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.

Bayesian Inference for Acoustic Direction of Arrival Analysis Using Spherical Harmonics

“…Figure 4a illustrates the sound energy distributions derived from experimentally measured data using Equations (8)- (12), while Figure 4b shows the predicted results using Equations (13)- (15) to visualize the sound field distribution around the spherical microphone array in Cartesian coordinates. The grid resolution for these two-dimensional maps is 3.6 • × 3.6 • with grid points of K × J = 100 × 50 across azimuth and elevation range as expressed in Equation (25). Figure 5 illustrates Bayes factor estimations over the different models H S from Equation (15).…”

“…A number of other recent investigations also exist using spherical harmonics either for sound radiation [20], sound field reconstruction [21], estimation of oblique incident surface impedance [22], in noise analysis [23], and capturing sound intensities, [24]. Rules-of-thumb [25] are also discussed on how the estimation precision for an incident source's azimuth-polar DoA depends on the number of identical isotropic sensors. A solution [26] has been suggested to avoid ill-conditioned singularity when solving least-squares and eigenvalue problems to estimate the DoAs.…”

Variational Principle of Least Psychomotor Action: Modelling Effects on Action from Disturbances in Psychomotor Work Involving Human, Cyborg, and Robot Workers

“…Most importantly, the framework is appropriate for engineering design, where there are several alternative production options to be considered, and the exact performance of each option cannot be measured accurately in advance. This lack of performance details precludes meaningful application of simulation tools, but does not preclude application of rule-of-thumb heuristics [ 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 ].…”

“…That is avoid the production of an analysis that corresponds too exactly to a particular set of data, and may therefore fail to fit additional data or predict future observations reliably [ 26 , 27 ]. Rule-of-thumb heuristics are widely used across science [ 28 , 29 , 30 ] and engineering [ 31 , 32 , 33 ], including to address diverse non-trivial problems involving measurement [ 34 , 35 , 36 ]. Rule-of-thumb heuristics are appropriate in engineering design when there are several alternative options to be considered, and the exact performance of each option cannot be measured accurately in advance [ 37 , 38 , 39 , 40 ].…”

Designing a Geodesic Faceted Acoustical Volumetric Array Using a Novel Analytical Method

“…Conformal volumetric arrays on a doubly curved surface similar to a spherical array (SA) are regarded as the most effective geometric configuration for applications requiring a hemispherical beam coverage [ 4 , 5 ]. Consequently, the design of an SA has been a subject of interest to many researchers for decades and numerous studies related to that have been reported in the literature [ 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 ].…”