CHAPTER 7

WAVELETS

**7.1 OVERVIEW**

**7.1.1 Chapter Outline**

In the last chapter we defined filter banks and started referring to “discrete wavelet transforms” in the finite-dimensional case. In this chapter we’ll make clear exactly what wavelets are and what they have to do with filter banks. This chapter contains a certain amount of unavoidable but elementary analysis. Our goal is to provide a basic understanding of wavelets, how they relate to filter banks, and how they can be useful. We state and provide examples concerning the essential truths about wavelets, and some rigorous proofs. But we provide only references for other more technical facts concerning wavelets, such as the existence of scaling functions, convergence of the cascade algorithm for computing scaling functions, and some of the properties of wavelets.

**7.1.2 Continuous from Discrete**

Suppose that by some miracle we had developed the discrete Fourier transform for sampled signals but had no notion of the underlying theory for the continuous case, that is, Fourier series. We know from Chapter 1 that any vector in C^{N} can be constructed as a superposition of the basic discrete waveforms **E**_{N, k} defined in equation (1.22). If we plot the components of **E**_{N,k} for some fixed *k* and increasing *N*, the vectors **E**_{N,k} seem to stabilize on some “mysterious” underlying function. Of course, this is none other than *e*^{2πikt} sampled at times *t* = *m*/*N* as illustrated in Figure 7.1 for *k* = 3. Even if we had no knowledge of the basis functions ...