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

Abstract

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.