)= 2 ] x^,(n)x(n)= 0n = - 0 3 I 1 - - C O

co co

Do : x - ( n) - ^ [ x^ (n ) + x ( n ) f n = - c o n = - c o

= X x / ( n ) + ^ \


26/279


(a) y(n) = cos [x(n)]

(b) y ( n ) = X x (k )k = -x

(c )v (n ) ^ x(n) cos((j()n).

( d ) y ( n ) - x ( - n + 2 ).

(e) y(n) Trun [x(n)J, vi Trun[\( ii)] biu th cho phn nmiyn cua \{n). c 'c b

ct bt.

(0 v(n) ^ Round[x(n)], vi Round[x(n)] biu th cho phn niiuyn cua x(n). c c bim

cch lm trn.

Ch V.' Cc l hng rog phn (e) v (f) c lng ho hn 0

0 nu x( n)


27/279

(j) Il tlim clnu. lLi>cn tnli. khiu bt bicn. khni nhn qua v n nli.

(k) thnii nli. khm iLiyn tnh, bt bin, nhn qua v n nh.

(I) thnu tnh, tuycn lili. bt bien. khniz Iihn qu v n dnh.

(ni) ihnii t nh, kl inu iLiyi l i ih. bt bin, nhn qua v n nh

(n) l tlnu tnh, tuycn tnl. bl bin, nhn qu v n nli.

1.31. Hv tnh tch chp y( n) - \ ( n ) * h ( n ) ca cc tn hiu v kini tra s chnh xc ca kt qu

b nu \' ic kiiii tra biu thc: y \ ( n ) h ( n ) = y y (n )

1) x ( n ) - | 1 . 2 , 4 . , h ( n ) - | l . l . L K

2 ) x(n ) = l . 2 , ^^ - l | , h (n)^x(n)

3 ) x ( n ) - . - 2, 3 , h ( n ) - . O . l . l . u

4) x(n) = , 1. 2 , h( n) = Li(n)

5) x ( n ) = | u ( n ) , h ( n ) = [ ; j ] u ( n )V-- / \ /

Li iai:

y(n) = h ( k ) x ( n - k)k

n 11 k k n -- x

(1) y(n) = h(n) *x(n) = 11.3.7JJ 6 , 4 | (.0

0 ((0 ) , ^ ( - 1 ) " x {n )z '" .

(2) y(n) - I.4.2,-4,1

J ^ y ( n ) - 4 . J ] h (k ) = 2 . ^ x ( k ) = 2

II k k (3) y (n )- 0,0J.-1.2.2J,3

^ y ( n ) = = 8 , ^ h ( n ) = 4 , ^ x ( n ) =2!\ n n

(4) y( n ) L(n) + Li(n - 1) + 2u( n - 2)

^ y ( n ) = x . ^ h ( n ) ^ , ^ x ( n ) = 4

Ch ng 1: T in h iu v h thng r i rc 29

(5) y ( n ) = 2 (- -) " | u ( n )2 4 1

11 n [1


28/279


1,32. Xc nh v biu din php chp tn hiu sau:

0 < n < 6

0 n khc

x ( n ) =

1 __- n3

h ( n ) =- 2 < n < 2

n khc

Theo p hong php s.

Theo phng php phn tch.

L(ri i:

(a) x(n)

h ( n ) = l , l , , l , l

y(n) = x ( n) * h(n) ={ i , , 2 , y , 5 , y , 6 , 5 , y , 2

(b) x(n) = - n u ( n ) - u ( n - 7 )

h(n) = u(n + 2) - u(n - 3)

y(n) = x(n) *h(n)

== - n u (n ) - u ( n - 7 ) * [u (n + 2 ) - u ( n - 3)

u(n) * u(n + 2) - u(n ) * u(n - 3) - u(n - 7) * u(n + 2) + u(n - 7) u(n - 3)

y(n) = - (n +l) + (n) + 2 (n-l ) + (n-2) + 5(n-3) + (n-4) +

6(n-5) + 5(n-6) + ~ (n-7)+ (n-8).

L33. Thirc hin tch chp y(n) ca tn hiu:

a " - 3 < n < 5x ( n ) .

h ( n ) =

0 n khc

1 0 < n < 4

0 n khc

Li gi i:

y(n) = 2 ] h ( k ) x ( n - k )k = 0

x (n ) = a \ a ^ % a ' , l , a .....,a'^|


29/279

Ch n g 1: T in h iu v h thng r i rc 31

h ( n ) - , l , U

Do :

y ( - 3 ) - a " '

y ( - 2 ) - + a"

y ( - l ) = + a + a ^

y(0) = + a ' + a ' + 1

y(l) = + a ^ + a ' + 1+ a

y(2 ) - a + a ' + a + + a + a

y(3) ~ a^' + 1+ a 4- cx" +

y(4) + a ^ + a a -fl

y(5) a'" 4- + a~ + a

y(6 ) ~ a'" + a"* + a' ' + a '

y(7) = a" + a"* + a

y(8)==a^ +a^

y(9) = a '

1.34. Hy tm p ni> xung ca h thng tuyn tnh b bin c p ng ra l:

y ( n ) = x (n ) + x ( n - l ) + . x (n - m) + ... m > 0

Nhn xt t nh nhn qu v ti nh n nh ca h thng.

Li gid i:

x ( n ) ^ 5 ( n ) = ^ y ( n ) = h (n )

Suy ra; h(n) = 8 (n) + - 8 ( n - 5(n " m) + ^ - ( n - m ) = ^

^ v5y m=oV5yXt tnh nhn qu ca h thng:

* m > 0 thi h(n) - 0 vi n < 0: h t hng l nhn qu

X n n n h ca h thng:

r 1 Y'* m > 0 : h thng l nhn qu, ta c: h ( n ) = - u ( n )

\ 5 /

+CO

s = h ( n ) - ^

l=-c 11=0

Suy ra: h thiig l n nh.


30/279

1.35. Cho h thorm: y(n) ny(n - 1) + x(n)

Xt tnh tuvn tnh, bt bin v n nh ca h thnii.

Li gi i:

Nu: y,(n ) - nyi (n -1) + x, (n)

y^ln) ^ n \s (n -1) + X2 (n )

D o X (n) = X|(n) + X2 (n) a vo h thim, ta c u ra;

y(n) = nv(n - ) + x(n)

v'i V (n) ^ a \ ' i(n ) + by:(n )

Vy. h thne l tuyn t nh. Neu u vo l x(n - 1) ta c;

y(n - 1) - (n - l)y(n -2 ) + x(n - 1)

M y(n - 1) = n y(n -2) + x(n - 1)

Vy h thns l khng bt bii. Nu x(n) u(n), do |x(n)| < 1. Nhinm do u vo b uii

hn nn u ra l;

y ( 0 ) - l : y ( l ) - l + = 2 ; y ( 2 ) = ^ 5 . . .

Ng hal u ra khng b \i hn. Vv h thng khng n nh.

1.36. Gii phoim trinh sai phn sau vi p niz u vo bngkh ns (x{n) 0)4

v(n - 1) + y(n ~ 1) ~ x(n)

Li di:

V'i x(n) ^ 0, ta c:

4y ( n - l ) + ~ y ( n - ) = ()

y{~-l) = ..~ y{-~2 )

y(0 ) = ( - ^ ) - y ( - 2 )

y{i) = ( - | ) V ( - 2 )

32 ii bi tp x l tn hiu s v M atlab

y(k) = ( - -^-) '^-y(-2 )

1.37. Gii phng trnh sai phn:

y ( n ) - - ^ y ( n - 1) + y ( n - 2 ) = x(n)6 6

vi x(n) 2 'u(n) v cc iu kin u: y(-2 ) = y(-l) = 0 .Li g i i:

Xt plurcTng trnh thun nht;


31/279

5 y( n ) - y( n ~ ) - - \ '( n 2) -0

6 (>'

5 1PhuxvnLi t r inh dc t r i rng la : X - X r ~ ~

(>

n u h i m : /.2 3

Nehii n ricnu CLiaphcTim rinh riL vcVi dii \ i x(n) 2''u{n) i:

Vp(n) --- k (2 ') u (n )

l ' ha y nu i i i i n \ \ o ph i r o ' i L i i r i ! h ba n d u l a c i i c :

5 , 1k ( 2 ) " u ( n ) - k ) ( 2 ) '' ' Li{n - - 1 ) - - k ( - ) { 2 ) ' - ' U n - 2 ) = 2 ' ' u ( n )

6 ()

\ ' ' i 11 2:

5 k


32/279


Ta c:

kn4" u(n) - 3k(n -1 )4 "' u(n -1) - 4k(n - 2)4"- - u(n - 2) = 4" u{n) + 2(4)"'' u(n ^ 1)

Vi n = 2, k(32-12) = 4 ' + 8 = 2 4 - > k - - ^

[ 6 1Ngh im tng qut l: y(n) = y(n) + yi,(n) = n(4)" + C| (4)" + Ct ( - 1)" Li(n)

tm C| v C2 ta gi s y(-2) = y(- l ) = 0. Khi :

y ( 0 ) = l v y ( l ) = 3 y( 0 ) + 4 + 2 = 9

Ta c: C| + C = ]

v 4c :- Ci = 21/5

Nn:26 1

C| = v c, =------' 2 5 ' 2 5

Nghi m tng qut l:

y(n) = u(n)

1.39. Tm p ng xung ca h thng cho bi pho ng trnh sai phn:

y(n) ~ 3y( n - 1) - 4y (n - 2) == x(n) + 2 x(n - 1)

L i g i i:

Theo bi 1.38, nghim c trng l: X -1, X2 = 4

Do : y i ,( n )- Ci4" + c . ( - l f

Khi x(n) = (n) ta tim iRTC v(0) =="1 v v (l) ^ 5

Ta c: Ci + C = 1 v 4C| - C2= 5.

Tm c: C 6/5; C2 = -1/5

Vv p nu xuh; h(n) =5 5

u(n)

1.40. Tm p ng xung v p ng nhy on v ca cc h thn c m t di y:

a) y(n) - 0,4y( n - 1) + 0,03y( n - 2) = x(n)

b) y(n) - 0,7y(n - 1) + 0,1 y(n - 2) = 2x(n) - x(n - 2)

L (/ i g i i:

a) y(n) - 0,4y(n - 1) + 0,03y(n - 2) = x(n)

Pliong trnh c trng l: -0, 4 \ + 0,03 = 0

x = 0 , l , 0 , 3 .

Vy:

Vi x(n) = 5{n), iu kin ban u l:

y (0 )= l ; y ( l ) - 0 . 4 y ( 0 ) - 0 . 4


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Chng 1: Tn hiu v h thng ri rc 35V \ : C | + C : = l

V 0,lC| + 0,3c: = 0.4

Suy ra: c, =- 1/ 2. C: = 3/2

Do h(n) = [ - l ( ^ y + ^ ( A ,n u(n)

p im nhv on v:

s(n) = ^ h ( n - k ) . n >0k=0

n

k=0

3 . 3 , , ^ 1 1

2 10 2 10

n-k 5 r . 1

9 l 10

1_5

7( A ) - ' _1

10u(n)

b) y(n) - 0 ,8y(n - 1) + 0,15y(n - 2 ) = 2x(n) - x(n - 2 )Phirng trnh c trng:

X' -0,8X + 0,15 = 0

X, =0 ,3 , \2= 0,5.

Do : y,,(n) = C| ( ) " + C:

Vi x(n) ^ (n), ta c;

y(0 ) - 2 ; y ( l ) - 0 .8y( 0 ) - 1.6

Do :

-'> C] 5 , Ct 3

h(n) =

C| + Ct = 2 v C) + C2 == 1.62 10

2 10u(n)

II

p ng nhv on v: s(n) = ^ h(n - k )k-o

= 5 ( i ) - " - 3 T '" " = 5 ( i ) - t ( 2 ) k=(l ^ k=() ^ k=0 k=0

= 5 (-V(2"" - 1)9

u ( n ) - " 3 Y n o

V.IO,

\ IHI

1 V * 90 ^ 10 1- 1 -, _

2j 21 UoJ

V 3

u ( n )

-1 u(n)

1.41. Cho h thnu vi p ng xune:

/ i \ I

h ( n ) H . 2 ]0 < n < 4

n


34/279

36 ii b i p X / ' / t i n hiu s v Maab

X c diil d u \' o x(n) \''i 0 ; 1 1 < 8 i m \ i d u ra l

ytn)= -. 2. 2 . 5. 3. 3. 2. 1.0!

. i ^i: h(n)' 2 ' 4 8 '16 '

y(n)- 1,2,2.5.3.3.3.2.1.01

x( 0 )h(0 ) = y(0 ) x( 0 ) =1

v( 1) => x( 1) = -)

h c hi n icp qu Irinh nay la di ic :

1 3 3 7 3 ix(n)=1. ..........

0 ..1 0 !^ 2 2 4 2

1.42. Xe m kt ni li ih nu . '! 1 nhir 0' hinh v.

(a) Biu dicn dp niz xiinu tieo li](n). h^(n). h;.(n) v h^(n)

(b) Xc dinh h{n) khi:

h , ( n ) = l - i . - i . ~ !

h : ( n ) = h ( n ) - ( n t 1)u(n)

l i 4 (n) = ( n 2 )

(c) Xc dnh p im h lliim troim cu (b) nu

,\( n) = ( n i 2) 13 (n - 1) - 4 ( n - 3)

y (n)

l i n h L I U

L i


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(c) x(n)- | l . ( ) .0 .3,0,-4 'V / I - I

> - . - , I

y(n ) = 1 2 . - . -- 2 .1 2 4 4 : :

1 .43 . ( ' c h t i im sau c l \ c p\v\ l a ! u \ ' cn l i nh hay k i inu ' /

a) y ( n ) = T | x ( n ) | = x ' ( n )

b) y ( n ) = 1] X ( n ) = n. \ ( n )

L() ai:

a)

T-a X, (n)- -a .x,(n) =[a ,x,(n) + a ,x, (n)]a, \ , ( n ) i ' a ,a . \ | ( n ) x . (n ) + a x (n )

a ^y j ( n) - i - U KX, ( n ) x , ( n ) + a y . ( n )

^^a Vi ( n ) i - a . y , ( n )

S u y ra h llim khnu tiivcn iih.

b)

T a, Xj ( n ) a.x , ( n ) V UX (n) f a . \ . (n)

- UHXi (n ) i- a.nx- , ( n )

: a V , ( n ) i a . y . ( n )

Suy,ra h ihn l luycn lini.

1.44. ( c h li ini sau dv c p hai a ba ICH Ihco n hay kh nu '/

a) y ( n ) = r | x ( n ) [ = x - ( n )

b) y (n ) = T x ( n ) = n x (n )

Li :

a) v(n k) ^ t X(n - k ) ~ X' ( n k ) h thnu bt bin.

T X(n - k )| n \ (n k)

y (n - k) (n - k )x (n - k) T X(n ~ k)

Suy ra h thng l kh n bt bin.

1.45. Cho 2 h ihnii, tiivn lnh bl bien c p rnu xung lni ng l h | ( n ) - 2 " Vn v

r 1 V' . ,h , ( n ) - u ( n )u h p n i t ipnhau.v5 /

a) Hv l m p n i xun u h(n) cua h hni> tim qut.

Chng 1: Tn hiu v h thng ri rcic 37


36/279

38 ii bi tp x i tin hiu s v Ma tlab

b) Hy nh n xt tnh nh n qu ca h th ng h| (n) ,h . (n) v h(n) .

Li gi i:

+ CC

a) h (n )= ^ h , ( k ) h , ( n - k ) = ^ h 2 ( k ) h | ( n - k )k=-co k = ~co

1 1 0 .

10

b) n < 0 ,h, (n) 0=> h thng c p ng xng h| (n ) l h th n2, khnu nhn qu.

n < (n) - 0=>h thng c p ng xung h, (n)l h thne nhn qua.

h( n) = 2 '\ V n , do vi n h thng tng qut c p ng xung

h(n)l h thng khng nhn qu.

1.46. Gi s e(n) l tn hiu ri rc c dng hm m: e( n )= ^a vi V n ,a :h a n g s .

x(n) v y(n) l cc tn hiu bt k.

Chng minh rng; ' e (n )x( n)" * ' e (n )y( n)" =e (n ) 'x ( n) * y( n) j

Li gi i: t:

X, (n) - a " x ( n )

y , ( n ) = a " y { n )

V T ^ [ e (n ) x ( n ) ] * [ e (n ) y ( n )] = x , (n ) * y , (n ) = g x , ( k ) y , ( n - k )k = ~ c o

+CO +C0

k= - 0 0 k = - c o

, " [ x ( n ) * y ( n ) ] = e ( n ) [ x ( n ) * y ( n ) ;= a = VP

1.47. Cho 2 h thng tuyn tnh bt bin c p ng xung tong ng l h, (n ) v (n )g h p ni

tip nhau.

h, (n) = h , (n) u(n) - u (n - 5)

a) Hy tm p ng xung h(n) ca h th ng tng qut.

b) Hy nhn x t t nh n nh v nhn qu ca h thng hj (n ) , (n ) v h(n) .

Li gii :

\ \ \ \ ^ 0 < n < 4a) h,(n)=::h2 (n) = u ( n ) - u ( n - 5 ) = [0 n ^

h ( n ) - h , ( n ) * h , ( n )


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Chng 1: Tn hiu v h thng ri rc 39

V i n < 0, hoc n > 9: h(n) ^ 0

h ( 0 ) - h ( 8 ) = l

l . ( l ) = h(7 ) = 2

h ( 2 ) = l,(6 ) = 3

h(3 ) = h(5 ) = 4

h(4) = 5

H thim l nhn qua.

s = 2 h(k) = h ( k ) = l - r 2+ 3 + 4 + 5 + 4 + 3 + 2 + l = 2 5 < o tk-o

Suy ra h thng l n nh.

1.48. Cho (n ), X. (n ), X3 (n) l 3 chui tun hon vi chu k CO' s- tng im l T, T 2, T 3 . Mt

chui l kt qu kt hp tuyn tnh cua 3 chui tun hon ny c phai l mt chui tun hon

khng? Nu l mt chui tun hon th chu ki CO' s ca n l bao nhiu?

L gii: Mt chui l kt qu kt hp tuvn tnh cua 3 chui tn hon ny cne l mt chui tun

hon. Chu ki ca chui mi bng BSCNN ca tt c cc chu ki.

Chu k ca chui mi = BSCNN(T|, T2. T 3).

V d: nu T, - 5, T. - 3, T, - 6 thi T - BSCNN(5,3,6) - 30

1.49. Hv tnh nn^ l-ng ca chui c chicu di N/. \ .-. ^Tkn i

X ( n ) - c o s--------1, 0 < n < N - I

Lii: x ( n ) - c o s27ikn

" i r)

N

, 0 1:: n 1: N 1 , suy ra:

N-l I \ - l _ N-!E, = 2] co s ' (2 7r kn /N ) = - ^ ^ 1-f CCS (47k n / N)J + ~ ^ c o s ( 4 j r k n / N)

1 1 = 0 ^ I i = ( ) I I -- 0

t: c = ^cos(4Tikn / N) , s = ^sin(47rkn / N ) , ta c:1 1 = 0 I I - 0

N -

C + j S = X '11= 0

Do : c = R e { c + jS} = 0 => ^

1.50. Hy xc nh nng luns v cnu sut trunu bnh ca cc chui sau:

a) X, (n) = nu ( n)

b) x , ( n ) = Ae' '' '

c) X, (n) = Asin2:n

M+ (p


38/279

40 Gii bi tp x ly tin hiu s va Matlab

L i ^ii: a ) \ , ( ! 1 ) - n u ( n ).

-r. - /.

n ^ - - r 1 - - 3 0

1\ , - Hm ..' ' K - - 2 K.-

- V n u( n) ]" = lim ' V n ' =c/;

b) x , (n ) .= A. ,c' -"

; A e - ' - ' - !Ar 11 - - / . n

1\ , Hm ...- V jA,.c"'''" 1 = l im y A| j ' = limn --- K n -: - K

2 K +

c) X, ( n ) /Vsin ---cp 1^ troni : co.^ .n / KA ^ \A ^ ' 7 9

p dim CP^ ihc t nh nn lni v cng sut rurm binh ca mt chui nhir phn a) v b)

ta c:

, -- a:x3

I^3 = A ^ f =

4

1.5L MI chui tn hin sin lin lc theo thi gian x, (t ) - cos(o) ,jn l' ) u'c ly mu li

t --- n ' . - co < n1 ) = ct)s (o)(j nT). Vi i tr

no cua T th x(n) l mt chui tun hon'? Clu k c SO' ca chuoi x(n) l bao nhiu nu

(0,J 16. ' --- 71/ 6

Li ai:

d iu i x (n ) x. (n 'r ) = cos((Ofn ) l chui lun hon vi tt c cc ei tr T tho mn iu

kin cO(/rN , vi r v N l cc songuyn. Do 0),T ==2 r / N v r/N l mt s liLi t nnc) jT phi l mt s hu t. Vi o.)g - 16. T ~ k / 6 , la c N=2r/3, suy ra i tr nh nht ca N = 3,

xay ra khi r 3.

l o i . Xt cc chui sau:

i) X, ( n ) = 2 8 ( n - l ) ~ 0 . 5 S ( n - 3 )

Ii) X. (n) - -36(n ~ ) + 8( n + 2)

lii) hj (n) 2 (n ) + (n - 1)-- 3S(n -3 )

iv) (n) = -S (n ~ 2) ~0,5 (n ~ l) + 3(n - 3)Xc nh cc chui l kt qu ca vic nhn mt cp chui trn,

a) y , ( n ) = x , ( n ) * h , ( n )


39/279

b) y , ( n ) - x , ( n ) * h , ( n )

c ) y , ( n ) - x , ( n ) * h , ( n )

d ) y , ( n ) = x , ( n ) * h , ( n )

L i d i: ' l a c ; 5 ( n -- r ) * 5 ( n - s ) = ^ {111 - r )cS( n - s - m ) = 6 ( n - r s )

a ) y , ( n ) = X, ( n ) * h , (n ) = (2 ( n - 1) - 0 .5 ( n - 3)) * (2 ( n ) f


40/279

r h o y ( n ) l u ra ca u vo x ( n ) , ta c:

y( n) x( n) - y '( n - 1) + y(n - 1) .Do , i v-i mt u v o x ( n n,j) , nu au ra l

v (n - n, ) , ta c mi qua n h eiCra u vo - u ra nh sau:

y (n - = x ( n - y" ( n - - ) ^n " 0 ^ thim l bt bien theo thi ian

Vv i vi u vo x (n ) = a j .i (n ), u ra y (n ) hi t ti me)thaim s K khi n 00 .

Pho-ng trinh sai phn trn khi n -~>(X), tr- thnh K= a - K + K hay K' =a , ngha l K ^ \ la .

1.54. Cho x(n) l chui c chiu dihu hn c xc nhi vi N, < n < , vi

Toim t, cho h ( n ) l chui c chiu di hu hn 'c xcnh i v iM, < n < M , , vi

M. > M . Ta nh ngha: y( n) = x (n )* h (n )

(a) Xc nh chiu di ca y(n),

(b) Phm vi ca h s n m y(n) c nh ngha.

Li gii:

y ( " ) = x ( m ) h ( n - m ) . v, h ( n - m ) 'c nh n eh a i v'i M| < n ~ m < M . .m= N|

Do , i vi m = N, , h ( n - m ) irc nh ngha oi vi Mj < n - N | < M . . hoc tiroim iro'ngi vi Mj + N j < n < M ^ + N 2 .

(a) Chiu d ca y (n ) l M, + N. Mj - Nj + 1

(b) Phm vi cua h s n m y(n) ^ 0 l:

min (M, + N| ,M , + N . ) < n < m ax (M j + N,,1V1\ + N , ), ngha l: M| + Nj < n < M, + N,

1 .5 5 . Cho y (n ) = X| (n )* (n ) v v (n ) = X, (n - N j ) * X. (n - N;,) . Biu din v(n)do-i dng


y(n). -f-ccLi gi i: y(n ) = X | ( n ) * X 2 ( n ) = ^ X | ( n - k ) x 2 ( n )

M v(n) = X | ( n - N | ) * x , ( n - N 2 ) = X N , ) .k=;

t k - N, = m , khi v(n ) = ^ X, (n ~N | - N, - m ) x 2 (m) = y ( n - N | N j)i n = - i x )

1.56. Cho g ( n ) = x ( n ) * x , ( n ) * x 3 (n) v h ( n ) = Xj (n - N | ) * X, (n - N . ) * X3 (n - N 3) . Biud i n h ( n ) d i d n g g ( n ) .


41/279

L i i:

( n ) - X| ( n ) * X , ( n ) * X 3 ( n ) . t r o i m d v ( n ) = X, ( n ) * X , ( n ) . 'Fa c :

v( n) = X, (n - N )* X, (n ~ N . ). do d h ( n ) = v( n)* X3(n - N 3) . ' kct quabi1.55,tacv( n) - y(n Nj ) . Do , h( n) = y(n ~ N| N 3) * X3 (n N 3). S dnu kt qubi1.55ln

nCra ta 'c: h (n ) g (n - N; - N. - N 3)

1.57. Chim minh rng tch chp ca mt chui c chiii di M vi mt chui c chiu di N l

mt chui c chiu di bn^ (M+N~l)

Urii: Gia s chui x(n) c chiu di N, chui h(n) c chiu di M.

y( n) ~ x (n )* h (n ) = ^ x ( n - k ) h ( k ) . Do h(k) c chiu di l M v i'c xc nh trongk="--o::'

M- l

k h o a n g 0 < k < M ~ 1, t ng t ch c h p tr n tr t h nh y ( n ) = ^ x ( n - k ) h ( k ) .

k-o

y(n) l khc khng i vi tt ca cc gi tr cua n v k m n-k tho mn iu kin

0 < n - k < N - l .

G tr nh nht l n - k 0 v xv ra vi n nh nht ti n 0 v k = 0. Gi tr l'n nht l n -

k ^ N - 1 v xy ra khi k t cc i ti M-1 . Do n - k ^ M - 1, => n = N + M - 2. Suy ra tng s

cc mu khc kling bng (N + M ' 1).

1.58. x(n) v h(n) l hai chui c chiu di N. c cho ah sau:1. 0 < n < N - l

0 n ^

Chng 1: Tn hiu v h thng ri rc 43

x(n ) =

h ( n ) =N + 1, 0 < n < N -1

0 n r/:

Hy xc nh v tr v gi tr ca mu doTig '! nht ca y ( n ) - x ( n ) h ( n ) m k h ng cn

phi tnh tch chp.

+CO N-l

L i i: y (n ) = x ( n ) * h (n ) = ^ x ( n - k ) h ( k ) = ^ . \ ( n - k)h (k) . Gi tr cc i ca y(n) xyk=~cr k-o

ra li n ^ N-1 khi tt c cc s hng tronu tim chp l khc khng. Gi tr bng;

k = () k = ] ^

1.59. Xt 2 chui s thc l h(n) v g(n) 'c biu din d'i dns phn chn v phn l, ngha l

h (n ) = h^,(n) + h (n ) v g( n) = (n ) + ( n ) . i vi cc chui sau, hy xc nh xem chng l

chn hay l.

a ) h , ( n ) * g , ( n )

b) h ( n ) *g , , (n )

c ) h ( n ) *g ( n )


42/279

44 Gii bi tp x y in hiti s a Matlab

Li (i\ a) y ( n ) ^ h , (n ) 0


43/279

Chng 1: i n hiu v h thng ri rc 45

s( I'! !- 5) - s( n !())*' \ i ( n T5) - ------u (n - 10)a a I

l'Lr hi ii h Vct r n la c: y ( n ) X ( n ) h( n ) \ ( n ) h ( n - 2)

II --6 1 I' 1 II - - 1 ! I

/ . a - , - ' / , \ n XS u v ra: v ( n ) ------- - - L i (n t ) - - L ( n - 1 0) - -------------u ( n - - 3 ) -r ---------

a 1 a a - 1a

1 . 6 2 . l l v x c d i n h m h nh lo i t rc l i p cho m i h l i nu . / n sau ;

2 y ( n ) - y ( n - ) 3> ()1 3) X( n) - 4 \ (n -5)

L i ( n ~ 1 2 )

n . 2

1 . 6 3 . M t h t h i m c i r c II t a o i huo im r inh sa i phan sau ;

y ( n ) -- a y (ii - l ) I b x ( n )

a) 1_\ xc diih b theo a sao ch) ^ h i i) - 1i;

b ) l v l n h d p n u buxVc r i m tl ii k h i m { / c r o - s t a t c s l c p r csp t>nsc ) s{n) c u a l th n i i v

c l in b sao c h o s(oC') 1

c) S( snh cc m tr cua !') liLi tirc (r a) \'a l) .

Li ai:

a) \ (i ) -V av(n - ] ) - b \ ( n )

h ( n ) b a " i i ( n )


44/279

46 ii bi tp x tin hiu s v Matlab

11=0 I - a

=> b = 1- a

b) s ( n ) : = ^ h ( n k) = bk-o

- a114 I

u(n)

1- a

c) b 1 - a t rong c hai t r i in h p a) v b)

1.64. Hy xc nh p ng y(n) , n > 0 ca h h n s c m t b'i phoTiutrnh sai phn bc 2

nh sau:

v (n )- 4 y ( n ~ l ) + 4 y ( n ~ 2 ) - x ( n ) x ( n - ' l ) vi u vo l:

x (n ) - (~ l ) " u(n ) v iu kin u l: y ( - l ) - y ( - 2 ) = 0

Lii: y ( n ) - 4y ( n - 1) + 4y ( n - 2 ) = x ( n ) x ( n - 1)

Phoig trinh c t rimg l: ." 4X + 4 = 0 => = 2

Suy ra: y j (n ) ==c, 2" + 0302

Nghim ri ng l: (n) k ( l)" Li(n)

Thay nghim ring ny vo phng trinh sai phn, ta irc:

k ( l)" Li(n) - 4 k ( l ) *u ( n - ) + 4 k ( - l ) " \ ( n - 2 ) - ( - 1 ) u ( n ) - ( - ) ' ' u ( n - 1 )

Khi n = 2, k( 1 + 4 + 4) - 2 => k - .

Phirng trinh tim qut;

y (n ) = c, 2 " + C . n 2 " + - ( - ! ) " u(n)

T iu kin ban u, ta c v( 0) - 1, y( l) - 2 . Do :

2 _ 7c, + = =>c, = ' 9 ' 9

1.65. Chng minh rng bt c tn hiu x(n) no cng c th biu din di dn^:

z [ x ( k ) - x ( k ~ l ) ] u ( n - k )k = - r .


45/279

Chng 1; Tn hiu v h thng ri rc 47

Li a:

x ( n ) - x ( n ) * ( n )

- x(n ) * ' u( n) ~ u(n - l )J

^ [ x ( n ) - x ( n - l ) ] n i { n )

x ( k ) ~ x ( k l) u ( n ~ k )k = --/:

1.66. Hy xc nh ph Tm trinh tna qut cho n > 0 ca phong trnh vi phn sau:

y ( n ) + O J y ( n - l ) - O .Oy(n - 2 ) - 2 " u( n)

vi ieu kin u v ( l) = 1. y ( - 2 ) = 0

Li ai:y( n) + 0.1y(n - l) - 0 . 0 6 y ( n 2) ^ 2 u( n) vi y( ~ ) ==1. y (- 2 ) = 0 . Ta c phong trinh

c t rng l; +0 , l / - ~0 ,0 6 = 0= >/ ., - - 0 .3 - 0 ,2 .

Suv ra y (n) - c, (- 0 ,3 ) + c ^( 0 ,2)"

Nghi m ri ng l: ( n ) - p 2 "

Thay nghim ring ny vo phone trinh vi phn cho, ta c:

P2 ' -H[3( 0 , 1) 2 " ' --(3( 0 , 0 6 ) 2 '" - 2 'u ( n ) . i v-i n - 0 . ta c;

200p + f ( 0 .1 )2 ' ~ p ( 0 , 0 6 ) 2 " - 1 hay = 0 ,96 62 .

Plng trinh lng qui: y ( n ) y , , ( n ) yp (n )- C| ( 0 ,3)" C\(0, 2)

M y( - l ) = c , (~0 ,3) ' +c , (0 .2 ) ' ' va

y (-"2 ) = c, ( 0 , 3 ) ^ + 03( 0, 2 ) 2 "= ( ) hay loTig oTig vcVi:

10 ^ 107 ^ 100 50----- c, + 5 c, - - v ----- -c, +2 5c .

3 207 9 207

Giai h phuxrne trnh la thu c nghim l c, =- 0, 10 17 v c. =0 ,0 35 6. Suy ra phong

Irnh im qut c dne nh sau:

y ( n ) - - 0 ,101 7( - 0 ,3 ) ' ' +0 .0356( 0 ,2 ) +0 ,9662( 2 )

1.67. Hy xc nh cc chui t tironi quan ca cc tn hiu sau y:

a) x ( n ) = l , 2 , 1. 1 :

b) y(n) = u , 2 , l


46/279

N cli ki luii cua bn/

Li gii :

R , , ( l ) = x ( n ) x ( n - l )

R , , ( - ^ 3 ) = x( 0 ) x ( 3 ) = l

R , , ( - 2 ) = x ( 0 ) x ( 2 ) + x ( l ) x ( 3 ) = 3

R, , (~- l ) = x(0) x( l ) + x( l )x( 2) i x (2)x(3) = 5

R , . , ( 0 ) = x ^ ( n ) = 7

iMtkhc: R, (--1) = R ,( l ) . D o d o ; R ^ (1) - l,3.5.7. 5.3.l

I ^ v v ( l ) = - V ( " ) y ( " 011 - - - X

48 Gii bi p x Iv in hu sv Malab

Ta co: R,, ( 1) = ]. 3 , 5 . 7 . 5 . 3 ,l

Ta thv rn v(n) = X(~n 4 3), diu nv tcim iroim vi vic lu ui chui \(n). Diu ny

khnu lm thav i chui l tircmL' quan.

c . BI TP NNC CAO1.6 8 . Tim chui t' iLion quan chun ho ca liii liiu x{n) sau y:

j l - N < n < NX n =

^ ' [() n 7:

1.69. ll y xc dnh chiii r Urone quan cua mi lin hiu sau dy v chiViii ininh 1'iii; no l chii i

chan tionu moi tiirni hp. 'ini \' Ir cun m tr cc di cua cliui lir tiron qLiaii Irone mi tnrim

hp:

a) X , ( n ) = a " u ( n )

, , M 0 < n < N - - lb ) x , n ) = ,

! 0 n =:

1.70. l!v xc dnh chui tliRvng quan v chu ki cua mi chui sau y:

a) Xi (n) - c o s ( 7[ n ' M ) , IronL* d M l s n miy n dironu

b) X2 (n) n modulo

1.71. Xc diii phni \ i cc ui tr cua tham s a sao cho h lim Xu yn tiili bt bin cc p im. , . a " n > 0 . nchn

xiiniz h ( n ) - < la 011 cih.0 n khc


47/279

r 1 Y'1.72. Ilv xc dni p na cua mt li i h o n n c dap iu x i inu h ( n ) = i u( n) vi cc t n h iu

V. 2J

u \ t) ln ll l:

u) x ( n ) = 2 ' ' u ( n )

b) \ { n ) = u ( - n )

1.73. Chim Iiinh rim iu kin cn v u cho mt l hni tuyn tnh bt bin (LTl) tr' thnh

mt i thiL n nh Bf30 l:

f X

^ ! h( n) < M,, < M J h n s l - /

1.74. C1io h thim ri rc irc m ta b ni Z quan h vo-ra nh sau:

y(n) =

Chng 1: T in h iu v h thng r i r c 49

y ( n - l ) ^

Tron g x ( n ) v y( n) toim nu l cc chui u vo v u ra. Chn g minh rng u ra

y( n) cua h thng trn i vi u vo x ( n) - cxu(n) vi y(~-l) ~ 1hi t UVi V a khi n co ( a

l mt s dong). H thng trn l tuvn tnh hav phi tuviV.^ L h thiii bt bin? Gii thch cu

tr li ca bn.

1.75. Chui cc s Pibonac ci f(n) l mt chui nhn qua irc nh ngh a bi:

f (n) = f (n - 1) + f (n 2 ), n > 2

vi f( 0) = 0, .

a) Xy dng mt cng thc chnh xc tnh f' (n) trc tiep cho bt c n no.

b) Chrm minh rnii f ( n ) l dp im xung ca mt h thng tuyn tinh bt bin nhn qu

irc m la bi phircng trnh sa] phn sau:

y(n) = y(n- - l ) + y(n * 2 ) ^ x ( n - l )

1.76. Xt mt b lc s phc bc 1 c cho bi mt ph ucng trnh sai ph n nh sau:y ( n ) ^ a y ( n - l) + x ( n )

Troim x(n) l chui u vo c m tr thc. y( n) - y,., (n ) + (n ) l chui u ra c gi

tr phc, vi (n ) v (n ) tirnu nu l phn thirc v phn o v a = a + jb l hng s phc.

Xv crng mt biu din ca phcrnu trnh sai phn thirc tonu oTig c mt li vo v 2 li ra ca

b lc s phc trn. CMR b lc s c mt li vo v 1 li ra trn c mi lin h gia (n) v

x(n) l theo mt phircng trnh vi phn bc 2 .

1.77. Chim minh rnu mt chui m hon lon c th tnh tHi 'c th c nng l'n hiu hn,nhimg mt chui c nri4 lirn hini hn thi c th khng hn ton tnh tng 0 .

1.78. Cho mt b lc s IIR nhn qua irc m t bi phircTni trnh VI phn sau y;


48/279

50 ii bi p x ly tn hiu s v Matlab

^ d , y ( n - k ) = 2 ] P k x ( n - k )k=:() k=()

Trong y( n) , X( n) toni n l cc chui u vo v u ra. Neu h( n) l p ne xunek

ca n, hy chni minh rnti P i ^ ' k = 0,l ......M11=0

T kt qu trn, chng minh rnu p( n) - h(n )(* )d

LI GII BI TP NNG CAO

1.68 .

2N + 1- f , - 2N < f < 2N

0 n khc

Rxx(0) = 2N + 1

Do , chui t tong quan chun ho ca tn hiu x(n) l:

1

Pxx(0-= 2N + 1

0

(2N 4- - ), ~ 2 N < f < 2 N

n khc

1.69 a) X, (n) = a u(n)

^xx( l)= z ( " )= a " u ( n) a" ' u ( n - r )n- - -< x 11= -0Q

a ' - ' = ^ f < 0h l - a =

-t-co

n- o +C0g a = - ' =

1 1 = 1

a1 - a '

l >0

_ a ' _ tt^'Ch , i vi f > 0 , R^^(f) =-------^ v (f) ^ ......f < 0 .Tha y f bng

1 a~ " ' ' 1 a

a '

- a 'f trong biu thc th 2 , ta c (~) =-------7 (f) nn (f) l hm chn i v'i f .

- a

Gi tr cc i ca (f ) xv ra ti f - 0 bi vi a ' l hi n suy


49/279

Ch ng 1: Tn hiu v h th ng r rc 51

n = -co n- -- ()

X : ( n ^ l ) =

1< n < N - 1+

0 n khc

Do ,

0 f < - ( N - l )

N + 1 - ( N -1)

N ~ 1

( f ) l hm ca mt tam ic theo bin s f , do ( ) l hiT chn vi gi tr cci ca N li f = 0 .

L70. a) Xi (n) ~ cos( 7i n / M ) , trong M l s nguyn dimng. Chu kv ca X| (n) l 2M, do

, 2M-1 i 2M-f^.v(')= z ("-') = 7^ z + = zI = -r/ n=011=0

cos=0 V

^7tn '' ^rt(n + 1) ^----- cos

[ m J . MV /

1 ^ nn\ cos

2 M ^ I m

Mt khc ta c: ^ COS"ti=()

' T n ^/ \

7 i n ^ 7f ' 7 n ^ 7lf ^ 1 711' ^2M-1

V c o s 'c o s _ _ cos - sin s i n ---- ^ = c o s ---- [ m j 1 v M y 2M L u v M J

^ Ttn' ' 2 M , -, 1 Ttn ^= = M . D o R , , (1 ^ - -C OS

. M , 0 U M ;

b) X : ( n ) = n modul o - 10,1,2,3,4,5 [, 0 < n < 5 . N l mt chui tun hon vi chu kv

1b ng 6 . Do d, f ) = X, ( n ) x , (n + f), 0 < < 5 , cng l chui tun hon v'i chu k l 6 .

6 ^

r ^ ^ j 0 ) - ~ r x 3 (0 ) x , ( 0 ) + x 2 ( f ) x , ( f ) + x . ( 2 ) x , ( 2 ) + x , ( 3 ) x . ( 3 ) + x , ( 4 ) x , ( 4 ) + x , ( 5 ) x , ( 5 )

^Nx(0 ^ r '^:(O)^'^2 { + ^ 2 (0 ^ 2 (2 ) + x . ( 2 ) x . ( 3 ) + x . ( 3 ) x J4 ) + x , ( 4 ) x . ( 5 ) + x , ( 5 ) x . ( 0 )L -

Tnh tong t ta c:

n 55

6

40

R . . ( 3 ) = f


50/279


1.71.

1.72

a)

N " ) = IH" = Zn=-3? 11=0.11chrn n=0

H thni n nh nu a < 1.

y (n )= ^ h ( k ) x ( n - k )k=-^

l / 1 II

= 7 2 - = 2 " Xk=0 V / K=()

2n

~ia

\k

- 2 " v 4 y

A

V2 u( n )

b) y ( n ) = f h ( k ) x ( n - k ) ^ h ( k ) ^ xk=-cc k-0k-0k-ov 2y

= 2, n < 0

y ( ) = | ; h w = | ; = 1 : 1 - g ik^ii k = ii V"^/ k=i()V / k=o V /

= 2 -

1 -v 2 .

= 2v 2 .

, n>0

1.73. Mt h thng l n nh BIBO nu v ch nu mt li vo c gii hn (kiunded Inpu) to ra

mt li ra cim c gii hn (Bounded Oulpit).+


51/279

Chng 1; Tn hiu v h thng ri rc 53

Di vi u vo x(n) = au ( n ) . du ra y(n)hi I uVi mt hnu s K khi n-> co .Phong

trinh VI phn trn khi n co. tro thnh i o K ~ c : > K = v .2 1 k J

De dng hv r nu h thim l phi tuvn. Gi thit y, (n ) l u ra cia du vo X| (n) .Ta c:

y , ( n ) = ' v ,2 [ y , ( n - l ) _

Nu x , ( ) = x ( - n . ) Ihi , ( n ) 4 | y , ( n - l ) + ^ ; ^2! y i ( n - l )

Suy ra y, (n ) = y( n n,, ) => h thnii trn l bt bin.

1.75. a) f (n) - f ( n l ) + f (n - 2) , n > 2 . t f ( n) =^ ar ' \ phucnu t rnh sai phn tr' thnh

ar " - ar " ' - a r ' 0 r" - r - - 0. c nehim l r - - . Do .

' \ ^ f 1+ V 5 ( n ) - a ,

B'i vi f (0) 0 nn a, + a . = 0 . Tirong t. l' (l) - 1. suv ra s/s ~ ^ = 1. Gii

h phong trinh trn la thu irc a. - a , .V5

Do , f(n)1

n / /- ^1 1- V 5I

J 7V. y

b) y ( n ) - y ( n - l) + y ( n - 2) + X(n - l)

Khi h thn l tuyn tnh bt bicii, cc iu kin u bng khnu.

t x(n ) ~ 5 ( n ) , a c y(n) = y(n - 1) + y(n - 2) + ( n ~ l ) . Suy ra:

y ( 0 ) = y ( - l ) + y ( - 2 ) = 0 v y(l] = l. i vi n>l phiroTig trnh sai ph n toTi g ng l

y ( n ) - y ( n - l ) + y ( n - 2 ) v i c c i u k i n u y ( 0 ) - 0 v y ( l ) 1, g i na v i c c i u k i n u

cho l'i gii chui Pibonacci. Do li gii khi n >1 'c cho b i

, 1 u S 1 i - s / ' "

f ( n ) l p i xung ca mt h thim LTI nhn qu 'c m t bi pho ng trnh sai

phn: y( n) = y ( n - l ) + y ( n ~ 2 ) + x ( n )

1.76. y ( n ) = a y ( n - l ) + x ( n ) . Vi y ( n ) - y , ( n) + jy ( n ) v a = a + jb tac :

Ykc (n) - jyin, (n) = (a + jb)| Yk,(n - l) + jy (n ~ l)] + x( n)


52/279

54 ii b i p x l tin hiu s v Matlab

Thc hin cn bng phn thc v phn o ring r v ch V vUi x (n) l thc ta c:

Y rc ( n ) = a yR . ( n - 1) b y, , ( n - l ) + X ( n ) ( 1)

y i n , ( n ) = b y K e ( n ~ l ) + a y , , ( n - ~ l )

Do , y (n -1 ) = - y (n) - - (n - 1)a a

Suy ra phoTg trnh sai phn mt u vo, hai u ra c dnu:

y Rc (i ) = a y R ^ . ( n - l ) - - y ( n ) + y ^ J n - l ) + x ( n )a a

Do , by (n - 1 ) = -ay^^, (n - l) + (a- + b (n - 2)y,, ( n ) + ax (n - l)

Thay th phong trnh trn vo phong trnh (1) ta c:

y R e ( n ) - 2 a y , , , ( n - l ) - ( a - + b ) y j , J n - 2 ) - a x ( n - l ) + x ( n )

Ph ong trnh ny l phoTg trnh V I phn bc 2 biu din yj^ , ( n ) the o hm ca x ( n ) .

+-J

1.77. a) Cho ^ x(n) < co . p dng bt ng thc Schwartz, ta cn=-cc

+CO +CO \ +CO N " ) f - N " ) I N " )n = -co = J \ w = - c o

b) Xt x ( n ) =n > 1

n < 1

Min hi t ca mt chui c chiu di v hn c th kim tra thng qua biu thc tch phn.

t a,, =f(x), trong f(x)l hm lin tc, dong v gim dn i vi mi X >1. Do c

chui phn f ( x ) d x u hi t hoc u phn k. Vi a,, " , f ( x ) = ta c:n = ! I ^ ^

dx = ln (x) = o o - 0 = oo . Suy ra ^ x( n) l khng hi t, v kt qu l x( n) kh n g

^ n = - 0 D n = i ^

hon ton tnh tng c. e chng minh chui x (n )c tng bnh phong, ta p dng cho

1 1- L -f 1 rr 1 , Suy ra / - y l hi t, hay

x ( n ) - c tng b nh pho ig.n

1.78. 2 ] d , y ( n - k ) - j ^ p , x ( n - k )k=0 k=0

00


53/279

\1 \

Dt u vo ca h thng l \ ( n) = (n) . Khi d ^ P k S ( n - k ) = d | ^ h ( n - k ) . Do k-o k--()

N'

p . " ^ d ^ h ( r k) . V h thn u ny c ia thit l nhn qu, h ( r - k) ~ 0, Vk > r .k-o

k--() k=()

T kt qu trn, ta suy ra p(n) = h(n)(*)d,j

D. B TP MATLAB

Hy vit mt chng trinh MATLAB o ra cc dy sau y v v cc dy ny s dng

h m sem: a) dy xun g n v , b) dy nhay o'! i v. c) dy ch nht , d) dy dc an v , e) dy hm

: _ ! . ^ '

m llic X = 0 , 2 ( 1,5 )" v 0 dy hm m phc y = 2e' , 0 < n < 40 .

Li gi:

% Chuong trinh Ml_l

% a) Tao day xung don vi

c l f ;

% Tao mot vecto tu -10 den 20

n = - 1 0 : 2 0 ;% Tao day xung don vi

delta = [2eros(l,1 0 ) 1 zeros(1,20}];

% Ve day xung don vi

subplot(321);

stem(n,delta);

xlabel{'Thoi gian roi rac n ');ylabel('Bien do');

title 'Day xung don vi') ;

axis([-10 2 0 0 1.2]);

% b) Tao day nhay don vi

u= [zeros(1,10) ones (1,21)];

subplot(322);

stem(n,u);

xlabel('Thoi gian roi rac n );ylabel('Ben do')/

title('Day nhay don v i');

axis([-10 2 0 0 1.2]);

% c) Tao day xung ch nhat co chieu da L

L=10 ;rec-[ zeros (1, 10) ones(l,L) zeros (1, 2 0 - L-f 1) ] ;

subplot(323);

s t e m ( n , r e c ) /

Ch n g 1: Tn h iu v h hng r i rc 55


54/279

xlabel('Thoi gian roi rac n ');ylabel('Bien do');

title('Day xung chu nhat');

axis ( [-10 20 0 1.2]);

% d) Tao day doc don vi

n = - 5 : 1 0 ;

m= [zeros(1,5) 0:10];

u= [zeros(1,5) ones(1,11)] ;

r=m.*u;

subplot(324);

stem(n,r);

xabel('Thoi gian roi rac n ');ylabel(Bien do');

title('Day doc don vi');

axis( [-5 10 0 10.2]) ;

% e) Tao day ham mu thuc

n = 0:30; a = 1.5; K = 0.2;

X = K* a. ^n;

subplot(325);

stem(n,x);

xlabelThoi gian roi rac n ');ylabel{'Bien do');

title(Day ham mu thuc');

figure;% f) Tao day ham mu phuc

s = - (1/12) + (pi/6)*i;

K = 2 ;

n 0:40;

X = K* exp(s * n ) ;

subplot(2,1,1);

stem(n,real(x));

xlabel{'Thoi gian roi rac n ');ylabel('Bien do'};

title('Phan thuc');

subplot(2,1,2);

stem(n,imag(x));

xlabel{'Thoi gian roi rac n ');ylabel(Bien do');

title('Phan ao'};



55/279

Ch ng 1: T in h iu v h th ng r i rc 57

Day xung don vi Day nhay don vi

I 0 5in

4 , Thi*>i ni-an rnt r^ ,n______" r ^ 5 ,0

Thoi gian roi rac n

0 10 2D 30Thoi gian roi rac n

Dy hm m phc;

Phan huc20

1o

o

c 0s?03-1

-e-o

o9?

J______________I----------------------1______________ L

1G 15 20 25 30 35 40

Thoi dan roi rac nPhan ao

o

O0

0

Q

1__ . . 1.

0 6 10 15 2 25 30 35 40

Thoi gian ro rac n

Ml_2. Hy vit int chng trnh MATLAB dng cc hm sawto()th v square to ra cc dy

xung vung v dy xung rng ca sau y v v cc dy ny s dng hm stem. Cc thng s sau

y c th thay i bi ngi lp trinh; chiu di dy (L), bin (A), chu k (N). Ngoi ra i vi

dy xun g vung ta cn quan t m n rng xung - l phn trm chu k m tn hiu c gi tr

dng. Hy dng chng trinh ny to ra 100 mu u tin ca mi loi dy xung trn vi tn

s ly mu 20kHz, bin nh A ^ 3; mt chu kv l 15 v rng xung vung l 60%.Li g i i:

% Chuong trinh MI 2


56/279


h,% a) Tao day xung vuong va day xung rang cua tuan hoan co chieu dai

% bien do dinh A, chu ky N

clf ;% Nhap cac thong so cua day xung tu ban phim

A=input('Bien do dinh =');

L=inputc'chieu dai day = );

N=input{'Chu ky cua day =');

Fs=input('Tan so lay mau mong muon =');

DRX=input{'Do rong cua xung vuong =');

% Tao cac day xung

Ts=l/Fs;

t=0:L-1;

x=A*sawtooth(2*pi*t/N);

y=A*square(2*pi*t/N, DRX);

subplot(211);

stem(t,x);

xlabel( [ 'Thoi gian ' ,num2str{Ts) , ' giay'] ) /ylabel('Bien do') ;

title('Day xung rang cua.');

subplot(212);stem(t,y);

xlabel( ['Thoi gian ',num2str(Ts) , ' giay']) ;ylabel{'Bien do') ;

title('Day xung vuong');

Day xung rang cua

Tho gian 5e


57/279

M l_ 3 . a) Hy vit mt c hm trinh VIA LAB to dy tn hiu hinh sin y (n ) ~ Acos(co jn -i- )

v v dy ny sir dim l im SCL Cc thng s ca dy sau y c th nhp t bn phm: chiu di

dy (L), bin (A) , ti s


58/279

60 ii bi tp x tin hiu s v Matlab

ylabel('Bien do')/

title('Day tin hieu dieu bien');

a) V d A = 2; L = 50; CQ 0 ,2 p i ; = Ota c kt qu nh sau:

2.5

2

1.5

1

0.5

E 0)CD

-0,5

-1

-1.5

-2

-2.S

Day sin tuan hoan I---------------------------------1----p -

o

o o

......

o

o o

.......----

o

J_____________\______L

o

J_______l0 5 10 15 20 25 30 35 40 45 50

Thi gian roi rac n

b) V d d y tn hiu iu bi n vi h s iu bin m = 0,3; tn s sng ma ng fc = 0.2 Hz

n s ca tn hiu f =0,0! Hz; chi u di dy L = 100.

Day in hieu dieu ben

Ml_4. Hy vit mt chong trnh MATLAB dng hm mpz tnh ton v v p ng xung ca

mt h thng ri rc theo thi gian c chiu di hu hn c dng ti qut nh sau:N - ! ^ 1

^ a ^ y ( n k) = ^ b ^ x ( n - r ) . Cc thng s u vo ca chng trinh l chiu di ca p ngk-o r = 0

XUI (L), cc hng s ai l v { b j c a phon trnh sai phn. Hy kim tra tnh n dnili ca mt

h tlin.a .


59/279

Chng 1: Tn hiu v h thng ri rac 61L' gi i:

% Chuong trinh Ml_4

% Tinh toan va ve dap ung xung cua rnot he thong roi rac theo thoi

gian co% chieu dai huu han va xet tinh on dinh cua he thong

% Nhap cac thong so cua tin hieu tu ban phim

M^input{'Chieu dai dap ung xung mong muon ='};

input('Nhap cac gia tri cua vecto p = );

d^input('Nhap cac gia tri cua vecto d =');

[h,t]=impz(p,d,N);

disp(h)

n=0:N-1;

stem(n,h);xlabel('Thoi gian roi rac n');

ylabel { 'Bien do'} ;

title{'Dap ung xung cua mot he thong roi rac chieu dai huu han');

% Xet tinh on dinh cua he thong

sum-0;

for k=l:N+l;

sum^sum+abs(h(k));

if abs(h(k))


60/279


Ml_ 5. Hy vit mt choim trinh MATLAB tnh tch chp ca 2 dy c chiu di hru hn. Gi

tr ca cc dy c nhp t bn phm. Kt qu ca tch chp sau c trc i d mu (^i r ca

d 'c nhp t bn phm). V (n)=x(n)*h(n) v y(n-d).

Li i:% Chuong trinh Ml_5

% Tinh toan va ve tich chap cua 2 da co chieu dai huu han

% Nhap cac cac day tin hieu tu ban phim

x = i n p u t ( ' N h a p d a y t i n h i e u X = ' ) ;

h=input('Nhap dap ung xung h =');

d=input('Nhap gia tri tre d =');

y=conv{x,h);

nx^lengthx); % chieu dai cua day X

nh=length(h); % chieu dai cua day h

ny=nx+nh-l; % chieu dai cua day y

dsp(y)

n=0 : ny-1;

subplot{211} ;

stem(n,y);

xlabel('Thoi gian roi rac n');

ylabel{* Bien do} ;

tit:le('Tich chap cua 2 day co chieu dai huu han' ) ;

subplot (212) ;

yd=[zeros(l,d) y] ;md=ny-i-d ;

m=0:md-1;

stem{m,yd);

xlabel('Thoi gian roi rac n );

ylabel{'Bien do');

title('Day tre');

Tich chap cua 2 day co chieu dai huu han20

10oc0)s

-1 0

-2 0

' (

o)

^ < 1 >

? ________....................... . '

................. , ................ ................ ... ................ ?

)1 t4 6 8

Thoi gian rpi rac na tr

10 12

* 0 ,

Ir ti co nu I io < 0

z > 1

u ( - n - l )

n u

z

- az

z | < l

z > 1

z > a

na "(n)

- a " u ( - n - 1)

c o s ( c o n ) u ( n )

sin( a

a"sin(co,,n)u(n) az" sino),

1- 2az" C O S O ) . + a z

z > a

z > z pk

z < z'pk

m!> 'p k


71/279

72 ii b i tp x ly tn h iu s v Ma t lab

( - v ) '

2.7, Biu din h thng trong min /

c trn cho h thng trong min z l hm truyn t H(z). Hm truyn t c vai tr nh

p n^ xung l(n) ca h thn tronu min thi ian ri rc.

Hm truyn t H(z) c hiu theo hai khi nim;

- Hm truyn t H(z) l tv s ca bin i z tn hiu ra Irn bin i z tn hiu vo.

Y(z)H( z ) =

x (z )

- Hm ruyn t H(z) l bin i z ca p n xuni h(n).2.8. Lin h gia bin i z v phoig trnh sai phn

Bin i z hai v ca phng trnh sai phn tuyn tnh h s hn:

^ a , y ( n - k ) = f ] b , x ( n - r )k= 0

ta thu c:

M

Y ( z ) ,H ( z ) - --------- nn nh' lun chun ho ao = 1 d v so thc hin.

X(z)N

k = l

-k

Cc phn t thc hin h thng ong min z cng ging nh trong min thi gian ri rc n: phn

t cng, nhn, nhn vi hng s. Phn t tr D trong min n khi sang min z tr thnh phn t z ' \

C 3 dnu cu trc thng thng ca h thng: song song, ni tip, hi tip. Cch xc inh

hm truyn t h thng tng qut tong ng nh sau:

- Neu N h thH mc song song vi nhau th hm truyn t ca h thni tng qut l:

H(z )= X H , ( z)i = l

- Nu c N h thng mc ni tip vi nhau thi hm truyn t ca h thng tng qut l:

H ( z ) = f H i ( z )i = I

- Neu H2(z) mc hi tip vi H|(z) thi hm truyn t ca h thng tnu qut s bng:

1 - H , ( 2 ) . H , ( z )

2.9. S n nh ca h thng trong min z

Mt h thng TTBB nhn qu trong min z mun n nh phi tho mn:

Tt c cc im cc ca hm truyn t H(z) phi nm bn trong vng trn on v tc l:

V 'p k


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N (z )I a c FI(z) ^ . Khi bc N cua h inu. tc l bc cua a tluVc c trirnii D(z) l'n hn

2 thi ta phai dnu tu chun Jurv xt lih II nh.

Mt s php ton cn nam vng lm bi tp roi chong ny:

* Cc khi nim v chui, chui hi t.

- C'c php ton v s phc.

- Tim cp s nhn.

Nu a < 11 I

Ch n g 2: Biu din h thng v tn h iu r i rc t rong min z 73

11=0

t i=()

- a

N + l1- a . :----- - Nu a > I

1- a

B. BI TP CO BN

2.1. Xc nh bieii i z ca cc tn hiu hu lin sau:

( a ) x , { n ) = {l 3 5 7 0 2}

( b ) x , ( n ) = l 3 5 7 0 2

( c ) x , ( n ) = O 0 1 3 5 7 0 2

( c i ) x , ( n ' ) - j 2 5 6 0 l

( e ) x , ( n ) = - 2 0 0 1 7 l

Li i:Theo nh ngha ta c:

(a) X (z ) = 1+ 3z"' + + 7z + 2 z " ' . ROC^' ca mt pbng z , tr z ==0 .

(b) X. (z ) - Z' + 3z + 5 + 7z"' + 2?/^, ROC: c in phng z , tr z - { ) v z ~

(c) X 3(z) ==z~~ + + 5z + l z ' + 2 z ' , ROC : c mt phni z , tr Z.^Q .

(d) (z) ^ 2z + 1+ 5z'' + 6 z" + z \ ROC: c mt pling z , tr z = 0 v 7.=

(e) (z) - + z' + 7 + z" , ROC: c mt phng z , tr z = 0 v z - c o

1 2 , Xc nh bin i z ca cc tn hiu hu hn sau:

(a) X, ( n ) - s ( n )

( b) vX, ( n ) = s ( n - k ) , k > 0

( c ) X, ( n ) = ( n + k ) , k > 0

Lidi:Theo nh ngha ta c:

(a) X ( z) = l [nuha l, (n)!], ROC; c mt phng z. (2.1)

co

co


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74 ii b i tp x l tin h iu s v Ma t lab

( b ) X . ( z ) = 7/^ [ngha l. ( n - k)z k > 0 , ROC: c mtphn u z , t r z= 0 . (2.2)

(c) X, (z) - 7 [nuha l, (n + k) 0,ROC:ca mtphi mz . tr z= CC'.(2.3

T v d ta d thy rn ROC ca mt tn hiu c chiu di hu hn l c mt phnu z.

ngoi tr z = 0 v hoc z = co vi z ^ ( k > 0 ) s r thnh v hn khi z = X) v z ' ^ ( k > 0 ) s tr-

nn v hn khi z - 0 .

Theo quan nim ton hc, bin i z l mt biu din ca tn hiu. iu ny 'c minh ho

' bi tp 2 .1. y ta nhn thv rim h s z" , trong bin i cho, l m tr ca tn hiu ti

thi im n . N khc i, s m ca z cha ng thim tin v thi ian m chng ta cn nhn

d n g cc mu ca tn hiu.

Trong nhiu tr'iig lp, ta c th biu din tng ca cc chui luTu hii lioc v hn i vi

bi n i z theo m t biu thc dng gn ng. Tr on g cc tr ng hp y bin i z c xem nh

mt biu din thay th rt gn ca tn hiu.

2.3. Xc nh bin i z ca tn hiu:

^ ( " ) = 3 ^ ( )V^ y

U r : T n h iu X(n) bao gm mt s v hn cc mu khc 0:

x ( n ) - <

1r o

3 n

v3;

, . . .

v3y

Bin i z ca x(n) l chui cng sut v hn:

X(z) = l + i z - ' +3

X / 1 /

n=oV'- y n=()l z ' ' "3

y l mt chui hinh hc v hn. Ta c th vit li:

1

Bi vy, i vi

x ( z ) =

nu A < 1

- z3

- A

< 1 hoc tong ng, vi > - , x ( z ) hi t n:

ROC:1

> -3

Ta thy rng trong trng hp ny, bin i z cho ta mt biu din ihay th nn gn ca

tn hiu x(n) .

2,4. Xc nh bin i z ca tn hiu:

x ( n ) = a ' ^ u ( n ) -n > 0

n < 0

Li gi i: Theo nh ngha ta c:


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Ch ng 2: Biu din h thng v t n h iu r i rc t rong min z 75

X i z ) = a - z " . {f = ( ) !1= 0

a z (2.4)

Nu a z < 1 ho c tirng im z > a . th chui cnu sut ny hi t n 1/ (l - a z ' ' ) .Nh vy, ta s c c p bin i z .

x ( n ) a u (n ) < -> x ( z ) ROC':- a z

(2.5)

Hnh 2.1: T hii hi ha nhn qu a . r ( / / )= a"u{) (a)

Re(z)

ROC l min nn ngoi ng trn c bn knli Ijt . llinl 2.1 l d th ca tn hiu x (n ) v

ROC tng ng ca n. Lu rng, ni chung, a cn khng phi l s thc.

Ncu ta thay 1 v o (2.4), ta s c bin i z ca tn hiu nhy bc n v.

x ( n ) = u ( n ) < ^ X ( z ) = ^ R O C :

2.5. Tm bin i z ca tn hiu:

x ( n) = - a " u ( - n - l ) -0

- a

n > 0

n < - l

Li gi i: Theo nh ngha ta c:

x ( z ) =n = - / m =

' y m - - n . D ng c ng thc;

A -f- A ' + A'"' + .... Al + A -f A~ ^1 - A

Khi A < l t a c : X ( z ) = -a z

1- a z 1 - a z


75/279

76 ii b i tp x ly tn h iu s M at lab

VcVi a" 'z hoc tuxTim im a . Nh vy:

x (n ) = - a " u ( - n - l ) < ^ X ( z ) =- a z

R O C : |z| < a (2.6)

ROC by gi' l min trong 'n trn bn knh a . i u nv c cli ra ' hinh 2.2.

Cc bi tp 2.4 v 2.5 trnh by hai chui rt quan trnu. Chui th nlit lin quan en tnh

duy nhl cua bin i z . T (2.5) v (2.6) ta nhn thv rani tn liu nhn qua a" Li(n) v tn hiu

khng nhn qu x (n ) - ~a" u ( n ~ ]) c dni biu din gn im uiim nhau, nsh a l:

Z T a" u (n ) = Z T - a " u ( - n - l) - '

H l 2.2: Tn iii kh ig nhn qua x (n ) - - a ' u ( " i - 1 ) (a) v RO C CU hicri c1(Yi z ca (h)

iu nv ni ln rnu dnu biu din gn nu cua bin i z khng chi ra irc mt cch

duy nht tn hiu trong min tli'i gian. S khng r rng ny c th irc |a|. Chui cng thc th hai cng

li t nu bz < 1 hoc z < b .


76/279

Ch ng 2: Biu din h th ng v tn h iu r i rac trong min z 77

ln(;:

H w 2 . R O C c h o h i n i z

xc nh SI' lii li c a x ( z ) . a x t hai ru nu hp k hc nhau:

Trurv^ lp I: b| < a : Tron e lrn> hp nv hai ROC 0' trn khng trng nhau nh chi

hinh 2.3.a. Bo'i v\\ ta khng th tim uc cc ui tr ca z m i vi chim c hai chui cng

sut nu lli hi t. R rnu trong trirnu hp ny, x ( z ) khng tn ti.

Tru p 2: b| > a : i vcVi trircrnii lip Iiy c mt vng kh uvn trn mt phng z, y

c hai chui cim sut nii thi hi t. nh clii troni, hnh 2.3.b. Khi ta c:

_________1__b - al - a / r ' 1-117. ' a + b ~ z ab?; '

ROC' ca X(z) l |a| < |z| < b

2.7. a) rim bin cti z v min hi t cua cc chui sau y:

I) X, ( n ) = ( -^ () . 7) " u ( n )

II) x , ( n ) = ( 0.2)" u ( n - 3 )

lii) x,(n)=^ 0 .6 ' u ( - n - 2 )

b) T ROC ca cc chui ' phn a), hy xc nh RO C cua cc chui sau v:

I) y , ( n ) = :x , ( n ) + x , ( n )

II) y 3 ( n ) = x , ( n ) + x , ( n )

III) y , ( n ) - - x , ( n ) + x , ( n )

L i i : a) The o nh niha bin i ta c:

I) x , ( z ) = ( - 0, 7) " u( n )z =( - (1 .77. )" = -n = - ' 11 =0 1 + u, 17.

ROCca x , (z ) l R , : | z >0 ,7

>0 ,7


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78 ii b i tp x l tin h iu s v M at lab

t m = n - 3, ta c: X, ( z) = ^ ( o , 2 z ' ' ) = ( 0 ,2 z ' ' ) ^ ( 0 , 2 zm=0 111=0

~ l - 0 , 2 z ' ~ l - 0 , 2 z '

ROC ca X2(z) l R, : z > 0,2

in) x , ( z ) = ^ 0 , 6" u (~ n -2 )z -" = ( 0, 67/ ' ) "1 1 = - / n = ~ /

t m= -n - 2, ta c: X 3(z) = (0 ,6z ' y'"" = (0 ,6z ' ) " (0 ,6^' z)ni=0 in=(t

0.2

(Q.6z~') ' 0^6"-z-~ 1 - 0 , 6 ' z ~ 1 - 0 , 6 ^ ' z

< 0,6

ROC ca X3(z ) l R , : | z |0 ,7 v ROC ca x , ( z ) l R, : |z |> 0, 2. Suy ra

RO Cc a Y,( z) l R, n : |z| > 0,7

ii) V ROC ca X |( z) l R : |z |> 0, 7 v ROC ca x , ( z ) l R, : | z | 0, 2 v ROC ca x , ( z ) l R, ; |z


78/279

Ch ng 2: Biu din h thng v tn h iu r i rc t rong min z 79

Ta c: ZT -5(0,3)" u(n) :zj > 0,3

ZT -5(0.6)" u ( - n - l )5

0.6zDo :

5 5 _5(l -0 .3 z ' - l + 0 , 6z - ' ) _ K5z^'

l - 0 , 6 z " ' l - 0 , 3 z ' " (l - ' 0 , 6 z "' ) ( l -0 , 3 z ^ ' ) ( l - 0 , 6 z ' ) ( l - 0 , 3 z ' )

v i 0 , 3 < z < 0 , 6

ROC ca X3(z) l 0,3


79/279

80 ii b i tp x l t n h iu s v Ma t lab

X, (z ) = = 7--" ^ w . H>lPll ~ a z ' ' l~ P z - ' ( l - a z - ' ) ( l - P z - ' ) (l - a z ' ) ( l - P / - ' ) '

b) x , ( n ) = a " u ( - n - l ) + P" u( n)

Ta c: ZT a " u ( - n ~ l ) i

z < |aI - a z '

1

ZT P"u(n)1- |3z

Do cc min |z< a v |z > |3| khng giao nhau nn bin i z cua .\2(n) khng hi t

(khnu tn ti).

c) X , ( n) = a " u ( n ) - 3 " u ( - n - )

Ta c: ZT

ZT

x , ( z ) =

a " u ( n )1 1

1 zl >i a1 az 1

1 - P z --I ';;

9 V ; ^X, (n) = ( c o s o i n ) u ( n ) - - e'"''u(n) + (n)

X,( z) = j Z T e ' ....u(n) + i z T j e - ~ - i , ( n )


80/279

Chng 2: Biu din h thng v t n h iu r i rc t rong min z 81

Nc u thav a - e a - e " I \ () (2.5), ta c:

e ' " u ( n ) R 0 ( : / i >

v e ' " ' " u n A - R O C : !z >11 - e z

Vy x , ( z ) - ---------^ r R O ( ;: z |> l

Sau mt vi bin i i s on uian, ta c kt qu inonu mun l:

(cos)^jn)u(n)\~- z cosco,.n

1- 2z"' cosco., z "R O C : zi > (2.7)

b) Theo cng thc Eul er

X, (n ) ( s inco , jn)u(n ) (n) ~ (n)

I I . . \ s in(0,nHay: (sin (0n ju( nj< ->

1- 2 z ' ' coso ,\ , + z

2.12, Tim bin i z ca cc tn hiii:

a) Xj (n ) a" (coso-\ ,n) Li(n)

b) X, (n) a" (sina )n ) u (n )

Li 1

ROC: (2.8)

a" (coso)n)u(n )1- az cosco,

------- 2az COSO)., -f a z

> !a

"^rcvng l. tlieo (2.5) v (2.8) ta c

a ( s i n c o n ) u ( n )az sinco,

1- 2az ' C0S(!), + a z(2.!0)

2.13. (a) Ti m bin i 2 ca tn hiu:

x ( n ) a ' ' |a| < 1

(b) Ti m bien oi z ca tn hiu hnu s x(n) = 1 . -o) < n < -KX)

Li g i i:

(a) X(z)--= X x (n )z " = X- = " + ^2 {^)n = - i 1 1: =- '- 1 1 = - ' ' n = 0

11---' ii i-0 IH--0

(2.9)


81/279

82 ii b i tp x l t in h iu s v Mat lab

az

1~az - az1/la

x , ( z ) = X a " z " ' = I K t=0 11--0

1- az> a

V'i iu kin u bi cho |a| < 1, ta c: x ( z ) = X| (z) + X. (z ) =az

1- az 1- az

ROC: a < 2 o)mu [ngha

l x (n ) x (n - k ) J l ton nu vcVi vic nhn tt c cc s hng ca bin i z vi z ^ . H s

z ' tr't hnh h s

2.17. Xc nh bin i 2 ca tn hiu:

x( n) = rect^.(n) (2 .11)

J)'ii: x ( n ) rcct^. (n) ===

x(z) = | ; i . z - " = i + zn-o

( ) < n i : N n cn li

N Z = 1

l - z7.:? 1

(2.12)

v x (n ) l hu hn, nn ROC ca n l c mt phn z , tr z = 0 .

Ta hy thc hin bin i ny bng cch dng tnh cht tuyn tnh v dch. Ch rng x(n)

c th c biu din theo cc s iiii ca hai tn hiu nhv bc n v:

x ( n) - u ( n ) - u ( n - N)

Do :

X (z ) = z { u ( n ) | - z i u { n - N ) | = ( l - z - " ) z | u ( n ) | (2,13).

M: ZT u ( n ) | = 1 z ROC : lz| > 1, kt hp v (2.13) suy ra (2.12)

2.18. Tnh tch chp x(n) ca cc tn liii:

x , (n ) = ( n ) - 2 S ( n - ] ) - r 8 ( n - 2 )


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84 ii b i tp x l t in h iu s v Mat lab

L i g i i:

. l 0 < n < 4x j n ) = ; .

^ ' 0 n cnlai

X , ( z ) = 1 - 2 7 / ' + z '

X, (z) = 1+ z"' + + 7/* +

x ( z ) = x , ( z ) x , ( z ) = l - z - ' - z " ' + z^'

Suy ra: x (n ) = ( n ) - 5 ( n - l ) - ( n - 5 ) + ( n - 6 )

2.19. Tnh tch chp x(n) ca cc tn hiu:

x , ( n ) = {l 2 3 4 5}

x , ( n ) = {5 4 3 2 1}

L i g i i:

X| (z) = 1+ 2z^' + 3z^' + 4z" + 5z^*

X, (z) = 5 + 4z' + 3z + 2z +z~"'

X (z ) = X , ( z ) X , ( z ) - 5 + 14z ' + 2 6 z " + 40z r + 552" + 40z^' + 26z^' + 14z' + 5z

ra: x ( n ) = {5 14 26 40 55 40 26 14 5}

2.20. Tm bin i z v min hi t ca tn hiu:

-8Suy

Li g i i:

(n) =

(n) =

J 0 ) n n > 0

n < 0

n > 0

n < 0

(o: tham s

(ii: tham s

x ( z ) = ! * ( " > ! "l i : : : :- ' , n=^0

11=0

, ROC:

2.21. Xc nh bin i z mt pha ca tn hiu hng s x(n) ^ 1 , - c o < n < 4-co

Li gi i:

X ' ( z ) = x ( n K " = I z - = | z| >ln-0 11=0 i z

2.22. Tnh tch chp x(n) ca cc tn hiu:

Li g i i:

^ i ( z ) - x , ( n ) z " = xn=() n=0


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Ch n g 2: Biu d in h thng v tin h iu r i rc t rong m in z 85

1-7

|z |>

= = x | n-0 n='i \ /

X( z) = X, (z ) .X, (z ) = -

7;" =

1- zs

2

2.23. Tnh tch chp x(n) ca cc tn hiu sau bng cch s dng bin i z;

f

(a) X | ( n ) - - u ( n - l ) , x , ( n )4 U y u( n )

(b) x , (n ) = u ( n ) . X ,(n ) = (n ) + u (n )

L/ iii:

1- z4

- -z4

K 4 n ) =

x , ( z ) =

II

1 f

^ 2 ,

u(n)

- z7

x(z) = x,(z).x,(z)_ 4 1

= + + 1

4 2

Suy ra: x(n) =

( b) X | ( n ) = u ( n )

4 " 1n

---- __ + - + 3 v4y 3 v2 y

u(n)

> 4

\ -


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86 ii b i D x n h iu s v M at lab

x , ( n ) - 6 ( n ) + w L i ( n )

x , ( z ) = u . z > lI - ' z- ' -

X(z ) = X, (z ) , X,{z

2-- z'3

1- - z3

1 _----z i

3

i - z - z3

Suy ra: x(n) -2 ~ 2

u( n )

2.24. Tnh tch chp x(n) ca cc tn hiu sau bng cch s dng bin i z:

f \ ( a ) X | ( n ) = u ( n ) , x , ( n ) = c os (7 m )u (n )

(b) x , (n ) = n u (n ) , X, (n) = 2" u(n - 1)

L gi i:

x , w =l - l z

2

> 2

X, (n) = cos(Tcn)u(n)

= > X j ( z ) =14- z'

l + 2z'" +z '

X ( 2 ) = X , ( z ) .X , ( z )

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