EM-Type Algorithms for DOA Estimation in Unknown Nonuniform Noise

The expectation--maximization (EM) algorithm updates all of the parameter estimates simultaneously, which is not applicable to direction of arrival (DOA) estimation in unknown nonuniform noise. In this work, we present several efficient EM-type algorithms, which update the parameter estimates sequentially, for solving both the deterministic and stochastic maximum--likelihood (ML) direction finding problems in unknown nonuniform noise. Specifically, we design a generalized EM (GEM) algorithm and a space-alternating generalized EM (SAGE) algorithm for computing the deterministic ML estimator. Simulation results show that the SAGE algorithm outperforms the GEM algorithm. Moreover, we design two SAGE algorithms for computing the stochastic ML estimator, in which the first updates the DOA estimates simultaneously while the second updates the DOA estimates sequentially. Simulation results show that the second SAGE algorithm outperforms the first one.Comment: arXiv admin note: text overlap with arXiv:2208.0751

Efficient Computation of ML DOA Estimates Under Unknown Nonuniform Sensor Noise Powers

This paper presents an efficient method for computing Maximum Likelihood (ML) direction-of-arrival (DOA) estimates in scenarios in which the sensor noise powers are nonuniform and unknown. The method combines the Alternating Projection (AP) algorithm for coarsely locating additional DOAs and Newton iterations for finally obtaining the ML estimates. Compared with the existing approaches, the method reduces the computational burden significantly due to the small number of Newton iterations required and to the efficient computation of each iteration. Specifically, the iterations are computed in a small number of arithmetic operations thanks to the closed-form formulas for the gradient and Hessian of the ML cost functions presented in this paper. The method’s total computational burden is of just a few mega-flops (mega floating-point operations) in typical cases. We present the method for the deterministic and stochastic ML estimators. Then, an analysis of the deterministic ML cost function’s gradient reveals an unexpected drawback: its associated estimator is either degenerate or inconsistent. Finally, we assess the method’s root-mean-square (RMS) error and computational burden numerically and compare it with other relevant estimators in the literature.This work was supported by the Spanish Ministry of Science and Innovation, the State Agency of Research (AEI) and the European Funds for Regional Development (EFRD) under Project PID2020-117303GB-C22

A review of closed-form Cramér-Rao Bounds for DOA estimation in the presence of Gaussian noise under a unified framework

The Cramér-Rao Bound (CRB) for direction of arrival (DOA) estimation has been extensively studied over the past four decades, with a plethora of CRB expressions reported for various parametric models. In the literature, there are different methods to derive a closed-form CRB expression, but many derivations tend to involve intricate matrix manipulations which appear difficult to understand. Starting from the Slepian-Bangs formula and following the simplest derivation approach, this paper reviews a number of closed-form Gaussian CRB expressions for the DOA parameter under a unified framework, based on which all the specific CRB presentations can be derived concisely. The results cover three scenarios: narrowband complex circular signals, narrowband complex noncircular signals, and wideband signals. Three signal models are considered: the deterministic model, the stochastic Gaussian model, and the stochastic Gaussian model with the a priori knowledge that the sources are spatially uncorrelated. Moreover, three Gaussian noise models distinguished by the structure of the noise covariance matrix are concerned: spatially uncorrelated noise with unknown either identical or distinct variances at different sensors, and arbitrary unknown noise. In each scenario, a unified framework for the DOA-related block of the deterministic/stochastic CRB is developed, which encompasses one class of closed-form deterministic CRB expressions and two classes of stochastic ones under the three noise models. Comparisons among different CRBs across classes and scenarios are presented, yielding a series of equalities and inequalities which reflect the benchmark for the estimation efficiency under various situations. Furthermore, validity of all CRB expressions are examined, with some specific results for linear arrays provided, leading to several upper bounds on the number of resolvable Gaussian sources in the underdetermined case

Regularized sampling of multiband signals

This paper presents a regularized sampling method for multiband signals, that makes it possible to approach the Landau limit, while keeping the sensitivity to noise at a low level. The method is based on band-limited windowing, followed by trigonometric approximation in consecutive time intervals. The key point is that the trigonometric approximation "inherits" the multiband property, that is, its coefficients are formed by bursts of non-zero elements corresponding to the multiband components. It is shown that this method can be well combined with the recently proposed synchronous multi-rate sampling (SMRS) scheme, given that the resulting linear system is sparse and formed by ones and zeroes. The proposed method allows one to trade sampling efficiency for noise sensitivity, and is specially well suited for bounded signals with unbounded energy like those in communications, navigation, audio systems, etc. Besides, it is also applicable to finite energy signals and periodic band-limited signals (trigonometric polynomials). The paper includes a subspace method for blindly estimating the support of the multiband signal as well as its components, and the results are validated through several numerical examples.Comment: The title and introduction have changed. Submitted to the IEEE Transactions on Signal Processin