GATE 2023 BM – Question 26

Signals and Systems · Discrete-Time Systems and Z Transform · 1 mark · Multiple select

A causal discrete-time system obeys y[n]=0.5y[n−1]+x[n]. With initial rest, which statements are correct?

$h[n]=(0.5)^n u[-n]$
The system is stable
The impulse response is infinite in duration
The impulse response is finite in duration

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Show answer and explanation

Correct answer: (B) The system is stable; (C) The impulse response is infinite in duration

Explanation

With an impulse input and initial rest, h[n]=(0.5)ⁿu[n], so A has the wrong sidedness. It is nonzero for arbitrarily large nonnegative n, giving an infinite impulse response, C. The absolute sum is 1/(1−0.5)=2, finite, so it is BIBO stable, B. D is false.

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