The GATE Grind

GATE 2015 EC – Question 55

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is $h[n]$. If $h[0] = 1$, we can conclude

A $z$-plane with the unit circle. There are four poles, at $0.5 + 0.5j$, $0.5 - 0.5j$, $-0.5 + 0.5j$ and $-0.5 - 0.5j$, and a zero of multiplicity 4 at the origin.
  1. $h[n]$ is real for all $n$
  2. $h[n]$ is purely imaginary for all $n$
  3. $h[n]$ is real for only even $n$
  4. $h[n]$ is purely imaginary for only odd $n$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $h[n]$ is real for all $n$

Explanation

The poles come in complex conjugate pairs, so the transfer function has real coefficients. A causal system with real coefficients and $h[0] = 1$ has a real impulse response for all $n$.