[Octave-bug-tracker] [bug #33581] Error in fftshift's documentation

[John W. Eaton]

Follow-up Comment #2, bug #33581 (project octave):

I think the text box ate some of your comments because of mismatched verbatim
tags.  Here is what I think is the full text, reconstructed from the email
notification I received:

I believe there could be a mistake in fftshift's documentation. It is claimed
that, after fftshift, the analog frequency vector is


f = ((1:N) - ceil(N/2)) / N / Dt


where N is the number of samples and Dt is the time resolution.
Consider the case N = 6 and Dt = 1. The digital frequencies are


> fk = (0:5)/N
ans =
  0.00000   0.16667   0.33333   0.50000   0.66667   0.83333


(The corresponding analog frequencies are the same, since Dt=1). As expected,
we see the Nyquist frequency (0.5). fftshift will arrange things so that the
FFT value corresponding to this frequency is aliased to -0.5. However, the
formula for f in fftshift's documentation produces:


> f = ((1:6) - ceil(6/2)) / 6
f =
 -0.33333  -0.16667   0.00000   0.16667   0.33333   0.50000


This is incorrect, the 0.5 term was not aliased. The error can be
verified graphically by running:


x=fft(cos(2*pi/3*(0:5)));
f=((1:6)-ceil(6/2))/6;
stem(f,abs(fftshift(x)));


where it will be clear that the spectrum is not symmetric around 0Hz. (See
also http://www.mathworks.com/matlabcentral/newsreader/view_thread/172590)

(By the way, the documented formula is correct for odd N). I believe a
correct
formula for the frequencies is


f = [ -(ceil((N-1)/2):-1:1)*df 0 (1:floor((N-1)/2))*df ];


where df=1/Dt. It may be expressed more succinctly, but I believe it is more
didactic expressed this way.


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