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Azimuthal centrosymmetry in an array grid is typically associated with arrays that are circular, concentric, cylindrical, spherical, or hemispherical. However, a recently proposed alternative combines an azimuthal circular array with a linear vertical array. For this elegantly simple new array grid's use in the direction-of-arrival estimation, this article advances array-design insights to meet a given estimation–precision threshold, by examining the tradeoff between the azimuth-angle Cramer–Rao bound vis-a-vis the polar-angle Cramer–Rao bound in a proposed two-step design procedure.

The linear array's one-dimensional spatial geometry is simple but suffices for univariate direction finding, i.e., is adequate for the estimation of an incident source's direction-of-arrival relative to the linear array axis. However, this nominal one-dimensional ideality could be often physically compromised in the real world, as the constituent sensors may dislocate three-dimensionally from their nominal positions. For example, a towed array is subject to ocean-surface waves and to oceanic currents [Tichavsky and Wong (2004). IEEE Trans. Sign. Process. 52(1), 36-47]. This paper analyzes how a nominally linear array's one-dimensional direction-finding accuracy would be degraded by the three-dimensional random dislocation of the constituent sensors. This analysis derives the hybrid Cramér-Rao bound (HCRB) of the arrival-angle estimate in a closed form expressed in terms of the sensors' dislocation statistics. Surprisingly, the sensors' dislocation could improve and not necessarily degrade the HCRB, depending on the dislocation variances but also on the incident source's arrival angle and the signal-to-noise power ratio.

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 Cramer–Rao bounds (CRB) are found to follow these elegantly simple approximations, useful for array design: (i) For the azimuth arrival angle: 2 π ( R / λ ) ( σ s / σ n ) 2 L M CRB ( ϕ ) sin ( θ ) ≈ ( L e 1 / 14 ) − 1 + 3 → L → ∞ 3 , ∀ ( ϕ , θ ); and (ii) for the polar arrival angle: 2 π ( R / λ ) ( σ s / σ n ) 2 L M CRB ( θ ) ≈ 3 − ( L e 6 / 7 ) − 1 → L → ∞ 3 , ∀ ( ϕ , θ ). Here, M denotes the number of snapshots, λ refers to the incident signal's wavelength, and ( σ s / σ n ) 2 symbolizes the signal-to-noise power ratio.

An array's constituent sensors could be spatially dislocated from their nominal positions. This paper investigates how such sensor dislocation would degrade a uniform circular array (UCA) of isotropic sensors (like pressure sensors) in their direction-finding precision. This paper analytically derives this direction finding's hybrid Cramer-Rao bound (HCRB) in a closed form that is expressed explicitly in terms of the sensors' dislocation parameters. In the open literature on UCA direction finding, this paper is the first to be three-dimensional in modeling the sensors' dislocation. Perhaps unexpectedly to some readers, sensor dislocation could improve and not necessarily degrade the HCRB; these opposing effects depend on the dislocation variances, the incident source's arrival angle, and the signal-to-noise power ratio—all analyzed rigorously in this paper. Interesting insights are thereby obtained: (a) The HCRB is enhanced for the impinging source's polar arrival angle as the sensors become more dislocated along the impinging wavefront due to aperture enlargement over the stochastic dislocation's probability space. (b) Likewise, the HCRB is improved for the azimuth arrival angle as the sensors become more dislocated on the circular array's plane, also due to aperture enlargement. (c) In contrast, sensor dislocation along the incident signal's propagation direction can only worsen the HRCBs due to nuisance-parameter effects in the Fisher information. (d) Sensor dislocation orthogonal to the array plane must degrade the HCRB for the azimuth arrival angle but could improve the HCRB for the polar arrival angle.An array's constituent sensors could be spatially dislocated from their nominal positions. This paper investigates how such sensor dislocation would degrade a uniform circular array (UCA) of isotropic sensors (like pressure sensors) in their direction-finding precision. This paper analytically derives this direction finding's hybrid Cramer-Rao bound (HCRB) in a closed form that is expressed explicitly in terms of the sensors' dislocation parameters. In the open literature on UCA direction finding, this paper is the first to be three-dimensional in modeling the sensors' dislocation. Perhaps unexpectedly to some readers, sensor dislocation could improve and not necessarily degrade the HCRB; these opposing effects depend on the dislocation variances, the incident source's arrival angle, and the signal-to-noise power ratio—all analyzed rigorously in this paper. Interesting insights are thereby obtained: (a) The HCRB is enhanced for the impinging source's polar arrival angle as the sensors become more dislocat...

Consider azimuth-elevation direction finding by a uniform circular array of isotropic sensors. In the real world, the sensors may dislocate from their nominal positions. These dislocations could be modeled as random variables having an a priori known distribution. This paper investigates how the dislocations would affect azimuth-elevation direction finding by deriving the corresponding hybrid Cramer-Rao bounds. Maximum a posteriori estimators are derived and Monte Carlo simulations are conducted to validate the derived hybrid Cramer-Rao bounds.