Complex Fourier Series

Abstract

This chapter paves the way for the Fourier transform which most often utilizes complex exponentials for basis functions. It is also the starting point for using complex numbers and analysis in the text. We show in this chapter how to decompose an arbitrary periodic signal in terms of summation of weighted complex exponentials of the form $f(t) = \sum _n a_n e^{j \omega _n t}$. The expansion coefficients are calculated by integrating the target function against the complex exponentials, for each frequency. We stress that even though the used basis functions are complex they are able to reproduce real functions simply by isolating the real or imaginary parts of the complex exponentials. We also derive the relation between the complex Fourier coefficients to those of the real Fourier series. We demonstrate the process generating the complex Fourier series on a few examples, including the periodic pulse and dwell on the meaning of negative frequencies. We also reexamine the signal spectrum and expand it now to include negative frequencies. The complex Fourier series is the starting point for the complex Fourier transform.