Stefano Fortunati

Single snapshot DOA estimation using compressed sensing

This paper deals with the problem of estimating the Directions of Arrival (DOA) of multiple source signals from a single observation of an array data. In particular, an estimation algorithm based on the emerging theory of Compressed Sensing (CS) is analyzed and its statistical properties are investigated. We show that, unlike the classical Fourier beamformer, a CS-based beamformer (CSB) has some desirable properties typical of the adaptive algorithms (e.g. Capon and MUSIC). Particular attention will be devoted to the super-resolution property. Theoretical arguments and simulation analysis are provided in order to prove that the CSB can achieve a resolution below the classical Rayleigh limit.

Single-snapshot DOA estimation by using Compressed Sensing

This paper deals with the problem of estimating the directions of arrival (DOA) of multiple source signals from a single observation vector of an array data. In particular, four estimation algorithms based on the theory of compressed sensing (CS), i.e., the classical ℓ1 minimization (or Least Absolute Shrinkage and Selection Operator, LASSO), the fast smooth ℓ0 minimization, and the Sparse Iterative Covariance-Based Estimator, SPICE and the Iterative Adaptive Approach for Amplitude and Phase Estimation, IAA-APES algorithms, are analyzed, and their statistical properties are investigated and compared with the classical Fourier beamformer (FB) in different simulated scenarios. We show that unlike the classical FB, a CS-based beamformer (CSB) has some desirable properties typical of the adaptive algorithms (e.g., Capon and MUSIC) even in the single snapshot case. Particular attention is devoted to the super-resolution property. Theoretical arguments and simulation analysis provide evidence that a CS-based beamformer can achieve resolution beyond the classical Rayleigh limit. Finally, the theoretical findings are validated by processing a real sonar dataset.

Least Squares Estimation and Cramér–Rao Type Lower Bounds for Relative Sensor Registration Process

This paper concerns the study of the Cramér-Rao type lower bounds for relative sensor registration (or grid-locking) problem. The theoretical performance bound is of fundamental importance both for algorithm performance assessment and for prediction of the best achievable performance given sensor locations, sensor number, and accuracy of sensor measurements. First, a general description of the relative grid-locking problem is given. Afterwards, the measurement model is analyzed. In particular, the nonlinearity of the measurement model and all the biases (attitude biases, measurement biases, and position biases) are taken into account. Finally, the Cramér-Rao lower bound (CRLB) is discussed and two different types of CRLB, the Hybrid CRLB (HCRLB) and the Modified CRLB (MCRLB), are calculated. Theoretical and simulated results are shown.

Maximum likelihood covariance matrix estimation for complex elliptically symmetric distributions under mismatched conditions

This paper deals with the maximum likelihood (ML) estimation of scatter matrix of complex elliptically symmetric (CES) distributed data when the hypothesized and the true model belong to the CES family but are different, then under mismatched model condition. Firstly, we derive the Huber limit, or sandwich matrix expression, for a generic CES model. Then, we compare the performance of mismatched and matched ML estimators to the Huber limit and to the Cramer-Rao lower bound (CRLB) in some relevant study cases.

On the identifiability problem in the presence of random nuisance parameters

This paper concerns with the identifiability of a vector of unknown deterministic parameters. In many practical applications, the data model is affected by additional random parameters whose estimation is not strictly required, the so-called nuisance parameters. In these cases, the classical definition of identifiability, which requires the calculation of the Fisher Information Matrix (FIM) and of its rank, is often difficult or impossible to perform. Instead, the Modified Fisher Information Matrix (MFIM) can be computed. We generalize the main results on the identifiability problem to take the presence of random nuisance parameters into account. We provide an alternative definition of identifiability that can be always applied but that is weaker than the classical definition, and we investigate the relationships between the identifiability condition and the MFIM. Finally, we apply the obtained results to the identifiability in presence of nuisance parameters to the relative grid-locking problem for netted radar system.

The Misspecified Cramér-Rao Bound and Its Application to Scatter Matrix Estimation in Complex Elliptically Symmetric Distributions

This paper focuses on the application of recent results on lower bounds under misspecified models to the estimation of the scatter matrix of complex elliptically symmetric (CES) distributed random vectors. Buildings upon the original works of Q. H. Vuong [Cramér-Rao Bounds for Misspecified Models, Div. of the Humanities and Social Sci., California Inst. of Technol., Pasadena, CA, USA, Working Paper 652, Oct. 1986] and Richmond-Horowitz [“Parameter Bounds on Estimation Accuracy Under Model Misspecification,” IEEE Trans. Signal Process., vol. 63, no. 9, pp. 2263-2278, May 2015], a lower bound, named misspecified Cramér-Rao bound (MCRB), for the error covariance matrix of any unbiased (in a proper sense) estimator of a deterministic parameter vector under misspecified models, is introduced. Then, we show how to apply these results to the problem of estimating the scatter matrix of CES distributed data under data mismodeling. In particular, the performance of the maximum likelihood (ML) estimator are compared, under mismatched conditions, with the MCRB and with the classical CRB in some relevant study cases.

The Constrained Misspecified Cramér–Rao Bound

The aim of this letter is to provide a constrained version of the misspecified Cramér-Rao bound (MCRB). Specifically, the MCRB is a lower bound on the error covariance matrix of any unbiased (in a proper sense) estimator of a deterministic parameter vector under misspecified models, i.e., when the true and the assumed data distributions are different. Here, we aim at finding an expression of the MCRB for estimation problems involving continuously differentiable equality constraints. Our proof generalizes the derivation of the classical constrained CRB (CCRB) by showing that the constrained MCRB (CMCRB) can be obtained by exploiting the building blocks of its unconstrained counterpart and a basis of the null space of the constraints Jacobian matrix. The conditions for the existence of the CMCRB are also discussed.

Matched, mismatched, and robust scatter matrix estimation and hypothesis testing in complex t-distributed data

Scatter matrix estimation and hypothesis testing are fundamental inference problems in a wide variety of signal processing applications. In this paper, we investigate and compare the matched, mismatched, and robust approaches to solve these problems in the context of the complex elliptically symmetric (CES) distributions. The matched approach is when the estimation and detection algorithms are tailored on the correct data distribution, whereas the mismatched approach refers to the case when the scatter matrix estimator and the decision rule are derived under a model assumption that is not correct. The robust approach aims at providing good estimation and detection performance, even if suboptimal, over a large set of possible data models, irrespective of the actual data distribution. Specifically, due to its central importance in both the statistical and engineering applications, we assume for the input data a complex t-distribution. We analyze scatter matrix estimators derived under the three different approaches and compare their mean square error (MSE) with the constrained Cramér-Rao bound (CCRB) and the constrained misspecified Cramér-Rao bound (CMCRB). In addition, the detection performance and false alarm rate (FAR) of the various detection algorithms are compared with that of the clairvoyant optimum detector.

Impact of Flight Disturbances on Airborne Radar Tracking

This correspondence is devoted to the study of the impact of flight disturbances due to the atmospheric turbulence on airborne radar tracking. The performance of a turbulence-ignorant tracking filter is assessed in a turbulent simulated scenario. The turbulence is modelled according to the Dryden model, a correlated, zero-mean, Gaussian random process. A quasi-ellipsoidal racetrack course is chosen for the air platform that carries the radar. The performance of the turbulence-ignorant tracking algorithm is analyzed in terms of mean value and standard deviation of the estimation errors for each component of the position and velocity vectors and compared with the posterior Cramer-Rao lower bound (PCRLB), evaluated for the ideal case of absence of platform vibrations.

A lower bound for the Mismatched Maximum Likelihood estimator

A lower bound on Mean Square Error (MSE) of the estimate of a real deterministic parameter vector under misspecified model is proposed in this paper. In particular, a lower bound on the MSE of the Mismatched Maximum Likelihood (MML) estimator is derived in closed form and its relation with the Huber limit is investigated. Two simple illustrative examples are provided. Finally, the proposed framework is applied to the estimation of the scatter matrix in the Complex Elliptically Symmetric (CES) distribution family.