Spherical Random Arrays With Application to Aerial Collaborative Beamforming

Abstract

Spherical volumetric arrays with randomly distributed elements are studied using geometrical probability theory. The array elements are considered as points of an isotropic homogeneous binomial point process (BPP), randomly located within a 3-D ball. The presented results are based on the distance distribution of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> th nearest element from the center of the ball. These elements are assumed to be the randomly distributed nodes of a wireless aerial network that cooperate to produce a beam toward a desired distant receiver. An alternative array elements’ distribution is also examined by placing a constraint to the distance of the farthest element and providing novel results on the average array factor, the average power pattern, and the directivity of the array. The half power beamwidth (HPBW), the peak sidelobe level (SLL), and the location of the peaks and nulls of the power pattern are examined in terms of the number of the array elements and the radius of the sphere. Finally, the impact of the location estimation errors on the performance of open-loop collaborative beamforming is examined both theoretically and using empirical simulation results.