The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the Causal Complementation Algorithm. This provides a causal (i.e., polynomial, hence realizable) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight 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 causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.

翻译：本文利用多项式环上二元线性丢番图方程的基本理论，为因果双通道有限冲激响应完美重构传递矩阵及小波变换构建因果提升分解（初等矩阵分解）。该丢番图方法生成的因果分解满足特定的多项式降阶不等式，从而形成一种称为“因果补全算法”的新型分解策略。这为Daubechies和Sweldens使用洛朗多项式扩展欧几里得算法开发的非因果提升方案，提供了一种因果性（即多项式化，从而是可实现的）替代方案。新方法用高斯消元法取代欧几里得算法，并采用多项式除法的轻微推广形式，确保商的存在唯一性且其余数满足用户指定的可除性约束。研究表明，因果补全算法比欧几里得算法方法的因果版本更具一般性，能够生成超出多项式欧几里得算法可获得范围的额外因果提升分解。