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13
Multichannel Blind Deconvolution: Fir Matrix Algebra And Separation Of Multipath Mixtures
, 1996
"... A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and mat ..."
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Cited by 74 (0 self)
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A general tool for multichannel and multipath problems is given in FIR matrix algebra. With Finite Impulse Response (FIR) filters (or polynomials) assuming the role played by complex scalars in traditional matrix algebra, we adapt standard eigenvalue routines, factorizations, decompositions, and matrix algorithms for use in multichannel /multipath problems. Using abstract algebra/group theoretic concepts, information theoretic principles, and the Bussgang property, methods of single channel filtering and source separation of multipath mixtures are merged into a general FIR matrix framework. Techniques developed for equalization may be applied to source separation and vice versa. Potential applications of these results lie in neural networks with feedforward memory connections, wideband array processing, and in problems with a multiinput, multioutput network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...
Levinson and Fast Choleski Algorithms for Toeplitz and Almost Toeplitz Matrices
 Internal Report, Lab of Elec., MIT
, 1984
"... In this paper, we review Levinson and fast Choleski algorithms for solving sets of linear equations involving Toeplitz or almost Toeplitz matrices. The LevinsonTrenchZohar algorithm is first presented for solving problems involving exactly Toeplitz matrices. A fast Choleski algorithm is derived by ..."
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Cited by 21 (0 self)
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In this paper, we review Levinson and fast Choleski algorithms for solving sets of linear equations involving Toeplitz or almost Toeplitz matrices. The LevinsonTrenchZohar algorithm is first presented for solving problems involving exactly Toeplitz matrices. A fast Choleski algorithm is derived by a simple linear transformation. The almost Toeplitz problem is then considered and a Levinsonstyle algorithm is proposed for solving it. A set of linear transformations converts the algorithm into a fast Choleski method. Symmetric and band diagonal applications are considered. Formulas for the inverse of an almost Toeplitz matrix are derived. The relationship between the fast Choleski algorithms and a Euclidian algorithm is exploited in order to derive accelerated "doubling " algorithms for inverting the matrix. Finally, strategies for removing the strongly nonsingular constraint
Matrixvalued NevanlinnaPick interpolation with complexity constraint: An optimization approach
 IEEE Trans. Automat. Contr
, 2003
"... Abstract—Over the last several years, a new theory of Nevanlinna–Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrixvalued case, also allowing for multiple interpolation points. We parameterize a class of in ..."
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Cited by 16 (5 self)
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Abstract—Over the last several years, a new theory of Nevanlinna–Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrixvalued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of “most interpolants ” of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional controllers, we demonstrate the advantage of the proposed method. Index Terms—Complexity constraint, control, matrixvalued Nevanlinna–Pick interpolation, optimization, spectral
A new algorithm for optimal filtering of discretetime stationary processes
 SIAM J. Control
, 1974
"... Abstract. An algorithm (which does not involve the usual Riccatitype equation) for computing t/e gain matrices of the Kalman filter is presented. If the dimension k of the state space is much larger than that of the observation process, the number of nonlinear equations to be solved in each step is ..."
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Cited by 14 (10 self)
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Abstract. An algorithm (which does not involve the usual Riccatitype equation) for computing t/e gain matrices of the Kalman filter is presented. If the dimension k of the state space is much larger than that of the observation process, the number of nonlinear equations to be solved in each step is of order k rather than k as by the usual procedure. 1. Introduction. Let {x.
Relationships between digital signal processing and control and estimation theory
 Proceedings of the IEEE
, 1978
"... The purpose of this paper is to explore several current research directions in the fields of digital signal processing and modern control and estimation theory. We examine topics such as stability theory, linear prediction, and parameter identification, system synthesis and implementation, twodimens ..."
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Cited by 8 (4 self)
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The purpose of this paper is to explore several current research directions in the fields of digital signal processing and modern control and estimation theory. We examine topics such as stability theory, linear prediction, and parameter identification, system synthesis and implementation, twodimensional filtering, decentralized control and estimation, and image processing, in order to uncover some of the basic similarities and differences in the goals, techniques, and philosophy of the two disciplines.
Computationally Efficient TwoDimensional Capon Spectrum Analysis
"... We present a computationally ecient algorithm for computing the 2D Capon spectral estimator. The implementation is based on the fact that the 2D data covariance matrix will have a ToeplitzBlockToeplitz structure, with the result that the inverse covariance matrix can be expressed in closed form ..."
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Cited by 3 (0 self)
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We present a computationally ecient algorithm for computing the 2D Capon spectral estimator. The implementation is based on the fact that the 2D data covariance matrix will have a ToeplitzBlockToeplitz structure, with the result that the inverse covariance matrix can be expressed in closed form by using a special case of the GohbergHeinig formula that is a function of strictly the forward 2D prediction matrix polynomials. Furthermore, we present a novel method, based on a 2D lattice algorithm, to compute the needed forward prediction matrix polynomials and discuss the dierence in the soobtained 2D spectral estimate as compared to the one obtained by using the prediction matrix polynomials given by the WhittleWigginsRobinson algorithm. Numerical simulations illustrate the clear computational gain in comparison to both the wellknown classical implementation and the method recently published by Liu et al. This work was supported in part by the Swedish Foundation for Strateg...
Assessment of linear and nonlinear EEG synchronization measures for
"... evaluating mild epileptic signal patterns ..."
Analyzing eventrelated EEG data with multivariate autoregressive parameters
"... Abstract: Methods of spatiotemporal analysis provide important tools for characterizing several dynamic aspects of brain oscillations that are reflected in the human scalpdetected electroencephalogram (EEG). The search to identify the dynamic connectivity of brain signals within different frequenc ..."
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Abstract: Methods of spatiotemporal analysis provide important tools for characterizing several dynamic aspects of brain oscillations that are reflected in the human scalpdetected electroencephalogram (EEG). The search to identify the dynamic connectivity of brain signals within different frequency bands, in order to uncover the transient cooperation between different brain sites, converges at the potential of multivariate autoregressive (MVAR) models and their derived parameters. In fact, MVAR parameters provide a whole battery of socalled coupling measures including classic coherence (COH), partial coherence (pCOH), imaginary part of coherence (iCOH), partialdirected coherence (PDC), directed transfer function (DTF), and full frequency directed transfer function (ffDTF). All of these approaches have been developed to quantify the degree of coupling between different EEG recording positions, with the specific aim to characterize the functional interaction between neural populations within the cortex. This work addresses the application of MVAR models to eventrelated brain processes, including different statistical approaches, and reviews most relevant findings in the expanding field of coupling analysis. Finally, we present several examples of coupling patterns associated with certain types of movement imagery.
NorthHolland Publishing Company TIMEVARYING PARAMETRIC MODELING OF SPEECH*
, 1982
"... Abstract. For linear predictive coding (LPC) of speech, the speech waveform is modeled as the output of an allpole filter. The waveform is divided into many short intervals (1030 msec) during which the speech signal is assumed to be stationary. For each interval the constant coefficients of the al ..."
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Abstract. For linear predictive coding (LPC) of speech, the speech waveform is modeled as the output of an allpole filter. The waveform is divided into many short intervals (1030 msec) during which the speech signal is assumed to be stationary. For each interval the constant coefficients of the allpole filter are estimated by linear prediction by minimizing a squared prediction error criterion. This paper investigates a modification of LPC, called timevarying LPC, which can be used to analyze nonstationary speech signals. In this method, each coefficient of the allpole filter is allowed to be timevarying by assuming it is a linear combination of a set of known time functions. The coefficients of the linear combination of functions are obtained by the same least squares error technique used by the LPC. Methods are developed for measuring and assessing the performance of timevarying LPC and results are given from the timevarying LPC analysis of both synthetic and real speech. Zusammenfassung. Bei der Linearen Pr~idiktion (LPC) von Sprache wird die Sprachzeitfunktion modellhaft als Ausgangssignal eines AllpoleFilters aufgefaJ3t. Die Zeitfunktion wird dabei in zahlreiche kurze Intervalle von 10 bis 30 ms Dauer unterteilt, in denen das Signal als station~ir betrachtet werden kann. Fiir jedes Intervall werden die konstanten Koeffizienten des AllpoleFilters durch Lineare Pr~idiktion ermittelt, wobei ein quadratisches Pr~idiktionsFehlermaJ ~ minimisiert wird. In der vorliegenden Arbeit wird eine Modifikation des LPCVerfahrens vorgestelltdas sog. Zeitvariante LPCVerfahren mit dessen Hilfe es m6glich ist, nichtstation~ire Sprachsignale zu analysieren. Bei diesem Verfahren dfirfen die Koeffizienten