3) The impulse response of a certain linear system is h(t) = u(t) + 8(t). Determine the zero stateresponse of the system to the input, u(t) e“ , using convolution.

The zero state response for t≥0 is: yt=ht*ute-t =ut+δt*ute-t =∫-∞∞uτ+δτut-τe-t-τdτ…

Answered: 3) The impulse response of a certain…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3) The impulse response of a certain linear system is h(t) = u(t) + 8(t). Determine the zero state
response of the system to the input, u(t) e“ , using convolution.

Expert Solution

Step 1

The zero state response for $t \geq 0$ is:

$y (t) = h (t) * u (t) e^{- t} = (u (t) + δ (t)) * u (t) e^{- t} = \int_{- \infty}^{\infty} (u (τ) + δ (τ)) u (t - τ) e^{- (t - τ)} d τ = \int_{- \infty}^{\infty} (u (τ) u (t - τ) e^{- (t - τ)} + δ (τ) u (t - τ) e^{- (t - τ)}) d τ = \int_{- \infty}^{\infty} u (τ) u (t - τ) e^{- (t - τ)} d τ + u (t) e^{- t} = \int_{0}^{t} e^{τ - t} d τ + u (t) e^{- t} = e^{t - t} u (t) - e^{0 - t} u (t) + u (t) e^{- t} = u (t) - e^{- t} u (t) + u (t) e^{- t} = u (t)$

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