Tri-Cardioid Co-Centered Co-Planar Array—Its Direction-Finding Cramér-Rao Bound and Design Guidelines

Cardioid microphones/hydrophones have a highly directional gain pattern of $[ \alpha + (1 - \alpha) \cos (\beta) ] \cos ^{k}(\beta)$, where $k$ refers to the sensor's directivity order, $\alpha$ denotes the same sensor's cardioidicity index, and $\beta$ represents an impinging signal's incident direction of arrival relative to the cardioid sensor's axis. Three such cardioids organized in orthogonal orientation in three Cartesian spatial dimensions and in spatial colocation as one sensing unit—such a 3-D triad has already attracted much recent attention in the research literature. However, not all three Cartesian coordinates have equal importance to many acoustical applications, which focus alternatively on the azimuthal direction defined on a flat plane but less on the elevation direction normal to that plane. So, this article will instead analyze a 2-D planar configuration of three colocated/cocentered cardioids differently oriented azimuthally apart by $120^\circ$. This aforesaid coplanar triplet conforms to a flat supporting surface more than a Cartesian tridimensionally perpendicular triad can. For such a coplanar triplet composed of cardioids preset at any specific $(k, \alpha)$, the triplet's polar/azimuthal direction-finding Cramér–Rao lower bounds will be analytically derived in closed forms here in this article. Those bounds will be uncovered to exhibit intricate mathematical structures, which will be dissected in detail to yield refined insights, producing simple “actionable” rules-of-thumb to guide the system engineer to choose the appropriate values for $(k, \alpha)$.