Signal Processing and Linear Systems-B.P.Lathi copy

k 0 123 k f ig p i06 8 b is t he o tft found

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Unformatted text preview: rm • T able 1 1.1: ( Unilateral) z -Transform P airs i [k] z z -1 2 u[k] 3 ku[k] 5 k 3 u[k] z (z2 6 "(k-1u[k - 1] 7 "(ku[k] 10 z -,,( z z -,,( (z - (z m n1 , n 1. + 1) "(ku[k] "()2 h lk c os .8k u[k] l Ib h lk s in .8k u[k] 12a r hlk c os (.8k + 8)u[k] 12b r hlk c os (.8k + 8)u[k] r hlk c os (.8k F[zJ = _ 19 6 .8 = t an- 1 z -3 ( _z_) z -2 +~ 3 ( _z_) z -3 (11.12b) T he r eader c an verify t hat t his a nswer is equivalent t o t hat i n Eq. (11.12a) by c omputing i[kJ i n b oth cases for k = 0, 1, 2, 3, . .. , a nd t hen c omparing t he r esults. T he f orm in E q. (11.12b) is m ore c onvenient t han t hat i n Eq. (11.12a). For t his r eason, we s hall a lways e xpand F[zJ/z r ather t han F[zJ i nto p artial f ractions a nd t hen m ultiply b oth sides by z t o o btain modified p artial f ractions o f F[z], w hich have a factor z i n t he n umerator. (O.5re- i8 )z + B) + 2az + h l 2 2 F rom P airs 1 a nd 7 i n T able 11.1, i t follows t hat +~--"""'--- z- +~ ( b) "(* F[z) = z(2z 2 - 11z + 12) (z - l)(z - 2)3 z (Az z2 a nd F[zJ A 2h·12+B2-2AaB 1"YI L a 2 8 + hl 2 r z[z c os 8 - 1"(1 c os (.8 - 8)] z2 - (21"(1 c os .8)z + h l 2 J "Y z ( z - "()m+l z - "( z -2 M ultiplying b oth s ides b y z yields Z2 - (2hl c os .8)z + 8)u[k] .8 = c os- 1 fiT, "()3 (O.5re i8 )z + ( 3/2) + ( 5/3) z z hl s in "( = h leiil ( 11.l2a) 8 z - 19 = ( -19/6) z(z - hi c os .8) Z2 - (2hl c os .8)z + h l 2 Ua + 5 (3)k-l] u[k - IJ z (z - 2)(z - 3) , ,(z(z+,,() k(k - 1)(k - 2)···(k - - F[zJ z " (z k 2"(k u [k] r- + 4z + 1) (z - 1)4 k"(ku[k] z- 3 I f we e xpand r ational F[zJ i nto p artial f ractions directly, we shall always o btain a n a nswer t hat is multiplied b y u[k - IJ b ecause o f t he n ature o f P air 6 i n T able 11.1. T his f orm is r ather a wkward a s well a s i nconvenient. We prefer t he form t hat is m ultiplied b y u[kJ r ather t han u[k - 1). A g lance a t T able 11.1 shows t hat t he z -transform o f e very signal t hat is multiplied by u[kJ h as a f actor z i n t he n umerator. T his o bservation s uggests t hat we e xpand F[zJ i nto modified p artial f ractions, where e ach t erm h as a f actor z i n t he n umerator. T his g oal c an b e a ccomplished by e xpanding F[zJ/z i nto p artial f ractions a nd t hen m ultiplying b oth s ides by z . We shall d emonstrate t his p rocedure b y r eworking p art ( a) in E xample 11.3. F or t his c ase (z - 1)3 "( 12c i[kJ = [ 3(2)k-l + 1) z (z + _ 5_ F rom T able 11.1, P air 6, we o btain z k 2 u[k] 8z - 19 = _ 3_ (z - 2)(z - 3) z- 2 F[zJ = (z - 1)2 4 9 2 z(3z + 17) ( c) (z _ 1)(z2 - 6z + 25) 2 ( b) z (2z - 11z + 12) (z - 1)(z - 2)3 8z - 19 (z - 2)(z - 3) ( a) E xpanding F[zJ i nto p artial f ractions yields b[k - j] 8 E xample 1 1.3 F ind t he inverse z -transform o f ( a) F[z] 675 T he Z - Transform 1 1.1 Z A a-B A Jh12-a2 2Z2 - 11z + 12 ( z-l)(z-2)3 k ao al a2 = --+---+---+-z - 1 (z - 2)3 (z - 2)2 ( z - 2) 676 11 D i...
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