用于完全重构多速率滤波器组提升分解的因果补全算法
The Causal Complementation Algorithm for Lifting Factorization of Perfect Reconstruction Multirate Filter Banks
AI总结：
针对两通道完全重构FIR滤波器组，提出因果补全算法实现因果提升分解，解决原有方法非因果的缺陷，可生成更多降阶因果分解，并给出提升分解的正式定义。
摘要：
本文针对任意两通道完全重构FIR滤波器组，提出了一种本质因果的提升分解方法——因果补全算法（Causal Complementation Algorithm）。该方法解决了Daubechies与Sweldens基于Laurent多项式扩展欧几里得算法的离散小波变换分解策略固有的非因果性工程缺陷。因果补全算法可复现因果版欧几里得算法方法生成的所有提升分解，还能生成因果欧几里得算法无法得到的额外因果分解，这些分解具备的降阶特性是欧几里得算法所提供特性的推广。新方法未采用欧几里得算法，而是使用矩阵多项式中的高斯消元法，对多项式长除法做了小幅推广。研究表明，特定的多项式降阶条件是利用因果补全算法得到因果初等矩阵分解的充要条件，由此给出了Daubechies与Sweldens的工作中缺失的“提升分解”的正式定义。
英文摘要：
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.
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