Arbitrary-Order Superdirective Beamforming for Three-Layer Concentric Circular Sensor Arrays

Superdirectivity can yield large directivity using a small array aperture, but it is sensitive to random errors. An originally proposed superdirectivity model decomposes the optimal beampattern into eigenbeams with different orders, and robust superdirective beampatterns can be obtained by summing eigenbeams with appropriately selected orders. However, the superdirectivity model is only suitable for circular arrays with one layer or two layers. This paper aims at extending the original model to three-layer concentric circular sensor arrays. It is found that the noise cross-correlation matrix of three-layer concentric circular sensor arrays in isotropic noise shows a block-circulant property when the matrix entries are properly arranged. By utilizing this property, the optimal beampattern, directivity factor, and error sensitivity function can be then expressed as the sums of eigenbeams and their directivity factors and error sensitivity functions, respectively, like circular arrays with one layer or two layers. Robust beampatterns can be similarly synthesized by summing some appropriately selected eigenbeams, and the maximum order of the those eigenbeams can be an arbitrary real number in its range. Simulation results show the proposed method can achieve flexible superdirectivity results for three-layer concentric circular sensor arrays.