Factoring Perfect Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine Approach

Abstract

The theory of linear Diophantine equations in two unknowns over polynomial rings is used to construct causal lifting factorizations for causal twochannel FIR perfect reconstruction multirate filter banks and wavelet transforms. The Diophantine approach generates causal lifting factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new lifting factorization strategy called the Causal Complementation Algorithm. This provides an alternative to the noncausal lifting scheme based on the Extended Euclidean Algorithm for Laurent polynomials that was developed by Daubechies and Sweldens. The new approach, which can be regarded as Gaussian elimination in polynomial matrices, utilizes a generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the Extended Euclidean Algorithm approach by generating causal lifting factorizations not obtainable using the polynomial Euclidean Algorithm.