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## Why use fftshift(fft(fftshi​ft(x))) in Matlab instead of fft(x)?

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Explain why we use fftshift(fft(fftshift(x))) in Matlab instead of fft(x)

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Explain why we use fftshift(fft(fftshift(x))) in Matlab instead of fft(x). An example is given. The example and Matlab codes are partially copied from Daniele Disco ‘s work in "A guide to the Fast Fourier Transform, 2nd Edition".

Bruce Liu

Nicolas Bigler

fancytriumph

### fancytriumph (view profile)

ifftshift is same as fftshift only for even length sequence of signal or spectrum.

fftshift(fft(ifftshift(signal))) or fftshift(ifft(ifftshift(spectrum))).

fancytriumph

### fancytriumph (view profile)

ifftshift is not same as fftshift for even length sequence of signal or spectrum.

fftshift(fft(ifftshift(signal))) or fftshift(ifft(ifftshift(spectrum))).

Hongyu Li

### Hongyu Li (view profile)

Thank you very much,Kan Wu!

Junqiu Qi

Kan Wu

### Kan Wu (view profile)

Hi Yuji,

ifftshift is to correct the bug in fftshift when dealing with the sequence with odd length (that is the number of elements in the sequence is odd). So it is safe to always use: fftshift(fft(ifftshift(sig))) or fftshift(ifft(ifftshift(spectrum))).

Yuji Zhang

### Yuji Zhang (view profile)

hi Kan,Shalin,

Thanks for the code and the discussion - nice and helpful.

A book I'm reading says it should be
fftshift(fft(fftshift(sig)))
and
ifftshift(ifft(ifftshift(spectrum)))

I'm confused. Could you explain where to use ifftshift when N=odd?

Thank you!
(The book is "Numerical Simulation of Optical Wave Propagation with examples in Matlab")

zhang lanqing

### zhang lanqing (view profile)

thanl you very much !

Shalin Mehta

### Shalin Mehta (view profile)

You're welcome. ifftshift is the same as fftshift for even length sequence, but different for odd length. So if one uses fftshift(fft(ifftshift(...))) things work well. By the way, scaling by dt and df to correct for scaling introduced by FFT algorithm is neat trick.

Kan Wu

### Kan Wu (view profile)

To Mehta, yes you r right. ifftshift is used for sequence with odd length. Thank you for the correction!

Shalin Mehta

### Shalin Mehta (view profile)

There is a problem with above recipe. It fails when you have sequence of odd length. Correct recipe is:
fftshift(fft(ifftshift(sig))) or fftshift(ifft(ifftshift(spectrum))).

A description of this can be found on my submission on fftshift, ifftshift.

To observe that above is true, run the following code with fftshift and ifftshift on inner call for computing Xfinal.
-------------------------
Bx = 50;
A = sqrt(log(2))/(2*pi*Bx);
fs = 500; %sampling frequency
dt = 1/fs; %time step
T=1; %total time window
t = -T/2:dt:T/2; %time grids
df = 1/T; %freq step
Fmax = 1/2/dt; %freq window
f=-Fmax:df:Fmax; %freq grids, not used in our examples, could be used by plot(f, X)

x = exp(-t.^2/(2*A^2));
Xan = A*sqrt(2*pi)*exp(-2*pi^2*f.^2*A^2); %X(f), analytical Fourier transform of x(t), real
Xfft = dt * fft(x); %directly using fft()
Xfftshift = dt * fft(fftshift(x)); %using fftshift() before fft()
Xfinal = dt * fftshift(fft(ifftshift(x))); %identical with analytical X(f), also note dt
subplot(211); plot(f,Xan,f,real(Xfinal),'--');
subplot(212); plot(f,imag(Xfinal));

Matt Fetterman

### Matt Fetterman (view profile)

Very interesting!

Kan Wu

### Kan Wu (view profile)

##### MATLAB Release
MATLAB 7.6 (R2008a)