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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 L-shaped and V-shaped arrays of first-order cardioid microphones for direction finding are presented in this paper. A comparative study of the direction of arrival estimation performance of the arrays was carried out by analytically deriving, in closed-form, and comparing the Cramer-Rao bounds of an incident signal’s direction-of-arrival (DoA) azimuth and polar angles for these arrays. The maximum-likelihood estimator is used to verify the correctness of derived bounds. This investigation reveals that for direction finding, the L-shaped array of cardioid microphones would generally outperform the V-shaped array of cardioid microphones in more sub-regions of the DoA polar-azimuth angle space. However, in regions where the V-shaped array of cardioid microphones outperforms the L-shaped array of cardioid microphone, the variance ratios are usually higher in favor of the V-shaped array.

A rectangular array of sensors is often used in direction finding, due to the geometric regularity in its spatial rectangular grid. The sensor positions may be confined to the rectangle's perimeter (as in a “rim” array), or may span over the rectangle's entire interior as well (as in a “filled” array). This paper compares these two array grids by their precision in direction finding, by pioneering Cramer-Rao bound expressions for both array grids above, in closed forms and explicitly in terms of the array parameters.