# FFT of the average vs average of the FFT

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Gianmarco Venditti on 13 Apr 2022
Edited: Paul on 13 Apr 2022
Hello everyone,
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?

Paul on 13 Apr 2022
Edited: Paul on 13 Apr 2022
For a matrix input, fft() works down the columns. Maybe this:
rng(100);
S = rand(5);
fft(mean(S,1)) % mean for each column, fft of resulting row
ans =
2.3586 + 0.0000i -0.0250 + 0.2397i -0.1065 - 0.2300i -0.1065 + 0.2300i -0.0250 - 0.2397i
mean(fft(S.').',1) % fft of each row S, mean down the columns
ans =
2.3586 + 0.0000i -0.0250 + 0.2397i -0.1065 - 0.2300i -0.1065 + 0.2300i -0.0250 - 0.2397i

R2020b

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