On the Relationship Between Robust Minimum Variance Beamformers With Probabilistic and Worst-Case Distortionless Response Constraints

An interesting relationship between the prohability-constrained and worst-case optimization based robust minimum variance (MV) beamformers has been discovered. It is shown that both in the cases of circularly symmetric Gaussian and worst-case distributions of the steering vector mismatch. the probability-constrained robust MV beamforming problem can he tightly approximated as a convex second-order cone programming (SOCII) problem. The latter problem is mathematically equivalent to that resulting from the deterministic worst-case approach and, therefore. probability-constrained beamformers can be interpreted and implemented using their deterministic worst-case counterparts. However, an important advantage of the developed probability-constrained MV beamformers with respect to their standard worst-case counterparts is that the former approaches enable to explicitly, quantify the parameters of the uncertainty, region in terms of the beamformer outage probability..