**1.1.3 Uncertainty set constraint approaches for general mismatch**

Very recently, many approaches have been proposed for improving the robustness of the standard minimum variance distortionless response (MVDR) beamformer. Their main ideas are based on the definition of the uncertainty set and the worst-case performance optimization, but these algorithms are all classified to the diagonal loading technique.

Jian Li et al propose the robust Capon beamformer under the constraint of steering vector uncertainty set [20], then the constraint of steering vector norm is imposed and the doubly constraint robust Capon beamformer is proposed [22]. For the two beamformers, although they give the exact weight vectors, and the methods of finding the optimal loading level, but their performance improvements are not obvious. Actually, the constraint of uncertainty set is the essence of the two robust beamformers, and the two beamfomer have the same robustness characteristic. F.Vincent and O.Besson also analyze the performance of the beamformer under the uncertainty set constraint approximatively, but they can't give the exact loading level [36].

S. A. Vorobyov et al propose a robust beamformer in the presence of an arbitrary unknown signal steering vector mismatch [19], although they prove the proposed approach equivalent to the loading sample matrix inversion (LSMI) algorithm, but they can't give the direct 88 Fourier Transform Applications

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

Several other approaches are known to provide the improved robustness against more general types of mismatches, for example, the algorithms that use the diagonal loading of the sample covariance matrix [14][16], the eigenspace-based beamformer [11][30][31], and the covariance matrix taper (CMT) approach [32]-[34]. For the diagonal loading method, a serious drawback is that there is no reliable way to choose the diagonal loading level, F.Vincent and O.Besson propose the method to select the optimal loading level with a view to maximizing the signal-to-noise ratio (SNR) in the presence of steering vector errors and it is shown that the loading is negative, but they can't give the exact solution, instead of the approximate solution, moreover, they can't give the expression of steering vector errors [35]. The eigenspace-based approach is essentially restricted in its performance at low SNR and when the dimension of the signal-plus-interference subspace is high, and the dimension must be known in the latter technique [31]. The CMT approach is known to provide an excellent robustness in scenarios with nonstationary interferers, however, its robustness against mismatches of the desired signal array response may be unsatisfactory, furthermore,

Very recently, many approaches have been proposed for improving the robustness of the standard minimum variance distortionless response (MVDR) beamformer. Their main ideas are based on the definition of the uncertainty set and the worst-case performance optimization, but these algorithms are all classified to the diagonal loading technique.

Jian Li et al propose the robust Capon beamformer under the constraint of steering vector uncertainty set [20], then the constraint of steering vector norm is imposed and the doubly constraint robust Capon beamformer is proposed [22]. For the two beamformers, although they give the exact weight vectors, and the methods of finding the optimal loading level, but their performance improvements are not obvious. Actually, the constraint of uncertainty set is the essence of the two robust beamformers, and the two beamfomer have the same robustness characteristic. F.Vincent and O.Besson also analyze the performance of the beamformer under the uncertainty set constraint approximatively, but they can't give the

S. A. Vorobyov et al propose a robust beamformer in the presence of an arbitrary unknown signal steering vector mismatch [19], although they prove the proposed approach equivalent to the loading sample matrix inversion (LSMI) algorithm, but they can't give the direct

uncertainty range is the critical problem.

**1.1.2 Robust approaches for general mismatch** 

it can also be explained as the diagonal loading [33].

exact loading level [36].

**1.1.3 Uncertainty set constraint approaches for general mismatch** 

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 converted to that in [19] equivalently.

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 is not up to optimal.
