Hybrid Cramer-Rao bound of direction finding, using a triad of cardioid sensors that are perpendicularly oriented and spatially collocated

Cardioid microphones/hydrophones are highly directional acoustical sensors, which enjoy easy availability via numerous commercial vendors for professional use. Collocating three such cardioids in orthogonal orientation to each other, the resulting triad would be sharply directional yet physically compact, while decoupling the incident signal's time-frequency dimensions from its azimuth-elevation directional dimensions, thereby simplifying signal-processing computations. This paper studies such a cardioid triad's azimuth-elevation direction-of-arrival estimation accuracy, which is characterized here by the hybrid Cramer-Rao bound. This analysis allows the cardioidicity index (alpha) to be stochastically uncertain, applies to any cardioidic order (k), and is valid for any real-valued incident signal regardless of the signal's time-frequency structure.