Laurent Polynomial Inverse Matrices and Multidimensional Perfect Reconstruction Systems

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, multipleinput multiple-output systems, and multirate systems. Given an N × P polynomial matrix H(z) of degree at most k, we want to find a P × 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 ×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 × P matrices. Lastly we study the generalized inverse on polynomial (resp. : Laurent polynomial) matrices.