Question

Find the inverse Z- Transform of \((\frac{z}{z-a})^3\).

(a) \(\frac{1}{2} . (n+1) (n-2) a^{n-2} U(n) \)

(b) \(\frac{1}{2} . (n-1) (n-2) a^{n-3} U(n) \)

(c) \(\frac{1}{2} . (n-1) (n+2) a^{n-1} U(n) \)

(d) \(\frac{1}{2} . (n+1) (n+2) a^n U(n) \)

I got this question at a job interview.

Question is from Linear Difference Equations and Z topic in section Fourier Integral, Fourier Transforms and Integral Transforms of Engineering Mathematics

Answer 1

Correct choice is (d) \(\frac{1}{2} . (n+1) (n+2) a^n U(n) \)

For explanation I would say: The inverse Z-Transform of \(\frac{z}{z-a} \) is \(a^n\). The inverse Z-Transform of \((\frac{z}{z-a})^2\) is the convolution of a^n and a^n. Now, the inverse Z-Transform of \((\frac{z}{z-a})^3\) is the convolution of the result of the previous step with a^n a^n. Thus we get the answer \(\frac{1}{2} . (n+1) (n+2) a^n U(n) \)