Perfect Reconstruction in Non-biorthonormal Filter Banks using Vector Space Approach

A formulation is proposed for construction of PRFB from a given non-PRFB and is described using vector space framework for filter banks. To construct PRFB, a transmultiplexer (TMUX) structure is inserted into the subband such that the synthesis and analysis parts of the TMUX are biorthonormal to analysis and synthesis bank of the given filter bank. The TMUX is a represented by transformation matrix. In addition to PR, in this paper, another objective is to study and exploit the properties of transformation matrix corresponding to non-PR TMUX. The transformation matrix is portioned into distinct subblocks. In case of uniform filter bank (UFB) it is shown that each subblock of transformation matrix has convolution matrix structure. Whereas in case of nonuniform filter bank (NUFB) it is shown that each of these subblock has a structure consisting of interspersed convolution matrices. Implementation of these matrices using discrete time FIR or IIR filters are also shown in this paper. It is also shown that implementation of convolution matrices involve linear time invariant filters whereas interspersed convolution matrices involve the time varying filters. During the implementation of transformation matrix it is also found that some of the blocks can be derived by using implemented blocks. By inserting one or more TMUXs in the subbands and merging with subchannels of UFB we can obtained NUFB from given UFB. There are various application of NUFB over UFB such as speech and audio signal processing where nonuniform division of bands is important