Re-understanding of the Fourier Expansion and Signal Expansion
Why do we make Fourier transform to signals? The first thing that came to my mind for this question is that the Fourier transform has many nice features, it can transfer complex convolution in the time domain into a simple product in the frequency domain, it can simplify the filter design in frequency domain, its linearity, symmetry property etc. But those nice features can’t explain well why we need it but not other orthogonal expansions. While I’m looking for the answer to this question, my understanding of Fourier expansion gets refreshed and also extended to some applications of signal expansion in other fields.