9.1 • THE Z-TRANSFORM 353Find the inverse z-transform of each termyn ( l =yo3-l[ --z I +--3 ab _^11 --^1 l ---3 b - 1! --^1 I
z-a a - 1 z - a a - 1 z - l
=Yo( an)+ ~(a"-^1 u[n - 1]) - _b_(u[n - l ]).a-1 a - 1
When n = 0 we get y (O] = Yo(a^0 ) + 0 - 0 = Yo, and when n 2'. 1 the
expression for y(n] simplifies to bean- 1
y[n] = yoa" + --
1
b.
a -(c) Start with the formula for the z-transform that we found in part (b):
Y(z) = b(~ffS::~·. Then use Corollaries 9.1 and 9.2 and residues to
find 3-^1 [Y(z)].y[n] = 3-^1 (Y(z)] = Res{Y(z)z"-^1 , 1] + Res(Y(z)zn-^1 ,a]
R [bz - zyo + z
(^2) yo n-l l ] R [bz - zyo + z (^2) yo n - 1 ]
= es ( z -1) ( z -a ) z , + es ( z - 1 ) (z - a ) z ,a
= lim(z - l)X(z)z"-^1 + lim(z -a)X(z)z"-^1
i:~l z-a= lim(z - 1) bz - zyo + z2yo zn-1 +Jim bz- zyo + z2yo X(z)z"- 1
z - 1 (z - l)(z -a) z-a (z - l)(z - a)I. bz- zyo+z^2 yo n - l 1· bz-zyo+z^2 yo n - 1
= 1m z + 1m z
z~l z -a z-a z - 1
= b -Yo+ Yo 1 n-1 + ba-ayo + a^2 Yo an-1
1-a a-1
b a-l+n (ab-ayo + a^2 yo)
= --+ ----'------~
1-a -l+ a
a" - 1
=yoa"+--b
a - 1