[Numpy-discussion] FFT definition

Warren Focke focke@slac.stanford....
Mon Feb 5 09:27:08 CST 2007


The frequencies produced by the two recipies are not the same.  But the
DFT is periodic in both frequency and time.  So whether you think that the
number in bin in n/2 should correspond to frequency n/2 or -n/2, it's the
same number.

w

On Mon, 5 Feb 2007, Hanno Klemm wrote:

>
> Hi there,
>
> I have a question regarding the definitions surrounding FFTs. The help
> to numpy.fft.fft says:
>
> >>> help(N.fft.fft)
> Help on function fft in module numpy.fft.fftpack:
>
> fft(a, n=None, axis=-1)
>     fft(a, n=None, axis=-1)
>
>     Will return the n point discrete Fourier transform of a. n
> defaults to the
>     length of a. If n is larger than a, then a will be zero-padded to
> make up
>     the difference. If n is smaller than a, the first n items in a will be
>     used.
>
>     The packing of the result is "standard": If A = fft(a, n), then A[0]
>     contains the zero-frequency term, A[1:n/2+1] contains the
>     positive-frequency terms, and A[n/2+1:] contains the
> negative-frequency
>     terms, in order of decreasingly negative frequency. So for an 8-point
>     transform, the frequencies of the result are [ 0, 1, 2, 3, 4, -3,
> -2, -1].
>
>     This is most efficient for n a power of two. This also stores a
> cache of
>     working memory for different sizes of fft's, so you could
> theoretically
>     run into memory problems if you call this too many times with too many
>     different n's.
>
> >>>
>
> However, the help to numpy.fft.helper.fftfreq says:
>
> >>> help(N.fft.helper.fftfreq)
> Help on function fftfreq in module numpy.fft.helper:
>
> fftfreq(n, d=1.0)
>     fftfreq(n, d=1.0) -> f
>
>     DFT sample frequencies
>
>     The returned float array contains the frequency bins in
>     cycles/unit (with zero at the start) given a window length n and a
>     sample spacing d:
>
>       f = [0,1,...,n/2-1,-n/2,...,-1]/(d*n)         if n is even
>       f = [0,1,...,(n-1)/2,-(n-1)/2,...,-1]/(d*n)   if n is odd
>
> >>>
>
> So one claims, that the packing goes from [0,1,...,n/2,-n/2+1,..,-1]
> (fft) and the other one claims the frequencies go from
> [0,1,...,n/2-1,-n/2,...-1]
>
> Is this inconsistent or am I missing something here?
>
> Hanno
>
> --
> Hanno Klemm
> klemm@phys.ethz.ch
>
>
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