Convolution

**Ulysses** · June 12th 2011, 12:16 PM

Hi there. I must calculate the convolution between these functions
$f(t)= e^{-t} H(t)$
$g(t)=e^t H(t)$ H(t) the unit step Heaviside function.
So I have to find: $f \star g$

This is what I did:
$f \star g=\displaystyle\int_{-\infty}^{\infty}e^{-\lambda}e^{t-\lambda}H(\lambda)H(t-\lambda)d\lambda=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} \left [ H(\lambda)-H(\lambda-t) \right ] d\lambda=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda}H(\lambda)d\lambda-\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} H(\lambda-t)d\lambda=\displaystyle\int_{0}^{\infty}e^{t-2\lambda}d\lambda-\displaystyle\int_{t}^{\infty}e^{t-2\lambda} d\lambda=\displaystyle\int_{0}^{t}e^{t-2\lambda} d\lambda=-2e^{-t}+2e^{t}$

Is this fine?

**ojones** · June 14th 2011, 05:09 PM

In the last line, you were meant to divide by 2, not multiply. Everything else looks OK.

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