I am working on a book with the imaginative and creative title of “Intuitive Guide to Fourier Analysis”. This is an old topic, but forever confounding in all its myriad forms.
The book will have some of my DFT, FFT material, plus Matlab code. It will also have chapters on spectral estimation which is ultimately the reason why we do the FFTs.
The DTFT we all know and love is mathematically valid only for deterministic signals. Real life signals are random, even when they have some deterministic components, such as a carrier. Per Dirichlet, we can not just take the DTFT of a random signal. That is a no-no. What we need to do is to take the auto-correlation first and then the DTFT of the Auto-Correlation Function (ACF). There are the issues of windowing and the length of the ACF. Then we have parametric and non-parametric estimation methods. Spectral estimation is a complex topic and there is little agreement on how to do it right. There are numerous complexities in understanding what the hardware is producing, how to best match it in simulation, and, most importantly, if the results are valid. I got carried away writing my last chapter on spectral estimation and even then never got to the parametric estimation which is a whole another world of mathematics, used mostly for describing finance, environmental, and biological phenomena.
I am working on this book with my son Victor as a co-author. He had just completed his masters in EE at Georgia Tech and is looking for a job in wireless or embedded systems. If anyone is looking for a smart young engineer, here is a link to his page.