Higher-order figure-8 microphones/hydrophones collocated as a perpendicular triad-Their "spatial-matched-filter" beam steering.

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.

Three-dimensional dislocations in a uniform linear array's isotropic sensors-Direction finding's hybrid Cramér-Rao bound.

The linear array's one-dimensional spatial geometry is simple but suffices for univariate direction finding, i.e., is adequate for the estimation of an incident source's direction-of-arrival relative to the linear array axis.

Higher-order figure-8 sensors in a pair, skewed and collocated-Their azimuthal "spatial matched filter" beam-pattern.

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.

Introduction to the Special Issue on Acoustic Source Localization.

Spatial localization based on acoustic observations is a rich field of interest in acoustic signal analysis. This special issue takes a close look at the diverse and growing range of problems in this area and the broad perspectives and methodologies that are presently being developed to solve them. The collection of articles presents recent advances in localization in complex and uncertain environments across a wide range of acoustic disciplines, from animal bioacoustics and acoustic signal processing in underwater environments to in air environments, architectural acoustics, and acoustic transduction.

Hybrid Cramér-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 Cramér-Rao bound.

A uniform circular array of isotropic sensors that stochastically dislocate in three dimensions-The hybrid Cramér-Rao bound of direction-of-arrival estimation.

An array's constituent sensors could be spatially dislocated from their nominal positions. This paper investigates how such sensor dislocation would degrade a uniform circular array (UCA) of isotropic sensors (like pressure sensors) in their direction-finding precision. This paper analytically derives this direction finding's hybrid Cramér-Rao bound (HCRB) in a closed form that is expressed explicitly in terms of the sensors' dislocation parameters.

Bivariate direction finding using two perpendicular bi-directional ("figure-8") sensors of (possibly) unequal orders.

A "figure-8" sensor is so labeled because its spatial pattern resembles the character "8" with regard to the sensor's axis. This figure-8 pattern narrows as the sensor's order increases. Using two such figure-8 directional sensors of higher order, oriented perpendicularly to each other-this paper pioneers closed-form signal-processing algorithms to estimate an incident signal's azimuth-elevation bivariate direction-of-arrival.

Rules-of-thumb to design a uniform spherical array for direction finding-Its Cramér-Rao bounds' nonlinear dependence on the number of sensors.

This paper discovers rules-of-thumb on how the estimation precision for an incident source's azimuth-polar direction-of-arrival (ϕ,θ) depends on the number (L) of identical isotropic sensors spaced uniformly on an open sphere of radius R. This estimation's corresponding Cramér-Rao bounds (CRB) are found to follow these elegantly simple approximations, useful for array design: (i) For the azimuth arrival angle: 2π(R/λ)(σ/σ)2LMCRB(ϕ) sin(θ)≈(Le)+3→L→∞3, ∀(ϕ,θ); and (ii) for the polar arrival angle: 2π(R/λ)(σ/σ)2LMCRB(θ)≈3-(Le)→L→∞3, ∀(ϕ,θ). Here, M denotes the number of snapshots, λ refers to the incident signal's wavelength, and (σ/σ) symbolizes the signal-to-noise power ratio.

Cardioid microphones/hydrophones in a collocated and orthogonal triad-A steerable beamformer with no beam-pointing error.

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.

A higher-order "figure-8" sensor and an isotropic sensor-For azimuth-elevation bivariate direction finding.

A "p-u probe" (also known as a "p-v probe") comprises one pressure-sensor (which is isotropic) and one uni-axial particle-velocity sensor (which has a "figure-8" bi-directional spatial directivity). This p-u probe may be generalized, by allowing the figure-8 bi-directional sensor to have a higher order of directivity. This higher-order p-u probe has not previously been investigated anywhere in the open literature (to the best knowledge of the present authors).

Beamforming pointing error of a triaxial velocity sensor under gain uncertainties.

A "triaxial velocity sensor" consists of three uniaxial velocity sensors, which are nominally identical, orthogonally oriented among themselves, and co-centered at one point in space. A triaxial velocity sensor measures the acoustic particle velocity vector, by its three Cartesian components, individually component-by-component, thereby offering azimuth-elevation two-dimensional spatial directivity, despite the physical compactness that comes with the collocation of its three components. This sensing system's azimuth-elevation beam-pattern has been much analyzed in the open literature, but only for an idealized case of the three uniaxial velocity sensors being exactly identical in gain.

Near-field/far-field array manifold of an acoustic vector-sensor near a reflecting boundary.

The acoustic vector-sensor (a.k.a.

Azimuth-elevation direction finding using a microphone and three orthogonal velocity sensors as a non-collocated subarray.

An acoustic vector-sensor consists of three identical but orthogonally oriented acoustic particle-velocity sensors, plus a pressure sensor-all spatially collocated in a point-like geometry. At any point in space, this tri-axial acoustic vector-sensor can sample an acoustic wavefield as a 3 × 1 vector, instead of simply as a scalar of pressure. This vector, after proper self-normalization, would indicate the incident wave-field's propagation direction, and thus the incident emitter's azimuth-elevation direction-of-arrival.

A directionally tunable but frequency-invariant beamformer on an acoustic velocity-sensor triad to enhance speech perception.

Herein investigated are computationally simple microphone-array beamformers that are independent of the frequency-spectra of all signals, all interference, and all noises. These beamformers allow the listener to tune the desired azimuth-elevation "look direction." No prior information is needed of the interference.