Reproducing the Acoustic Velocity Vectors in a Circular Listening Area

TL;DR

This work introduces radial independent cylindrical-harmonic velocity coefficients (CHV-indR) for circular listening areas, derived from the pressure cylindrical-harmonic coefficients via the sound-field translation formula, enabling reconstruction of acoustic velocity vectors throughout the area from circular microphone-array measurements. By formulating velocity as a linear transformation of pressure coefficients and defining CHV-indR through matrices that map beta to zeta, the authors enable velocity-matching reproduction (VM) that requires only circular-array data and scales to 2D height-invariant fields. Compared with pressure-based reproduction, CHV-indR VM yields lower direction errors in the reproduced velocity vectors at low frequencies, improving localization where velocity cues dominate. The approach offers a practical path to accurate 2D spatial sound field reproduction with off-the-shelf circular microphones, while avoiding boundary-velocity measurements and sweet-spot limitations.

Abstract

Acoustic velocity vectors are important for human's localization of sound at low frequencies. This paper proposes a sound field reproduction algorithm, which matches the acoustic velocity vectors in a circular listening area. In previous work, acoustic velocity vectors are matched either at sweet spots or on the boundary of the listening area. Methods based on sweet spots experience performance degradation when the listener moves away from sweet spots, whereas measuring the acoustic velocity vectors on the boundary requires complicated measurement setup. This paper proposes the radial independent cylindrical harmonic coefficients of the acoustic velocity vectors (CHV-indR coefficients) in the circular listening area, which are calculated from the cylindrical harmonic coefficients of the pressure in the circular listening area by using the sound field translation formula. The cylindrical harmonic coefficients of the pressure can be measured by a circular microphone array, which can be bought off-the-shelf. By matching the CHV-indR coefficients, the acoustic velocity vectors are reproduced throughout the listening area. Simulations show that at low frequencies, where the acoustic velocity vectors are the dominant factor for localization, the proposed reproduction method based on matching the CHV-indR coefficients results in higher accuracy in reproduced acoustic velocity vectors when compared with traditional method based on matching the cylindrical harmonic coefficients of the pressure.

Figures (4)

• Figure 1: Setup of the geometric model. The listening region is bounded by the blue circle. $\mathbf{r}_{\boldsymbol{q}}$ is a point in the listening region.
• Figure 2: Real part of the acoustic velocity vectors at 500 Hz. (a) The source is a plane wave with incident direction $\phi_{\text{pw}} = 8\pi/9$ rad. (b) The source is a point source at $\mathbf{r_{s}} = (1 \text{ m}, 8\pi/9 \text{ rad})$.
• Figure 3: (a) Setup of the reproduction system. The listening area bounded by the red circle has radius 0.5 meters. Black crosses denote loudspeakers. (b) Condition numbers of $\mathbf{H}(k)$ in VM and $\mathbf{G}(k)$ in PM.
• Figure 4: Real part of the reproduced pressure and the reproduced acoustic velocity vectors at 500 Hz. The desired sound field is a plane wave with incident direction $8\pi/9$ rad. (a) and (c) - VM; (b) and (d) - PM.