The Causal Complementation Algorithm for Lifting Factorization of Perfect Reconstruction Multirate Filter Banks

TL;DR

This paper introduces the Causal Complementation Algorithm, a new method for causal lifting factorization of perfect reconstruction FIR filter banks, overcoming noncausality issues in previous approaches and providing a formal definition of lifting factorization.

Contribution

It develops a causal algorithm for filter bank factorization that generalizes the Euclidean Algorithm using matrix polynomial Gaussian elimination, enabling new causal factorizations.

Findings

Reproduces all causal factorizations from the Euclidean Algorithm

Generates additional causal factorizations with degree-reducing properties

Provides a formal definition of lifting factorization for FIR filter banks

Abstract

An intrinsically causal approach to lifting factorization, called the Causal Complementation Algorithm, is developed for arbitrary two-channel perfect reconstruction FIR filter banks. This addresses an engineering shortcoming of the inherently noncausal strategy of Daubechies and Sweldens for factoring discrete wavelet transforms, which was based on the Extended Euclidean Algorithm for Laurent polynomials. The Causal Complementation Algorithm reproduces all lifting factorizations created by the causal version of the Euclidean Algorithm approach and generates additional causal factorizations, which are not obtainable via the causal Euclidean Algorithm, possessing degree-reducing properties that generalize those furnished by the Euclidean Algorithm. In lieu of the Euclidean Algorithm, the new approach employs Gaussian elimination in matrix polynomials using a slight generalization of polynomial long division. It is shown that certain polynomial degree-reducing conditions are both necessary and sufficient for a causal elementary matrix decomposition to be obtainable using the Causal Complementation Algorithm, yielding a formal definition of ``lifting factorization'' that was missing from the work of Daubechies and Sweldens.