GATE EE 2022 Set 1 — Question 47
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Signals & Systems → Fourier Analysis & Sampling → CTFT & DTFT
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Question
Let an input is
passed through an LTI system having an impulse response,
.
The output of the system is
passed through an LTI system having an impulse response,
.
The output of the system is
Correct answer
(C) 7 sin(42π t) + 4cos(45π t)
Solution
The input signal is .
The frequencies present in the input signal are , , , and .The impulse response is .
We need to find the frequency response of the system.
Let . Its Fourier Transform is .
This is a rectangular pulse with amplitude 1 for and 0 otherwise.
Let . Its Fourier Transform is .Using the multiplication property in time domain, , so in frequency domain:
.Convolving with :
. This shifts the rectangular pulse to be centered at . The pulse will be from to .
. This shifts the rectangular pulse to be centered at . The pulse will be from to .So, .
This means the system acts as an ideal bandpass filter with passbands from to and from to , with a gain of 1.Now, let's check which frequencies from fall into these passbands:
.Thus, the correct option is (C).
The frequencies present in the input signal are , , , and .The impulse response is .
We need to find the frequency response of the system.
Let . Its Fourier Transform is .
This is a rectangular pulse with amplitude 1 for and 0 otherwise.
Let . Its Fourier Transform is .Using the multiplication property in time domain, , so in frequency domain:
.Convolving with :
. This shifts the rectangular pulse to be centered at . The pulse will be from to .
. This shifts the rectangular pulse to be centered at . The pulse will be from to .So, .
This means the system acts as an ideal bandpass filter with passbands from to and from to , with a gain of 1.Now, let's check which frequencies from fall into these passbands:
1.: . This component is blocked.
2.: . This component is blocked.
3.: . This component passes through with unity gain.
4.: . This component passes through with unity gain.
Therefore, the output will be:.Thus, the correct option is (C).
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