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Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2008
"... We study the invertibility of Mvariate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multipleinput multipleoutput systems, and multirate systems. Given an N × P Laurent polynomial matrix H(z1,..., zM) of degree at mos ..."
Abstract

Cited by 5 (4 self)
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We study the invertibility of Mvariate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multipleinput multipleoutput systems, and multirate systems. Given an N × P Laurent polynomial matrix H(z1,..., zM) of degree at most k, we want to find a P × N 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 paper is to prove that H(z) is generically invertible when N −P ≥ M; whereas when N −P < M, then H(z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us.
LAURENT POLYNOMIAL INVERSE MATRICES AND MULTIDIMENSIONAL PERFECT RECONSTRUCTION SYSTEMS
, 2008
"... We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput systems, and multirate systems. Given an N × P polynomial matrix H ..."
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We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput 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 Nchannel 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
GENERIC INVERTIBILITY OF MULTIDIMENSIONAL FIR MULTIRATE SYSTEMS AND FILTER BANKS
"... We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput systems, and multirate systems. The main result of this paper is ..."
Abstract
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We study the invertibility of Mvariate polynomial (respectively: Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multipleinput multipleoutput systems, and multirate systems. The main result of this paper 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. As a result, we can have an alternative approach in design of the multidimensional systems.
Multivariate
"... generalized sampling in shiftinvariant spaces and its approximation properties ..."
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generalized sampling in shiftinvariant spaces and its approximation properties