In practice, it is more efficient to compute the discrete Fourier transform with the FFT than with the DFT matrix. The FFT also uses less memory. The two procedures give the same result.
x = 1:256;
y1 = fft(x);
n = length(x);
y2 = x*dftmtx(n);
norm(y1-y2)
A discrete Fourier transform matrix is a complex
matrix whose matrix product with a vector computes the discrete Fourier transform of the
vector. dftmtx takes the FFT of the identity matrix to generate the
transform matrix.
For a column vector
x,
y = dftmtx(n)*x
is
the same as y = fft(x,n). The inverse discrete Fourier transform matrix
is
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