In this paper,wepresent the theoretical basics and implementation strategies for sound field reproduction using circular and spherical loudspeaker arrays. The presented approach can be seen as an analytical formulation of what is known as higher order Ambisonics. It relies on the assumption of acontinuous distribution of secondary sources on which sampling is performed to yield the loudspeaker driving signals for real-world implementations. We present the theoretical derivation of the loudspeaker driving signals and investigate the properties of the actual reproduced wave field, whereby the focus lies on the consequences of the spatial discretization of the secondary source distribution. PACS no. 43.38.Md, 43.60.Fg, 43.60.Sx, 43.60.Tj 988 ©S.Hirzel Verlag · EAA Ahrens, Spors: Sound field reproduction ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 94 (2008)paper,numerical methods such as [3,5,6,7,8] give only little insight into the general properties of the reproduced wave field. NomenclatureThe following notational conventions are used: Forscalar variables lower case denotes the time domain, upper case the temporal frequencyd omain. Ve ctors are denoted by lower case boldface. The three-dimensional position vector in Cartesian coordinates is givena sx = [ xyz ] T . The Cartesian coordinates are linked to the spherical coordinates via x = r cos α sin β, y = r sin α sin β,a nd z = r cos β. α denotes the azimuth, β the elevation. Confer also to Figure 1. Forfunctions dependent on spatial coordinates, we use the notations F (x)r espectively F (r, α, β) to emphasize agiven coordinate system.The acoustic wavenumber is denoted by k.Itisrelated to the temporal frequencyb yk 2 = ( ω/c) 2 with ω being the radial frequencyand c the speed of sound. Propagating monochromatic plane wavesand outgoing spherical waves are denoted by e −ik T pw x and (1/r)e −irω/c respectively,w ith k T pw = (ω/c)[cos θ pw sin φ pw sin θ pw sin φ pw cos φ pw ] T and (θ pw ,φ pw )d enoting the propagation direction of the plane wave.iisthe imaginary unit (i = √ −1). Due to the continuous formulation, we will not refer to loudspeakers butrather to secondary sources respectively their distributions and also to secondary source driving functions rather than to loudspeaker signals.The term driving function as referred to in this paper, denotes the function D(ω)bywhich an input signalŜ(ω) has to be filtered to yield the secondary source driving signal S D (ω). In the temporal frequencydomain this formulation reads 989 ACTA ACUSTICA UNITED WITH ACUSTICA Ahrens, Spors: Sound field reproduction Vol. 94 (2008)

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Abstract-In this paper, we consider physical reproduction of sound fields via planar and linear distributions of secondary sources (i.e., loudspeakers). The presented approach employs a formulation of the reproduction equation in spatial frequency domain which is explicitly solved for the secondary source driving signals. Wave field synthesis (WFS), the alternative formulation, can be shown to be equivalent under equal assumptions. Unlike the WFS formulation, the presented approach does not employ a far-field approximation when linear secondary source distributions are considered but provides exact results. We focus on the investigation of the spatial truncation and discretization of the secondary source distribution occurring in real-world implementations and present a rigorous analysis of evanescent and propagating components in the reproduced sound field.

The acoustic theory for multichannel sound reproduction systems usually assumes free-field conditions for the listening environment. However, their performance in real-world listening environments may be impaired by reflections at the walls. This impairment can be reduced by suitable compensation measures. For systems with many channels, active compensation is an option, since the compensating waves can be created by the reproduction loudspeakers. Due to the time-varying nature of room acoustics, the compensation signals have to be determined by an adaptive system. The problems associated with the successful operation of multichannel adaptive systems are addressed in this contribution. First, a method for decoupling the adaptation problem is introduced. It is based on a generalized singular value decomposition and is called eigenspace adaptive filtering. Unfortunately, it cannot be implemented in its pure form, since the continuous adaptation of the generalized singular value decomposition matrices to the variable room acoustics is numerically very demanding. However, a combination of this mathematical technique with the physical description of wave propagation yields a realizable multichannel adaptation method with good decoupling properties. It is called wave domain adaptive filtering and is discussed here in the context of wave field synthesis.