0.2 A multiresolution formulation of wavelet systems

Both the mathematics and the practical interpretations of wavelets seem to be best served by using the concept of resolution [link] , [link] , [link] , [link] to define the effects of changing scale. To do this, we will start with a scaling function $\phi \left(t\right)$ rather than directly with the wavelet $\psi \left(t\right)$ . After the scaling function is defined from the concept of resolution, the wavelet functionswill be derived from it. This chapter will give a rather intuitive development of these ideas, which will be followed by more rigorousarguments in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients .

This multiresolution formulation is obviously designed to represent signals where a single event is decomposed into finer and finer detail, but itturns out also to be valuable in representing signals where a time-frequency or time-scale description isdesired even if no concept of resolution is needed. However, there are other cases where multiresolution is not appropriate, such as for theshort-time Fourier transform or Gabor transform or for local sine or cosine bases or lapped orthogonal transforms, which are all discussedbriefly later in this book.

Signal spaces

In order to talk about the collection of functions or signals that can be represented by a sum of scaling functions and/or wavelets, we need someideas and terminology from functional analysis. If these concepts are not familiar to you or the information in this section is not sufficient, youmay want to skip ahead and read Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients or [link] .

A function space is a linear vector space (finite or infinite dimensional) where the vectors are functions, the scalars are real numbers(sometime complex numbers), and scalar multiplication and vector addition are similar to that done in [link] . The inner product is a scalar $a$ obtained from two vectors, $f\left(t\right)$ and $g\left(t\right)$ , by an integral. It is denoted

with the range of integration depending on the signal class being considered. This inner product defines a norm or “length" of a vector which is denoted and defined by

which is a simple generalization of the geometric operations and definitions in three-dimensional Euclidean space. Two signals (vectors)with non-zero norms are called orthogonal if their inner product iszero. For example, with the Fourier series, we see that $sin\left(t\right)$ is orthogonal to $sin\left(2t\right)$ .

A space that is particularly important in signal processing is called $L^2\left(\mathbf{R}\right)$ . This is the space of all functions $f\left(t\right)$ with a well defined integral of the square of the modulus of the function. The “L" signifies aLebesque integral, the “2" denotes the integral of the square of the modulus of the function, and $\mathbf{R}$ states that the independent variable of integration $t$ is a number over the whole real line. For a function $g\left(t\right)$ to be a member of that space is denoted: $g\in L^2\left(\mathbf{R}\right)$ or simply $g\in L^2$ .

Although most of the definitions and derivations are in terms of signals that are in $L^2$ , many of the results hold for larger classes of signals. For example, polynomials are not in $L^2$ but can be expanded over any finite domain by most wavelet systems.