FFT of the average vs average of the FFT
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I've a (probably naive and simple question):
I've a NxM matrix (S): N measures, with M data point
I do the average along N and than compute the FFT =>
A = FFT(mean(S) )
on the other side I do first the FFT of each of the N measure and than I average along N:
B = mean(FFT(S))
Now my A and B are different, and look like that the average and FFT are not abelian operations.
However from my memory using the linearity of the Fourier transform and the Fubini-Tonelli theorem (you can switch sum and integral if every integral is finite) A and B should be the same.
I mean: the fourier transform of the average of a set of signals should be the average of the fourier transforms of eahc signals
Am I missing something? Should I expect my A and B to be the same? And if not, why?
Thanks in advance