Two Perpendicularly Colocated Cardioid Sensors of Different Cardioidicity Indices/Orders– Their Design Guidelines Based on Their Polar-Azimuthal Direction-Finding Cramér–Rao Bound

A cardioid sensor has a remarkably directional gain pattern of $[ \alpha + (1 - \alpha) \cos (\beta) ] \cos ^{k}(\beta)$, where $\alpha$ represents the sensor’s cardioidicity index, $k$ symbolizes the sensor’s directivity order, and $\beta$ refers to the incident angle with respect to the cardioid sensor’s axis. Cardioid microphones/hydrophones are commercially available through numerous manufacturers and widely adopted in workaday acoustics. Nonetheless, this letter is first in the open literature to consider any array comprising cardioid sensors of (possibly) different cardioidicity indices/orders. That is, this analysis pioneers the versatile possibility of $(k, \alpha)$ being different between the two cardioids that constitute the pair. These two foresaid cardioids are perpendicularly oriented, but spatially colocated in this article. First, their dissimilar orientations facilitate the estimation of an incident signal’s 2-D direction-of-arrival over the azimuth and the elevation, despite the deployment of simply two sensors. Second, their spatial colocation decouples the incident signal’s time/frequency dimensions from the signal’s azimuth/elevation directional dimensions, thus simplifying any subsequent signal-processing computations. This article will analytically explore such a pair’s azimuth/elevation direction-of-arrival estimation precision, as measured by the Cramér–Rao lower bound, producing action-able insights for the system engineer to choose what combination of $(k_{x}, \alpha _{x})$ and $(k_{y}, \alpha _{y})$ for the two cardioids forming the pair.