Adaptive subspace estimation-application to moving sources localization and blind channel identification

Riou, Christian; Chonavel, Thierry; Cochet, Philippe

doi:10.1109/icassp.1996.544121

Cited by 8 publications

(7 citation statements)

References 9 publications

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“…Consequently, it is claimed in[62] that if the norm of each estimated eigenvector is set to one at each iteration, the stability of the algorithm is ensured. The simulations presented in[61] confirm this intuition.…”

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confidence: 71%

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Subspace Tracking for Signal Processing

Delmas

2010

Adaptive Signal Processing

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Research in subspace and component-based techniques were originated in Statistics in the middle of the last century through the problem of linear feature extraction solved by the Karhunen-Loève Transform (KLT). Then, it application to signal processing was initiated three decades ago, and has met considerable progress. Thorough studies have shown that the estimation and detection tasks in many signal processing and communications applications such as data compression, data filtering, parameter estimation, pattern recognition, and neural analysis can be significantly improved by using the subspace and component-based methodology. Over the past few years new potential applications have emerged, and subspace and component methods have been adopted in several diverse new fields such as smart antennas, sensor arrays, multiuser detection, time delay estimation, image segmentation, speech enhancement, learning systems, magnetic resonance spectroscopy, and radar systems, to mention only a few examples. The interest in subspace and component-based methods stems from the fact that they consist in splitting the observations into a set of desired and a set of disturbing components. They not only provide new insight into many such problems, but they also offer a good tradeoff between achieved performance and computational complexity. In most cases they can be considered to be low-cost alternatives to computationally intensive maximum-likelihood approaches. In general, subspace and component-based methods are obtained by using batch methods, such as the eigenvalue decomposition (EVD) of the sample covariance matrix or the singular value decomposition (SVD) of the data matrix. However, these two approaches are not suitable for adaptive applications for tracking nonstationary signal parameters, where the required repetitive estimation of the subspace or the eigenvectors can be a real computational burden because their iterative implementation needs O(n 3) operations at each update, where n is the dimension of the vector-valued data sequence. Before proceeding with a brief literature review of the main contributions of adaptive estimation of subspace or eigenvectors, let us first classify these algorithms with respect to their computational complexity. If r denotes the rank of the principal or dominant) or minor subspace we would like to estimate, since usually r ≪ n, it is classic to refer to the following classification. Algorithms requiring O(n 2 r) or O(n 2) operations by update are classified as high complexity;

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confidence: 71%

“…Alternatively, a stochastic gradient-like algorithm denoted Direct Adaptive Subspace Estimation (DASE) has been proposed in [61] with a direct parametrization of the eigenvectors by means of their coefficients.…”

Section: A Rayleigh Quotient-based Methodsmentioning

confidence: 99%

Subspace Tracking for Signal Processing

Delmas

2010

Adaptive Signal Processing

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“…We are now in position to solve the Lyapunov equation in the new parameter defined in the previous subsection. The stochastic equation governing the evolution of this vector parameter is obtained by applying the transformation Vec to the original (8).…”

Section: E Solution Of the Lyapunov Equationmentioning

confidence: 99%

Performance analysis of an adaptive algorithm for tracking dominant subspaces

Delmas

Cardoso

1998

IEEE Trans. Signal Process.

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This paper provides a performance analysis of a least mean square (LMS) dominant invariant subspace algorithm. Based on an unconstrained minimization problem, this algorithm is a stochastic gradient algorithm driving the columns of a matrix W W W to an orthonormal basis of a dominant invariant subspace of a correlation matrix. We consider the stochastic algorithm governing the evolution of W W WW W W H to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution. A closed-form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this LMS type algorithm. In particular, it is shown that even though the algorithm does not constrain W W W to have orthonormal columns, there is deviation from orthonormality of the first order. We also give a closed-form expression of the asymptotic covariance of DOA's estimated by the MUSIC algorithm applied to the adaptive estimate of the projector. It is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion. Finally, the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a "small" step size but in a very large domain.

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“…An analysis of the parametrized stochastic gradient algorithm by Regalia [18] was sketched out in [6] and [7]. Finally, a deflation algorithm for tracking dominant or minorant eigensubspaces [19] and some algorithms tracking dominant eigensubspaces from a least square-like approach (see [23,24]) were presented and studied by the same tools. The main aim of this paper is to study the convergence and performances of a parametrized adaptive algorithm that gives a canonic orthonormal eigenbasis by introducing the necessary methodology and exploiting some of the results that can be derived therefrom.…”

Section: Introductionmentioning

confidence: 99%

“…Putting all the pieces together, we get the expressions (17), (18) and (19), where Q ( , )"*q ( , )/* with…”

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confidence: 99%

Performances analysis of a Givens parametrized adaptive eigenspace algorithm

Delmas¹

1998

Signal Processing

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In this paper, we address an adaptive estimation method for eigenspaces of covariance matrices. We are interested in a gradient procedure based on coupled maximizations or minimizations of Rayleigh quotients where the constraints are replaced by a Givens parametrization. This enables us to provide a canonic orthonormal eigenbasis estimator. We study the convergence of this algorithm with the help of the associated ordinary differential equation (ODE), and propose a performance evaluation by computing the variances of the estimated eigenvectors and of the estimated projection matrices on eigenspaces for fixed gain factors. In particular, we show that these misadjustments depend on whether the successive analyzed vector signals are correlated or not, and thus greatly depend on the origin of the covariance matrices of interest (spatial, temporal, spatio-temporal). More precisely, we show that these misadjustments can be smaller in the case of correlated observations than in the case of independent observations. Finally, we show that performance can be improved when the symmetric-centrosymmetric property of some of those covariance matrices is exploited. 1998 Elsevier Science B.V. All rights reserved. ZusammenfassungIn diesem Artikel wird eine adaptive Methode zur Scha¨tzung von Eigenra¨umen von Kovarianzmatrizen behandelt. Wir interessieren uns fu¨r ein Gradientenverfahren, welches auf gekoppelten Maximierungen oder Minimierungen von Rayleigh-Quotienten beruht, wobei die Nebenbedingungen durch eine Givens-Parametrisierung ersetzt werden. Dies ermo¨glicht es, einen kanonischen Scha¨tzer fu¨r orthonormale Eigenbasen anzugeben. Wir studieren die Konvergenz dieses Algorithmus mit Hilfe der zugeho¨rigen gewo¨hnlichen Differentialgleichung. Zur Beurteilung der Leistungsfa¨higkeit schlagen wir vor, die Varianzen der gescha¨tzten Eigenvektoren und der gescha¨tzten Eigenraum-Projektionsmatrizen bei festen Versta¨rkungsfaktoren zu berechnen. Insbesondere zeigen wir, da{ diese Fehlanpassungen davon abha¨ngen, ob die nacheinander analysierten Vektorsignale korreliert sind, wodurch sich eine starke Abha¨ngigkeit von der Herkunft der Kovarianzmatrizen (ra¨umlich, zeitlich, ra¨umlich-zeitlich) ergibt. Wir zeigen konkret, da{ diese Fehlanpassungen im Fall korrelierter Beobachtungen kleiner sein ko¨nnen als im Fall unabha¨ngiger Beobachtungen. Schlie{lich zeigen wir, da{ die Leistungsfa¨higkeit erho¨ht werden kann, wenn die Symmetrie-Zentrosymmetrie-Eigenschaft einiger dieser Kovarianzmatrizen ausgenu¨tzt wird.1998 Elsevier Science B.V. All rights reserved. Re´sumeŃ ous conside´rons dans cet article une me´thode d'estimation de sous espaces propes de matrices de covariance. Nous nous inte´ressons a`une me´thode de gradient base´e sur des minimisations ou des maximisations de quotients de Rayleigh dans lesquelles les contraintes sont remplace´es par une parame´trisation de Givens. Cela permet de fournir de facon structurelle un estimateur orthonorme´de bases orthonorme´es. Nous e´tudions la convergence de cet algorithme graˆ...

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Adaptive subspace estimation-application to moving sources localization and blind channel identification

Cited by 8 publications

References 9 publications

Subspace Tracking for Signal Processing

Subspace Tracking for Signal Processing

Performance analysis of an adaptive algorithm for tracking dominant subspaces

Performances analysis of a Givens parametrized adaptive eigenspace algorithm

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