Matrix Factorization Theory for Multidimensional Systems Applications

二维系统稳定控制器的参数化在1985年首次被报道，从那时起，即使对于某些可稳定传递矩阵不具有左/右质因数分解的模型，也取得了相当大的进展。多元矩阵分解结果的影响在多维滤波器设计中得到了体现，实现了不可分离小波的构造。潜在的应用涉及分析；在带宽受限的通信信道上重构多维多媒体信号的处理、编码和压缩。本文提出的研究进一步发展了多元分解理论及其变体，以适应和应用于多维系统问题的具有挑战性的应用。本文将研究多元多项式矩阵原始分解存在的充分必要条件，并将其与非模补全的泛化问题联系起来。在运用算法代数中的最新工具，特别是多项式维情况下的Grobner基理论，保证了因子分解的存在性后，将注意实际构造因子分解，研究左右因子分解的存在和构造条件，特别是当约化子式的零共质约束不成立时。一元多项式环上的任何平方单模矩阵都可以表示为Suslin稳定性定理派生的初等矩阵的乘积，这一非常无限制的条件为生物正交多带滤波器组的实现提供了机制，利用这种结构的阶梯进行完美的重构，目的是推广它们在滤波器组和小波构造中的应用。一般来说，当执行完全重构和线性相位约束（或其他类似的准线性）时，处理这种情况仍然是一个悬而未决的问题。最近给出的两波段n-D情况的解决方案，使用Grobner碱基，将被研究，以充分了解其推广到多波段n-D情况的能力的局限性，特别注意各种可能提供完整的建设性解决方案的专业化。

Parameterization of stabilizing controllers for two- dimensional systems was first reported in 1985 and since that time considerable progress has been made even for models in which some stabilizable transfer matrices do not have right/left coprime factorizations. The impact of the multivariate matrix factorization results has recently been demonstrated in the design of multidimensional filter the construction of non- separable wavelets have been realized. Potential applications of interest involved the analysis; processing, coding and compression for reconstruction of multidimensional multimedia signals over bandwidth constrained communication channels.The research proposed here develops further the theory of multivariate factorization and its variants for adaptation and use in challenging applications in multidimensional systems problems. Research will be undertaken towards the of necessary and sufficient conditions for the existence of primitive factorization of multivariate polynomial matrices by relating the problem to generalization of unimodular completion for which construction algorithms are now available. Attention will be directed towards actually constructing the factorization after its existence is guaranteed by applying recent tools in algorithmic algebra, particulary Grobner basis theory over polynomial dimensional cases, the conditions for existence and construction of right and left factorizations will be investigated, especially when the zero coprimenss constraint on the reduced minors fail to hold.The very nonrestrictive conditions under which any square unimodular matrix with entries in a multivariate polynomial ring can be expressed as a product of elementary matrices derivable from Suslin's Stability Theorem provides the machinery for biothogonal multiband filter bank realization for perfect reconstruction using the ladder of such structures with the objective of popularizing their use in filter bank as well as wavelet construction.The tackling of the case when both perfect reconstruction and linear phase constraints (or other like paraunitary) are enforced, is still, in general, an open problem. The recent solution given for the two-band n-D case, using Grobner bases, will be studied to understand fully the limitations of its capability for generalization to the multiband n-D case with particular attention to various specializations that may offer complete constructive solutions.