The choices shade regions of convergence in the z plane. For x[n]=2 n , -<n<, which choice gives its z-transform ROC?

GATE 2016 EC (EC3) – Question 41

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

The choices shade regions of convergence in the z plane. For $x[n]=2^{|n|}$, $-\infty<n<\infty$, which choice gives its z-transform ROC?

Diagram for GATE 2016 EC (EC3) question 41

Option A in the figure
Option B in the figure
Option C in the figure
Option D in the figure

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) Option D in the figure

Explanation

For n≥0 the terms are $2^nz^{-n}$, whose geometric series converges only for |z|>2.

For n<0 put m=−n>0. The terms become $2^mz^m$, which require |z|<1/2. No z satisfies both conditions.

Therefore the bilateral z-transform has **no ROC, D**. The two pole radii do not create an annulus because the required inequalities point away from each other.

GATE 2016 EC (EC3) – Question 41

Explanation

More Discrete-time Signals questions