GATE EE 2026 Set 1 — Question 12
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Signals & Systems → Signal Representation → Periodic & Aperiodic Signals
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Question
Let be any continuous-time periodic signal with period . It is sampled uniformly with a sampling period where , resulting in the discrete sequencewhere is an integer.
Which one of the following statements is correct about ?
Which one of the following statements is correct about ?
Correct answer
(D) x[n] will be periodic if and only if T/Tₛ is a rational number
Solution
A discrete-time signal is periodic if there exists a positive integer such that for all .
For the sampled signal , the condition for periodicity is:
Since is a continuous-time periodic signal with period , this condition is satisfied if the time interval is an integer multiple of the fundamental period :
for some integer .Rearranging this gives:
Since and are both integers, their ratio is a rational number. Therefore, is periodic if and only if the ratio of the signal period to the sampling period, , is a rational number. Thus, option (D) is correct.
For the sampled signal , the condition for periodicity is:
Since is a continuous-time periodic signal with period , this condition is satisfied if the time interval is an integer multiple of the fundamental period :
for some integer .Rearranging this gives:
Since and are both integers, their ratio is a rational number. Therefore, is periodic if and only if the ratio of the signal period to the sampling period, , is a rational number. Thus, option (D) is correct.
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