GATE EC 2023 Set 1 — Question 47
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Signals & Systems → Discrete-Time Signals → DFT & FFT
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Question
Consider a discrete-time periodic signal with period . Let the discrete-time Fourier series (DTFS) representation be , where and . The value of the sum is
Correct answer
(A) -10
Solution
Given a discrete-time periodic signal with period . Its DTFS representation is .
The coefficients are .
We need to find the value of the sum .First, express in terms of complex exponentials: .
So, .Substitute this into the sum:
Recall the DTFS analysis equation: .
This implies .For the first sum, :
We can write .
Comparing with , we have and .
Since the coefficients are defined for , we map to its equivalent in the range by adding : .
So, .For the second sum, :
Comparing with , we have and .
So, .Substitute these back into the expression for :
.Given and .
.
.Thus, the value of the sum is -10.The final answer is .
The coefficients are .
We need to find the value of the sum .First, express in terms of complex exponentials: .
So, .Substitute this into the sum:
Recall the DTFS analysis equation: .
This implies .For the first sum, :
We can write .
Comparing with , we have and .
Since the coefficients are defined for , we map to its equivalent in the range by adding : .
So, .For the second sum, :
Comparing with , we have and .
So, .Substitute these back into the expression for :
.Given and .
.
.Thus, the value of the sum is -10.The final answer is .
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