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46
A note on the stochastic realization problem
 Hemisphere Publishing Corporation
, 1976
"... Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizati ..."
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Cited by 121 (24 self)
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Abstract. Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density such that (oo) is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations)require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steadystate KalmanBucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families Of external realizations defined on the same probability space and totally ordered with respect to state covariances. 1. Introduction. One
Oversampled Filter Banks
 IEEE Trans. Signal Processing
, 1998
"... Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a neces ..."
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Cited by 120 (2 self)
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Perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in ` (Z). These frames are the subject of this paper. First, necessary and sufficient conditions on a filter bank for implementing a frame or a tight frame expansion are established, as well as a necessary and sufficient condition for perfect reconstruction using FIR filters after an FIR analysis. Complete parameterizations of oversampled filter banks satisfying these conditions are given. Further, we study the condition under which the frame dual to the frame associated with an FIR filter bank is also FIR and give a parameterization of a class of filter banks satisfying this property. Then, we focus on nonsubsampled filter banks. Nonsubsampled filter banks implement transforms similar to continuoustime transforms and allow for very flexible design. We investigate relations of these filter banks to continuoustime filtering and illustrate the design flexibility by giving a procedure for designing maximally flat twochannel filter banks that yield highly regular wavelets with a given number of vanishing moments.
Frametheoretic analysis of oversampled filter banks
 IEEE Trans. Sign. Proc
"... Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given over ..."
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Cited by 86 (5 self)
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Abstract—We provide a frametheoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (IIR) uniform filter banks (FB’s). Our analysis is based on a new relationship between the FB’s polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FB’s providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frametheoretic procedure for the design of paraunitary FB’s from given nonparaunitary FB’s is formulated. We show that the frame bounds of an FB can be obtained by an eigenanalysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented. Index Terms — Filter banks, frames, oversampling, polyphase representation.
Precoding in MultiAntenna and MultiUser Communications
"... In this paper, TomlinsonHarashima precoding for multipleinput/multipleoutput systems including multipleantenna and multiuser systems is studied. It is shown that nonlinear preequalization offers significant advantages over linear preequalization which increases average transmit power. Moreover ..."
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Cited by 85 (2 self)
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In this paper, TomlinsonHarashima precoding for multipleinput/multipleoutput systems including multipleantenna and multiuser systems is studied. It is shown that nonlinear preequalization offers significant advantages over linear preequalization which increases average transmit power. Moreover, it outperforms decisionfeedback equalization at the receiver side which is applicable if joint processing at the receiver side is possible, and which suffers from error propagation. A number of aspects of practical importance are studied. Loading, i.e., the optimum distribution of transmit power and rate is discussed in detail. It is shown that the capacity of the underlying MIMO channel can be utilized asymptotically by means of nonlinear precoding.
FIR Filter Design via Spectral Factorization and Convex Optimization
, 1997
"... We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Usin ..."
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Cited by 46 (6 self)
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We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Using a change of variables and spectral factorization, we can pose such problems as linear or nonlinear convex optimization problems. As a result we can solve them efficiently (and globally) by recently developed interiorpoint methods. We describe applications to filter and equalizer design, and the related problem of antenna array weight design.
Optimization Problems over Positive PseudoPolynomial Matrices
 SIAM J. MATRIX ANAL. APPL
, 2000
"... The Nesterov characterizations of positive pseudopolynomials on the real line, the imaginary and the unit circle are extended to the matrix case. With the help of these characterizations, a class of optimization problems over the space of positive pseudopolynomial matrices is considered. These pro ..."
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Cited by 11 (1 self)
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The Nesterov characterizations of positive pseudopolynomials on the real line, the imaginary and the unit circle are extended to the matrix case. With the help of these characterizations, a class of optimization problems over the space of positive pseudopolynomial matrices is considered. These problems can be solved in an efficient manner due to the inherent block Toeplitz or block Hankel structure induced by the characterization in question. The efficient implementation of the resulting algorithms is discussed in details. In particular, the real line setting of the problem leads naturally to illconditioned numerical systems. However, adopting a Chebyshev basis instead of the natural basis for describing the polynomial matrix space yields a restatement of the problem and of its solution approach with much better numerical properties.
Computation of General InnerOuter and Spectral Factorizations
 IEEE TRANS. AUTO. CONTR
, 2000
"... In this paper we solve two problems in linear systems theory: the computation of the innerouter and spectral factorizations of a continuoustime system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a stan ..."
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Cited by 11 (4 self)
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In this paper we solve two problems in linear systems theory: the computation of the innerouter and spectral factorizations of a continuoustime system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic Riccati equation of order usually much smaller than the McMillan degree of the transfer function matrix of the system. The proposed procedures are completely general being applicable for a polynomial /proper/improper system whose transfer function matrix could be rank deficient and could have poles/zeros on the imaginary axis or at infinity. As an application we discuss the extension to rational matrices of the complete orthogonal decomposition of a constant matrix. Numerical refinements are discussed in detail. To illustrate the proposed approach several numerical examples are also given.