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.
Fourier series are discussed in Section 8.2. The idea is similar to that of Maclaurin series – we rewrite the given function in terms of basic elements. While the family of functions xn, n = 0, 1, … serve as building blocks for Maclaurin series, complex exponentials eikω, k ∈ ℤ are used to construct Fourier series. Also included in Section 8.2 are several useful properties obeyed by Fourier series. We connect Fourier series with convolution and filters in Section 8.3.
We have constructed (bi)orthogonal wavelet filters/filter pairs in Chapters 4, 5 and 7. The constructions were somewhat ad hoc and in the case of the biorthogonal filter pairs, limiting. Fourier series opens up a completely new and systematic way to look at wavelet filter construction. It is important you master the material in this chapter before using Fourier series to construct (bi)orthogonal wavelet filters/filter pairs in Chapter 9.
8.1 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. Let’s start with the imaginary number
It immediately follows that
In Problem 8.2 you will ...