Theoretical and Practical Solutions for High-Order Superdirectivity of Circular Sensor Arrays

An analytical and closed-form optimal solution expressed in elementary functions for superdirectivity of a circular sensor array is proposed in this paper. By utilizing the circulant property of the data covariance matrix for circular sensor arrays, such solutions are derived to accurately calculate the optimal beamforming vector, optimal beam pattern, corresponding directivity factor, and eigenbeams. Based on these solutions, it is proved that such array possesses several important properties which facilitate an eigenbeam decomposition and synthesis approach for practical implementations. Experiments conducted result in superdirective beams which contain zeroth- to fifth- and sixth-order eigenbeams. As a typical result, the superdirective beam which contains zeroth- to fifth-order eigenbeams has a 28^° beamwidth and a 12-dB directivity index with a small circular array radius of 0.267 wavelength.