GATE EC 2026 Set 1 — Question 16
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Signals & Systems → Discrete-Time Signals → Causality & Stability
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Question
The response of a discrete time system obeys the following relation:The input to the system is .Which of the following options is TRUE for ?
Correct answer
(A) Stable and causal response
Solution
Given the difference equation:
Taking the Z-transform on both sides:
The transfer function is:
The input is , so its Z-transform is:
The Z-transform of the output is:
has a single pole at .
Taking the Z-transform on both sides:
The transfer function is:
The input is , so its Z-transform is:
The Z-transform of the output is:
has a single pole at .
1.For a causal response, the Region of Convergence (ROC) is . Since this ROC includes the unit circle (), the response is stable. In this case, .
2.For a non-causal (specifically anti-causal) response, the ROC is . This ROC does not include the unit circle, so the response would be unstable. In this case, .
Standard systems described by such difference equations are typically assumed to be causal. Therefore, the response is stable and causal.Continue learning with Success Tracker
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