Oversampled Filter Banks

Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in t(2)(Z), These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a. necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given, Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIE and give a parameterization of a class of filter banks satisfying this property, Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuous-time transforms and allow for very flexible design. We investigate relations of these filter banks to continuous-time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two-channel filter banks that yield highly regular wavelets with a given number of vanishing moments.