# CHAPTER 4

# COMPLEX NUMBERS AND FOURIER SERIES

Complex numbers and Fourier series play vital roles in digital and signal processing. In this chapter we introduce the complex plane and complex numbers and discuss arithmetic with complex numbers. In Section 4.2 we introduce the complex exponential via Euler’s famous formula.

As you progress through the book, you will learn why Fourier series are so important. The idea is similar to that of MacLaurin series — we rewrite the given function in terms of basic elements. While the family of functions *x*^{n}, *n* = 0, 1, … serve as building blocks for MacLaurin series, the complex exponentials *e*^{ikω}, *k* are used to construct Fourier series. Fourier series coefficients hold much information about the transformations we develop in this book. As you will see in Chapters 8 and 10, once we arrive at a set of desirable conditions for constructing our transformations, it is natural to reformulate the conditions in terms of Fourier series! Thus, a mastery of the material in Section 4.3 is essential for understanding transformation construction that follows later in the text.

## 4.1 THE COMPLEX PLANE AND ARITHMETIC

In this section we review complex numbers and some of their basic properties. We will also discuss elementary complex arithmetic, modulus, and conjugates. ...