Introduction

In subsequent chapters we will make substantial use of some basic results from the Fourier theoryof sequences and – to a lesser extent – functions, and we will find that filters playa central role in the application of wavelets. This chapter is intended as a self-contained guide tosome key results from Fourier and filtering theory. Our selection of material is intentionallylimited to just what we will use later on. For a more thorough discussion employing the samenotation and conventions adopted here, see Percival and Walden (1993). We also recommend Briggs andHenson (1995) and Hamming (1989) as complementary sources for further study.

Readers who have extensive experience with Fourier analysis and filters can just quickly scanthis chapter to become familiar with our notation and conventions. We encourage others to study thematerial carefully and to work through as many of the embedded exercises as possible (answers areprovided in the appendix). It is particularly important that readers understand the concept ofperiodized filters presented in Section 2.6 since we use this idea repeatedly in Chapters 4 and5.

Complex Variables and Complex Exponentials

The most elegant version of Fourier theory for sequences and functions involves the use ofcomplex variables, so here we review a few key concepts regarding them (see, for example, Brown andChurchill, 1995, for a thorough treatment). Let i ≡ √–1 sothat i2 = –1 (throughout the book, we take‘≡’ to mean ‘equal by definition’).