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

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?

Answer 1

[Matt J]

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

[Adrian Lomeli Martin]

This worked perfectly! Thank you!

Answer 2

[Bjorn Gustavsson]

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...)

[Adrian Lomeli Martin]

Thank you! I checked the function you mentioned and it'll surely be useful down the line. Thank you again for taking the time to write this down.