0.1 Introduction to wavelets

This chapter will provide an overview of the topics to be developed in the book. Its purpose is to present the ideas, goals, and outline ofproperties for an understanding of and ability to use wavelets and wavelet transforms. The details and more careful definitions are given later inthe book.

A wave is usually defined as an oscillating function of time or space, such as a sinusoid. Fourier analysis is wave analysis. It expandssignals or functions in terms of sinusoids (or, equivalently, complex exponentials) which has proven to be extremely valuable in mathematics,science, and engineering, especially for periodic, time-invariant, or stationary phenomena. A wavelet is a “small wave", which has its energy concentrated in time to give a tool for the analysis of transient,nonstationary, or time-varying phenomena. It still has the oscillating wave-like characteristic but also has the ability to allow simultaneoustime and frequency analysis with a flexible mathematical foundation. This is illustrated in [link] with the wave (sinusoid) oscillating with equal amplitude over $-\infty \le t\le \infty$ and, therefore, having infinite energy and with the wavelet in [link] having its finite energy concentrated around a point in time.

We will take wavelets and use them in a series expansion of signals or functions much the same way a Fourier series uses the wave or sinusoidto represent a signal or function. The signals are functions of a continuous variable,which often represents time or distance. From this series expansion, we will develop a discrete-time version similar to the discrete Fouriertransform where the signal is represented by a string of numbers where the numbers may be samples of a signal, samples of another string of numbers,or inner products of a signal with some expansion set. Finally, we will briefly describe the continuous wavelet transform where both the signaland the transform are functions of continuous variables. This is analogous to the Fourier transform.

Wavelets and wavelet expansion systems

Before delving into the details of wavelets and their properties, we need to get some idea of their general characteristics and what we aregoing to do with them [link] .

What is a wavelet expansion or a wavelet transform?

A signal or function $f\left(t\right)$ can often be better analyzed, described, or processed if expressed as a linear decomposition by

where $\ell$ is an integer index for the finite or infinite sum, $a_{\ell }$ are the real-valued expansion coefficients, and ${\psi }_{\ell }\left(t\right)$ are a set of real-valued functions of $t$ called the expansion set. If the expansion [link] is unique, the set is called a basis for the class of functions that can be so expressed. If the basis is orthonormal, meaning

then the coefficients can be calculated by the inner product

One can see that substituting [link] into [link] and using [link] gives the single $a_k$ coefficient. If the basis set is not orthogonal, then a dual basis set ${\tilde{\psi }}_k\left(t\right)$ exists such that using [link] with the dual basis gives the desired coefficients. This will be developed in Chapter: A multiresolution formulation of Wavelet Systems .