Mathematica Fourier Transform not responding

Question

I am trying to do Fourier Transformation on the following function Izzn, which is essentially just a lot of trigonometric function, whose Fourier transformation are all well defined. However, when I type FourierTransform[Izzn,t,f], mathematica takes an unreasonably long time to evaluate. By unreasonable, I mean that it never gives the result but keeps evaluating. Some typical terms of Izz are posted below

 0.272441 Cos[1519.09 t]^2 + 2.69272 Cos[1519.22 t]^2 + 
 0.383058 Cos[1519.22 t]^2 - 
 0.0419625 Cos[1519.22 t] Cos[1519.22 t] + 
 0.0494935 Cos[1519.22 t]^2 + 
 0.910475 Cos[1519.22 t] Cos[1519.24 t] + 
 0.407475 Cos[1519.22 t] Cos[1519.24 t] + 2.42896 Cos[1519.24 t]^2 + 
 0.127629 Cos[1519.22 t] Cos[1519.25 t] - 
 0.0659875 Cos[1519.22 t] Cos[1519.25 t] - 
 0.351161 Cos[1519.24 t] Cos[1519.25 t] + 1.05013 Cos[1519.25 t]^2 - 
 0.5246 Cos[1519.22 t] Cos[1519.25 t] + 
 0.0886742 Cos[1519.22 t] Cos[1519.25 t] + 
 2.17117 Cos[1519.24 t] Cos[1519.25 t] - 
 2.97282 Cos[1519.25 t] Cos[1519.25 t] + 5.7118 Cos[1519.25 t]^2 - 
 1.68453 Cos[1519.22 t] Cos[1519.26 t] - 
 0.181303 Cos[1519.22 t] Cos[1519.26 t] - 
 5.27604 Cos[1519.24 t] Cos[1519.26 t] + 
 0.928094 Cos[1519.25 t] Cos[1519.26 t]

Answer 1

I have had a play with your data. I don't know what your data means but, as Daniel Huber points out, you can take the FourierTransform.

  ex = 0.272441 Cos[1519.09 t]^2 + 2.69272 Cos[1519.22 t]^2 + 
   0.383058 Cos[1519.22 t]^2 - 
   0.0419625 Cos[1519.22 t] Cos[1519.22 t] + 
   0.0494935 Cos[1519.22 t]^2 + 
   0.910475 Cos[1519.22 t] Cos[1519.24 t] + 
   0.407475 Cos[1519.22 t] Cos[1519.24 t] + 
   2.42896 Cos[1519.24 t]^2 + 
   0.127629 Cos[1519.22 t] Cos[1519.25 t] - 
   0.0659875 Cos[1519.22 t] Cos[1519.25 t] - 
   0.351161 Cos[1519.24 t] Cos[1519.25 t] + 
   1.05013 Cos[1519.25 t]^2 - 0.5246 Cos[1519.22 t] Cos[1519.25 t] + 
   0.0886742 Cos[1519.22 t] Cos[1519.25 t] + 
   2.17117 Cos[1519.24 t] Cos[1519.25 t] - 
   2.97282 Cos[1519.25 t] Cos[1519.25 t] + 5.7118 Cos[1519.25 t]^2 - 
   1.68453 Cos[1519.22 t] Cos[1519.26 t] - 
   0.181303 Cos[1519.22 t] Cos[1519.26 t] - 
   5.27604 Cos[1519.24 t] Cos[1519.26 t] + 
   0.928094 Cos[1519.25 t] Cos[1519.26 t];
FourierTransform[ex, t, f]

As you can see it is full of delta functions which you may not want (or you may). There is a post somewhere that explains why FourierTransform does this.

If this is a time series analysis I suggest you use LaplaceTransform and then convert to Fourier. Thus

lt = LaplaceTransform[ex, t, s];
ft = lt /. s -> I ω // Chop

I don't know if this helps. I had some fun working out the poles of your expression thus

b = List @@ ft;
den = Denominator[#] & /@ b;
Solve[# == 0, ω] & /@ den

So you have lots of poles at high frequencies and low.

Hope that helps.

Answer 2

f[t_] = .272441 Cos[1519.09 t]^2 + 2.69272 Cos[1519.22 t]^2 + 
   0.383058 Cos[1519.22 t]^2 - 
   0.0419625 Cos[1519.22 t] Cos[1519.22 t] + 
   0.0494935 Cos[1519.22 t]^2 + 
   0.910475 Cos[1519.22 t] Cos[1519.24 t] + 
   0.407475 Cos[1519.22 t] Cos[1519.24 t] + 
   2.42896 Cos[1519.24 t]^2 + 
   0.127629 Cos[1519.22 t] Cos[1519.25 t] - 
   0.0659875 Cos[1519.22 t] Cos[1519.25 t] - 
   0.351161 Cos[1519.24 t] Cos[1519.25 t] + 
   1.05013 Cos[1519.25 t]^2 - 0.5246 Cos[1519.22 t] Cos[1519.25 t] + 
   0.0886742 Cos[1519.22 t] Cos[1519.25 t] + 
   2.17117 Cos[1519.24 t] Cos[1519.25 t] - 
   2.97282 Cos[1519.25 t] Cos[1519.25 t] + 5.7118 Cos[1519.25 t]^2 - 
   1.68453 Cos[1519.22 t] Cos[1519.26 t] - 
   0.181303 Cos[1519.22 t] Cos[1519.26 t] - 
   5.27604 Cos[1519.24 t] Cos[1519.26 t] + 
   0.928094 Cos[1519.25 t] Cos[1519.26 t];

FourierTransform[f[t], t, w]

Is this the result you expect? And it is lightning fast.

However, you do not actually need a Fourier Transform. Every Cos term will give 2 Delta functions and every Cos^2 term 3 Delta Functions and Cos Cos 4 Delta FunctionsE.g.:

FourierTransform[ Cos[a  t], t, w]
FourierTransform[Cos[a  t]^2, t, w]
FourierTransform[Cos[a  t] Cos[b t], t, w]

Therefore, as the Fourier Transform is linear, you can get the result by replacing

Note, due to numerical accuracy with machine numbers, you do not get exactly the same result as from Fourier Transform. For this you would need to use accurate numbers.