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Digital Signal Processing Handbook P35

Filter Banks and Wavelets The methods of designing bases that we will employ draw on ideas first used in the construction of multirate filter banks. The idea of such systems is to take an input system and split it into subsequences using banks of filters. This simplest case involves splitting into just two parts using a structure such as that shown in Fig. 35.1. This technique has a long history of use in the area of subband coding: first of speech [1, 2] and more recently of images [3, 4]. In fact, the most successful image coding schemes are based on. | Cormac Herley. Wavelets and Filter Banks. 2000 CRC Press LLC. http www.engnetbase.com . Wavelets and Filter Banks Cormac Herley Hewlett Packard Laboratories 35.1 Filter Banks and Wavelets Deriving Continuous-Time Bases From Discrete-Time Ones Two-Channel Filter Banks and Wavelets Structure of Two-Channel Filter Banks Putting the Pieces Together References 35.1 Filter Banks and Wavelets The methods of designing bases that we will employ draw on ideas first used in the construction of multirate filter banks. The idea of such systems is to take an input system and split it into subsequences using banks of filters. This simplest case involves splitting into just two parts using a structure such as that shown in Fig. 35.1. This technique has a long history of use in the area of subband coding first of speech 1 2 and more recently of images 3 4 . In fact the most successful image coding schemes are based on filter bank expansions 5 6 7 . Recent texts on the subject are 8 9 10 . We will consider only the two-channel case in this section. If X z X z then the filter bank has the perfect reconstruction property. FIGURE 35.1 Maximally decimated two-channel multirate filter bank. It is easily shown that the output X z of the overall analysis synthesis system is given by X z 1 z _ _ 2 Go z Gi z Ho z Ỷ Ho -z H1 z Ỷ H1 -z X z X -z 35.1 1 2 Ho z Go z C H1 z Gi z X z 1999 by CRC Press LLC 1 C -H -z Go z C H1 -Z G1 Z U X -z . Call the above 2 x 2 matrix Hm z . This gives that the unique choice for the synthesis filters is Go z Ho z Ho z 1 2 Gi z _ _ Hi z Hi z _ 0 2 r Hi z Am z Ho z 35.2 where 1m z det Hm z . Ifwe observe that 1m z -1m -z anddefine P z 2-Hq z Hi -z 1m z Hq z Gq z it follows from 35.2 that G _ z H _ z 2 Hi z Hq -z 1m -z P -z . We can then write that the necessary and sufficient condition for perfect reconstruction 35.1 is P z C P z 2. 35.3 Since this condition plays an important role in what follows we will refer to any function having this property as valid. The .