An overview of beamforming from a signal-processing perspective is provided, with an emphasis on recent research. Data-independent, statistically optimum, adaptive, and partially adaptive beamforming are discussed. Basic notation, terminology, and concepts are included. Several beamformer implementations are briefly described. >

Beamforming: a versatile approach to spatial filtering

1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.

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Spectral analysis of signals

Array processing involves manipulation of signals induced on various antenna elements. Its capabilities of steering nulls to reduce cochannel interferences and pointing independent beams toward various mobiles, as well as its ability to provide estimates of directions of radiating sources, make it attractive to a mobile communications system designer. Array processing is expected to play an important role in fulfilling the increased demands of various mobile communications services. Part I of this paper showed how an array could be utilized in different configurations to improve the performance of mobile communications systems, with references to various studies where feasibility of apt array system for mobile communications is considered. This paper provides a comprehensive and detailed treatment of different beam-forming schemes, adaptive algorithms to adjust the required weighting on antennas, direction-of-arrival estimation methods-including their performance comparison-and effects of errors on the performance of an array system, as well as schemes to alleviate them. This paper brings together almost all aspects of array signal processing.

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Application of antenna arrays to mobile communications. II. Beam-forming and direction-of-arrival considerations

Introduction to linear and nonlinear programming

Adaptive beamforming algorithms can be extremely sensitive to slight errors in array characteristics. Errors which are uncorrelated from sensor to sensor pass through the beamformer like uncorrelated or spatially white noise. Hence, gain against white noise is a measure of robustness. A new algorithm is presented which includes a quadratic inequality constraint on the array gain against uncorrelated noise, while minimizing output power subject to multiple linear equality constraints. It is shown that a simple scaling of the projection of tentative weights, in the subspace orthogonal to the linear constraints, can be used to satisfy the quadratic inequality constraint. Moreover, this scaling is equivalent to a projection onto the quadratic constraint boundary so that the usual favorable properties of projection algorithms apply. This leads to a simple, effective, robust adaptive beamforming algorithm in which all constraints are satisfied exactly at each step and roundoff errors do not accumulate. The algorithm is then extended to the case of a more general quadratic constraint.

Robust adaptive beamforming

The probability hypothesis density (PHD) recursion propagates the posterior intensity of the random finite set (RFS) of targets in time. The cardinalized PHD (CPHD) recursion is a generalization of the PHD recursion, which jointly propagates the posterior intensity and the posterior cardinality distribution. In general, the CPHD recursion is computationally intractable. This paper proposes a closed-form solution to the CPHD recursion under linear Gaussian assumptions on the target dynamics and birth process. Based on this solution, an effective multitarget tracking algorithm is developed. Extensions of the proposed closed-form recursion to accommodate nonlinear models are also given using linearization and unscented transform techniques. The proposed CPHD implementations not only sidestep the need to perform data association found in traditional methods, but also dramatically improve the accuracy of individual state estimates as well as the variance of the estimated number of targets when compared to the standard PHD filter. Our implementations only have a cubic complexity, but simulations suggest favorable performance compared to the standard Joint Probabilistic Data Association (JPDA) filter which has a nonpolynomial complexity.

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Analytic Implementations of the Cardinalized Probability Hypothesis Density Filter

It is shown analytically that the multitarget multiBernoulli (MeMBer) recursion, proposed by Mahler, has a significant bias in the number of targets. To reduce the cardinality bias, a novel multiBernoulli approximation to the multi-target Bayes recursion is derived. Under the same assumptions as the MeMBer recursion, the proposed recursion is unbiased. In addition, a sequential Monte Carlo (SMC) implementation (for generic models) and a Gaussian mixture (GM) implementation (for linear Gaussian models) are proposed. The latter is also extended to accommodate mildly nonlinear models by linearization and the unscented transform.

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The Cardinality Balanced Multi-Target Multi-Bernoulli Filter and Its Implementations

Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

In this paper a class of linear constraints, also termed as derivative constraints, which is applicable to broad-band element space antenna array processors, is presented. The performance characteristics of the optimum processor with derivative constraints are demonstrated by computer studies involving two types of array geometries, namely linear and circular arrays. As a consequence of derivative constraints, the beam width in the look direction can be made as broad as desired and the beam spacings can be selected without fear of substantial signal suppression in the event of signal arrivals between beams. However, this increased beam width is achieved at the price of reducing array gain.

Derivative constraints for broad-band element space antenna array processors

This paper presents a novel and mathematically rigorous Bayes' recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter. Our Bayesian formulation is mathematically well-founded due to our use of a consistent likelihood function derived from random finite set theory. It is established that under certain assumptions, the proposed Bayes' recursion reduces to the cardinalized probability hypothesis density (CPHD) recursion for a single target. A particle implementation of the proposed recursion is given. Under linear Gaussian and constant sensor field of view assumptions, an exact closed-form solution to the proposed recursion is derived, and efficient implementations are given. Extensions of the closed-form recursion to accommodate mild nonlinearities are also given using linearization and unscented transforms.

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Antonio Cantoni

Papers

Analytic Implementations of the Cardinalized Probability Hypothesis Density Filter

The Cardinality Balanced Multi-Target Multi-Bernoulli Filter and Its Implementations

Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

Derivative constraints for broad-band element space antenna array processors

Bayesian Filtering With Random Finite Set Observations