**1.1 Robust beamforming overview**

Beamforming is a ubiquitous task in array signal processing with applications, among others, in radar, sonar, acoustics, astronomy, seismology, communications, and medical imaging. Without loss of generality, we consider herein beamforming in array processing applications. The introduction to beamforming can be found in [1]-[9] and the references herein.

The traditional approach to the design of adaptive beamformers assumes that the desired signal components are not present in training data, and the robustness of beamformer is known to depend essentially on the availability of signal-free training data. However, in many important applications such as mobile communications, passive location, microphone array speech processing, medical imaging, and radio astronomy, the signal-free training data cells are unavailable. In such scenarios, the desired signal is always present in the training snapshots, and the adaptive beamforming methods become very sensitive to any violation of underlying assumptions on the environment, sources, or sensor array. In fact, the performances of the existing adaptive array algorithms are known to degrade substantially in the presence of even slight mismatches between the actual and presumed array responses to the desired signal [10]-[12]. Similar types of degradation can take place when the array response is known precisely but the training sample size is small, namely the mismatch between the actual and the estimated covariance matrix [13]-[15]. Therefore, robust approaches to adaptive beamforming appear to be of primary importance in these cases [16][17].

Many approaches have been proposed to improve the robustness of the adaptive beamformer during the past three decades. Indeed, the literatures on the robust adaptive beamformer are quite extensive. We provide a brief review as fellows. For more recent detailed critical reviews, see [18] and [19]-[25].

## **1.1.1 Robust approaches for signal direction mismatch**

For the specific case of the signal direction mismatch, there are several efficient methods have been developed. Representative examples of such techniques are the linearly constrained minimum variance (LCMV) beamformer [26], which is also denoted as the linearly constrained minimum power (LCMP) beamformer in other references [27] and this chapter, signal blocking-based algorithms [10][28], and Bayesian beamformer [29]. Although

Robust Beamforming and DOA Estimation 89

method to compute the optimal weight vector, and the second-order cone (SOC) programming-based approach is used to solve the original problem. Ayman Elnashar et al make use of the diagonal loading technique to implement the robust beamformer [24], but the optimal value of diagonal loading level is not solved exactly, alternatively, the diagonal loading technique is integrated into the adaptive update schemes by means of optimum variable loading technique. R. G. Lorenz and S. P. Boyd also solve the similar beamformer by the Lagrange multiplier techniques [23], but they express the weight vector and the array manifold as the direct sum of the corresponding real and imaginary components. Almir Mutapcic et al show that worst-case robust beamforming with multiplicative uncertainty in the weights can be cast as a tractable convex optimization problem [37], but they can't give the solving method, In fact, the proposed robust beamformer with uncertain weights can be

S. Shahbazpanahi et al consider the general-rank signal model, and the robust beamformer is proposed for the distributed sources [21], therein, an elegant closed-form solution is given, but its performance improvement depends on the constraint parameter severely, and

Jian Li et al propose a Capon beamforming approach with the norm inequality constraint (NIC) to improve the robustness against array steering vector errors and noise [22], although the exact solution is given, and optimal loading level can be computed via the proposed method, but by analysis and simulation, its efficiency is not as good as expectation. Since the constraint parameter determines its robustness, but how to select the constraint parameter is

Quadratic inequality constraints (QIC) on the weight vector of LCMP beamformer can improve robustness to pointing errors and random perturbations in sensor parameter [27]. The weights that minimize the output power subject to linear constraints and an inequality constraint on the norm of the weight vector have the same form as that of the optimum LCMP beamformer with diagonal loading of the data covariance matrix. But the optimal loading level cannot be directly expressed as a function of the constraint in a closed form, and cannot be solved exactly. Hence, its application is restricted by the optimal weight vector finding. So that some numerically algorithms are proposed to implement the QICLCMP, such as Least Mean Squares (LMS) or Recursive Least Squares (RLS) [27], but the

This chapter is organized as follows [38]. First, the norm inequality constraint Capon beamformer (NICCB) is introduced and analyzed particularly. Second, the choice of the norm constraint parameter and the selecting bound is discussed. Third, the norm equality constraint Capon beamformer (NECCB) is proposed and is solved effectively. Finally, the

The Capon beamformer can experience significant performance degradation when there is a mismatch between the presumed and actual characteristics of the source or array. The goal of NICCB is to impose an additional inequality constraint on the Euclidean norm of **w** for

**1.1.4 Weight norm constraint approaches for general mismatch** 

converted to that in [19] equivalently.

application effect isn't good as the expectation.

simulation analyses and the conclusion are given.

**1.2 Capon beamformer under norm inequality constraint (NICCB)** 

is not up to optimal.

not discussed.

all these methods provide excellent robustness against the signal direction mismatch, they are not robust against other types of mismatches caused by poor array calibration, unknown sensor mutual coupling, near-far wavefront mismodeling, signal wavefront distortions, source spreading, and coherent/incoherent local scattering, as well as other effects [17].

Chun-Yang Chen and P.P.Vaidyanathan consider a simplified uncertainty set which contains only the steering vectors with a desired uncertainty range of direction of arrival (DOA) [25], although the closed-form solution is given, and the diagonal loading level can be computed by the iteration method systematically, but how to determine the DOA uncertainty range is the critical problem.
