GATE EC 2014 Set 3 — Question 54
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Question
Let denote the impulse response of a causal system with transfer function . Consider the following three statements.
S1: The system is stable.
S2: is independent of for .
S3: A non-causal system with the same transfer function is stable.
For the above system,
S1: The system is stable.
S2: is independent of for .
S3: A non-causal system with the same transfer function is stable.
For the above system,
Correct answer
(A) only S1 and S2 are true
Solution
The transfer function is , which has a pole at .
S1: For a causal system, the Region of Convergence (ROC) is . Since the ROC includes the axis, the system is stable. Thus, S1 is true.
S2: The impulse response for the causal system is . For , .This is a constant and independent of . Thus, S2 is true.
S3: For a non-causal (specifically anti-causal) system with the same transfer function, the ROC would be . This ROC does not include the axis, so the system is unstable. Thus, S3 is false.
Therefore, only S1 and S2 are true.
S1: For a causal system, the Region of Convergence (ROC) is . Since the ROC includes the axis, the system is stable. Thus, S1 is true.
S2: The impulse response for the causal system is . For , .This is a constant and independent of . Thus, S2 is true.
S3: For a non-causal (specifically anti-causal) system with the same transfer function, the ROC would be . This ROC does not include the axis, so the system is unstable. Thus, S3 is false.
Therefore, only S1 and S2 are true.
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