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 ($f_s/2$ ), which can be taken to be either plus or minus $f_s/2$ since they are the same point on the unit circle in the $z$ plane. Elements 2 and 4 are plus and minus $f_s/4$ , 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 $-f_s/4$ , followed by dc and $f_s/4$ .

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 $f_s/2$ 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 $\pm f_s/2$ , so the result

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

is the only reasonable answer, corresponding to frequencies $-f_s/3, 0, f_s/3$ , respectively.