Rules-of-thumb to design a uniform spherical array for direction finding : its Cramér–Rao bounds' nonlinear dependence on the number of sensors

Abstract

This paper discovers rules-of-thumb on how the estimation precision for an incident source's azimuth-polar direction-of-arrival (ϕ,θ) depends on the number (L) of identical isotropic sensors spaced uniformly on an open sphere of radius R. This estimation's corresponding Cramér-Rao bounds (CRB) are found to follow these elegantly simple approximations, useful for array design: (i) For the azimuth arrival angle: 2π(R/λ)(σs/σn)2LMCRB(ϕ) sin(θ)≈(Le1/14)-1+3→L→∞3, ∀(ϕ,θ); and (ii) for the polar arrival angle: 2π(R/λ)(σs/σn)2LMCRB(θ)≈3-(Le6/7)-1→L→∞3, ∀(ϕ,θ). Here, M denotes the number of snapshots, λ refers to the incident signal's wavelength, and (σs/σn)2 symbolizes the signal-to-noise power ratio.