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,

\begin{displaymath}
X[k,:] := \sum_{j=0}^{n-1} e^{-2\pi j k \sqrt{-1}/n} X[j,:],
\qquad k=0,\ldots,n-1.
\end{displaymath}

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

\begin{displaymath}
X[k,:] :=
\frac{1}{n} \sum_{j=0}^{n-1} e^{2\pi j k \sqrt{-1}/n} X[j,:],
\qquad k=0,\ldots,n-1.
\end{displaymath}