Robust Adaptive Beamforming Based on Steering Vector Estimation With as Little as Possible Prior Information

Abstract

A general notion of robustness for robust adaptive beamforming (RAB) problem and a unified principle for minimum variance distortionless response (MVDR) RAB techniques design are formulated. This principle is to use standard MVDR beamformer in tandem with an estimate of the desired signal steering vector found based on some imprecise prior information. Differences between various MVDR RAB techniques occur only because of the differences in the assumed prior information and the corresponding signal steering vector estimation techniques. A new MVDR RAB technique, which uses as little as possible and easy to obtain imprecise prior information, is developed. The objective for estimating the steering vector is the maximization of the beamformer output power, while the constraints are the normalization condition and the requirement that the estimate does not converge to any of the interference steering vectors and their linear combinations. The prior information used is only the imprecise knowledge of the antenna array geometry and angular sector in which the actual steering vector lies. Mathematically, the proposed MVDR RAB is expressed as the well known non-convex quadratically constrained quadratic programming problem with two constraints, which can be efficiently and exactly solved. Some new results for the corresponding optimization problem such as a new algebraic way of finding the rank-one solution from the general-rank solution of the relaxed problem and the condition under which the solution of the relaxed problem is guaranteed to be rank-one are derived. Our simulation results demonstrate the superiority of the proposed method over other previously developed RAB techniques.Index Terms-Output power maximization, robust adaptive beamforming (RAB), quadratically constrained quadratic programming (QCQP), semi-definite programming (SDP) relaxation, steering vector estimation.

Robust Adaptive Beamforming Against Array Calibration Errors

“…We can decompose the error betweenâ(θ d ) and a(θ d ) into two components, i.e., e ⊥ which is orthogonal toâ(θ d ) and e which is parallel toâ(θ d ). We only search for e ⊥ , because any scaling of the steering vector does not impact the SINR [11,13]. e ⊥ can be solved by the following optimization problem:…”

“…Note that the steering vectorã(θ d ) can also be estimated by other methods, such as the method in [13], and by using the reconstructed interference-plus-noise matrixR v can improve the performance of the beamformer in [13] by using R x . The comparison will be shown in Section 3.…”

“…The convex problem will dominate the computational cost, and the computational complexity of (12) is o(N 3.5 ). Therefore, the proposed method has complexity higher than the conventional Capon beamformer (o(N 3 )) but has comparable complexity with other robust adaptive beamforming algorithms [11,13].…”

“…Thus, many techniques have been proposed to improve the robustness of adaptive beamformers in the past decades ( [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], and many references therein). One popular and widely used approach is the so-called diagonal loading technique [5], where the array covariance matrix is diagonally loaded with a scaled identity matrix.…”

“…Many state-of-the-art robust adaptive beamformers will degrade their performance greatly when large calibration errors exist. One promising technique used to process the presumed steering vector of the desired signal is to estimate the actual steering vector in a convex formulation [13]. To prevent the steering vector converge to the steering vector associated with one of the interfering sources, a positive definite matrix is built as:C Φ a(ϕ)a H (ϕ)dϕ, where the angular sectorΦ is the complement sector of Φ, and Φ is the angular sector in which the desired signal is located.…”