0.1 Introduction to wavelets  (Page 3/5)

The operations of translation and scaling seem to be basic to many practical signals and signal-generating processes, and their use is one of thereasons that wavelets are efficient expansion functions. [link] is a pictorial representation of the translation and scaling of a single mother wavelet described in [link] . As the index $k$ changes, the location of the wavelet moves along the horizontal axis. This allows the expansion to explicitlyrepresent the location of events in time or space. As the index $j$ changes, the shape of the wavelet changes in scale. This allows a representation of detail or resolution. Note that as the scale becomes finer ( $j$ larger), the steps in time become smaller. It is both the narrower wavelet and the smaller steps that allow representation ofgreater detail or higher resolution. For clarity, only every fourth term in the translation ( $k=1,5,9,13,\cdots$ ) is shown, otherwise, the figure is a clutter.What is not illustrated here but is important is that the shape of the basic mother wavelet can also be changed. That is done during the designof the wavelet system and allows one set to well-represent a particular class of signals.

For the Fourier series and transform and for most signal expansion systems, the expansion functions (bases) are chosen, then the propertiesof the resulting transform are derived and

analyzed. For the wavelet system, the desired properties or characteristics are mathematicallyrequired, then the resulting basis functions are derived. Because these constraints do not use all the degrees of freedom, other properties can berequired to customize the wavelet system for a particular application. Once you decide on a Fourier series, the sinusoidal basis functions arecompletely set. That is not true for the wavelet. There are an infinity of very different wavelets that all satisfy the above properties. Indeed,the understanding and design of the wavelets is an important topic of this book.

Wavelet analysis is well-suited to transient signals. Fourier analysis is appropriate for periodic signals or for signals whose statisticalcharacteristics do not change with time. It is the localizing property of wavelets that allow a wavelet expansion of a transient event to be modeledwith a small number of coefficients. This turns out to be very useful in applications.

Haar scaling functions and wavelets

The multiresolution formulation needs two closely related basic functions. In addition to the wavelet $\psi \left(t\right)$ that has been discussed (but not actually defined yet), we will need another basic function called the scaling function $\varphi \left(t\right)$ . The reasons for needing this function and the details of the relations will be developed in the next chapter, but here wewill simply use it in the wavelet expansion.

The simplest possible orthogonal wavelet system is generated from the Haar scaling function and wavelet. These are shown in [link] . Using a combination of these scaling functions and wavelets allows a largeclass of signals to be represented by