19th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 COMPARISON OF HIGHER-ORDER AMBISONICS AND WAVE FIELD SYNTHESIS WITH RESPECT TO SPATIAL ALIASING ARTIFACTS

Wave field synthesis (WFS) and higher-order Ambisonics (HOA) are two sound reproduction techniques that facilitate a high number of reproduction channels to overcome the sweet-spot limitation known from conventional stereophonic reproduction techniques. HOA is based on a representation of the reproduced wave field in terms of spatial harmonics, WFS is based on the KirchhoffHelmholtz integral. It has been shown that HOA and WFS are comparable when applying a near-field correction to HOA and assuming an infinite number of loudspeakers around the listening area. However, both methods diverge with respect to their spatial aliasing properties for a finite number of loudspeakers. This contribution investigates the difference of HOA and WFS in terms of spatial aliasing artifacts for two-dimensional reproduction systems. INTRODUCTION Various massive multichannel reproduction systems have been proposed to enable spatial rendering of sound. The best-known of which are wave field synthesis (WFS) and higher-order Ambisonics (HOA). Although both of them are derived on the basis of the wave equation, the approaches come from different directions and their relationship is not clear. The work presented in [3, 5] has revealed a strong connection between WFS and HOA under certain conditions. These conditions are: (1) an infinite number of loudspeakers and (2) a near-field correction for HOA. However, a practical system will only have a limited number of loudspeakers. This might result in spatial aliasing artifacts present in the reproduced wave field. It was already noted in [3] that WFS and HOA show different performance w.r.t. spatial aliasing. Here, we will extend the comparison of WFS and HOA to the behavior of HOA and WFS in terms of their spatial aliasing properties. This paper is organized as follows: We first introduce the theoretical basics of WFS and HOA. We then apply specializations that allow for a comparison of both reproduction techniques. For WFS this specialization is a circular distribution of secondary sources and sampling of the secondary source distribution, for HOA it is a near-field compensation. We will then dedicate a section to the emphasis of the similarities and differences of WFS and HOA with respect to spatial sampling artifacts and illustrate the observations with numerical simulations. Nomenclature We will restrict the descriptions in this paper to two dimensional reproduction. Two dimensional in this context means that an observed sound field is independent from one of the spatial coordinates, e.g. P (x, y, z, ω) = P (x, y, ω). The following conventions are used: For scalar variables lower case denotes the time domain, upper case the temporal frequency domain. Vectors are denoted by lower case boldface. The Cartesian and polar coordinate system are used. The 0 S(x, ω) ∂V V virtual source n