Habti Abeida

Cramer-Rao bounds of DOA estimates for BPSK and QPSK Modulated signals

This paper focuses on the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive circular complex Gaussian noise. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform are given. These CRBs are compared, on the one hand, with those obtained with different a priori knowledge and, on the other hand, with CRBs under the noncircular and circular complex Gaussian distribution and with different deterministic CRBs. It is shown in particular that the CRBs under the noncircular [respectively, circular] complex Gaussian distribution are tight upper bounds on the CRBs under the BPSK [respectively, QPSK] distribution at very low and very high signal-to-noise ratios (SNRs) only. Finally, these results and comparisons are extended to the case of two independent BPSK or QPSK distributed sources where an explicit expression of the CRB for the DOA parameters alone is given for large SNR.

Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing

This paper presents a series of user parameter-free iterative Sparse Asymptotic Minimum Variance (SAMV) approaches for array processing applications based on the asymptotically minimum variance (AMV) criterion. With the assumption of abundant snapshots in the direction-of-arrival (DOA) estimation problem, the signal powers and noise variance are jointly estimated by the proposed iterative AMV approach, which is later proved to coincide with the Maximum Likelihood (ML) estimator. We then propose a series of power-based iterative SAMV approaches, which are robust against insufficient snapshots, coherent sources and arbitrary array geometries. Moreover, to overcome the direction grid limitation on the estimation accuracy, the SAMV-Stochastic ML (SAMV-SML) approaches are derived by explicitly minimizing a closed form stochastic ML cost function with respect to one scalar paramter, eliminating the need of any additional grid refinement techniques. To assist the performance evaluation, approximate solutions to the SAMV approaches are also provided for high signal-to-noise ratio (SNR) and low SNR scenarios. Finally, numerical examples are generated to compare the performances of the proposed approaches with those of the existing ones.

SAR imaging via efficient implementations of sparse ML approaches

High-resolution spectral estimation techniques are of notable interest for synthetic aperture radar (SAR) imaging. Several sparse estimation techniques have been shown to provide significant performance gains as compared to conventional approaches. We consider efficient implementation of the recent iterative sparse maximum likelihood-based approaches (SMLAs). Furthermore, we present approximative fast SMLA formulation using the Quasi-Newton approach, as well as consider hybrid SMLA-MAP algorithms. The effectiveness of the discussed techniques is illustrated using numerical and experimental examples.

Fast implementation of sparse iterative covariance-based estimation for source localization

Fast implementations of the sparse iterative covariance-based estimation (SPICE) algorithm are presented for source localization with a uniform linear array (ULA). SPICE is a robust, user parameter-free, high-resolution, iterative, and globally convergent estimation algorithm for array processing. SPICE offers superior resolution and lower sidelobe levels for source localization compared to the conventional delay-and-sum beamforming method; however, a traditional SPICE implementation has a higher computational complexity (which is exacerbated in higher dimensional data). It is shown that the computational complexity of the SPICE algorithm can be mitigated by exploiting the Toeplitz structure of the array output covariance matrix using Gohberg-Semencul factorization. The SPICE algorithm is also extended to the acoustic vector-sensor ULA scenario with a specific nonuniform white noise assumption, and the fast implementation is developed based on the block Toeplitz properties of the array output covariance matrix. Finally, numerical simulations illustrate the computational gains of the proposed methods.

Fast implementation of sparse iterative covariance-based estimation for array processing

Fast implementations of the SParse Iterative Covariance-based Estimation (SPICE) algorithm are presented for source localization in passive sonar applications. SPICE is a robust, user parameter-free, high-resolution, iterative and globally convergent estimation algorithm for array processing. SPICE offers superior resolution and lower sidelobe levels for source localization at the cost of a higher computational complexity compared to the conventional delay-and-sum beamforming method. It is shown in this paper that the computational complexity of the SPICE algorithm can be reduced by exploiting the Toeplitz structure of the array output covariance matrix using the Gohberg-Semencul factorization. The fast implementations for both the hydrophone uniform linear array (ULA) and the vector-sensor ULA scenarios are proposed and the computational gains are illustrated by numerical simulations.

Direct Derivation of the Stochastic CRB of DOA Estimation for Rectilinear Sources

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 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 letter 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.

Fast implementation of SAR imaging using sparse ML methods

High-resolution sparse spectral estimation techniques have recently been shown to offer significant performance gains as compared to most conventional estimation approaches, although such methods typically suffer the drawback of being computationally cumbersome. In this paper, we seek to alleviate this drawback somewhat, examining computationally efficient implementations of the recent iterative sparse maximum likelihood-based approaches (SMLA), exploiting the inherent rich structure of these estimators. The derived implementations reduce the resulting computational complexity with at least one order of magnitude, while still yielding exact implementations. The effectiveness of the discussed techniques are illustrated using experimental examples.

Survey and some new results on performance analysis of complex-valued parameter estimators

Recently, there has been an increased awareness that simplistic adaptation of performance analysis developed for random real-valued signals and parameters to the complex case may be inadequate or may lead to intractable calculations. Unfortunately, many fundamental statistical tools for handling complex-valued parameter estimators are missing or scattered in the open literature. In this paper, we survey some known results and provide a rigorous and unified framework to study the statistical performance of complex-valued parameter estimators with a particular attention paid to properness (i.e., second order circularity), specifically referring to the second-order statistical properties. In particular, some new properties relative to the properness of the estimates, asymptotically minimum variance bound and Whittle formulas are presented. A new look at the role of nuisance parameters is given, proving and illustrating that the noncircular Gaussian distributions do not necessarily improve the Cramer-Rao bound (CRB) with respect to the circular case. Efficiency of subspace-based complex-valued parameter estimators that are presented with a special emphasis is put on noisy linear mixture.