Robust adaptive nonlinear beamforming by kernels and projection mappings

This paper introduces a novel adaptive nonlinear beamforming design by using the wide frame of Reproducing Kernel Hilbert Spaces (RKHS). The task is cast in the framework of convex optimization. A collection of closed convex constraints is developed that describe: (a) the information dictated by the training data and, (b) the required robustness against steering vector errors. Since a time recursive solution is sought, the task is equivalent with the problem of finding a point, in a Hilbert space, that satisfies an infinite number of closed convex constraints. An algorithm is derived using projection mappings. Numerical results show the increased resolution offered by the proposed approach, even with a few antenna elements, as opposed to the classical Linearly Constrained Minimum Variance (LCMV) beamformer, and to a nonlinear regression approach realized by the Kernel Recursive Least Squares (KRLS) method.