Analysis of Planar Random Arrays With Stochastic Geometry Tools

The performance of 2-D random arrays is analyzed using the distance distributions of an isotropic homogeneous binomial point process. The array factor is expressed as the superposition of the patterns of equal-amplitude isotropic elements uniformly randomly distributed in a disk area. The distribution of the distance to the
$k$
th nearest element from the center of the disk is utilized in the summation, providing a novel representation of the array factor in terms of ordered distances. Furthermore, an alternative distribution is examined by setting a constraint to the distance of the farthest element from the origin. The mean range of the elements in the disk and a numerical calculation of the mean interelement distance is provided. The statistical average and the variance of the array factor as well as the average power pattern are derived and compared to previous results. The 3 dB beamwidth, directivity, nulls, and sidelobe peaks are discussed in terms of the number of elements and the array density. Simulation results for various random arrays and comparison of ensemble-averaged parameters with those of deterministic and sparse arrays from the literature are provided. The results are applicable to collaborative beamforming for distributed wireless ad hoc networks.