1. ## Convolution

Hi,

I have the following problem:

$\displaystyle h(t)=(sin t+cost)\theta(t)$
$\displaystyle w(t)=\theta(t)-\theta(t-2\pi)$

How do I compute the following convolution::
$\displaystyle h(t)\ast w(t)$

I know that the definition for convolution is as follow:

(1)

But how do I apply (1) to the given problem?

Do I distribute the parentheses

$\displaystyle w(t)=\theta(t)-\theta(t-2\pi)=\theta(t)-\theta(t)-\theta2\pi$

Any help would be greatly appreciate it.
Thank you

2. If you apply the definition of convolution then you get $\displaystyle \displaystyle(h*w)(t) = \int_{-\infty}^\infty\!\!\!(\sin\tau + \cos\tau)\theta(\tau)\bigl(\theta(t-\tau) - \theta(t-\tau-2\pi)\bigr)\,d\tau.$ You cannot simplify that any further unless you know something about the nature of the function $\displaystyle \theta$.

3. Originally Posted by Opalg
If you apply the definition of convolution then you get $\displaystyle \displaystyle(h*w)(t) = \int_{-\infty}^\infty\!\!\!(\sin\tau + \cos\tau)\theta(\tau)\bigl(\theta(t-\tau) - \theta(t-\tau-2\pi)\bigr)\,d\tau.$ You cannot simplify that any further unless you know something about the nature of the function $\displaystyle \theta$.
How do I integrate this integral?

Kind regards

4. I guess that integration is solved by partialintegration.

$\displaystyle \int udv=uv-\int vdu$

My question is this:

what is $\displaystyle u$ and $\displaystyle dv$ in this integral: $\displaystyle \displaystyle \int_{-\infty}^\infty\!\!\!(\sin\tau + \cos\tau)\theta(\tau)\bigl(\theta(t-\tau) - \theta(t-\tau-2\pi)\bigr)\,d\tau.$

$\displaystyle u=...$
$\displaystyle dv=....$

5. May be $\displaystyle \theta(t)$ is the Heaviside step function ?