An Analytical Approach to Sound Field Reproduction Using Circular and Spherical Loudspeaker Distributions

Abstract

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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