COMPLEX NUMBERS AND FOURIER SERIES

Complex numbers and Fourier series play vital roles in digital and signal processing. This chapter begins with an introduction to complex numbers and complex arithmetic. It discusses elementary complex arithmetic, modulus, and conjugates. The chapter then introduces Fourier series, which are some of the most useful tools in applied mathematics. In general people can always build the Fourier series of a function that is expressed as a translation of another function. The chapter considers the notions of lowpass and highpass filters in terms of Fourier series. It also explains via the convolution theorem about an important connection between convolution and Fourier series. The convolution theorem provides further motivation for the construction of discrete wavelet transformations. The convolution theorem is an important result that characterizes convolution in the Fourier domain. An immediate observation the authors make is that the complicated operation of convolution turns into simple multiplication in the Fourier domain.