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 michaelbarrett May 25th, 2012 10:59 AM

Fourier Transform Tricky Integral

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Hi I am trying to analytically calculate the Fourier transform below.

I am getting really stuck with the integral, can anyone help?

The signal I need to fourier transform is:

f(t)=e^-(t/t0) cos(w0(1-qt)t) (also attached with attempted working)

Any help much appreciated!

 Dougy May 25th, 2012 11:40 AM

Re: Fourier Transform Tricky Integral

1) replace your cosine part by a complex exponential, so the result would be the real part at the end. (that is the tip they gave you)
2) Then you have to integrate an exponential that you need to rewrite as a rescale gaussian. Use identities like for instance $t^2+xt=(t+x/2)^2-x^2/4$ and finally the integral they gave you at the end that does not depend on x.
3) Finally your function is only defined in a certain domain right? So you need to use that your fourier transform would be the convoluate of your solution with the Fourier transform of some step function (which is a sine cardinal). Or perhaps easier, just try to calculate the integral given at then end on the domain of definition, that might then depend on x...

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