#### Matlab/Octave fftshift utility

Matlab and Octave have a simple utility called fftshift that performs this bin rotation. Consider the following example:

octave:4>
fftshift([1 2 3 4])
ans =
3  4  1  2
octave:5>

If the vector [1 2 3 4] is the output of a length 4 FFT, then the first element (1) is the dc term, and the third element (3) is the point at half the sampling rate ( ), which can be taken to be either plus or minus since they are the same point on the unit circle in the plane. Elements 2 and 4 are plus and minus , respectively. After fftshift, element (3) is first, which indicates that both Matlab and Octave regard the spectral sample at half the sampling rate as a negative frequency. The next element is 4, corresponding to frequency , followed by dc and .

Another reasonable result would be fftshift([1 2 3 4]) == [4 1 2 3], which defines half the sampling rate as a positive frequency. However, giving to the negative frequencies balances giving dc to the positive frequencies, and the number of samples on both sides is then the same. For an odd-length DFT, there is no point at , so the result

octave:4>
fftshift([1 2 3])
ans =
3  1  2
octave:5>

is the only reasonable answer, corresponding to frequencies , respectively.

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