Robust Adaptive Beamforming for General-Rank Signal Model with Positive Semi-Definite Constraint via POTDC

The robust adaptive beamforming (RAB) problem for general-rank signal model with an additional positive semi-definite constraint is considered. Using the principle of the worst-case performance optimization, such RAB problem leads to a difference-of-convex functions (DC) optimization problem. The existing approaches for solving the resulted non-convex DC problem are based on approximations and find only suboptimal solutions. Here we solve the non-convex DC problem rigorously and give arguments suggesting that the solution is globally optimal. Particularly, we rewrite the problem as the minimization of a one-dimensional optimal value function whose corresponding optimization problem is non-convex. Then, the optimal value function is replaced with another equivalent one, for which the corresponding optimization problem is convex. The new one-dimensional optimal value function is minimized iteratively via polynomial time DC (POTDC) algorithm.We show that our solution satisfies the Karush-Kuhn-Tucker (KKT) optimality conditions and there is a strong evidence that such solution is also globally optimal. Towards this conclusion, we conjecture that the new optimal value function is a convex function. The new RAB method shows superior performance compared to the other state-of-the-art general-rank RAB methods.Comment: 29 pages, 7 figures, 2 tables, Submitted to IEEE Trans. Signal Processing on August 201

Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems

Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur as well in other signal processing applications and are typically solved using different modifications of the branch-and-bound method. This method, however, does not have any polynomial time complexity guarantees. In this paper, we show that a class of DC programming problems, to which the sum-rate maximization in two-way MIMO relaying belongs, can be solved very efficiently in polynomial time, and develop two algorithms. The objective function of the problem is represented as a product of quadratic ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial-Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We also derive an upper-bound for the optimal values of the corresponding optimization problem and show by simulations that this upper-bound can be achieved by both algorithms. The proposed methods for maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing in Nov. 201

Twenty-Five Years of Advances in Beamforming: From Convex and Nonconvex Optimization to Learning Techniques

Beamforming is a signal processing technique to steer, shape, and focus an electromagnetic wave using an array of sensors toward a desired direction. It has been used in several engineering applications such as radar, sonar, acoustics, astronomy, seismology, medical imaging, and communications. With the advances in multi-antenna technologies largely for radar and communications, there has been a great interest on beamformer design mostly relying on convex/nonconvex optimization. Recently, machine learning is being leveraged for obtaining attractive solutions to more complex beamforming problems. This article captures the evolution of beamforming in the last twenty-five years from convex-to-nonconvex optimization and optimization-to-learning approaches. It provides a glimpse of this important signal processing technique into a variety of transmit-receive architectures, propagation zones, paths, and conventional/emerging applications

An alternative to diagonal loading for implementation of a white noise array gain constrained robust beamformer

Diagonal loading is one of the most popular methods of robust adaptive beamforming, and the solution to many different problems aimed at producing beamformers which are robust to finite samples effects or/and steering vector errors. Among the latter, constraining the white noise array gain (WNAG) is a meaningful approach. However, relating the loading level to the desired WNAG is not straightforward. In this communication, using a generalized sidelobe canceler structure of the beamformer, we prove that the WNAG constraint can be encoded directly in the beamformer, and the latter can be obtained in a rather simple way from a specific eigenvector and without going through the diagonal loading step

Principles of minimum variance robust adaptive beamforming design

Robustness is typically understood as an ability of adaptive beamforming algorithm to achieve high performance in the situations with imperfect, incomplete, or erroneous knowledge about the source, propagation media, and antenna array. It is also desired to achieve high performance with as little as possible prior information. In the last decade, several fruitful principles to minimum variance distortionless response (MVDR) robust adaptive beamforming (RAB) design have been developed and successfully applied to solve a number of problems in a wide range of applications. Such principles of MVDR RAB design are summarized here in a single paper. Prof. Gershman has actively participated in the development and applications of a number of such MVDR RAB design principles