GATE EC 2015 Set 1 — Question 55
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Signals & Systems → LTI Systems → Pole-Zero Stability & Causality
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Question
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 . If , we can conclude
Correct answer
(A) h[n] is real for all n
Solution
From the pole-zero plot, the poles are located at and . The zeros are at the origin with multiplicity 4. Since the poles and zeros occur in complex conjugate pairs, the transfer function has real coefficients. For a system with real coefficients, the impulse response is real for all . Specifically, . Expanding this gives , which is real for all and satisfies .
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