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.

翻译：本文针对任意双通道完美重构FIR滤波器组，发展了一种本质上因果的提升分解方法，称为因果补全算法。该方法解决了Daubechies和Sweldens基于洛朗多项式扩展欧几里得算法分解离散小波变换时固有非因果策略的工程缺陷。因果补全算法不仅复现了因果版本欧几里得算法方法所产生的所有提升分解，还能生成额外的因果分解——这些分解无法通过因果欧几里得算法获得，并具有推广欧几里得算法所提供性质的降阶特性。新方法采用矩阵多项式的高斯消元法，辅以多项式长除法的轻微推广，替代了传统的欧几里得算法。研究证明，某些多项式降阶条件既是必要也是充分的，以确保因果初等矩阵分解可通过因果补全算法获得，从而为Daubechies和Sweldens工作中缺失的“提升分解”概念提供了形式化定义。