Fourier Transform for a Convolution

Question

Alright, so I am using the Convolution property of Fourier Transforms to find a function $f(x)$. So the obvious equation: $h(x) = f(x) \ast g(x)$.

Definitions:

$$g(x)=Rect\left[\frac x w \right]$$

h(x) =  al*exp(-((abs((x-b1)./c1).^d)))+a2
a2 =  1.205e+004  ;
al =  1.778e+005  ;
b1 =       94.88  ;
c1 =       224.3  ;
d =       4.077  ;

That is, $$h(x)=a_1 \exp\left[-\left(\frac{|x-b_1|}{c_1}\right)^d \right]+a_2$$ with the constants defined above.

So I want to find $f(x)$ by fourier transforming everything. The only prblem is that I can not find the fourier transform of h(x). I have tried to use fft() in matlab, FourierTransform[h,x,$\omega$] in mathematica. In matlab, when I apply fft to both $h(x)$ and the Rect[] function, I do not end up with a reasonable result after ifft (most likely due to the zeros of sinc). However, in Mathematica, I can not even get a result for the FT of $h(x)$. The computer just sits and does nothing. So I am really stuck. I do not have enough math background to try and find the FT by hand. So if anyone has any advice (or a really fast computer that will actually perform the FT). Thank you!

Answer 1

The description of Mathematica command FourierTransform says

gives the symbolic Fourier transform

It rather unlikely that the Fourier transform of your $h$ has a sensible symbolic expression, due to fractional power $d$. But you can transform $h$ numerically, as explained in Numerical Fourier transform of a complicated function.

Also, I am not surprised that you don't like the result of $\mathcal F^{-1}(\mathcal F h/\mathcal F g)$. Deconvolution is not a straightforward operation.