The ideal system has h i(t)=e- tu(t). The real system has delayed response h r(t)=e-(t-T)u(t-T) plus auxiliary path h a(t)=[u(t)-u(t-5T)]. For impulse input, choose β so the outputs agree at t=4T.

$e^{-3\alpha T}(1-e^{-2\alpha T})$

$-e^{-\alpha T}(1-e^{-3\alpha T})$

$e^{-2\alpha T}(1-e^{-2\alpha T})$

GATE 2022 BM – Question 40

Signals and Systems · LTI Systems and Convolution · 2 marks · Multiple choice

The ideal system has $h_i(t)=e^{-\alpha t}u(t)$. The real system has delayed response $h_r(t)=e^{-\alpha(t-T)}u(t-T)$ plus auxiliary path $h_a(t)=\beta[u(t)-u(t-5T)]$. For impulse input, choose β so the outputs agree at t=4T.

$e^{-3\alpha T}(1-e^{-2\alpha T})$
$-e^{-\alpha T}(1-e^{-3\alpha T})$
$-e^{-3\alpha T}(1-e^{-\alpha T})$
$e^{-2\alpha T}(1-e^{-2\alpha T})$

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Correct answer: (C) $-e^{-3\alpha T}(1-e^{-\alpha T})$

Explanation

At four T, the ideal output is e⁻⁴ᵅᵀ, the delayed output e⁻³ᵅᵀ, and the auxiliary rectangular path contributes β. Thus β=e⁻⁴ᵅᵀ−e⁻³ᵅᵀ=−e⁻³ᵅᵀ(1−e⁻ᵅᵀ), C. This negative correction removes the excess caused by the delay.

GATE 2022 BM – Question 40

Explanation

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