TABLE OF CONTENTS Wavelet transforms are a relatively recent development in functional analysis that have attracted a great deal of attention from the signal processing community (Daubechies, 1991). The wavelet transform of a function belonging to 2{ ?}, the space of the square integrable functions, is its decomposition in a base formed by expansions, compressions, and translations of a single mother function ?( t), called a wavelet.
The applications of wavelet transforms range from quantum physics to signal coding. It can be shown that for digital signals the wavelet transform is a special case of critically decimated filter banks (Vetterli & Herley, 1992). In fact, its numerical implementation relies heavily on that approach. In what follows, we give a brief introduction to wavelet transforms, emphasizing their relation to filter banks. In the literature there is plenty of good material analyzing wavelet transforms from different points of view. Examples are Daubechies (1991), Vetterli & Kova?evi? (1995), Strang & Nguyen (1996), and Mallat (1998).
The cascading of 2-band filter banks can produce many different kinds of critically decimated decompositions. For example, one can make a 2 k-band uniform decomposition, as depicted in Figure 9.47a, for k = 3. Another common type of hierarchical decomposition is the octave-band decomposition, in which only the lowpass band is further decomposed. In Figure 9.47b, one can see a 3-stage octave-band decomposition. In these figures, the synthesis bank is not drawn, because it is entirely analogous...
