Question

What is the ROC of the system function H(z) if the discrete time LTI system is BIBO stable?

(a) Entire z-plane, except at z=0

(b) Entire z-plane, except at z=∞

(c) Contain unit circle

(d) None of the mentioned

The question was asked during an interview for a job.

My doubt is from Z Transform in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct choice is (c) Contain unit circle

The best explanation: A discrete time LTI is BIBO stable, if and only if its impulse response h(n) is absolutely summable. That is,

\(\sum_{n=-\infty}^{\infty}|h(n)|<\infty\)

We know that, H(z)= \(\sum_{n=-\infty}^{\infty}h(n)z^{-n}\)

Let z=e^jω so that |z|=|e^jω|=1

Then, |H(e^jω)|=|H(ω)|=| \(\sum_{n=-\infty}^{\infty}\)h(n) e^-jωn|≤\(\sum_{n=-\infty}^{\infty}\)|h(n) e^-jωn|

=\(\sum_{n=-\infty}^{\infty}\)|h(n)|<∞

Hence, we see that if the system is stable, then H(z) converges for z=e^jω. That is, for a stable discrete time LTI system, ROC of H(z) must contain the unit circle |z|=1.