Xjyuk

The function defined below has three frequencies 2, 4 and 10. How to use the Fourier transform (FFT) to show these frequencies?

 myfun[t_] = 
 1/(2 Sqrt[
 2]) ([Sqrt]Abs[
 3 + Cos[4 t] - 
 4 Cos[2 t] (-1 + 
 E^(-0.018` t) (Cos[9.99999594999918` t] + 
 0.0009000003645002213` Sin[9.99999594999918` t])^2) + 
 4 E^(-0.018` t) (Cos[9.99999594999918` t] + 
 0.0009000003645002213` Sin[9.99999594999918` t])^2])

Edit: The suggested duplicate deals with a simple function. It really doesn't work for this complicated scenario. Would appreciate if someone gets it done.

First lets plot your function to see what it looks like.

Plot[myfun[t], t, 0, 100]

This is only between 0 to 100 seconds. What we see is some very high frequencies but also a constant frequency which we can estimate at about 0.3 Hz. If we plot at later times

Plot[myfun[t], t, 500, 600]

We can see that the starting high frequencies have died away and the frequency at about 0.3 Hz remains.

Now we can do a Fourier analysis. As you have some high frequencies we need to sample at a very fast sample rate. I have guessed at the sample rate but hope it is good enough.

sr = 100; (* Sample rate *)
data = Table[myfun[t], t, 0, 1000, 1/sr];
nn = Length@data

Plotting again shows that I seem to have captured the high frequency data

ListLinePlot[data[[1 ;; Round[100 sr]]]]

You should probably check by looking at a shorter time interval.

Now for the Fourier analysis.

ft = Fourier[data, FourierParameters -> -1, -1];
freqs = Table[f, f, 0, sr (nn - 1)/nn, sr/nn];

The second line of code is to make the frequencies that correspond to the data in the Fourier transform. See here for details.

ListLogPlot[Transpose[freqs, Abs[ft]][[1 ;; 10000]], 
 PlotRange -> All, Joined -> True]

We can see a dominant peak at about 0.3 Hz and lots of other peaks that make up your high frequencies.

Hope that helps.

Periodogram and Spectrogram are also useful:

data = Table[myfun[t], t, 0, 100000, .1];
Periodogram[data, SampleRate -> 10, PlotRange -> All, Frame -> True, 
 FrameLabel -> "Hz", "dB"]

Spectrogram[data, SampleRate -> 10, PlotRange -> 0, .5, 
 FrameLabel -> "Time", "Hz"]