Signals and Systems - Electrical Engineering

9.5 One-Sided Z-Transform Inverse 557

rational. That term equals

B( 1 −0.5z−^1 )+Cz−^1

( 1 −0.5z−^1 )^2

=

B

1 −0.5z−^1

+

Cz−^1

( 1 −0.5z−^1 )^2

which is very similar to the expansion for multiple poles in the inverse Laplace transform. Once

we find the values ofBandC, the inverse Z-transforms are obtained from the Z-transforms table. A

simple method to obtain the coefficientsBandCis to first obtainCby multiplying the two sides

of Equation (9.39) by( 1 −0.5z−^1 )^2 to get

Y(z)( 1 −0.5z−^1 )^2 =B( 1 −0.5z−^1 )+Cz−^1

and then lettingz−^1 =2 on both sides to find that

C=

Y(z)( 1 −0.5z−^1 )^2

z−^1

∣

∣z− (^1) = 2 =−0.5
TheBvalue is then obtained by choosing a value forz−^1 that is different from 1 or 0.5 to compute
Y(z). For instance, assume you choosez−^1 =0 and that you have foundAandC, then
Y(z)|z− (^1) = 0 =A+B= 1
from whichB=−3. The complete response is then
y[n]= 4 u[n]− 3 (0.5)n−0.5n(0.5)nu[n]
n
nExample 9.19
Find the complete response of the difference equation
y[n]+y[n−1]− 4 y[n−2]− 4 y[n−3]= 3 x[n] n≥ 0
y[−1]= 1
y[−2]=y[−3]= 0
x[n]=u[n]
Determine if the discrete-time system corresponding to this difference equation is BIBO stable or
not, and the effect this has in the steady-state response.
Solution
Using the time-shifting and linearity properties of the Z-transform, and replacing the initial
conditions, we get
Y(z)[1+z−^1 − 4 z−^2 − 4 z−^3 ]= 3 X(z)+[− 1 + 4 z−^1 + 4 z−^2 ]