A Unified Approach to Structured Covariances: Fast Generalized Sliding Window RLS Recursions for Arbitrary Basis

This paper extends the existing fast RLS recursions originally intended to exponentially windowed problems for general models, to a generalized sliding window formulation (GSWRLS). From a matrix algebra perspective, we show explicitly how the displacement rank of the underlying inverse covariance ma...

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Published in:	IEEE transactions on signal processing Vol. 61; no. 23; pp. 6060 - 6075
Main Author:	Merched, Ricardo
Format:	Journal Article
Language:	English
Published:	New York, NY IEEE 01.12.2013 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Applied sciences Covariance matrices Data structures Detection, estimation, filtering, equalization, prediction Exact sciences and technology Information, signal and communications theory lattice Matrix decomposition orthonormal model Polynomials regularized least-squares Signal and communications theory Signal processing Signal processing algorithms Signal, noise Sliding window RLS algorithm Telecommunications and information theory Breakpoint Recurrence orthonormal model Matrix algebra Generating function Equalizer Covariance matrix Recursive algorithm Kernel method Covariance Inverse matrix Least squares method First order lattice regularized least-squares Signal processing Sliding window RLS algorithm Data structure
ISSN:	1053-587X, 1941-0476
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Abstract	This paper extends the existing fast RLS recursions originally intended to exponentially windowed problems for general models, to a generalized sliding window formulation (GSWRLS). From a matrix algebra perspective, we show explicitly how the displacement rank of the underlying inverse covariance matrix associated to any operator is defined as a function of the number of window breakpoints and how the fast GSWRLS calculates these rank factors in a fast manner. The recursions hold regardless of the (first order) data structure induced and show that fast fixed order and order recursive RLS algorithms can still be obtained for unwindowed data matrices exhibiting a fixed arbitrary relation between successive regressors. Our approach highlights the existence of a certain degree of freedom inherent to structured data matrices induced by general models, showing that efficient representations of their inverse covariances are not limited to factor circulants, but rather constructed from any arbitrary operator. These Bezoutians, usually expressed via reproducing kernel relations, can be represented exactly in matrix form, along with a precise correspondence to variables of a GSWRLS. As a fallout, we obtain a vector relation stating the so-called minimality property, for extended models and windows, as opposed to analogous generating function arguments normally seen in original approaches. These results pave the way to a more general framework of polynomial Vandermonde covariance decompositions which arise naturally via a proper choice of recurrence related polynomials. This has further impact on several signal processing applications, including superfast realization of equalizers in communications scenarios.
AbstractList	This paper extends the existing fast RLS recursions originally intended to exponentially windowed problems for general models, to a generalized sliding window formulation (GSWRLS). From a matrix algebra perspective, we show explicitly how the displacement rank of the underlying inverse covariance matrix associated to any operator is defined as a function of the number of window breakpoints and how the fast GSWRLS calculates these rank factors in a fast manner. The recursions hold regardless of the (first order) data structure induced and show that fast fixed order and order recursive RLS algorithms can still be obtained for unwindowed data matrices exhibiting a fixed arbitrary relation between successive regressors. Our approach highlights the existence of a certain degree of freedom inherent to structured data matrices induced by general models, showing that efficient representations of their inverse covariances are not limited to factor circulants, but rather constructed from any arbitrary operator. These Bezoutians, usually expressed via reproducing kernel relations, can be represented exactly in matrix form, along with a precise correspondence to variables of a GSWRLS. As a fallout, we obtain a vector relation stating the so-called minimality property, for extended models and windows, as opposed to analogous generating function arguments normally seen in original approaches. These results pave the way to a more general framework of polynomial Vandermonde covariance decompositions which arise naturally via a proper choice of recurrence related polynomials. This has further impact on several signal processing applications, including superfast realization of equalizers in communications scenarios.
Author	Merched, Ricardo
Author_xml	– sequence: 1 givenname: Ricardo surname: Merched fullname: Merched, Ricardo email: r.merched@gmail.com organization: Dept. of Electron. & Comput. Eng., Fed. Univ. of Rio de Janeiro, Rio de Janeiro, Brazil
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Keywords	Breakpoint Recurrence orthonormal model Matrix algebra Generating function Equalizer Covariance matrix Recursive algorithm Kernel method Covariance Inverse matrix Least squares method First order lattice regularized least-squares Signal processing Sliding window RLS algorithm Data structure
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References	ref35 ref13 ref12 ref37 ref36 ref14 ref31 merched (ref2) 2012 ref33 ref11 ref32 ref10 eidelman (ref24) 2007; 420 ref39 maouche (ref30) 1995; 1 ref17 ref38 ref19 ref18 sayed (ref1) 2003 ref45 ref23 ref26 ref25 ref20 ref42 bella (ref27) 0 ref41 ref22 ref21 ammar (ref44) 1990; iii ref43 szeg (ref34) 1975 bella (ref28) 2010; 199 heinig (ref29) 1984 ref8 ref7 ref9 ref4 ref3 ref6 ref5 ref40 b zout (ref16) 1764 euler (ref15) 1748
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SubjectTerms	Applied sciences Covariance matrices Data structures Detection, estimation, filtering, equalization, prediction Exact sciences and technology Information, signal and communications theory lattice Matrix decomposition orthonormal model Polynomials regularized least-squares Signal and communications theory Signal processing Signal processing algorithms Signal, noise Sliding window RLS algorithm Telecommunications and information theory
Title	A Unified Approach to Structured Covariances: Fast Generalized Sliding Window RLS Recursions for Arbitrary Basis
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