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Incorrect fftshift documentation


From: Miguel Bazdresch
Subject: Incorrect fftshift documentation
Date: Thu, 16 Jun 2011 14:26:35 -0500

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.

-- 
Miguel Bazdresch


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