How can I obtain the Fourier Transformation as a product of matrices?

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Adrian Lomeli Martin 2021-9-15

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评论： Adrian Lomeli Martin 2021-9-15

Hello everyone!

Before I being, I'd like to say I already know about the fft() command. My goal is to compare the resulting FT matrix obtained by it with the one obtained via Matrix Multiplication. I have the following code:

N = [0:255];

X = (2*cos(((15.8*pi)/128)*N)) + (3*sin(((27.3*pi)/128)*N));

fX = fft(X);

How can I write the values for 'X' as a vector and the other vector 'x' so that I may multiply them to get the Fourier Transformation?

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Answer 1

Matt J 2021-9-15

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编辑：Matt J 2021-9-15

在 MATLAB Online 中打开

N = (0:255).';
X = (2*cos(((15.8*pi)/128)*N)) + (3*sin(((27.3*pi)/128)*N));
F=fft( eye(numel(X)), [],1);
fX = fft(X); %function implementation
FX = F*X; %matrix implementation
Difference = norm(fX-FX)
Difference = 4.8302e-13

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Adrian Lomeli Martin 2021-9-15

This worked perfectly! Thank you!

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Answer 2

Bjorn Gustavsson 2021-9-15

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Have a look at the help and documentation for dftmtx. That function should return the discrete Fourier-transform matrix and give you the matrix you need. If you need to figure out how the Fourier-transform-component is an inner product between a function and a basis-function, you might gain some understanding by looking at that matrix (or its real and imaginary components) and have a think about what the matrix-multiplication does. Also just write down the integral for that Fourier-component beside the inner-product and look at how they are "similar enough". (Caveat: written by a physicist not a mathematician...)