GATE EC 2018 Set 1 — Question 23
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Signals & Systems → LTI Systems → Magnitude & Phase Response
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Question
A discrete-time all-pass system has two of its poles at and . Which one of the following statements about the system is TRUE?
Correct answer
(B) It is stable only when the impulse response is two-sided.
Solution
An all-pass system is defined by having a constant magnitude response for all frequencies .
For the ROC to include , it must be the ring .
An ROC that is a ring (bounded by poles on both sides) corresponds to a non-causal, two-sided impulse response.
Therefore, the system is stable only if the impulse response is two-sided.
1.Pole-Zero Relationship: For an all-pass system, if there is a pole at , there must be a corresponding zero at . Option (A) incorrectly suggests that the reciprocals are poles, whereas they would actually be zeros.
2.Phase Response: All-pass systems have a frequency-dependent phase response. They do not have constant phase response over all frequencies or the entire -plane. Thus, options (C) and (D) are false.
3.Stability and ROC: A discrete-time system is stable if and only if its Region of Convergence (ROC) includes the unit circle ().
The given poles are (inside the unit circle, ) and (outside the unit circle, ).For the ROC to include , it must be the ring .
An ROC that is a ring (bounded by poles on both sides) corresponds to a non-causal, two-sided impulse response.
Therefore, the system is stable only if the impulse response is two-sided.
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