MATLAB: Does the FFT2 function not return a principal diagonal matrix when applied to a circulant matrix in the Signal Processing Toolbox

circulantsdiagonalfft2matrixprincipalSignal Processing Toolbox

The FFT2 function, when applied to a circulant matrix, is supposed to give a principal diagonal matrix.
A circulants matrix is something like
 x =
      4     1     1
      1     4     1
      1     1     4
and the process of two dimensional fft on it should give:
 FFT2(x)=    18     0     0
                    0     9     0
                    0     0     9
However, when I use the FFT2 function in MATLAB, the result is:
 fft2(x)=    18     0     0
    0     0     9
    0     9     0
This is not a principal diagonal matrix.

Best Answer

FFT2 applies the FFT to the columns, then the rows. This is mentioned in the documentation for this function. You can find a copy of it online at:
http://www.mathworks.com/access/helpdesk/help/techdoc/ref/fft2.shtml
Thus
   fft2(x) =  fft(fft(x).').'
Note that the transpose is a NON-conjugate transpose.
The theory that you are referring to is the conjugate transpose. Thus
 Let 
   x = [ 4     1     1
         1     4     1
         1     1     4 ];
then
    fft(fft(x)')' 
 ans =
     18     0     0
      0     9     0
      0     0     9
which is a principal diagonal matrix.
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