Fourier Series and Periodic Functions

Abstract

This is the introductory chapter to spectral methods. The main theme is decomposing a rather arbitrary periodic signal in terms of sines/cosines. The result is the Fourier series and the main task at hand is to figure the expansion coefficients. Those are obtained by integrating the target function against the sine/cosine and give all of DC, sine, and cosine series coefficients. We apply the process on a multitude of signals and stress all the way the visual aspect of the analysis to convince the reader this method works. We see the signal gradually assume the desired shape by including more harmonics. We also learn about the spectrum of the signal which is a plot of the Fourier series coefficients versus frequency. As a reference case we take the periodic pulse and examine its spectrum as a function of period, pulse width, and time elongation and see the corresponding effect on the spectrum. Finally we touch on patching signals and superposition.