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Given two con...

Question

Given two continuous time signals $x (t) = e^{- t}$ and $y (t) = e^{- 2 t}$ which exist for $t > 0$ , the convolution z(t) = x(t) * y(t) is

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Solution

$X (s) = \frac{1}{s + 1}$ , $Y (s) = \frac{1}{s + 2}$

$Z(s) = X(s) . Y(s)$

$= \frac{1}{s + 1}$ $\times$ $\frac{1}{s + 2} = \frac{1}{s + 1} - \frac{1}{s + 2}$

$z (t) = e^{- t} - e^{- 2 t}$

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