fft vs nufft- scaling

I tried to do a fast fourier transform to a non-uniformly sampled data using nufft. The order of magnitude of the results was weird so I tried to compare the results of MATLAB's fft example. The results seem identical except for a scaling factor of about 550. see attahced script and attached picture of the results from the script.

where does this order of magnitude difference come from? how should I scale the nufft results? It seems pretty obvious that the nufft is the "problematic one" as the magnitude of the fft is approximatly equal to the multiplier (~0.7, 1) in the original signal.

many thanks

Nathan

Answer 1

[David Goodmanson]

Hi Nathan,

In the fft case you are dividing by N = 1500 and multiplying by 2. For nufft you aren't doing that. Hence the factor of about 750.

[Nathan Bblanc]

:) yup, that would explain that.

do you know the reason it is required to divide by N and multiply by 2?

I mean in order to acieve the original amplitude

many thanks

Nathan

[David Goodmanson]

There are two reasons. The factor of N is basic. Ignoring some off-by-one issues, suppose the signal is a complex oscillation at amplitude A, frequency n0. As a function of m = (1:N),

s(m) = A*e^(2*pi*i*n0*m/N)  m = 1...N

The fft does the calculation

g(n) = Sum{m=1,N} s(m) e^(-2*pi*i*n*m/N)  = Sum{m=1,N} A*e^(2*pi*i*(n0-n)*m/N)

If n~=n0, the sum is 0. If n=n0 you get N identical terms in the sum and g(n0) = A*N.

There is more than one way to normalize an fft, but if you want to get back the amplitude A, you need to divide by N.

The factor of 2 is boring by comparison. An fft spectrum has contributions at both positive and negative frequencies. For a real signal, those two contributions have the same size, as in

cos(2*pi*i*n0*m/N) = ( e^(+2*pi*i*n0*m/N) + e^(-2*pi*i*n0*m/N) ) / 2,

similarly for sine. So people take the absolute values of the fft at positive freqs, toss the negative frequencies, and multiply by a factor of 2 to make up for it and obtain the amplitude of the cosine wave. Good for plotting, really bad for doing anthing else.

[Nathan Bblanc]

Thanks again David