Homework Help: Help in performing convolution

It's super easy (though a little abstact) to prove using the convolution theorem.
[tex] F\{a \ast b \} = F\{a\} \cdot F\{b\}[/tex]
The Fourier transform of cosine gives you the sum of two delta peaks at [itex]-\omega_0[/itex] and at [itex]\omega_0[/itex]. The Fourier transform of 1/x gives you a sgn-function. You multiply those two together and you get the difference of the delta peaks. You inverse Fourier transform that and you get the sine.

You might as well compute the convolution directly, but I'm still thinking about how to carry out the integration.

If you want a rational argument, keep in mind that the convolution measures the two signals correlation where the argument of the convoluted function is the phase difference of your two input functions.

 

Okay, the sign is pretty simple. You know that the delta function is zero except for one certain point, right? So there is actually a very simple relation.
[tex] \delta(x-x_0) sgn(x) = \delta(x-x_0), x_0>0[/tex]
[tex] \delta(x-x_0) sgn(x) = -\delta(x-x_0), x_0<0[/tex]
I think you can figure that out by yourself.
About the factors... I'll look into it and come back to you, okay?
Btw: I think I figured out a way to do the convolution integral directly through contour integration. Do you wanna know that or are you happy with the convolution theorem?

 

It looks like they missed a factor in the exercise sheet.