How to build a Fourier matrix?

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Valentin
Valentin 2014-1-10
回答： Robert Kim 2017-6-28
How would I build a Fourier matrix in Matlab? Intuitively what is this matrix telling me and is there a difference between a Fourier matrix for a vector signal (x) vs. a Fourier matrix for an image (I) signal? My intuition is that it doesn't matter and that the matrix simply holds the frequencies I wish to capture. For example:
y = Fx,
where F is the Fourier matrix, x is a sparse vector and y are my signals. x could stand for my vectorized image, say x = I(:)
so far I have
F = dftmtx(numel(I))
Is this right?
p.s. Are there different names used to refer to this Fourier Matrix?
Thank you

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Bjorn Gustavsson
Bjorn Gustavsson 2014-1-10
Yes, it is right. F would be a 1-D Fourier matrix. A Fourier coefficient is the inner product between the signal and the corresponding Fourier waveform:
fftX_n = exp(-1i*pi*n*(0:length(f))/length(f))*f(:)
(if I got the parenthesises and signs right) So you can simply stack all of the Fourier base-functions, line by line into a Fourier matrix and calculate all the Fourier components in one matrix multiplication. This is for a 1-D dft, to calculate a 2-D you'd either have to do it step-wise column by column and then row by row - or use the powerful ':' operator matlab has and build a larger 2-D Fourier matrix. But why bother, there are fft2 and ifft2 functions?
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Bjorn Gustavsson
Bjorn Gustavsson 2014-1-11
编辑：Matt J 2014-1-11
Not exactly the same. You have to take into account that the image is 2-D. This is the 2-D dft-matrix for 8-by-8 images:
function DFTMTX2D = TwoDdftmtx(d)
% only for 8-by-8 d
DFTMTX = dftmtx(8);
DFTMTX2D = zeros(8^2*[1,1]);
for i1 = 1:8,
for i2 = 1:8,
DFTMTX2D(8*(i1-1)+[1:8],8*(i2-1)+[1:8]) = DFTMTX*DFTMTX(i1,i2);
end
end
Then you get the DFT like this:
DFTMTX2D = TwoDdftmtx(d);
ftD = 0*d;
ftD(:) = DFTMTX2D*d(:);
Matt J
Matt J 2014-1-11
If F is a 1D dft matrix then the 2D dft matrix can also be obtained as
DFTMTX2D = kron(F,F);

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Matt J
Matt J 2014-1-11
编辑：Matt J 2014-1-11
If the whole motivation for this thread is that MATLAB's fft2() can't handle sparse input images, then I would recommend implementing it this way for an NxN image
F=fft(eye(N)); %or dftmtx if you have the correct Toolbox
fft2d = F*Image*F.';
As I commented here, you can also do this as
fft2d = kron(F,F)*Image(:);
but this will be much slower and more memory-consuming.
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Valentin
Valentin 2014-1-11
@Matt Thanks for your posts. I am just reading through the answers. I'm learning about compressed sensing. My motivation is to do a test that measures an "incoherence property" between a sensing matrix (Phi) and a representation matrix (Psi). The literature states one option for the representation matrix is the Fourier matrix. My data is a 2D image that I subsample randomly and then I reconstruct using compressed sensing (and matrix completion)
SwishSwisher
SwishSwisher 2015-2-9
This is very helpful Matt! Thank you.

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Robert Kim
Robert Kim 2017-6-28
You can get both cos and sin part of Fourier basis. If you need the complex number, just remove the real, and imag parts. But if really need to use this for FFT purpose, just try matlab function "fft".
sigLen = 2;
AC_t = 0.001:0.001:sigLen;
AC = ones(sigLen/0.001, 1);
for n = 1:4
y = exp(2^n*i*pi/sigLen*AC_t)';
AC = [AC real(y) imag(y)];
end
figure
plot(AC_t, AC)
ylim([-1.2 1.2])

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