MATLAB Central Newsreader - Re: Strange FFT Behavior in MATLAB

Re: Strange FFT Behavior in MATLAB

robert bristow-johnson — Thu, 20 Sep 2007 16:56:05 -0400

On Sep 20, 8:01 am, "patrickjennings" <patrick.t.jenni...@gmail.com>
wrote:
> I think we can all agree that FFT { x*(n) } = X*(N-k),

actually the folks at The Math Works do not agree. as fatnbafan said,
MATLAB is hard-wired or hard-coded so that the indices of all arrays
begin with 1, not 0 as it should for the DFT or FFT.

so in MATLAB, if N=length(x); y = conj(x); X = fft(x); Y = =
fft(y); then

Y(k+1) = conj( X(mod(N-k+1, N)) ); % for 0 <= k < N

or stated so elegantly that it's amazing we all don't just sing the
praises of MATLAB,

Y(k) = conj( X(mod(N-k+2, N) ); % for 1 <= k <= N

gee, isn't that elegant?

r b-j

> or the FFT of the
> conj of x(n) is the conj of the reversed version of the FFT of x(n).
>
> But in MATLAB if a= [1+2j 3+4j 5+6j 7+8j] then
>
> fft(conj(a)) = [16+20j -8+0j -4-4j 0-8j]
>
> and
>
> conj(fliplr(fft(a))) = [-8+0j -4-4j 0-8j 16+20j]
>
> Any ideas? Another engineer and I spent most of a day looking at the
> model before finding the fundamental problem.
>
> Cheers
>
> /Patrick

Re: Strange FFT Behavior in MATLAB

Andy Robb — Sat, 16 Aug 2008 03:40:20 -0400

When analyzing (real) time series data, the complex
frequency spectrum is symmetric in the real part and
anti-symmetric in the imaginary part.

Way back in Matlab 3, when memory and performance were at a
premium, I formulated MEX files that used this symmetry to
reduce the data size.

It is possible to arrange a real time series into a complex
series of half the length and post process the fft output to
get the spectrum. Or the other way around - arranging half
the spectrum into a real series. For lengths of a power of
2, Matlab 3 used a fast Cooley Tukey algorithm which was
most efficient going from real to complex.

For even lengths that are not a power of 2, Matlab 3 only
used a complex FFT. In those cases, I took the real series
(either time series or half the spectrum), and arranged it
as a complex series of half the length. Again, I could take
half a frequency spectrum and rearrange it as a real series
then rearrange it again as a complex series of half the length.

It sounds more complicated than it was and it ran several
times faster.