Question

What is the causal signal x(n) having the z-transform X(z)=\(\frac{1}{(1+z^{-1})(1-z^{-1})^2}\)?

(a) [1/4(-1)^n+3/4-n/2]u(n)

(b) [1/4(-1)^n+3/4-n/2]u(-n-1)

(c) [1/4+3/4(-1)^n-n/2]u(n)

(d) [1/4(-1)^n+3/4+n/2]u(n)

The question was asked during an interview.

This key question is 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 (d) [1/4(-1)^n+3/4+n/2]u(n)

The explanation: The partial fraction expansion of X(z) is \(X(z) = \frac{z}{4(z+1)} + \frac{3z}{4(z-1)} + \frac{z}2{(z-1)^2}\)

When we apply the inverse z-transform for the above equation, we get

x(n)=[1/4(-1)^n+3/4+n/2]u(n).