Orthogonal time-varying filter banks and wavelets

The construction of time-varying orthogonal filter banks is considered. It is shown that implementing an orthogonal finite impulse response filter bank over a finite signal segment involves finding a set of orthogonal boundary filters, and that by carrying out a Gram-Schmidt orthogonalization procedure boundary filters are generated that necessarily remain localized in the region of the boundary. A complete constructive characterization of such boundaries is given for two-channel finite impulse response filter banks. These boundary constructions allow changing the topology of orthogonal subband trees at will, by growing or pruning branches at any time. The boundary filter case can be further generalized to give overlapping transition filters when changing between orthogonal structures. If the time-varying filter banks are used in an iterated scheme, they converge to continuous-time bases, much as in the non-time-varying case