Olivier Besson

On parameter estimation of MIMO flat-fading channels with frequency offsets

We address the frequency offsets and channel gains estimation problem for a multi-input multi-output (MIMO) flat-fading channel using a training sequence. The general case where the frequency offsets are possibly different for each transmit antenna is considered. The Cramer-Rao bound (CRB) for the problem at hand is derived. Additionally, we present a simple, closed-form expression for the large-sample CRB and show that it depends in a simple way on the channel parameters. Next, the parameters estimation issue is investigated. First, the maximum likelihood estimator (MLE), which entails solving an n-dimensional maximization problem where n is the number of transmit antennas, is derived. Then, we show that the likelihood function can be written as the product of n one-dimensional (1-D) functions if a suitable choice of the training sequence is made. Based on this fact, we suggest two computationally simpler methods. Numerical examples that illustrate the performance of the estimators and compare it with the CRB are provided.

Training sequence design for frequency offset and frequency-selective channel estimation

We consider the problem of data-aided frequency-offset and channel estimation in the case of frequency-selective channels. More precisely, we address the problem of training sequence selection with the goal of providing accurate frequency offset and channel estimates. Toward this end, we consider the Crame/spl acute/r-Rao bound (CRB), for which we derive a closed-form expression. Since the CRB is a complicated function of the training sequence and the channel parameters, a much simpler asymptotic CRB is derived. Two criteria for training sequence design based on the asymptotic CRB are proposed, and a minmax approach is presented to optimize them. Our main contribution is to show that a white sequence is minmax optimal for both criteria considered, and that the quest for a generally optimal sequence is hardly motivated.

Decoupled estimation of DOA and angular spread for a spatially distributed source

We consider the problem of estimating the parameters (direction-of-arrival (DOA) and angular spread) of a spatially distributed source, using a uniform linear array. A two-step procedure which enables decoupling the estimation of the DOA from that of the angular spread is proposed. This method combines a covariance matching algorithm with the use of the extended invariance principle (EXIP). More exactly, the first step makes use of an unstructured model for the part of the covariance matrix that depends on the angular spread. Then, the solution is refined by invoking the EXIP. Instead of a 2-D search, the proposed scheme requires two successive 1-D searches. Additionally, the DOA estimate is robust to mismodelling the spatial distribution of the scatterers. A statistical analysis is carried out and a formula for the asymptotic variance of the estimates is derived. Numerical examples illustrate the performance of the method.

Approximate maximum likelihood estimators for array processing in multiplicative noise environments

We consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. Since the exact maximum likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low signal to-noise ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML/sub 1/) estimator is derived. The performance of the AML/sub 1/ estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML/sub 2/ solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis.

A Bayesian Approach to Adaptive Detection in Nonhomogeneous Environments

We consider the adaptive detection of a signal of interest embedded in colored noise, when the environment is nonhomogeneous, i.e., when the training samples used for adaptation do not share the same covariance matrix as the vector under test. A Bayesian framework is proposed where the covariance matrices of the primary and the secondary data are assumed to be random, with some appropriate joint distribution. The prior distributions of these matrices require a rough knowledge about the environment. This provides a flexible, yet simple, knowledge-aided model where the degree of nonhomogeneity can be tuned through some scalar variables. Within this framework, an approximate generalized likelihood ratio test is formulated. Accordingly, two Bayesian versions of the adaptive matched filter are presented, where the conventional maximum likelihood estimate of the primary data covariance matrix is replaced either by its minimum mean-square error estimate or by its maximum a posteriori estimate. Two detectors require generating samples distributed according to the joint posterior distribution of primary and secondary data covariance matrices. This is achieved through the use of a Gibbs sampling strategy. Numerical simulations illustrate the performances of these detectors, and compare them with those of the conventional adaptive matched filter.

Parameter estimation for random amplitude chirp signals

We consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise, i.e., whose amplitude is randomly time-varying. Two methods for solving this problem are presented. First, an unstructured nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional (2-D) maximization problem. A theoretical analysis of the NLS estimator is presented, and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance that is very close to the Cramer-Rao bound. The second approach combines the principles behind the high-order ambiguity function (HBF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of the HAF-based estimator is also carried out, and closed-form expressions are derived for the asymptotic variance of the HAF estimators based on the data and on the squared data. Numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods.

Performance analysis of beamformers using generalized loading of the covariance matrix in the presence of random steering vector errors

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. In this paper, we present a theoretical analysis of the signal-to-interference-plus-noise ratio (SINR) for the class of beamformers based on generalized (i.e., not necessarily diagonal) loading of the covariance matrix in the presence of random steering vector errors. A closed-form expression for the SINR is derived that is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR.

On estimating the frequency of a sinusoid in autoregressive multiplicative noise

Abstract In classical spectral or frequency estimation, generally, the case of sinusoids embedded in additive noise is studied. Since this model is not relevant in some cases, this paper considers a new multiplicative model denoted ARCOS: a sinusoid amplitude modulated by a random autoregressive process. The spectral properties of such a model are derived and used to define a new frequency estimator. Modified versions of the corresponding algorithm are proposed and a theoretical formula for the asymptotic variance of the estimator is derived. Numerical simulations are presented to illustrate the specific nature of the model and the respective performances of the algorithms. An extension to AR( p ). AR( q ) processes, which generalizes the spectral properties of ARCOS, is also proposed.

Estimation of directions of arrival of multiple scattered sources

We consider the problem of estimating the directions of arrival (DOA) of multiple sources in the presence of local scattering. This problem is encountered in wireless communications due to the presence of scatterers in the vicinity of the mobile or when the signals propagate through a random inhomogeneous medium. Assuming a uniform linear array (ULA), we develop DOA estimation algorithms based on covariance matching applied to a reduced-size statistic obtained from the sample covariance matrix after redundancy averaging. Next, a computationally efficient estimator based on AR modelling of the coherence loss function is derived. A theoretical expression for the asymptotic covariance matrix of this estimator is derived. Finally, the corresponding Cramer-Rao bounds (CRBs) are derived. Despite its simplicity, the AR-based estimator is shown to possess performance that is nearly as good as that of the covariance matching method.

Sinusoidal signals with random amplitude: least-squares estimators and their statistical analysis

The asymptotic properties of constrained and unconstrained least-squares estimates of the parameters of a random-amplitude sinusoid are analyzed. An explicit formula for the asymptotic covariance matrix of the estimation errors is derived for both the constrained and unconstrained estimators. Accuracy aspects are investigated with the following main results. For a certain weighting matrix, which is shown to be the same for the constrained and unconstrained methods, the estimation errors achieve their lower bounds. It is proven that in the optimal case, the constrained method always outperforms the unconstrained method. It is also proven that the accuracy of the optimal estimators improves as the number of least-squares equations increases. A formula for the sample length needed for the asymptotic theory to hold is derived, and its dependence on the lowpass modulating sequence is stressed. Simulations provide illustrations of the difference between the constrained and unconstrained estimators as well as the difference between the optimal and basic estimates. The influence of the number of least-squares equations and the characteristics of the lowpass envelope on the estimation accuracy is also investigated.