Fourier Transform of One-Sided Convolution

[thrillhouse86]

Hi,

Can anyone tell me if there is a convolution theorem for the Fourier transform of:
$$\int^{t}_{0}f(t-\tau)g(\tau)d\tau$$

I know the convolution theorem for the Fourier Transform of:
$$\int^{\infty}_{-\infty}f(t-\tau)g(\tau)d\tau$$

But I can't seem to find (or proove!) anything about the first one.

 

[rrogers]

Have you looked at the LaPlace transform?
http://en.wikipedia.org/wiki/Laplace_transform
"Convolution" in the table.


Ray

 

[thrillhouse86]

Hey Ray,

yeah I've noticed that Laplace transform one, but I really need the Fourier transform of this one sided one. I was hoping that the heaviside function would kill the -ve bounds of my Fourier Transform so that it would look like a Laplace transform, but in order to do that I need to switch the order of integration, and the process of doing that gives me annoying bounds on the integral.

 

[rrogers]

I don't understand, but you can reconstruct an equivalent Fourier transform from a Laplace transform.
Ah I think I understand; maybe you want to convolve over the last three seconds in an ongoing stream? If that is right then letting I=intergal(fg,a..(a+t)) you can do I=intergal(fg,a+t)-integral(fg,a) which can be accomplisher by (1-exp(-it)).
But this seems too easy: (1-exp(i*t*s)) F(s)G(s)
Ray