in Maxima how to get Fourier transform identity ($\mathcal{F}^{-1}(\mathcal{F}(f)) = f$)?

Question

I'm currently trying to figure out something related to signal processing and wanted to draft Maxima for the task. As a starting point I tried to make Maxima evaluate the identity $\mathcal{F}^{-1}[\mathcal{F}[f]] = f$. Fruitlessly so far.

What I've come up so far with is

H(f):=integrate( integrate(f(t)*exp(-%i*2*%pi*(y*t)),t,-inf,inf) *exp(%i*2*%pi*(y*x)),x,-inf,inf);

If applied to some test function (sin in my case) the result is just the integrals written out, instead of just sin (the name H stems from the fact, that I want to turn this into the Hilbert transform).

Also I'm wondering if I'm barking up the wrong tree here, by explicitly writing down the FT.

Note that I'm not interested in DFT or FFT!

Answer 1

Maxima has the concept of nouns. If integrate fails for whatever reason, it returns the noun for what it would have done. Your integrates are failing and returning the noun output.

You can also specify a function as a noun with a preceding single quote. Then you can set up values and variables before calling the function with ev and telling it to evaluate nouns.

So why is your integrate failing? Probably because of infinite area under the sine function with the limits specified.

This example damps the cosine, giving a finite answer in terms of the angular frequency:

kill(all)$
assume(w>0)$
h(t):=cos(t)*%e^(-t)$

H1:'integrate(%e^(%i*w*t)*h(t),t,0,inf);
H2:'integrate(%e^(%i*w*t)*h(-t),t,minf,0);

i1:ev(H1,nouns);
i2:ev(H2,nouns);
ratsimp(i1+i2);

This will not work (and returns the noun):

H3:integrate(exp(%i*w*t)*h(t),t,minf,inf);

There are examples of use here:

http://web.csulb.edu/~woollett/mbe10fltrans.pdf

See section 10.3 and 10.3.3 in particular.

EDIT: I played a bit more...