GATE EC 2024 Set 1 — Question 21
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Signals & Systems → LTI Systems → Impulse Response & Convolution
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Question
A causal and stable LTI system with impulse response produces an output for an input signal . A signal is applied to another causal and stable LTI system with impulse response . The resulting output is
Correct answer
(A) 2y(0.5t)
Solution
For a Linear Time-Invariant (LTI) system, the output is the convolution of the input and the impulse response :
Now, a new system has an input signal and an impulse response . Let the resulting output be .Substitute the given scaled signals into the convolution integral:
To simplify this integral, let's use a substitution. Let . Then, .
When , .
When , .Substitute and into the integral:
The integral is the definition of the convolution , but evaluated at the scaled time .
So, .Therefore, the resulting output is:
The final answer is
Now, a new system has an input signal and an impulse response . Let the resulting output be .Substitute the given scaled signals into the convolution integral:
To simplify this integral, let's use a substitution. Let . Then, .
When , .
When , .Substitute and into the integral:
The integral is the definition of the convolution , but evaluated at the scaled time .
So, .Therefore, the resulting output is:
The final answer is
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