GATE EC 2025 Set 1 — Question 55
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Signals & Systems → Continuous-Time Signals → Fourier Series
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Question
Let be a periodic signal with fundamental period . Consider the signal , where .The Fourier series expansions of and are given byWhich of the following statements is/are TRUE?
Correct answer
(A) cₖ = dₖ for all k; (D) y(t) is periodic with a fundamental period T₀/α
Solution
1.Periodicity of :
Given has a fundamental period , it satisfies .For the signal , let be its fundamental period. Then:
This equality holds if , which gives .
Since , the signal is compressed in time, and the fundamental period becomes . Thus, statement (D) is TRUE and (B) is FALSE.
2.Fourier Series Coefficients:
The Fourier series expansion of is:Substituting for to find the expansion of :
The problem provides the expansion for as:
By comparing the two expressions, we can see that the coefficients must be identical: for all .
Thus, statement (A) is TRUE and (C) is FALSE.
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