Two-Channel Perfect Reconstruction FIR Filter Banks over Commutative Rings

The relation between ladder and lattice implementations of two-channel filter banks is discussed and it is shown that these two concepts differ in general. An elementary proof is given for the fact that over any integral domain which is not a field there exist causal realizable perfect reconstructing filter banks that can not be implemented with causal lifting filters. A complete parametrization of filter banks with coefficients in local rings and semiperfect rings is given. 1 Filter Banks Let A be a commutative ring. We assume that all signals and filters are elements of the Laurent polynomial ring B = A[z; z \Gamma1 ]. Note that B is isomorphic to the group algebra A[Z]. Therefore, we refer to the multiplication in B as a convolution or filter operation. We define a downsampling operation [ # 2] on B by [ # 2] a(z) = a e (z), where a(z) = a e (z 2 ) + za o (z 2 ). An upsampling operation [ " 2] on B is defined by [ " 2] a(z) = a(z 2 ). A two-channel filter bank con..