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Unequal “directivity orders” between two figure8 directional microphones/hydrophones are proposed and analyzed in this paper for beamforming. Specifically, two figure 8 sensors here, possibly of different “directivity orders”, are (i) orthogonally oriented to facilitate spatial filtering in both the azimuth and the elevation, and (ii) spatially colocated with an isotropic sensor (thereby decoupling the incident source's azimuth-elevation-radial dimensions from the impinging signal's time-frequency dimensions). This versatility (in the sensors’ diverse “directivity orders”) contrasts with all prior open literature on beamforming using a colocated triad of possibly directional sensors. Instead, this paper analytically explores how unequal directivity orders between two perpendicularly colocated figure-8 acoustic sensors would affect the triad array's “spatial matched filter” beam steering. The resulting “actionable” insights show which directivity-order combinations allow the beam of full maneuverability toward any azimuthal direction, and which directivity-order combinations only partial maneuverability.

Directional sensors, if collocated but perpendicularly oriented among themselves, would facilitate signal processing to uncouple the azimuth-polar direction from the time-frequency dimension—in addition to the physical advantage of spatial compactness. One such acoustical sensing unit is the well-known “tri-axial velocity sensor” (also known as the “gradient sensor,” the “velocity-sensor triad,” the “acoustic vector sensor,” and the “vector hydrophone”), which comprises three identical figure-8 sensors of the first directivity-order, collocated spatially but oriented perpendicularly of each other. The directivity of the figure-8 sensors is hypothetically raised to a higher order in this analytical investigation with an innocent hope to sharpen the overall triad's directionality and steerability. Against this wishful aspiration, this paper rigorously analyzes how the directivity-order would affect the triad's “spatial-matched-filter” beam's directional steering capability, revealing which directivity-order(s) would allow the beam-pattern of full maneuverability toward any azimuthal direction and which directivity-order(s) cannot.

Directional sensors, if collocated but perpendicularly oriented among themselves, would facilitate signal processing to uncouple the azimuth-polar direction from the time-frequency dimension—in addition to the physical advantage of spatial compactness. One such acoustical sensing unit is the well-known “tri-axial velocity sensor” (also known as the “gradient sensor,” the “velocity-sensor triad,” the “acoustic vector sensor,” and the “vector hydrophone”), which comprises three identical figure-8 sensors of the first directivity-order, collocated spatially but oriented perpendicularly of each other. The directivity of the figure-8 sensors is hypothetically raised to a higher order in this analytical investigation with an innocent hope to sharpen the overall triad's directionality and steerability. Against this wishful aspiration, this paper rigorously analyzes how the directivity-order would affect the triad's “spatial-matched-filter” beam's directional steering capability, revealing which directivity-order(s) would allow the beam-pattern of full maneuverability toward any azimuthal direction and which directivity-order(s) cannot.

Directional sensors, if collocated but perpendicularly oriented among themselves, would facilitate signal processing to uncouple the azimuth-polar direction from the time-frequency dimension—in addition to the physical advantage of spatial compactness. One such acoustical sensing unit is the well-known “tri-axial velocity sensor” (also known as the “gradient sensor,” the “velocity-sensor triad,” the “acoustic vector sensor,” and the “vector hydrophone”), which comprises three identical figure-8 sensors of the first directivity-order, collocated spatially but oriented perpendicularly of each other. The directivity of the figure-8 sensors is hypothetically raised to a higher order in this analytical investigation with an innocent hope to sharpen the overall triad's directionality and steerability. Against this wishful aspiration, this paper rigorously analyzes how the directivity-order would affect the triad's “spatial-matched-filter” beam's directional steering capability, revealing which directivity-order(s) would allow the beam-pattern of full maneuverability toward any azimuthal direction and which directivity-order(s) cannot.

Directional sensors, if collocated but perpendicularly oriented among themselves, would facilitate signal processing to uncouple the azimuth-polar direction from the time-frequency dimension—in addition to the physical advantage of spatial compactness. One such acoustical sensing unit is the well-known “tri-axial velocity sensor” (also known as the “gradient sensor,” the “velocity-sensor triad,” the “acoustic vector sensor,” and the “vector hydrophone”), which comprises three identical figure-8 sensors of the first directivity-order, collocated spatially but oriented perpendicularly of each other. The directivity of the figure-8 sensors is hypothetically raised to a higher order in this analytical investigation with an innocent hope to sharpen the overall triad's directionality and steerability. Against this wishful aspiration, this paper rigorously analyzes how the directivity-order would affect the triad's “spatial-matched-filter” beam's directional steering capability, revealing which directivity-order(s) would allow the beam-pattern of full maneuverability toward any azimuthal direction and which directivity-order(s) cannot.

Directional sensors, if collocated but perpendicularly oriented among themselves, would facilitate signal processing to uncouple the azimuth-polar direction from the time-frequency dimension—in addition to the physical advantage of spatial compactness. One such acoustical sensing unit is the well-known “tri-axial velocity sensor” (also known as the “gradient sensor,” the “velocity-sensor triad,” the “acoustic vector sensor,” and the “vector hydrophone”), which comprises three identical figure-8 sensors of the first directivity-order, collocated spatially but oriented perpendicularly of each other. The directivity of the figure-8 sensors is hypothetically raised to a higher order in this analytical investigation with an innocent hope to sharpen the overall triad's directionality and steerability. Against this wishful aspiration, this paper rigorously analyzes how the directivity-order would affect the triad's “spatial-matched-filter” beam's directional steering capability, revealing which directivity-order(s) would allow the beam-pattern of full maneuverability toward any azimuthal direction and which directivity-order(s) cannot.

Two figure-8 sensors, differently oriented but collocated, can facilitate azimuth- elevation two-dimensional beamforming that is invariant of the frequency spectrum of the incident signal. For such a pair of identical figure-8 sensors of any arbitrary directivity-order, their spatial matched filter beam-pattern is characterized analytically here in this paper. This paper shows that the beamformer would suffer a high pointing bias if the figure-8 sensors' directivity-order exceeds one. Other characteristics of the beam-pattern are also analytically explained herein.

Cardioid sensors offer low sidelobes/backlobes compared to figure-8 bi-directional sensors (like velocity-sensors). Three cardioid sensors, in orthogonal orientation and in spatial collocation, have recently been proposed by Wong, Nnonyelu, and Wu [(2018). IEEE Trans. Sign. Process. 66(4), 895–906] and such a cardioid-triad's “spatial matched filter” beam-pattern has been analyzed therein. That beam-pattern, unfortunately, suffers pointing error, i.e., the spatial beam's actual peak direction deviates from the nominal “look direction.” Instead, this paper will propose a steerable data-independent beamformer for the above-mentioned cardioidic triad to avoid beam-pointing error. Also analytically derived here (via multivariate calculus) is this beam-pattern's lobes' height ratio, beamwidth, directivity, and array gain.

This paper proposes a new configuration of acoustic sensors-the collocation of three cardioid sensors in perpendicular orientation, in order to increase the mainlobe-to-sidelobe height ratio (possibly to ∞). This paper will analyze such a proposed triad's “spatial matched filter” beam-pattern that is independent of the frequency/spectrum of the incident signals. Specifically, this paper will analytically derive the mainlobe's pointing error in azimuth-elevation, the mainlobe's two-dimensional beam “width,” the necessary and sufficient conditions for a sidelobe to exist, the mainlobe-to-sidelobe height ratio, and the array gain. These above characteristics depend on the cardioids' “cardiodicity parameter” and on the beam's nominal “look direction.”

The “tri-axial velocity-sensor” has three axes that are nominally perpendicular, but may be nonperpendicular in practice, due to real-world imperfections in manufacturing, deployment, or maintenance.This paper comprehensively investigates how such non-perpendicularity would affect the tri-axial velocitysensor’s azimuth-elevation beam-pattern in terms of the beam’s pointing direction. It was shown that thenon-perpendicular axes do not affect the overall shape of the beampattern, but would introduce a pointingbias.

A biaxial velocity-sensor (aka a v-v probe) measures two Cartesian components of an incidentacoustical wavefield’s particle velocity vector. Such biaxial velocity-sensors are sometimes usedas elements in an array. If these elements are not identically oriented, what would happen to thearray’s “spatial matched filter” beam-pattern? This paper investigates such a beam-pattern’spointing error, for a pair of non-identically oriented biaxial velocity-sensor. This pointing error isinvestigated here in terms of (i) the biaxial velocity-sensor’s misorientation skew angle, (ii) thebiaxial velocity-sensor’s spatial separation, (iii) the incident signal’s wavelength, and (iv) thebeamformer’s nominal look direction.