# Convolution math homework

#### 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:

#### benorin

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

Last edited:

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving