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

TL;DR

The paper develops the Causal Complementation Algorithm (CCA) to factorize causal two-channel FIR PR filter banks, addressing the noncausal limitations of the classic Extended Euclidean Algorithm (EEA) based lifting. By employing Gaussian elimination on matrix polynomials and a Slightly Generalized Division, the CCA produces degree-reducing, causal factorizations and reproduces all causal EEA factorizations while also yielding new factorization options, including left-degree-lifting cascades. It formalizes the CCT (a Bezout-like result in the causal setting) and the SGDA, enabling constructive and unique degree-reducing complements while managing irreducibility and downlifting to arrive at standard causal lifting form. The Cubic B-Spline filter bank CDF(7,5) serves as a detailed case study, demonstrating the CCA’s ability to generate well-conditioned factorizations and to produce factorization schemas beyond the reach of the causal EEA. The work further develops a theory of left degree-lifting factorizations, establishing intrinsic lifting signatures and a one-to-one correspondence between left degree-lifting cascades and degree-reducing right downliftings, with a compelling Daubechies 4-tap/4-tap example illustrating uniqueness and completeness of the approach.

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.

Figures (2)

• Figure 1: The polyphase-with-delay filter bank representation.
• Figure 2: Example of standard causal lifting form for a PR filter bank. The initial update characteristic in this example is lower-triangular, $\raisebox{1pt}{$\chi$}(\mathbf{U}_0)=1$, and the number of lifting steps, $N$, is odd.

Theorems & Definitions (8)

• Lemma 2.1: Coprimality
• Corollary 2.2: Linear Diophantine Degree-Reduction Corollary
• Theorem 2.3: Causal Complementation Theorem
• Lemma 2.4
• Lemma 4.1
• Theorem 4.2: Equivalence Theorem
• Theorem 4.3
• Theorem 4.4