Subspace-based algorithms that exploit the orthogonality between a sample subspace and a paramete... more Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The statistical performance of these subspacebased algorithms depends on the deterministic and stochastic statistical model of the noisy linear mixture of the data, the estimate of the projector associated with different estimates of the scatter/covariance of the data, and the algorithm that estimates the parameters from the projector. This chapter presents different complex circular (C-CES) and non-circular (NC-CES) elliptically symmetric models of the data and different associated non-robust and robust covariance estimators among which, the sample covariance matrix (SCM), the maximum likelihood (ML) estimate, robust M-estimates, Tyler's M-estimate and the sample sign covariance matrix (SSCM), whose asymptotic distributions are derived. This allows us to unify the asymptotic distribution of subspace projectors adapted to the different models of the data and to prove several invariance properties that have impacts on the parameters to be estimated. Particular attention is paid to the comparison between the projectors derived from Tyler's M-estimate and SSCM. Finally, asymptotic distributions of estimates of parameters characterized by the principal subspace derived from the distributions of subspace-based parameter estimates are studied. In particular, the efficiency with respect to the stochastic and semiparametric Cramér-Rao bound is considered.
This letter presents an extension of the wellknown Whittle's formula for the asymptotic Fisher in... more This letter presents an extension of the wellknown Whittle's formula for the asymptotic Fisher information matrix (FIM) on the power spectrum parameters of zeromean stationary Gaussian processes to compound Gaussian processes (CGP). The new formula includes a corrective factor that depends on the considered CG distribution, in addition to the usual Gaussian term.
This paper addresses the joint path delay and time-varying complex gain estimation for continuous... more This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (C-PM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closedform expression of the modified hybrid Cramér-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.
HAL (Le Centre pour la Communication Scientifique Directe), Jan 20, 2010
Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO... more Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO ». T T3 3. .2 2-R Ra ap pp po or rt t d de e s sp pé éc ci if fi ic ca at ti io on n d de es s a al lg go or ri it th hm me es s d de e d dé ét te ec ct ti io on n b ba as sé és s s su ur r l le e f fi il lt tr ra ag ge e p pa ar rt ti ic cu ul la ai ir re e ((v ve er rs si io on n f fi in na al le e)) .Date 20 janvier 2010
This paper mainly deals with an extension of the matrix Slepian-Bangs (SB) formula to elliptical ... more This paper mainly deals with an extension of the matrix Slepian-Bangs (SB) formula to elliptical symmetric (ES) distributions under the assumption that the arbitrary density generator depends on unknown parameters, aiming to rigorously quantify and understand the impact of this assumption on ES distributed parametric estimation models. This matrix SB formula is derived in a unified way within the framework of real (RES) and circular (C-CES) or noncircular (NC-CES) complex elliptically symmetric distributions, and then compared to the matrix SB formula obtained with fully known or completely unknown density generators. This new matrix SB formula involves a common structure to the existing one with a simple corrective coefficient. Closed-form expressions of this coefficient are given for Student's t and generalized Gaussian distributions and are each compared according to different knowledge of the density generator. This allows us to conclude that for an arbitrary parameterization, the Cramér-Rao bound (CRB) may be very sensitive to the knowledge of the density generator for super-Gaussian distributions contrary to sub-Gaussian distributions. Finally, we prove that for the parametrization with an unknown scale factor, the CRB for the estimation of the other parameters of the scatter matrix does not depend on the type of knowledge of the density generator. This latter result remains true for the specific noisy linear mixture data model where the parameter of interest is characterized by the range space of the mixing matrix.
Complex-valued data in statistical signal processing applications have many advantages over their... more Complex-valued data in statistical signal processing applications have many advantages over their realvalued counterparts. It allows us to use the complete statistical information of the signal thanks to its statistical property of non-circularity. This paper presents a general framework for developing asymptotic theoretical results on the distribution-free sample sign covariance matrix (SSCM) under circular complex-valued elliptically symmetric (C-CES) and non-circular CES (NC-CES) multidimensional distributed data. It extends some partial asymptotic results on SSCM derived for real elliptically symmetric (RES) distributed data. In particular closedform expressions of the first and second-order of the SSCM are derived for arbitrary spectra of eigenvalues for C-CES and NC-CES distributed data which facilitates the derivation of numerous statistical properties. Then, the asymptotic distributions of associated projectors are deduced, which are applied in the study of asymptotic performance analysis of SSCM-based subspace algorithms, followed by a comparison to the asymptotic results derived using Tyler's M estimate. However, a more in-depth analytical analysis of the efficiency of the SSCM relative to Tyler's M estimate is performed, yielding that the performances of the SSCM and Tyler's M estimate are close for a high-dimensional data and not too small dimension of the principal component space. We conclude therefore that, although the SSCM is inefficient relative to Tyler's M estimate, it is of great interest from the point of view of its lower computational complexity for high-dimensional data. Finally, numerical results illustrating the theoretical analysis are presented through the direction-of-arrival (DOA) estimation CES data models.
21st European Signal Processing Conference (EUSIPCO 2013), 2013
This paper focuses on carrier frequency offset (CFO) estimation in the presence of time-selective... more This paper focuses on carrier frequency offset (CFO) estimation in the presence of time-selective Rayleigh fading (i.e., Gaussian multiplicative noise) channel. The time-variant fading is modeled by considering the Jakes' and the first order autoregressive AR(1) correlation models. A high signal-to-noise-ratio maximum likelihood (ML) estimators based on the AR(1) correlation model and for slow-fading channels are derived when the channel statistics are unknown. The main objective is to reduce algorithm complexity to a single-dimensional search on the CFO parameter alone. Closed-form expressions of the Cramér-Rao lower bound (CRB) for the CFO parameter alone are derived for fast-fading and slow-fading channels. Approximate analytical expressions for the CRB are derived for low and high SNR that enable the derivation of a number of properties that describe the bound's dependence on key parameters such as SNR, channel correlation. Finally, simulation results illustrate the per...
The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR a... more The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR at which specific high-resolution algorithms are able to resolve two closely spaced far-field sources, allows to quantify and to compare sensors array performance in localizing remote targets. This paper generalizes and extends the expressions of the threshold ASNR given in the literature for the conventional and non-circular (NC) MUSIC direction-of-arrival (DOA) estimation algorithms in the context of uncorrelated stochastic circular or rectilinear Gaussian sources and circular complex Gaussian (C-CG) noise, in a more general stochastic framework. We assume that the sources are correlated with an arbitrary distribution, which is inherent in a context of multipath or smart jammers, and that the noise is circular complex elliptically symmetric (C-CES) distributed, which can model impulsive noise with heavy-tailed distributions. The C-CES and NC-CES distributed observations are also considered to quantify the gain in resolution provided when the sample covariance matrix (SCM) of the observations is replaced by M-estimates of this matrix. Asymptotic approaches and perturbation analysis have been performed to derive closed-form expressions of the mean null spectra of the two considered MUSIC algorithms for both observation models, which allow us to derive, for the first time, general unified explicit analytical expressions of the threshold ASNR along the Cox and the Sharman and Durrani criteria. These expressions allow us to quantify the impact of the non-Gaussianity of noise and observations, as well as of the phase and magnitude of the correlation on the resolution threshold, and to quantify the benefit provided when the SCM is replaced by M-estimates of this covariance matrix for CES observations. Finally, numerical illustrations are included to support our theoretical analysis.
Abstract:Direction-of-arrival (DOA) estimation algorithms in array processing applications have b... more Abstract:Direction-of-arrival (DOA) estimation algorithms in array processing applications have been developed under time-invariant wavefronts. In most applications this assumption is not realistic due to the nonhomogeneous propagation medium which can distort the wavefront received by the array. This paper extends the author’s previous work on signal-to-noise ratio (SNR) estimation, in developing a novel approach for estimating the DOA of a single narrow-band amplitude-distorted wavefront received by an arbitrary antenna array. The distorted-amplitude wavefront is assumed to vary according to the first order autoregressive AR(1) model with unknown coefficients. An approximate maximum-likelihood-based (ML-based) approach to estimate the DOA parameter is developed in the high SNR scenario. Compared with the classical ML method that requires computationally prohibitive multi-dimensional search, the proposed approach obtains the DOA estimate by maximizing a new cost function with respe...
2010 18th European Signal Processing Conference, 2010
This paper focuses on the data-aided (DA) direction of arrival (DOA) estimation of a single narro... more This paper focuses on the data-aided (DA) direction of arrival (DOA) estimation of a single narrow-band source in time-varying Rayleigh fading amplitude. The time-variant fading amplitude is modeled by considering the Jakes' and the first order autoregressive (AR1) correlation models. Closed-form expressions of the CRB for DOA alone are derived for fast and slow Rayleigh fading amplitude. As a special case, the CRB under uncorrelated fading Rayleigh channel is derived. A analytical approximate expressions of the CRB are derived for low and high SNR that enable the derivation of a number of properties that describe the bound's dependence on key parameters such as SNR, channel correlation. A high signal-to-noise-ratio maximum likelihood (ML) estimator based on the AR1 correlation model is derived. The main objective is to reduce algorithm complexity to a single-dimensional search on the DOA parameter alone as in the static-channel DOA estimator. Finally, simulation results ill...
Subspace-based algorithms that exploit the orthogonality between a sample subspace and a paramete... more Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameterdependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M-estimates of the orthogonal projection matrix is derived from those of the M-estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M-estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler's M-estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M-estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced.
This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with... more This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with respect to non-Gaussian noise distributions using perturbation expansions. Its purpose is twofold. It aims, first, to derive the asymptotic distribution of the estimated projector matrix obtained from the sample covariance matrix (SCM) for arbitrary distributions of the useful signal and the noise. It proves that this distribution depends only of the second-order statistics of the useful signal, but also on the second and fourth-order statistics of the noise. Second, it derives the asymptotic distribution of the estimated projector matrix obtained from any M-estimate of the covariance matrix for both real (RES) and complex elliptical symmetric (CES) distributed observations. Applied to the MUSIC algorithm for direction-of-arrival (DOA) estimation, these theoretical results allow us to theoretically evaluate the performance loss of this algorithm for heavy-tailed noise distributions when it is based on the SCM, which is significant for weak signal-to-noise ratio (SNR) or closely spaced sources. These results also make it possible to prove that this performance loss can be alleviated by replacing the SCM by an M-estimate of the covariance for CES distributed observations, which has been observed until now only by numerical experiments. Paper accepted to Signal Processing I. INTRODUCTION Subspace-based algorithms are obtained by exploiting the orthogonality between a sample subspace and a parameter-dependent subspace. They have been proved very useful in many applications, including array processing and linear system identification (see e.g., [1], [2]). The purpose of this paper is to complement theoretical results already available on subspace-based estimators. Among them, the asymptotic distribution of
Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the struct... more Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cramér Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood (ML) estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this paper is to derive this expression directly from the Slepian-Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are given.
-Cet article est consacré au calcul des bornes de Cramer-Rao (CR) déterministe et stochastique de... more -Cet article est consacré au calcul des bornes de Cramer-Rao (CR) déterministe et stochastique des directions d'arrivée (DOA) sous la connaissance a priori de signaux non circulaires rectilignes non corrélés en présence de bruit additif gaussien circulaire de matrice de covariance spatiale arbitraire. Des expressions analytiques de ces deux bornes sont données permettant de préciser le rôle primordial des phases de non circularité où plusieurs propriétés sont démontrées. Inférieures aux bornes générales ne prenant pas en compte ces informations a priori, des exemples numériques montrent que les bornes de CR déterministes et stochastiques avec a priori sont très proches entre elles mais très faibles par rapportà la borne de CR stochastique qui ne tient pas compte de ces a priori.
Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO... more Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO ». T T3 3. .2 2-R Ra ap pp po or rt t d de e s sp pé éc ci if fi ic ca at ti io on n d de es s a al lg go or ri it th hm me es s d de e d dé ét te ec ct ti io on n b ba as sé és s s su ur r l le e f fi il lt tr ra ag ge e p pa ar rt ti ic cu ul la ai ir re e ((v ve er rs si io on n f fi in na al le e)) .Date 20 janvier 2010
GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference, 2009
This paper addresses the joint path delay and time-varying complex gain estimation for continuous... more This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (C-PM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closedform expression of the modified hybrid Cramér-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.
2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 2008
This paper addresses the degree of second-order non-circularity or impropriety of complex random ... more This paper addresses the degree of second-order non-circularity or impropriety of complex random variables and its purpose is to complement previously available theoretical results. New properties of the non-circularity rate (also called circularity spectrum) are given for scalar and multidimensional complex random variables with a particular attention paid to rectilinear random variables, i.e., with maximum circularity spectrum. Finally, the maximum likelihood estimate of the circularity spectrum in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given.
Subspace-based algorithms that exploit the orthogonality between a sample subspace and a paramete... more Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The statistical performance of these subspacebased algorithms depends on the deterministic and stochastic statistical model of the noisy linear mixture of the data, the estimate of the projector associated with different estimates of the scatter/covariance of the data, and the algorithm that estimates the parameters from the projector. This chapter presents different complex circular (C-CES) and non-circular (NC-CES) elliptically symmetric models of the data and different associated non-robust and robust covariance estimators among which, the sample covariance matrix (SCM), the maximum likelihood (ML) estimate, robust M-estimates, Tyler's M-estimate and the sample sign covariance matrix (SSCM), whose asymptotic distributions are derived. This allows us to unify the asymptotic distribution of subspace projectors adapted to the different models of the data and to prove several invariance properties that have impacts on the parameters to be estimated. Particular attention is paid to the comparison between the projectors derived from Tyler's M-estimate and SSCM. Finally, asymptotic distributions of estimates of parameters characterized by the principal subspace derived from the distributions of subspace-based parameter estimates are studied. In particular, the efficiency with respect to the stochastic and semiparametric Cramér-Rao bound is considered.
This letter presents an extension of the wellknown Whittle's formula for the asymptotic Fisher in... more This letter presents an extension of the wellknown Whittle's formula for the asymptotic Fisher information matrix (FIM) on the power spectrum parameters of zeromean stationary Gaussian processes to compound Gaussian processes (CGP). The new formula includes a corrective factor that depends on the considered CG distribution, in addition to the usual Gaussian term.
This paper addresses the joint path delay and time-varying complex gain estimation for continuous... more This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (C-PM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closedform expression of the modified hybrid Cramér-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.
HAL (Le Centre pour la Communication Scientifique Directe), Jan 20, 2010
Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO... more Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO ». T T3 3. .2 2-R Ra ap pp po or rt t d de e s sp pé éc ci if fi ic ca at ti io on n d de es s a al lg go or ri it th hm me es s d de e d dé ét te ec ct ti io on n b ba as sé és s s su ur r l le e f fi il lt tr ra ag ge e p pa ar rt ti ic cu ul la ai ir re e ((v ve er rs si io on n f fi in na al le e)) .Date 20 janvier 2010
This paper mainly deals with an extension of the matrix Slepian-Bangs (SB) formula to elliptical ... more This paper mainly deals with an extension of the matrix Slepian-Bangs (SB) formula to elliptical symmetric (ES) distributions under the assumption that the arbitrary density generator depends on unknown parameters, aiming to rigorously quantify and understand the impact of this assumption on ES distributed parametric estimation models. This matrix SB formula is derived in a unified way within the framework of real (RES) and circular (C-CES) or noncircular (NC-CES) complex elliptically symmetric distributions, and then compared to the matrix SB formula obtained with fully known or completely unknown density generators. This new matrix SB formula involves a common structure to the existing one with a simple corrective coefficient. Closed-form expressions of this coefficient are given for Student's t and generalized Gaussian distributions and are each compared according to different knowledge of the density generator. This allows us to conclude that for an arbitrary parameterization, the Cramér-Rao bound (CRB) may be very sensitive to the knowledge of the density generator for super-Gaussian distributions contrary to sub-Gaussian distributions. Finally, we prove that for the parametrization with an unknown scale factor, the CRB for the estimation of the other parameters of the scatter matrix does not depend on the type of knowledge of the density generator. This latter result remains true for the specific noisy linear mixture data model where the parameter of interest is characterized by the range space of the mixing matrix.
Complex-valued data in statistical signal processing applications have many advantages over their... more Complex-valued data in statistical signal processing applications have many advantages over their realvalued counterparts. It allows us to use the complete statistical information of the signal thanks to its statistical property of non-circularity. This paper presents a general framework for developing asymptotic theoretical results on the distribution-free sample sign covariance matrix (SSCM) under circular complex-valued elliptically symmetric (C-CES) and non-circular CES (NC-CES) multidimensional distributed data. It extends some partial asymptotic results on SSCM derived for real elliptically symmetric (RES) distributed data. In particular closedform expressions of the first and second-order of the SSCM are derived for arbitrary spectra of eigenvalues for C-CES and NC-CES distributed data which facilitates the derivation of numerous statistical properties. Then, the asymptotic distributions of associated projectors are deduced, which are applied in the study of asymptotic performance analysis of SSCM-based subspace algorithms, followed by a comparison to the asymptotic results derived using Tyler's M estimate. However, a more in-depth analytical analysis of the efficiency of the SSCM relative to Tyler's M estimate is performed, yielding that the performances of the SSCM and Tyler's M estimate are close for a high-dimensional data and not too small dimension of the principal component space. We conclude therefore that, although the SSCM is inefficient relative to Tyler's M estimate, it is of great interest from the point of view of its lower computational complexity for high-dimensional data. Finally, numerical results illustrating the theoretical analysis are presented through the direction-of-arrival (DOA) estimation CES data models.
21st European Signal Processing Conference (EUSIPCO 2013), 2013
This paper focuses on carrier frequency offset (CFO) estimation in the presence of time-selective... more This paper focuses on carrier frequency offset (CFO) estimation in the presence of time-selective Rayleigh fading (i.e., Gaussian multiplicative noise) channel. The time-variant fading is modeled by considering the Jakes' and the first order autoregressive AR(1) correlation models. A high signal-to-noise-ratio maximum likelihood (ML) estimators based on the AR(1) correlation model and for slow-fading channels are derived when the channel statistics are unknown. The main objective is to reduce algorithm complexity to a single-dimensional search on the CFO parameter alone. Closed-form expressions of the Cramér-Rao lower bound (CRB) for the CFO parameter alone are derived for fast-fading and slow-fading channels. Approximate analytical expressions for the CRB are derived for low and high SNR that enable the derivation of a number of properties that describe the bound's dependence on key parameters such as SNR, channel correlation. Finally, simulation results illustrate the per...
The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR a... more The concept of threshold array signal-to-noise ratio (ASNR) which is defined as the minimal SNR at which specific high-resolution algorithms are able to resolve two closely spaced far-field sources, allows to quantify and to compare sensors array performance in localizing remote targets. This paper generalizes and extends the expressions of the threshold ASNR given in the literature for the conventional and non-circular (NC) MUSIC direction-of-arrival (DOA) estimation algorithms in the context of uncorrelated stochastic circular or rectilinear Gaussian sources and circular complex Gaussian (C-CG) noise, in a more general stochastic framework. We assume that the sources are correlated with an arbitrary distribution, which is inherent in a context of multipath or smart jammers, and that the noise is circular complex elliptically symmetric (C-CES) distributed, which can model impulsive noise with heavy-tailed distributions. The C-CES and NC-CES distributed observations are also considered to quantify the gain in resolution provided when the sample covariance matrix (SCM) of the observations is replaced by M-estimates of this matrix. Asymptotic approaches and perturbation analysis have been performed to derive closed-form expressions of the mean null spectra of the two considered MUSIC algorithms for both observation models, which allow us to derive, for the first time, general unified explicit analytical expressions of the threshold ASNR along the Cox and the Sharman and Durrani criteria. These expressions allow us to quantify the impact of the non-Gaussianity of noise and observations, as well as of the phase and magnitude of the correlation on the resolution threshold, and to quantify the benefit provided when the SCM is replaced by M-estimates of this covariance matrix for CES observations. Finally, numerical illustrations are included to support our theoretical analysis.
Abstract:Direction-of-arrival (DOA) estimation algorithms in array processing applications have b... more Abstract:Direction-of-arrival (DOA) estimation algorithms in array processing applications have been developed under time-invariant wavefronts. In most applications this assumption is not realistic due to the nonhomogeneous propagation medium which can distort the wavefront received by the array. This paper extends the author’s previous work on signal-to-noise ratio (SNR) estimation, in developing a novel approach for estimating the DOA of a single narrow-band amplitude-distorted wavefront received by an arbitrary antenna array. The distorted-amplitude wavefront is assumed to vary according to the first order autoregressive AR(1) model with unknown coefficients. An approximate maximum-likelihood-based (ML-based) approach to estimate the DOA parameter is developed in the high SNR scenario. Compared with the classical ML method that requires computationally prohibitive multi-dimensional search, the proposed approach obtains the DOA estimate by maximizing a new cost function with respe...
2010 18th European Signal Processing Conference, 2010
This paper focuses on the data-aided (DA) direction of arrival (DOA) estimation of a single narro... more This paper focuses on the data-aided (DA) direction of arrival (DOA) estimation of a single narrow-band source in time-varying Rayleigh fading amplitude. The time-variant fading amplitude is modeled by considering the Jakes' and the first order autoregressive (AR1) correlation models. Closed-form expressions of the CRB for DOA alone are derived for fast and slow Rayleigh fading amplitude. As a special case, the CRB under uncorrelated fading Rayleigh channel is derived. A analytical approximate expressions of the CRB are derived for low and high SNR that enable the derivation of a number of properties that describe the bound's dependence on key parameters such as SNR, channel correlation. A high signal-to-noise-ratio maximum likelihood (ML) estimator based on the AR1 correlation model is derived. The main objective is to reduce algorithm complexity to a single-dimensional search on the DOA parameter alone as in the static-channel DOA estimator. Finally, simulation results ill...
Subspace-based algorithms that exploit the orthogonality between a sample subspace and a paramete... more Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameterdependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M-estimates of the orthogonal projection matrix is derived from those of the M-estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M-estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler's M-estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M-estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced.
This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with... more This paper addresses the theoretical analysis of the robustness of subspace-based algorithms with respect to non-Gaussian noise distributions using perturbation expansions. Its purpose is twofold. It aims, first, to derive the asymptotic distribution of the estimated projector matrix obtained from the sample covariance matrix (SCM) for arbitrary distributions of the useful signal and the noise. It proves that this distribution depends only of the second-order statistics of the useful signal, but also on the second and fourth-order statistics of the noise. Second, it derives the asymptotic distribution of the estimated projector matrix obtained from any M-estimate of the covariance matrix for both real (RES) and complex elliptical symmetric (CES) distributed observations. Applied to the MUSIC algorithm for direction-of-arrival (DOA) estimation, these theoretical results allow us to theoretically evaluate the performance loss of this algorithm for heavy-tailed noise distributions when it is based on the SCM, which is significant for weak signal-to-noise ratio (SNR) or closely spaced sources. These results also make it possible to prove that this performance loss can be alleviated by replacing the SCM by an M-estimate of the covariance for CES distributed observations, which has been observed until now only by numerical experiments. Paper accepted to Signal Processing I. INTRODUCTION Subspace-based algorithms are obtained by exploiting the orthogonality between a sample subspace and a parameter-dependent subspace. They have been proved very useful in many applications, including array processing and linear system identification (see e.g., [1], [2]). The purpose of this paper is to complement theoretical results already available on subspace-based estimators. Among them, the asymptotic distribution of
Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the struct... more Several direction of arrival (DOA) estimation algorithms have been proposed to exploit the structure of rectilinear or strictly second-order noncircular signals. But until now, only the compact closed-form expressions of the corresponding deterministic Cramér Rao bound (DCRB) have been derived because it is much easier to derive than the stochastic CRB (SCRB). As this latter bound is asymptotically achievable by the maximum likelihood (ML) estimator, while the DCRB is unattainable, it is important to have a compact closed-form expression for this SCRB to assess the performance of DOA estimation algorithms for rectilinear signals. The aim of this paper is to derive this expression directly from the Slepian-Bangs formula including in particular the case of prior knowledge of uncorrelated or coherent sources. Some properties of these SCRBs are proved and numerical illustrations are given.
-Cet article est consacré au calcul des bornes de Cramer-Rao (CR) déterministe et stochastique de... more -Cet article est consacré au calcul des bornes de Cramer-Rao (CR) déterministe et stochastique des directions d'arrivée (DOA) sous la connaissance a priori de signaux non circulaires rectilignes non corrélés en présence de bruit additif gaussien circulaire de matrice de covariance spatiale arbitraire. Des expressions analytiques de ces deux bornes sont données permettant de préciser le rôle primordial des phases de non circularité où plusieurs propriétés sont démontrées. Inférieures aux bornes générales ne prenant pas en compte ces informations a priori, des exemples numériques montrent que les bornes de CR déterministes et stochastiques avec a priori sont très proches entre elles mais très faibles par rapportà la borne de CR stochastique qui ne tient pas compte de ces a priori.
Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO... more Contribution du Gipsa lab au projet ANR LURGA « Localisation d'Urgence Reconfigurable par GALILEO ». T T3 3. .2 2-R Ra ap pp po or rt t d de e s sp pé éc ci if fi ic ca at ti io on n d de es s a al lg go or ri it th hm me es s d de e d dé ét te ec ct ti io on n b ba as sé és s s su ur r l le e f fi il lt tr ra ag ge e p pa ar rt ti ic cu ul la ai ir re e ((v ve er rs si io on n f fi in na al le e)) .Date 20 janvier 2010
GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference, 2009
This paper addresses the joint path delay and time-varying complex gain estimation for continuous... more This paper addresses the joint path delay and time-varying complex gain estimation for continuous phase modulation (C-PM) over a time-selective slowly varying flat Rayleigh fading channel. We propose an expectation-maximization (EM) algorithm for path delay estimation in a Kalman smoother framework. The time-varying complex gain is modeled by a first order autoregressive (AR) process. Such a modeling yields to the representation of the problem by a dynamic bayesian system in a state-space form that allows the application of EM algorithm in the context of unobserved data for obtaining an estimate of the path delay. This is used with Kalman smoother for state estimation. We derive analytically a closedform expression of the modified hybrid Cramér-Rao bound (MHCRB) for path delay and complex gain parameters. Finally, some numerical examples are presented to illustrate the performance of the proposed algorithm compared to the conventional generalized correlation method and to the MHCRB.
2008 IEEE International Conference on Acoustics, Speech and Signal Processing, 2008
This paper addresses the degree of second-order non-circularity or impropriety of complex random ... more This paper addresses the degree of second-order non-circularity or impropriety of complex random variables and its purpose is to complement previously available theoretical results. New properties of the non-circularity rate (also called circularity spectrum) are given for scalar and multidimensional complex random variables with a particular attention paid to rectilinear random variables, i.e., with maximum circularity spectrum. Finally, the maximum likelihood estimate of the circularity spectrum in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given.
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Papers by Habti Abeida