Arbitrary-order superdirectivity of circular sensor arrays

Superdirectivity can yield large directivity using a small array aperture, but it suffers from high sensitivity to random errors. In a previously proposed model, robust Nth-order superdirectivity was obtained by combining low-order superdirectivity sub-solutions with orders from 0 to N. However, the orders N were exclusively positive integers, which may lead to an over-improved robustness with an excessively degraded directivity. This paper presents an arbitrary-order superdirectivity model of circular sensor arrays, which can achieve a more flexible compromise between directivity and robustness than the previous model. A coefficient η is weighted at the (N + 1)th-order entry of the optimal weighting vector and the new weighting vector will consist of the 0th- to Nth-order and the weighted (N + 1)th-order entries. The beampattern, the directivity factor, and the error sensitivity function can then be expressed as functions of the coefficient η. Moreover, broadband superdirective beampatterns with a desired directivity factor or error sensitivity function can be readily synthesized using the closed-form solutions derived from the arbitrary-order superdirectivity model. Simulations and experimental results show that the proposed model can easily synthesize superdirective beampatterns with arbitrary orders, and it is more flexible than the previous integer-order model. It is also demonstrated that the performance of the proposed method is similar to that of the optimization-based method, but it uses a simple closed-form weighting vector rather than a numerical algorithm.