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Consider a continuous time LTI system described by $\dfrac {dy(t)}{dt} + 2y=x(t). $

Using the Fourier transform, find out output to each of the following input signals.

$i) \space\space x(t) = e^{-t} u(t) \\ ii) \space x(t) = u(t)$

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 10

Year : DEC 2014

1 Answer
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Given

$\dfrac {dy(t)}{dt}+2y=x(t)$

By Laplace transform,

$s\space y(s) + 2 y(s) = x (s) \\ y (s) (s+2) = x (s) \\ \dfrac {y(s)}{x(s)} = \dfrac 1{(s+2)} \\ H(s) = \dfrac 1{(s+2)} \\ (i)\space x(t) = e^{-t} u(t) \\ X (s) =\dfrac 1{(s+1)} $

By Laplace transform,

$Y (s) = x (s) \times H (s) \\ =\dfrac 1{(s+2)}\dfrac 1{(s+1) } $

By partial fraction,

$Y (s) = \dfrac A{(s+1)} + \dfrac B{(s+2)} $

enter image description here

$Y (s) = \dfrac 1{(s+1)} -\dfrac 1{(s+2)} $

By Inverse Laplace transform,

$ y(t) = e^{-t} u(t)-e^{-2t} u(t). \\ (ii) \space x(t) = u(t) \\ X (s) = \dfrac 1s $

By Laplace transform,

$Y (s) = x (s) \times H (s) \\ = \dfrac 1{(s+2)}\dfrac 1s$

By partial fraction,

$Y (s) =\dfrac As + \dfrac B{(s+2) } $

enter image description here

$Y (s) = \dfrac {1/2}s+ \dfrac {1/2}{(s+2)}$

By Inverse Laplace transform, $\rightarrow y (t) =1/2( u(t)+e^{-2t} u(t)).$

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