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Q.2 We know that the output of a continuo us-time LTI system, whose input x(t) and mpukse response h(t) are nonzero for -0

Q.2 We know that the output of a continuo us-time LTI system, whose input x(t) and mpukse response h(t) are nonzero for -0

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Q.2 We know that the output of a continuous-time LTI system, whose input x(t) and impuse response h(t) are nonzero for -00 <t< 0, is given by the convolut ion integral y(t) = x(t) * h(t) = L x(t – t)h(t)dr. %3D a) Now let h(t) be given as h(t) = e-5tu(t). Giving the mathematical justification, make the required change in the integral limits of the convolution integral given in (1) according to the signal h(t) given above and rewrite (1). b) Now let x(t) = 5u(t), the unit-step signal, which implies another change in the integral limits in (1). Giving the mathematical justification, make the change in the integral limits of (1) according to (2). c) Now substitute h(t) given above and (2) into (1) and find y(t) by taking the integral given in (1) with the modified limits and show that y(t) = (1– e-5t)u(t). Hint: u(1-r)=1 for 0STSt.
Transcribed Image Text:Q.2 We know that the output of a continuous-time LTI system, whose input x(t) and impuse response h(t) are nonzero for -00 <t< 0, is given by the convolut ion integral y(t) = x(t) * h(t) = L x(t – t)h(t)dr. %3D a) Now let h(t) be given as h(t) = e-5tu(t). Giving the mathematical justification, make the required change in the integral limits of the convolution integral given in (1) according to the signal h(t) given above and rewrite (1). b) Now let x(t) = 5u(t), the unit-step signal, which implies another change in the integral limits in (1). Giving the mathematical justification, make the change in the integral limits of (1) according to (2). c) Now substitute h(t) given above and (2) into (1) and find y(t) by taking the integral given in (1) with the modified limits and show that y(t) = (1– e-5t)u(t). Hint: u(1-r)=1 for 0STSt.
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