5.1 Discrete Fourier Transform

dft(X)

Replaces the columns of a dense complex matrix with their discrete Fourier transforms: if X has n rows,

        n∑-1      √--
X [k,:] :=    e-2πjk -1∕nX[j,:],   k = 0,...,n - 1.
        j=0

idft(X)

Replaces the columns of a dense complex matrix with their inverse discrete Fourier transforms: if X has n rows,

           n∑-1    √--
X [k,:] := 1   e2πjk -1∕nX [j,:],    k = 0,...,n - 1.
         n j=0

The separable discrete two dimensional Fourier transform first computes the corresponding one dimensional tranform along the columns of the matrix, followed by the one dimensional transform along the rows of the matrix.

dft2(X)

Replaces a dense complex matrix with the two dimensional discrete Fourier transform.

idft2(X)

Replaces a dense complex matrix with the inverse two dimensional discrete Fourier transform.