Question

What is the partial fraction expansion of X(z)=\(\frac{1+z^{-1}}{1-z^{-1}+0.5z^{-2}}\)?

(a) \(\frac{z(0.5-1.5j)}{z-0.5-0.5j} – \frac{z(0.5+1.5j)}{z-0.5+0.5j}\)

(b) \(\frac{z(0.5-1.5j)}{z-0.5-0.5j} + \frac{z(0.5+1.5j)}{z-0.5+0.5j}\)

(c) \(\frac{z(0.5+1.5j)}{z-0.5-0.5j} – \frac{z(0.5-1.5j)}{z-0.5+0.5j}\)

(d) \(\frac{z(0.5+1.5j)}{z-0.5-0.5j} + \frac{z(0.5-1.5j)}{z-0.5+0.5j}\)

I have been asked this question in my homework.

This intriguing question comes from Inversion of Z Transform in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Correct answer is (b) \(\frac{z(0.5-1.5j)}{z-0.5-0.5j} + \frac{z(0.5+1.5j)}{z-0.5+0.5j}\)

The best I can explain: To eliminate the negative powers of z, we multiply both numerator and denominator by z^2. Thus,

X(z)=\(\frac{z(z+1)}{z^{-2}-z+0.5}\)

The poles of X(z) are complex conjugates p1=0.5+0.5j and p2=0.5-0.5j

Consequently the expansion will be

X(z)= \(\frac{z(0.5-1.5j)}{z-0.5-0.5j} + \frac{z(0.5+1.5j)}{z-0.5+0.5j}\).