Meaning of each element inside a Fourier Matrix?

[silentwf]

Hi everyone,
Our digital image processing teacher is now teaching Fourier transforms. He's using MATLAB and using the FFT function to get the Fourier transforms of a signal, but he has also taught us that we can use a Fourier matrix. He finished going through how to form one, but on a homework assignment, he asks what each element inside a Fourier matrix means/represents. Also asks why the real elements are symmetric about the column X/2 + 1 (where x is even) (for example, a 32 point Fourier matrix would be symmetric about 17).

Through searching around the internet, I'm guessing that each row represents a frequency fraction of the signal, but i have no clue what each elements means. As for the symmetry problem, I'm guessing that it is because the Fourier matrix is based on Euler's equation. Since Euler's equation is periodic, it is symmetric around pi, which is exactly X/2+1.

Any thoughts on the problem or my guesses?

 

[gtoguy97]

Yes, your guesses are correct. Each element in the Fourier matrix represents a frequency fraction of the signal. The symmetry of the real elements is due to the fact that the Fourier matrix is based on Euler's equation, which is periodic and is symmetric around pi. This means that the Fourier matrix is also symmetric around X/2+1.