**5**

**MULTIRESOLUTION ANALYSIS**

In the previous chapter, we described a procedure for decomposing a signal into its Haas wavelet components of varying frequency (see Theorem 4.12). The Haar wavelet scheme relied on two functions: the Haar scaling function *ϕ* and the Haar wavelet *ψ*. Both are simple to describe and lead to an easy decomposition algorithm. The drawback with the Haar decomposition algorithm is that both of these functions are discontinuous (*ϕ* at *x =* 0, 1 and *ψ* at *x =* 0, 1*/*2, 1). As a result, the Haar decomposition algorithm provides only crude approximations to a continuously varying signal (as already mentioned, Figure 4.4 does not approximate Figure 4.3 very well). What is needed is a theory similar to what has been described in the past sections but with continuous versions of our building blocks, *ϕ* and *ψ*. In this chapter, we present a framework for creating more general *ϕ* and *ψ*. The resulting theory, due to Stéphane Mallat (Mallat, 1989, 1998), is called a *multiresolution analysis.* In the sections that follow, this theory will be used together with a continuous *ϕ* and *ψ* (to be constructed later) that will improve the performance of the signal decomposition algorithm with the Haar wavelets described in a previous section.

**5.1 THE MULTIRESOLUTION FRAMEWORK**

**5.1.1 Definition**

Before we define the notion of a multiresolution analysis, we need to provide some background. Recall that the Sampling Theorem (see Theorem 2.23) approximately reconstructs a signal *f* from samples taken ...