Pascal Larzabal

A New Barankin Bound Approximation for the Prediction of the Threshold Region Performance of Maximum Likelihood Estimators

It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain signal-to-noise ratio (SNR) or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the Cramer-Rao bound (CRB). The search of the SNR threshold value (where the CRB becomes unreliable for prediction of maximum likelihood estimator variance) can be achieved with the help of the Barankin bound (BB), as proposed by many authors. The major drawback of the BB, in comparison with the CRB, is the absence of a general analytical formula, which compels one to resort to a discrete form, usually the Mcaulay-Seidman bound (MSB), requesting the search of an optimum over a set of test points. In this paper, we propose a new practical BB discrete form that provides, for a given set of test points, an improved SNR threshold prediction in comparison with existing approximations (MSB, Abel bound, Mcaulay-Hofstetter bound) at the expense of the computational complexity increased by a factor les (P+1)3 , where P is the number of unknown parameters. We have derived its expression for the general Gaussian observation model to be used in place of existing approximations.

The Bayesian ABEL Bound on the Mean Square Error

This paper deals with lower bound on the mean square error (MSE). In the Bayesian framework, we present a new bound which is derived from a constrained optimization problem. This bound is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakai bound, and the Bayesian Cramer-Rao bound

Improving the threshold performance of maximum likelihood estimation of direction of arrival

We propose to improve the performance of some direction of arrival (DOA) estimators using array of sensors. We consider those maximum likelihood (ML) estimators that generate some DOA candidates and select one of them through an ML criterion. Our proposal modifies the candidate selection process substituting the traditional sample covariance matrix by a new one computed after filtering the received data with an optimum noise reduction filter. Simulation results indicate an improvement of the performance at low signal-to-noise ratios (SNR) and a considerable reduction of the threshold SNR. The computation of the new selection cost function implies in a small increase in the overall computational effort.

A parameterized maximum likelihood method for multipaths channels estimation

In paths localization, a resolution that goes beyond the classical Rayleigh beamwidth is of great interest. To improve the resolution, model based techniques have been introduced (high resolution methods), but they are very sensitive to noise correlation and they assume underlying data model. We develop a parameterized maximum likelihood (PML) technique, based on a knowledge of the transmitted signal. We develop the exact PML approach and present its implementation by a Gauss Newton procedure. Simulations on data sets are examined. The performances are compared to the Cramer Rao bound. Its superiority over the traditional matched filter (MF) and the conditional maximum likelihood (CML) is shown. The paper concludes with the improvements introduced by a knowledge of the transmitted signals.

Compressed Sensing with Basis Mismatch: Performance Bounds and Sparse-Based Estimator

Compressed sensing (CS) is a promising emerging domain that outperforms the classical limit of the Shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix. Unfortunately, in realistic scenario, the knowledge of this basis or equivalently of the entire dictionary is often uncertain, i.e., corrupted by a Basis Mismatch (BM) error. The consequence of the BM problem is that the estimation accuracy in terms of Bayesian Mean Square Error (BMSE) of popular sparse-based estimators collapses even if the support is perfectly estimated and in the high Signal to Noise Ratio (SNR) regime. This saturation effect considerably limits the effective viability of these estimation schemes. In the first part of this work, the Bayesian Cramér-Rao Bound (BCRB) is derived for CS model with unstructured BM. We show that the BCRB foresees the saturation effect of the estimation accuracy of standard sparse-based estimators as for instance the OMP, Cosamp or the BP. In addition, we provide an approximation of this BMSE threshold. In the second part and in the context of the structured BM model, a new estimation scheme called Bias-Correction Estimator (BiCE) is proposed and its statistical properties are studied. The BiCE acts as a post-processing estimation layer for any sparse-based estimator and mitigates considerably the BM degradation. Finally, the BiCE i) is a blind algorithm, i.e., is unaware of the uncorrupted dictionary matrix, ii) is generic since it can be associated to any sparse-based estimator, iii) is fast, i.e., the additional computational cost remains low, and iv) has good statistical properties. To illustrate our results and propositions, the BiCE is applied in the challenging context of the compressive sampling of non-bandlimited impulsive signals.

Large-System Estimation Performance in Noisy Compressed Sensing With Random Support of Known Cardinality—A Bayesian Analysis

Compressed sensing (CS) enables measurement reconstruction by using sampling rates below the Nyquist rate, as long as the amplitude vector of interest is sparse. In this paper, we first derive and analyze the Bayesian Cramér-Rao bound (BCRB) for the amplitude vector when the set of indices (the support) of its nonzero entries is known. We consider the following context: i) The dictionary is nonstochastic but randomly generated; ii) the number of measurements and the support cardinality grow to infinity in a controlled manner, i.e., the ratio of these quantities converges to a constant; iii) the support is random; and iv) the vector of nonzero amplitudes follow a multidimensional generalized normal distribution. Using results from random matrix theory, we obtain closed-form approximations of the BCRB. These approximations can be formulated in a very compact form in low and high SNR regimes. Second, we provide a statistical analysis of the variance and the statistical efficiency of the oracle linear mean-square-error (LMMSE) estimator. Finally, we present results from numerical investigations in the context of non-bandlimited finite-rate-of-innovation (FRI) signal sampling. We show that the performance of Bayesian mean-square error (BMSE) estimators that are aware of the cardinality of the support, such as OMP and CoSaMP, are in good agreement with the developed lower bounds in the high SNR regime. Conversely, sparse estimators exploiting only the knowledge of the parameter vector and the noise variance in form of a priori distributions of these parameters, like LASSO and BPDN, are not efficient at high SNR. However, at low SNR, their BMSE is lower than that of the former estimators and may be close to the BCRB.

Adaptive maximum likelihood algorithms for the blind tracking of time-varying multipath channels

Transmissions through multipath channels suffer from Rayleigh fading and intersymbol interference. This can be overcome by sending a (known) training sequence and identifying the channel (active identification). However, in a non-stationary context, the channel model has to be updated by periodically sending the training sequence, thus reducing the transmission rate. We address herein the problem of blind identification, which does not require such a sequence and allows a higher transmission rate. We have first proposed a two-stage algorithm (see Reference 2) for the blind identification of multipath channel. We investigate here the maximum-likelihood approach for the blind estimation of channel parameters. In order to track non-stationary channels, we have derived an adaptive (Kalman) algorithm which directly estimates the entire set of characteristic parameters. An original adaptive estimation of the noise model has been proposed for this investigation. Furthermore, the proposed method can easily cope with a model including Doppler shift, which is not directly possible with more common methods. Monte-Carlo simulations confirm the expected results and demonstrate the performance.

Lower bounds for non standard deterministic estimation

In this paper, non standard deterministic parameters estimation is considered, i.e. the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on additional random variables. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known deterministic lower bounds on the mean-squared-error (MSE). However an embedding mechanism allows to transpose all the known lowers bounds into modified lower bounds fitted with non-standard deterministic estimation, encompassing the modified Cramér-Rao/Bhattacharyya bounds and hybrid lower bounds.

Hybrid Barankin–Weiss–Weinstein Bounds

This letter investigates hybrid lower bounds on the mean square error in order to predict the so-called threshold effect. A new family of tighter hybrid large error bounds based on linear transformations (discrete or integral) of a mixture of the McAulay-Seidman bound and the Weiss-Weinstein bound is provided in multivariate parameters case with multiple test points. For use in applications, we give a closed-form expression of the proposed bound for a set of Gaussian observation models with parameterized mean, including tones estimation which exemplifies the threshold prediction capability of the proposed bound.

Recursive Hybrid Cramér–Rao Bound for Discrete-Time Markovian Dynamic Systems

In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. As a contribution to the hybrid estimation framework, we introduce a recursive hybrid Cramér-Rao lower bound for discrete-time Markovian dynamic systems depending on unknown deterministic parameters. Additionally, the regularity conditions required for its existence and its use are clarified.