0.1 Introduction to wavelets  (Page 5/5)

While some of these details may not be clear at this point, they should point to the issues that are important to both theory and application and givereasons for the detailed development that follows in this and other books.

The discrete wavelet transform

This two-variable set of basis functions is used in a way similar to the short-time Fourier transform, the Gabor transform, or the Wignerdistribution for time-frequency analysis [link] , [link] , [link] . Our goal is to generate a set of expansion functions such that any signal in $L^2\left(\mathbf{R}\right)$ (the space of square integrable functions) can be represented by the series

or, using [link] , as

where the two-dimensional set of coefficients $a_{j,k}$ is called the discrete wavelet transform (DWT) of $f\left(t\right)$ . A more specificform indicating how the $a_{j,k}$ 's are calculated can be written using inner products as

if the ${\psi }_{j,k}\left(t\right)$ form an orthonormal basis Bases and tight frames are defined in Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases. for the space of signals of interest [link] . The inner product is usually defined as

The goal of most expansions of a function or signal is to have the coefficients of the expansion $a_{j,k}$ give more useful information about the signal than is directly obvious from the signal itself. A second goalis to have most of the coefficients be zero or very small. This is what is called a sparse representation and is extremely important in applications for statistical estimation and detection, data compression,nonlinear noise reduction, and fast algorithms.

Although this expansion is called the discrete wavelet transform (DWT), it probably should be called awavelet series since it is a series expansion which maps a function of a continuous variable into a sequence of coefficients much the same way theFourier series does. However, that is not the convention.

This wavelet series expansion is in terms of two indices, the time translation $k$ and the scaling index $j$ . For the Fourier series, there are only two possible values of $k$ , zero and $\pi /2$ , which give the sine terms and the cosine terms. The values $j$ give the frequency harmonics. In other words, the Fourier series is also a two-dimensional expansion, butthat is not seen in the exponential form and generally not noticed in the trigonometric form.

The DWT of a signal is somewhat difficult to illustrate because it is a function of two variables or indices, but wewill show the DWT of a simple pulse in [link] to illustrate the localization of the transform. Other displays will be developed in thenext chapter.

The discrete-time and continuous wavelet transforms

If the signal is itself a sequence of numbers, perhaps samples of some function of a continuous variable or perhaps a set of inner products, theexpansion of that signal is called a discrete-time wavelet transform (DTWT). It maps a sequence of numbers into a sequence of numbers much thesame way the discrete Fourier transform (DFT) does. It does not, however, require the signal to be finite in duration or periodic as the DFT does.To be consistent with Fourier terminology, it probably should be called the discrete-time wavelet series, but this is not the convention. If thediscrete-time signal is finite in length, the transform can be represented by a finite matrix. This formulation of a series expansion of adiscrete-time signal is what filter bank methods accomplish [link] , [link] and is developed in Chapter: Filter Banks and Transmultiplexers of this book.

If the signal is a function of a continuous variable and a transform that is a function of two continuous variables is desired, the continuouswavelet transform (CWT) can be defined by

with an inverse transform of

where $w\left(t\right)$ is the basic wavelet and $a,b\in \mathbf{R}$ are real continuous variables. Admissibility conditions for the wavelet $w\left(t\right)$ to support this invertible transform is discussed by Daubechies [link] , Heil and Walnut [link] , and others and is briefly developed in Section: Discrete Multiresolution Analysis, the Discrete-Time Wavelet of this book. It is analogous to the Fourier transform or Fourier integral.

Exercises and experiments

As the ideas about wavelets and wavelet transforms are developed in this book, it will be very helpful to experiment using the Matlab programs inthe appendix of this book or in the Matlab Toolbox [link] . An effort has beenmade to use the same notation in the programs in Appendix C as is used in the formulas in the book so that going over the programs can help inunderstanding the theory and vice versa.

This chapter

This chapter has tried to set the stage for a careful introduction to both the theory and use of wavelets and wavelet transforms. We havepresented the most basic characteristics of wavelets and tried to give a feeling of how and why they work in order to motivate and givedirection and structure to the following material.

The next chapter will present the idea of multiresolution, out of which willdevelop the scaling function as well as the wavelet. This is followed by a discussion of how to calculate the wavelet expansion coefficientsusing filter banks from digital signal processing. Next, a more detailed development of the theory and properties of scaling functions, wavelets,and wavelet transforms is given followed by a chapter on the design of wavelet systems. Chapter: Filter Banks and Transmultiplexers gives a detailed development of wavelet theory in terms of filter banks.

The earlier part of the book carefully develops the basic wavelet system and the later part develops several importantgeneralizations, but in a less detailed form.