Design of an Optimum Superdirective Beamformer Through Generalized Directivity Maximization

Maximization of the directivity with a constraint on the lower bound of the white noise gain is a typical strategy used to attain robust performance in superdirective arrays. However, array imperfections and sensor mismatches significantly reduce the nominal directivity, even in robust designs. We focus on the generalized directivity (i.e., the directivity of the mean beam power pattern, computed using the probability density functions of the array errors) as a reliable metric to assess the average performance. We show that the generalized directivity is an extension of the nominal directivity, corresponds to the array gain when the array errors are considered, and is strictly related to the mean directivity. We demonstrate that, unlike the mean directivity, the generalized directivity of a linear array (under fair assumptions for gain, phase, and position errors) can be computed in closed form. Starting from this result, we propose a method to analytically compute, through the generalized eigenvalues, the weight vector that maximizes the generalized directivity. The numerical results confirm that maximizing the performance by setting a lower bound for the white noise gain a priori is a critical operation. Differently, maximization of the generalized directivity directly provides the optimum average performance. Notable advantages in designing maximum directivity arrays and in achieving the maximum degree of superdirectivity are reported, particularly for linear arrays steered at end-fire.