On adaptive wavelet packets

An algorithm is presented for fast implementation of time-varying wavelet packets maintaining perfect reconstruction throughout transitions. It is based on filter banks known as the extended lapped transforms in an association following the paths of a binary-tree. Methods for pruning or expanding the branches of the tree are presented and a discussion on adaptation issues is carried. Recently, concepts such as time-frequency representation and wavelet packets have gained a great deal of attention in the field of digital signal processing [l] because of the ability to trade time and frequency resolutions, maintaining orthogonality, and perfect reconstruction (PR). It is also well known that orthogonal wavelet packets can be designed by hierarchical association of PR paraunitary filter banks [2]. In these, synthesis filters are time reversed versions of the analysis ones. In the 2-band case, the two impulse responses of the analysis and synthesis and K is the overlapping factor. In this paper we will concentrate on binary trees. However, this concept can be easily extended to m-ary trees. The extended lapped transform (ELT) is certainly a good way of implementing hierarchical associations [3,4], since it (i) is paraunitary; (ii) has a fast implementation algorithm; (iii) can be defined for several overlapping factors and number of channels; (iv) is reasonably regular (mainly if we chose the parametric design [3]). Figure 1. The binary tree notation. a)2-band filter bank; b)its equivalent representation in a tree; c) labelling of nodes in the tree. If the same paraunitary filter bank is used as the analysis cell for each stage, it is sufficient to describe the paths of the tree to completely describe the whole analysis-synthesis system. With the aid of Fig. 1 we present a more convenient notation. In Fig. l a we see a 2-band maximally decimated filter bank with its low-and high-pass filters, as well a s subsamplers. This system will be represented here by tree nodes and branches (Fig. l b) , where the signal flows in the nodes and the branches represent filters and decimators. We label the nodes in the tree as ~ i j , where i is the stage number and j is the number of the node in that stage, just as indicated in Fig. IC, for 3-stage binary tree. The familiar parenthood notation is used, so that node qij is parent of nodes qi+1,2j and q i + 1 , 2 j + l 1 …