Primary Source Correction (PSC) in Wave Field Synthesis

The theory of wave field synthesis is traditionally derived based on the Kirchhoff-Helmholtz integral equation for bounded volumes with continuous secondary point source distributions on the boundary area. Usually, a secondary source distribution on a boundary contour is used in practical implementations and the theory is adapted by reducing the geometry to two dimensions, employing secondary line sources perpendicular to the listening area. Consequently, the sound fields that can be synthesized are restricted to fields independent of the Cartesian coordinate direction defined by the line source axes, explicitly excluding the synthesis of spherical waves. Usually, secondary point sources are employed instead of line sources. In this case, correct synthesis is possible at one reference position in the listening area if secondary source correction is applied. In this paper, the theory of wave field synthesis is revised, resulting in a global formulation of secondary source correction that allows for correct amplitude and phase reproduction at the reference position. Based on that framework, primary source correction (PSC) is introduced. This procedure permits at the reference position correct synthesis of spherical waves evolving at arbitrary distances in the plane defined by the listening area, especially including the correct inverse proportionality of level and source distance, which is not possible using current approaches to the synthesis of spherical waves. Introduction Wave field synthesis (WFS) is an audio playback method, aiming at synthesizing the reference sound field (primary field) within a finite spatial region (Berkhout 1988). Theoretically, based on the Kirchhoff-Helmholtz integral equation (KHI), the primary field can be synthesized within a listening volume using an infinite number of secondary monopole and dipole point sources distributed continuously on the listening volume boundary (threedimensional, 3D WFS, Vogel 1993). Typical primary fields are spherical waves (Boone et al. 1995). Current implementations attempt to reduce the number of secondary sources by degenerating the boundary area to a boundary contour (two-dimensional, 2D WFS, Spors et al. 2008). In this case, the situation is assumed to be independent of one arbitrary chosen Cartesian coordinate direction and the secondary point sources are replaced by secondary line sources with their axes along the coordinate direction selected (Spors and Ahrens 2010). The restriction to 2D situations prevents simulation of spherical wave fields, since no Cartesian coordinate direction can be found for a spherical wave to be independent of. This restriction is usually disregarded in the derivation of WFS, resulting in errors in the synthesized fields (Spors 2005, Spors et al. 2008). For typical implementations, the secondary line sources are replaced by point sources (2.5D WFS), requiring a correction term (Berkhout et al. 1993, Spors et al. 2008) and resulting in the fact that the synthesized field is correct at a reference position only for closed boundary contours (Start 1996) or on a reference line parallel to a linear secondary source distribution (de Vries 1996). All current secondary source corrections are based on far-field and high-frequency approximations, often disregarding phase relations, which results in erroneous synthetic fields, especially in the near-field and lowfrequency regions (Start 1997, Spors et al. 2008). In this paper, an updated and extended framework for WFS with continuous secondary source distributions is given, including a global formulation of secondary source correction. Furthermore, primary source correction (PSC) is introduced, which permits the correct synthesis of spherical waves evolving at arbitrary distances in the plane defined by the listening area and results in the correct inverse proportionality of level and distance for primary point sources at the correct absolute level, which is shown not to hold true for previous approaches of simulating primary spherical waves by means of WFS. The paper starts with a refinement of continuous 3D WFS in a listening volume, followed by the discussion of spherical and cylindrical primary sources. On that basis, the derivations of 2D and 2.5D WFS are revised, resulting in a general formulation of secondary source correction and the derivation of primary source correction (PSC), allowing for corrected reproduction of spherical waves. Finally, with regard to implementation, monopole only WFS is discussed. Time dependent variables are denoted by lower case, frequency dependent variables by upper case letters. 3D Wave Field Synthesis Following Williams (1999), equation 8.15, the KHI can be written so that it describes the homogeneous acoustic pressure field p(x) within the source-free volume V , with no field existing outside V (interior KHI). If P (x) denotes the temporal Fourier transform of p(x), the pressure field p(x) is given by the KHI on basis of the sound pressure spectrum P (x0) and its directional gradient ∂ ∂n (x0) = 〈∇P (x),n(x0)〉 ∣∣ x=x0 (1) (Bronstein et al. 2001, 13.34) on the surface with x0 ∈ S0 and the inward normal n(x0) to the surface S0 by