Parseval's Theorem Doesn't Work With Ifft

I have a temporal signal and I calculate its Fourier Transform to get the frequencial signal. According to Parseval's theorem, the two signals have the same energy. I successfully

Solution 1:

The numpy FFT procedures actually and in contrast to other software do adjust for the sequence length, so that you get

nf.ifft(nf.fft(gx)) == gx

up to some floating point error. If your dx and dk are computed the usual way, then dk*dx=(2*pi)/N which only works for unadjusted FFT routines.

You can test the behavior of numpy.fft using

In [20]: sum(abs(gx)**2.0)
Out[20]: 35.226587122763036

In [21]: gk = nf.fft(gx)

In [22]: sum(abs(gk)**2.0)
Out[22]: 35226.587122763049

In [23]: sum(abs(nf.ifft(gk))**2.0)
Out[23]: 35.226587122763014

which tells us that the fft is the usual unadjusted transform and ifft divides the result by sequence length N=num. The typical ifft can be emulated by

gxx = (nf.fft(gk.conj())).conj()

then you get that

gx == gxx/1000

up to floating point errors. Or you can reverse the adjustment using

#inverse FTgx_ = nf.ifft(nf.ifftshift(gkk))*(num*dk)/(2 * np.pi)**0.5