Two Cardioid Sensors in Perpendicular Colocation—Their “Spatial Matched Filter” Beam Steering

Cardioid microphones/hydrophones have been commonplace in practical acoustics. Indeed, cardioid sensors are widely obtainable in retail as diverse commercial models produced by numerous manufacturers. A cardioid sensor has a highly directional gain pattern of
$\alpha + {(}{1} - \alpha {)} \text {cos}{(}\gamma {)}$
, where
$\alpha $
represents the sensor’s cardioidicity index, and
$\gamma $
denotes the impinging wavefront’s spatial angle incident upon the cardioid sensor’s axis. This article is the first in the open literature to investigate the “spatial matched filter” beam-pattern obtainable from two such cardioids that lie in perpendicular colocation. Such a bi-axial cardioid pair is worth investigating because 1) their diverse orientation successfully realizes 2-D (azimuth-polar) directivity with only two sensors and 2) their spatial colocation is advantageous for physical compactness and also for the decoupling of the data’s time-and-frequency dimensions from the azimuth-and-elevation directional dimensions to simplify signal-processing computation. This present investigation, however, uncovers that this beam-pattern generally peaks away from the nominal preset “look direction,” but algorithmic mitigation is made possible by this article’s pre-steering formula derived here to judiciously pre-adjust the beamformer’s nominal “look direction.” Further action-able insights are provided in this article for the hardware system engineer to pre-select the sensors’ cardioidicity index
$\alpha $
for maximum steerability and freedom from sidelobes.