IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 25, NO. 5, MAY2017 1117 Improved Referencing Schemes for 2.5D Wave Field Synthesis Driving Functions Gergely Firtha, P´ eter Fiala, Frank Schultz, and Sascha Spors, Member, IEEE Abstract—Wave Field Synthesis allows the reconstruction of an arbitrary target sound field within a listening area by using a sec- ondary source contour of spherical monopoles. While phase cor- rect synthesis is ensured over the whole listening area, amplitude deviations are present besides a predefined reference curve. So far, the existence and potential shapes of this reference curve was not extensively discussed in the Wave Field Synthesis literature. This paper introduces improved driving functions for 2.5D Wave Field Synthesis. The novel driving functions allow for the control of the locations of amplitude correct synthesis for arbitrarily shaped— possibly curved—secondary source distributions. This is achieved by deriving an expressive physical interpretation of the stationary phase approximation leading to the presented unified Wave Field Synthesis framework. The improved solutions are better suited for practical applications. Additionally, a consistent classification of existing implicit and explicit 2.5D sound field synthesis solutions as special cases of the unified framework is given. Index Terms—Local wavenumber vector, stationary phase ap- proximation, wave field synthesis, 2.5D WFS. I. INTRODUCTION W AVE Field Synthesis (WFS) [1]–[4] is a well established sound field synthesis (SFS) technique for the reproduc- tion of an arbitrary target sound field over a listening area. Synthesis is performed using a loudspeaker array, termed as the secondary source distribution (SSD). The objective of WFS is to find the appropriate input signals for the SSD—termed as the driving functions—so that the resultant sound field coincides with the target sound field at each point of the listening area. WFS originates from the boundary integral representation of the target sound field [5]–[7] in a synthesis domain bounded by a smooth SSD surface. The Kirchhoff–Helmholtz integral rep- resentation of the target field implicitly contains the appropriate WFS driving functions. However, 3D surface SSDs are imprac- tical to implement, and 2D SSD contours are realized instead. This dimensionality reduction has two important consequences. Manuscript received August 1, 2016; revised January 14, 2017 and March 20, 2017; accepted March 26, 2017. Date of publication March 29, 2017; date of current version April 24, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Augusto Sarti. (Corresponding author: Gergely Firtha.) G. Firtha and P. Fiala are with the Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest 1117, Hungary (e-mail: firtha@hit.bme.hu; fiala@hit.bme.hu). F. Schultz is with the Audio Communication Group, Technische Universit¨ at Berlin, Berlin 10623, Germany (e-mail: franks.schultz@tu-berlin.de). S. Spors is with the Institute of Communications Engineering, Universit¨ at Rostock, Rostock 18119, Germany (e-mail: sascha.spors@uni-rostock.de). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASLP.2017.2689245 Phase correct synthesis is restricted to a 2D synthesis plane bounded by the SSD contour, and the synthesized field cannot be amplitude correct in the whole synthesis area, only at specific reference locations. This article focuses on WFS referencing, i.e., the analysis and control of positions where the synthesized field is amplitude correct. During the last three decades two main WFS methodologies have been developed, commonly referred to as traditional WFS and revisited WFS. The two approaches mainly differ in the way of the dimensionality reduction and have different referencing limitations. Traditional WFS aims to synthesize a virtual point source us- ing a 2D linear contour of secondary point sources. The method applies the 3D Rayleigh integral to the virtual point source field [8]–[11], and dimensionality reduction is performed by means of the stationary phase approximation (SPA) of rapidly oscillating integrals. The resulting so-called 2.5D Neumann Rayleigh integral [3] contains the appropriate driving functions. Note that the very initial derivation [1], [2] led to the 2.5D Dirichlet Rayleigh integral using a SSD containing secondary dipole sources, instead of spherical monopoles. Revisited WFS [12]–[14] can be used to synthesize a 2D vir- tual sound field with a 2D contour of secondary point sources. This approach stems from the 2D Neumann Rayleigh integral that assumes 2D secondary line sources. In order to obtain driv- ing functions for secondary point sources instead of line sources, the SPA is applied to the 2D Rayleigh integral. As 3D virtual fields are not unambiguously given on the SSD contour, revisited WFS theory is unable to synthesize 3D virtual fields. Theoretically, both WFS approaches are able to reconstruct the phase—i.e. the wavefront shape—of the virtual sound field over the listening area. However, due to different formulations of the dimensionality reduction, the approaches exhibit differ- ent amplitude deviations. The traditional WFS theory references the amplitude of the virtual point source typically to a reference line parallel to the SSD. The revisited WFS theory optimizes amplitude correct synthesis in a single reference point in the syn- thesis domain. This, as a side effect, results in amplitude correct synthesis along a reference curve which depends on the virtual source field. The exact positions of amplitude correct synthesis (PCS) in the revisited WFS theory have not been investigated in detail so far. In the present treatise we derive a unified WFS framework al- lowing the synthesis of arbitrary 2D and 3D virtual sound fields using a 2D SSD contour of point sources. The unified frame- work allows to optimize amplitude correct synthesis to an arbi- 2329-9290 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.