GATE EC 2023 Set 1 — Question 48
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Signals & Systems → Discrete-Time Signals → DTFT & Properties
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Question
Let an input having discrete time Fourier transform be passed through an LTI system. The frequency response of the LTI system is . The output of the system is
Correct answer
(C) δ[n] - δ[n-1] - (1)/(2)δ[n-2] + (5)/(2)δ[n-3] - δ[n-5]
Solution
Given the Discrete-Time Fourier Transform (DTFT) of the input signal :
Taking the inverse DTFT, we find the input signal :
Given the frequency response of the LTI system :
Taking the inverse DTFT, we find the impulse response :
For an LTI system, the output in the frequency domain is the product of the input DTFT and the system's frequency response:
Substitute the given expressions for and :
Expand the product:
Group terms with the same exponential:
Finally, take the inverse DTFT of to find :
Comparing this result with the given options, option (C) matches.The final answer is .
Taking the inverse DTFT, we find the input signal :
Given the frequency response of the LTI system :
Taking the inverse DTFT, we find the impulse response :
For an LTI system, the output in the frequency domain is the product of the input DTFT and the system's frequency response:
Substitute the given expressions for and :
Expand the product:
Group terms with the same exponential:
Finally, take the inverse DTFT of to find :
Comparing this result with the given options, option (C) matches.The final answer is .
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