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 feed-forward memory connections, wideband array processing, and in problems with a multi-input, multi-output network having channels between each source and sensor, such as source separation. Particular applications of FIR polynomial matrix alg...

In this paper, we review Levinson and fast Choleski algorithms for solving sets of linear equations involving Toeplitz or almost Toeplitz matrices. The Levinson-Trench-Zohar 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 Levinson-style 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

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 matrix-valued 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, matrix-valued Nevanlinna–Pick interpolation, optimization, spectral

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.

We present a computationally ecient algorithm for computing the 2-D Capon spectral estimator. The implementation is based on the fact that the 2-D data covariance matrix will have a Toeplitz-BlockToeplitz 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 2-D prediction matrix polynomials. Furthermore, we present a novel method, based on a 2-D lattice algorithm, to compute the needed forward prediction matrix polynomials and discuss the dierence in the so-obtained 2-D spectral estimate as compared to the one obtained by using the prediction matrix polynomials given by the Whittle-Wiggins-Robinson algorithm. Numerical simulations illustrate the clear computational gain in comparison to both the well-known classical implementation and the method recently published by Liu et al. This work was supported in part by the Swedish Foundation for Strateg...

evaluating mild epileptic signal patterns

Abstract: Methods of spatio-temporal analysis provide important tools for characterizing several dynamic aspects of brain oscillations that are reflected in the human scalp-detected 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 so-called coupling measures including classic coherence (COH), partial coherence (pCOH), imaginary part of coherence (iCOH), partial-directed 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 event-related 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.

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Abstract. For linear predictive coding (LPC) of speech, the speech waveform is modeled as the output of an all-pole filter. The waveform is divided into many short intervals (10-30 msec) during which the speech signal is assumed to be stationary. For each interval the constant coefficients of the all-pole filter are estimated by linear prediction by minimizing a squared prediction error criterion. This paper investigates a modification of LPC, called time-varying LPC, which can be used to analyze nonstationary speech signals. In this method, each coefficient of the all-pole filter is allowed to be time-varying 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 time-varying LPC and results are given from the time-varying LPC analysis of both synthetic and real speech. Zusammenfassung. Bei der Linearen Pr~idiktion (LPC) von Sprache wird die Sprachzeitfunktion modellhaft als Ausgangssignal eines Allpole-Filters 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 Allpole-Filters durch Lineare Pr~idiktion ermittelt, wobei ein quadratisches Pr~idiktions-FehlermaJ ~ minimisiert wird. In der vorliegenden Arbeit wird eine Modifikation des LPC-Verfahrens vorgestellt-das sog. Zeitvariante LPC-Verfahren- mit dessen Hilfe es m6glich ist, nicht-station~ire Sprachsignale zu analysieren. Bei diesem Verfahren dfirfen die Koeffizienten

Abstract — We present a new modified Burg-Like algorithm for spectral estimation and adaptive signal processing that yield the same prediction coefficients given by the Levinson algorithm for the solution of the normal equations. An equivalency proof is given for both the 1D signal and 2D signal cases. Numerical simulations illustrate the improved accuracy and stability in spectral power amplitude and localization; especially in the cases of low signal to noise ratio, and (or) augmenting the used prediction coefficients number for a relatively short data records. Also our simulations illustrate that for relatively short data records the unmodified version of Burg Algorithm fail to minimize the mean square residual error beyond certain Order, while the new algorithm continue the minimization with Order elevation. Index Terms—Adaptive Signal processing, lattice filters, image processing, multidimensional signal processing.