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. 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.

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. However, this nominal one-dimensional ideality could be often physically compromised in the real world, as the constituent sensors may dislocate three-dimensionally from their nominal positions. For example, a towed array is subject to ocean-surface waves and to oceanic currents [Tichavsky and Wong (2004). IEEE Trans. Sign. Process. 52(1), 36-47]. This paper analyzes how a nominally linear array's one-dimensional direction-finding accuracy would be degraded by the three-dimensional random dislocation of the constituent sensors. This analysis derives the hybrid Cramér-Rao bound (HCRB) of the arrival-angle estimate in a closed form expressed in terms of the sensors' dislocation statistics. Surprisingly, the sensors' dislocation could improve and not necessarily degrade the HCRB, depending on the dislocation variances but also on the incident source's arrival angle and the signal-to-noise power ratio.

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. Other characteristics of the beam-pattern are also analytically explained herein.

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. This analysis allows the cardioidicity index (α) to be stochastically uncertain, applies to any cardioidic order (k), and is valid for any real-valued incident signal regardless of the signal's time-frequency structure.

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. In the open literature on UCA direction finding, this paper is the first to be three-dimensional in modeling the sensors' dislocation. Perhaps unexpectedly to some readers, sensor dislocation could improve and not necessarily degrade the HCRB; these opposing effects depend on the dislocation variances, the incident source's arrival angle, and the signal-to-noise power ratio-all analyzed rigorously in this paper. Interesting insights are thereby obtained: (a) The HCRB is enhanced for the impinging source's polar arrival angle as the sensors become more dislocated along the impinging wavefront due to aperture enlargement over the stochastic dislocation's probability space. (b) Likewise, the HCRB is improved for the azimuth arrival angle as the sensors become more dislocated on the circular array's plane, also due to aperture enlargement. (c) In contrast, sensor dislocation along the incident signal's propagation direction can only worsen the HRCBs due to nuisance-parameter effects in the Fisher information. (d) Sensor dislocation orthogonal to the array plane must degrade the HCRB for the azimuth arrival angle but could improve the HCRB for the polar arrival angle.

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. Monte Carlo simulations verify these proposed algorithms' efficacy and statistical closeness to the corresponding Cramér-Rao bounds.

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)(σs/σn)2LMCRB(ϕ) sin(θ)≈(Le1/14)-1+3→L→∞3, ∀(ϕ,θ); and (ii) for the polar arrival angle: 2π(R/λ)(σs/σn)2LMCRB(θ)≈3-(Le6/7)-1→L→∞3, ∀(ϕ,θ). Here, M denotes the number of snapshots, λ refers to the incident signal's wavelength, and (σs/σn)2 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. 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.

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.

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). For such a sensing system, this paper is first (1) to develop closed-form eigen-based signal-processing algorithms for azimuth-elevation direction finding; (2) to analytically derive the associated Cramér-Rao lower bounds (CRB), which are expressed explicitly in terms of the two constituent sensors' spatial geometry and in terms of the figure-8 sensor's directivity order; (3) to verify (via Monte Carlo simulations) the proposed direction-of-arrival estimators' efficacy and closeness to the respective CRB. Here, the higher-order p-u probe's two constituent sensors may be spatially displaced.

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. If this nominal identity is violated among the three uniaxial velocity sensors, as may occur in practical hardware, what would happen to the corresponding "spatial matched filter" beam-pattern's peak direction? How would this effective peak direction deviate from the nominal "look direction"? This paper, by modeling each uniaxial velocity sensor's gain as stochastic, derives this deviation's statistical mean and variance, analytically in closed mathematical forms. This analytical derivation is verified by Monte Carlo simulations.

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

The acoustic vector-sensor (a.k.a. the vector hydrophone) is a practical and versatile sound-measurement device, with applications in-room, open-air, or underwater. It consists of three identical uni-axial velocity-sensors in orthogonal orientations, plus a pressure-sensor-all in spatial collocation. Its far-field array manifold [Nehorai and Paldi (1994). IEEE Trans. Signal Process. 42, 2481-2491; Hawkes and Nehorai (2000). IEEE Trans. Signal Process. 48, 2981-2993] has been introduced into the technical field of signal processing about 2 decades ago, and many direction-finding algorithms have since been developed for this acoustic vector-sensor. The above array manifold is subsequently generalized for outside the far field in Wu, Wong, and Lau [(2010). IEEE Trans. Signal Process. 58, 3946-3951], but only if no reflection-boundary is to lie near the acoustic vector-sensor. As for the near-boundary array manifold for the general case of an emitter in the geometric near field, the far field, or anywhere in between-this paper derives and presents that array manifold in terms of signal-processing mathematics. Also derived here is the corresponding Cramér-Rao bound for azimuth-elevation-distance localization of an incident emitter, with the reflected wave shown to play a critical role on account of its constructive or destructive summation with the line-of-sight wave. The implications on source localization are explored, especially with respect to measurement model mismatch in maximum-likelihood direction finding and with regard to the spatial resolution between coexisting emitters.

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. This "self-normalization" direction-of-arrival estimator is predicated on the spatial-collocation among the three particle-velocity sensors and the pressure-sensor. This collocation constriction is relaxed here by this presently proposed idea, to realize a spatially distributed acoustic vector-sensor, allowing its four component-sensors to be separately located. This proposed scheme not only retains the algorithmic advantages of the aforementioned "self-normalization" direction-of-arrival estimator, but also will significantly extend the spatial aperture to improve the direction-finding accuracy by orders of magnitude.

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. These beamformers deploy a physically compact triad of three collocated but orthogonally oriented velocity sensors. These proposed schemes' efficacy is verified by a jury test, using simulated data constructed with Mandarin Chinese (a.k.a. Putonghua) speech samples. For example, a desired speech signal, originally at a very adverse signal-to-interference-and-noise power ratio (SINR) of -30 dB, may be processed to become fully intelligible to the jury.