# Convolution math homework

1. Apr 15, 2008

### wildman

1. The problem statement, all variables and given/known data

Find $$R(\tau)$$ if a) $$S(\omega) = \frac{1}{(4+\omega^2)^2}$$

2. Relevant equations

I have given $$\frac{4}{4+\omega^2}$$ <==> $$e^{-2|\tau|}$$

3. The attempt at a solution
So $$S(\omega) = \frac{1}{(4+\omega^2)^2}= \frac{1}{16}\frac{4}{(4+\omega^2)}\frac{4}{(4+\omega^2)}$$

$$R(\tau)= \frac{1}{16} e^{-2|\tau|} * e^{-2|\tau|}$$

Where * is convolution

So

$$R(\Tau) = \frac {1}{8}\int_{0}^{\infty} e^{-2(\tau-\alpha)} e^{-2\alpha} d\alpha$$

But that turns out to be infinite. Does anyone have any idea where I went wrong?

Last edited: Apr 15, 2008
2. Apr 16, 2008

### benorin

The Laplace transform is $$\frac{a}{a^2+\omega^2} \Leftrightarrow \sin a\tau$$

Last edited: Apr 16, 2008