Signals and Systems - Electrical Engineering

568 C H A P T E R 9: The Z-Transform

9.18. Inverse Z-transform

Find the inverse Z-transform of

X(z)=

8 − 4 z−^1

z−^2 + 6 z−^1 + 8

and determinex[n]asn→∞. Assumex[n]is causal.

9.19. Z-transform properties and inverse transform

Sometimes the partial fraction expansion is not needed in finding the inverse Z-transform—instead the

properties of the transform can be used. Consider the function

F(z)=

z+ 1

z^2 (z− 1 )

(a) Determine whetherF(z)is a proper rational function as a function ofzand ofz−^1.

(b) Verify thatF(z)can be written as

F(z)=

z−^2

1 −z−^1

+

z−^3

1 −z−^1

Find the inverse Z-transformf[n]using the above expression.

9.20. Inverse Z-transform—MATLAB

We are interested in the unit-step solution of a system represented by the difference equation

y[n]=y[n−1]−0.5y[n−2]+x[n]+x[n−1]

(a) Find an expression forY(z).

(b) Do a partial expansion ofY(z).

(c) Find the inverse Z-transformy[n]and verify your results using MATLAB.

9.21. Pade approximation ́

Suppose we are given a finite-length sequenceh[n](it could be part of an infinite-length impulse response

from a discrete system that has been windowed) and would like to obtain a rational approximation for it.

This means that ifH(z)=Z[h[n]], a rational approximation of it would beH(z)=B(z)/A(z), from which

we get

H(z)A(z)=B(z)

Letting

B(z)=

M∑− 1

k= 0

bkz−k

A(z)= 1 +

N∑− 1

k= 1

akz−k

for some choice ofMandN, equations fromH(z)A(z)=B(z)should allow us to find theM+N− 1

coefficients{akbk}.