2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics October 20-23, 2013, New Paltz, NY FUNCTIONAL ANALYSIS GUIDED APPROACH FOR SOUND FIELD REPRODUCTION WITH FLEXIBLE LOUDSPEAKER LAYOUTS Wen Zhang and Thushara D. Abhayapala * The Australian National University Research School of Engineering Canberra, 2601, Australia {wen.zhang, thushara.abhayapala}@anu.edu.au Filippo M. Fazi University of Southampton Institute of Sound and Vibration Research Southampton, SO17 1BJ, UK ff1@isvr.soton.ac.uk ABSTRACT This paper proposes a design of multiple circular and partial circu- lar loudspeaker arrays for reproducing sound fields originated from a limited spatial region. We apply a functional analysis framework to formulate the sound field reproduction problem in closed form. Analytical solutions are derived for a circular secondary source ar- rangement, from which circular arc layouts are investigated and the design of placing multiple loudspeaker arrays over the limited re- gion of interest is proposed. Such a design allows for non-spherical and non-uniform loudspeaker placement and thus provides flexibili- ty to suit reproduction in real audio environments. The reproduction using the proposed method are illustrated by numerical simulations in comparison with the Least-squares based schemes. Index Terms— Circular loudspeaker arrays, partial layouts, sound field reproduction, 3D audio 1. INTRODUCTION Ambisonics is based on representation of a sound field as a sum of angular modes (cylindrical/spherical harmonics) and by mode- matching equations to derive loudspeaker driving signals for a nat- ural reproduction of sound over a large listening area [1]. Even though a few 3D reproduction systems were implemented in higher order Ambisonics [2, 3], there is a critical problem in terms of loud- speaker placement. To perfectly reconstruct incident sound fields, the mode-matching approach requires the placement of loudspeak- ers on a sphere that surrounds the target reproduction region [1, 4]. The deployment of a spherical loudspeaker array is impractical in reality. Non-spherical loudspeaker arrays have been deployed, such as a partial-sphere or a hemisphere [5] and multiple circular ar- rays [6], based on the Least-squares (LS) approach to match pres- sures recorded at multiple points in the target region and decom- posed by the spherical harmonics. The problem in most cases is ill-posed and Tikhonov regularization is the common method for obtaining loudspeaker weights with limited energy [7]. Using the compressive sensing idea, the LS formulation of the sound field re- production problem is regularized with the ℓ1 norm and solved us- ing the Least-absolute shrinkage and selection operator (Lasso) [8]. The assumption here is that the desired sound field can be repro- duced by a few loudspeakers, which are placed close to the direction of incidence and sparsely distributed in space. The functional analysis framework was firstly introduced into sound field reproduction by Fazi [2, 4]. A continuous distribution ∗ This work was supported under the Australian Research Councils Dis- covery Projects funding scheme (project no. DP110103369). of secondary sources and spatial sound fields are interrelated by an integral operator, from which a self-adjoint operator is construct- ed and singular value decomposition can be applied to modal de- compose the source distributions and sound fields with two sets of orthonormal functions (called as modes or modal basis functions). The construction of source distributions allows for a perfectly ac- curate reproduction if the full sets of source modes and sound field modes are included. As expected, the modal basis functions for source distributions and sound fields arranged on two concentric circles and spheres are cylindrical/spherical harmonics, respective- ly. This paper presents the design of a flexible scheme of placing multiple circular or partial circular loudspeaker arrays for sound field reproduction. We apply the functional analysis framework to analytically derive modal basis functions for each individual array at a particular colatitude and to determine the loudspeaker driving signals from the spherical spectrum of the desired sound field. The flexible geometry would suit sound field reproduction in real rooms and auditoria; in addition, by analyzing the reproduction efficien- cy of each array, we can design systems to focus on sound fields originated from a limited region of interest, i.e., only loudspeakers close to the virtual source positions are active—a feature imposed as constraints and achieved implicitly in the LS-based schemes. 2. PRELIMINARIES AND PROBLEM STATEMENT Consider a continuous secondary source distribution on the surface Λ0 that fully encloses a bounded region Λ, the generated sound field S(x) can be represented with an integral operator, S(x)=(Aρ)(x)=  Λ 0 ρ(y)G(x|y)dy, y ∈ Λ0, (1) where x is the observation point and y denotes a secondary source position. The control region considered here are the surface enclos- ing the entire reproduction region V 1 , i.e., x ∈ V0. The reproduc- tion region V is contained within the region Λ. G(x|y) is a Green’s function that represents a spatial-temporal transfer function from y to x; in free-field conditions for an omnidirectional point source it is defined as [9], G(x|y)= e −ik|y−x| 4π|y − x| , (2) 1 When the wave number is not one of the Dirichlet eigenvalues of the reproduction region, reproducing a sound field on the surface enclosing the region of interest ensures the exact sound field reproduction within the re- gion [9, 10].