Diffuse Sound Field Synthesis

TL;DR

This work develops a physics-grounded framework for synthesizing diffuse sound fields from uncorrelated surrounding sources by exploiting Green's functions, Gauss' law, and Newton's spherical shell theorem. It shows that ideal diffuse fields (constant energy density with vanishing intensity) are achievable in circular (2D) or spherical (3D) layouts when the radial decay matches the physical free-field law, $|G(r)|\propto 1/\sqrt{r}^{D-1}$, and, in some geometries, via modified decays $1/r^\beta$ analyzed with Gegenbauer polynomials. The authors identify optimal hulls (circles, spheres, opposing planes) and develop mode-matching strategies to extend diffuseness to non-circular/non-spherical shapes, including ellipsoids and cuboids, as well as discrete realizations through spherical $t$-designs. The work further assesses the practical implications for 2.5D WFS and perceptual relevance by connecting diffuseness to interaural cues (ILD, IC) using perceptual data and KU100 HRTFs. Overall, the paper provides theoretical and computational tools to guide loudspeaker-based diffusion synthesis, offering design principles and discretization guidelines with direct relevance to psychoacoustic outcomes and experimental setups.

Abstract

Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing average sound intensity, uncorrelated sound waves arriving isotropically from all directions. Does this require specific sources and geometries for surrounding 2D and 3D source layouts? As methods, we employ numeric simulations and undertake a series of calculations with uncorrelated circular/spherical source layouts, or such with infinite excess dimensions, and we point out relations to potential theory. Using a radial decay 1/r^b modified by the exponent b, the representation of the resulting fields with hypergeometric functions, Gegenbauer polynomials, and circular as well as spherical harmonics yields fruitful insights. In circular layouts, waves decaying by the exponent b=1/2 synthesize ideally extended, diffuse sound fields; spherical layouts do so with b=1. None of the layouts synthesizes a perfectly constant expected sound pressure level but its flatness is acceptable. Spherical t-designs describe optimal source layouts with well-described area of high diffuseness, and non-spherical, convex layouts can be improved by restoring isotropy or by mode matching for a maximally diffuse synthesis. Theory and simulation offer a basis for loudspeaker-based synthesis of diffuse sound fields and contribute physical reasons to recent psychoacoustic findings in spatial audio.

Figures (17)

• Figure 1: Horizontal map of sound energy density (contours), normalized intensity $\bm{I}/(c\,w)$ (arrows), and diffuseness $\psi$ (colors) with 100 regularly spaced sources on the azimuth for a circle (salmon), or for a sphere with 480 sources at Chebyshev-type nodes from Gräf's website graefurl, all playing statistically independent signals of uniform variance.
• Figure 2: Diffuseness $\psi$ (colormap, solid contour: $90\%$) and level $w$ normalized to origin (dotted 1dB contour) for a 2D, 3:2 ellipse / rectangle ($L_\mathsf{p}$-superellipse with $\mathsf{p}=10$) of 100 equi-angle uncorrelated vertical line sources (salmon): (a,c) with unity gain $\sigma=1$, (b,d) isotropy-enforcing gain $\sigma=\sqrt{R}$.
• Figure 3: Differential surface elements in $\mathbb{R}^3$ for symmetric arrangements of (left) infinite parallel planes with shift invariance in $yz$, (middle) an infinite cylinder with and rotational symmetry in $xy$ and shift invariance in $z$, and (right) a sphere with rotational symmetry in $xyz$, according to Newton's spherical shell theorem.
• Figure 4: Point source at $\bm x_\mathrm{s}=\bm 0$ with radial decay exponent modified from left to right $\beta=\{1,0.5,0\}$, at $f=1kHz$.
• Figure 5: Evaluation of sound energy density (a) and diffuseness (b) for $-1.25\leq\beta\leq1.25$ (vertical axis) within $-1\leq x\leq 1$ (horizontal axis; orange-red arrows display $I_\mathrm{x}/c\,w$ at $x=\pm 0.6$).