I recently attended the DSP Workshop sponsored by the IEEE Signal Processing Society. It focues on both signal processing and signal processing education. I spoke to a lot of the attendees. Some of the professors and instructors mentioned how important it is to teach the basics solidly first, before going into more depth. If the students don't get the fundamentals, then going further doesn't help.
With that context, I asked how they teach Fourier transforms, especially once they go from the analog domain to the digital. It turns out that most of them were unaware of some of the tools in Signal Processing Toolbox, such as dftmtx.
Here's the help.
h = help('dftmtx'); disp(h(1:744))
DFTMTX Discrete Fourier transform matrix. DFTMTX(N) is the N-by-N complex matrix of values around the unit-circle whose inner product with a column vector of length N yields the discrete Fourier transform of the vector. If X is a column vector of length N, then DFTMTX(N)*X yields the same result as FFT(X); however, FFT(X) is more efficient. The inverse discrete Fourier transform matrix is CONJ(DFTMTX(N))/N. An interesting example is D = dftmtx(4) which returns D = [1 1 1 1 1 -i -1 i 1 -1 1 -1 1 i -1 -i] which illustrates why no multiplications are necessary for 4-point DFTs. See also FFT and IFFT.
The help also mentions fft, the discrete Fourier transform, computed with a fast Fourier transform (fft) algorithm. With the information given for dftmtx, let's compare results with those of fft.
On further inspection, you may notice that there are comments in the file outlining the calculation. As an aside, the computation is actually done using the function fft in MATLAB.
Here's a brief description of the non-|fft| algorithm. Basically calculate frequencies on the unit circle and multiply by sqrt(-1) (w). Multiply these by integral steps for the number of points in the transform (w*x). And then calculate all the sin and cos terms using exp.
Explicitly, the calculation is:
f = 2*pi/n; % Angular increment. w = (0:f:2*pi-f/2).' * i; % Column. x = 0:n-1; % Row. D = exp(-w*x); % Exponentiation of outer product.
n = 1024; X = eye(n); Xf = fft(X); % fft result f = 2*pi/n; w = (0:f:2*pi-f/2).' * 1i; x = 0:n-1; Xd = exp(-w*x); % matrix representation norm(Xf-Xd)
ans = 1.6769e-011
If you know of web sites with good material for teaching discrete Fourier transforms, would you please post them here? Thanks.