The gain achieved by a superdirective beamformer operating in a diffuse noise-field is significantly higher than the gain attainable with conventional delay-and-sum weights. A classical result states that for a compact linear array consisting of N sensors which receives a plane-wave signal from the end-fire direction, the optimal superdirective gain approaches N2. It has been noted that in the near-field regime higher gains can be attained. The gain can increase, in theory, without bound for increasing wavelength or decreasing source-receiver distance. We aim to address the phenomenon of near-field superdirectivity in a comprehensive manner. We derive the optimal performance for the limiting case of an infinitesimal-aperture array receiving a spherical-wave signal. This is done with the aid of a sequence of linear transformations. The resulting gain expression is a polynomial, which depends on the number of sensors employed, the wavelength, and the source-receiver distance. The resulting gain curves are optimal and outperform weights corresponding to other superdirectivity methods. The practical case of a finite-aperture array is discussed. We present conditions for which the gain of such an array would approach that predicted by the theory of the infinitesimal case. The white noise gain (WNG) metric of robustness is shown to increase in the near-field regime.

The directivity factor (DF) of a beamformer describes its spatial selectivity and ability to suppress diffuse noise which arrives from all directions. For a given array constellation, it is possible to select beamforming weights which maximize the DF for a particular look-direction, while enforcing nulls for a set of undesired directions. In general, the resulting DF is dependent upon the specific look- and null directions. Using the same array, one may apply a different set of weights designed for any other feasible set of look- and null directions. In this contribution, we show that when the optimal DF is averaged over all look directions, the result equals the number of sensors minus the number of null constraints. This result holds regardless of the positions and spatial responses of the individual sensors and regardless of the null directions. The result generalizes to more complex wave-propagation domains (e.g., reverberation).

The optimal weights for a beamformer that provide maximum directivity, are often found to be severely lacking in terms of robustness. Although an ideal implementation of the beamformer with these weights provides high directivity, minor perturbations of the weights or of sensor placement cause severe degradation. Therefore, a robustness constraint is often imposed during the beamformer's design stage. The classical method of diagonal loading is commonly used for this purpose. There are known results in this field which pertain to an array consisting of sensors with identical directivity-patterns and orientations. We extend these results to account for sensors with nonidentical directivity patterns, and sensors which share placement errors. We show that in such cases, modification of the classical loading scheme to incorporate nonidentical diagonal elements and off-diagonal elements is beneficial.

The beampattern of an array consisting of$N$elements is determined by the beampatterns of the individual elements, their placement, and the weights assigned to them. For each look direction, it is possible to design weights that maximize the array directivity factor (DF). For the case of an array of omnidirectional elements using optimal weights, it has been shown that the average DF over all look directions equals the number of elements. The validity of this theorem is not dependent on array geometry. We generalize this theorem by means of an alternative proof. The chief contributions of this letter are a) a compact and direct proof, b) generalization to arrays containing directional elements (such as cardioids and dipoles), and c) generalization to arbitrary wave propagation models. A discussion of the theorem's ramifications on array processing is provided.

A vector-sensor consisting of a monopole sensor collocated with orthogonally oriented dipole sensors can be used for direction-of arrival (DOA) estimation. A method is proposed to estimate the DOA based on the direction of maximum power. Algorithms mentioned in earlier works are shown to be special cases of the proposed method. An iterative algorithm based on the principal of gradient ascent is presented for the solution of the maximum power problem. The proposed maximum-power method is shown to approach the Cramer-Rao lower bound (CRLB) with a suitable choice of parameter.