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Laurent polynomial inverse matrices and multidimensional perfect reconstruction systems.

机译：Laurent多项式逆矩阵和多维完美重构系统。

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摘要

We study the invertibility of M-variate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multiple-input multiple-output systems, and multirate systems. Given an N x P polynomial matrix H(z1,...,zM) of degree at most k, we want to find a P x N polynomial (resp.: Laurent polynomial) left inverse matrix G(z) of H(z) such that G(z)H(z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this thesis is to prove that when N -- P ≥ M, then H(z) is generically invertible; whereas when N -- P M, then H(z) is generically noninvertible. Based on this fact, we provide some applications and propose a faster algorithm to find a particular inverse of a Laurent polynomial matrix.;The next main topic we are interested is the theory and algorithms for the optimal use of multidimensional signal reconstruction from multichannel acquisition using a filter bank setup. Suppose that we have an N-channel convolution system in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we can first reduce the collected data set by an integer M x M sampling matrix D and still perfectly reconstruct the signal with a synthesis polyphase matrix. First, we determine the existence of perfect reconstruction systems for given finite impulse response (FIR) analysis filters with some sampling matrices and some FIR synthesis polyphase matrices. Second, we present an efficient algorithm to find a sampling matrix with maximum sampling rate and FIR synthesis polyphase matrix for given FIR analysis filters so that the system provides a perfect reconstruction. Third, we develop an algorithm to find a FIR synthesis polyphase matrix for given FIR analysis filters with pure delays allowed in each branch of analysis filters before a given downsampling. Next, once a particular synthesis matrix is found, we can characterize all synthesis matrices and find an optimal one by applying frame analysis and according to design criteria including robust reconstruction in the presence of noise.;Instead of focusing on the application, we are also interested in more theoretical setting. We discuss the conditions on density of the set of invertible (resp.: noninvertible) N x P matrices. Lastly we study the generalized inverse on polynomial (resp.: Laurent polynomial) matrices.
机译：我们研究了大小为N的M变量多项式（分别为：Laurent多项式）矩阵的可逆性。这些矩阵表示各种设置下的多维系统，包括滤波器组，多输入多输出系统和多速率系统。给定一个度数最多为k的N x P多项式矩阵H（z1，...，zM），我们想找到H（z）的一个逆矩阵G（z）的一个P x N多项式（分别是：Laurent多项式） ），这样G（z）H（z）=I。我们提供了可计算的条件来测试可逆性，并提出了寻找特定逆的算法。本论文的主要结果是证明当N-P≥M时，H（z）是可逆的。而当N-P

著录项

  • 作者

    Law, Ka Lung.;

  • 作者单位

    University of Illinois at Urbana-Champaign.;

  • 授予单位 University of Illinois at Urbana-Champaign.;
  • 学科 Mathematics.;Engineering Electronics and Electrical.
  • 学位 Ph.D.
  • 年度 2008
  • 页码 81 p.
  • 总页数 81
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;无线电电子学、电信技术;
  • 关键词

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