A higher-order figure-8 sensor and an isotropic sensor - For azimuth-elevation bivariate direction finding

A "p-u probe" (also known as a "p-v probe") comprises one pressure-sensor (which is isotropic) and one uni-axial particle-velocity sensor (which has a "figure-8" bi-directional spatial directivity). This p-u probe may be generalized, by allowing the figure-8 bi-directional sensor to have a higher order of directivity. This higher-order p-u probe has not previously been investigated anywhere in the open literature (to the best knowledge of the present authors). For such a sensing system, this paper is first (1) to develop closed-form eigen-based signal-processing algorithms for azimuth-elevation direction finding; (2) to analytically derive the associated Cramér-Rao lower bounds (CRB), which are expressed explicitly in terms of the two constituent sensors' spatial geometry and in terms of the figure-8 sensor's directivity order; (3) to verify (via Monte Carlo simulations) the proposed direction-of-arrival estimators' efficacy and closeness to the respective CRB. Here, the higher-order p-u probe's two constituent sensors may be spatially displaced.