Towards a Theory for Arbitrarily Shaped Sound Field Reproduction Systems Sascha Spors, Matthias Rath and Jens Ahrens Sascha.Spors@telekom.de Acoustics 08, Paris ===!" Laboratories AG Laboratories Aim of Sound Field Reproduction Perfect reproduction of virtual wave field S(x,ω) within listening area V by loudspeakers (secondary sources) placed on border virtual source S(x, ω) V bound and simply connected domain V with smooth border no sources and objects within V Focus of this contribution: Explicit solution of underlying physical problem w.r.t. loudspeaker driving signals. ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 1
Existing work boundary pressure control [Takane et al.,1999] higher-order Ambisonics (HOA), Wave Field Synthesis (WFS) operator theory for active noise control (ANC) [Loncaric et al.,2000] application of Functional Transformation Method (FTM) [Spors,2006] application of functional analysis [Fazi et al.,2007]... Aim of this contribution thorough review of the underlying physical problem, and its solution using methods from functional analysis ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 2 Fundamentals of Sound Field Reproduction The Kirchhoff-Helmholtz integral provides the solution of the homogeneous wave equation with respect to inhomogeneous boundary conditions ( G(x x 0, ω) n P(x 0, ω) P(x 0, ω) ) n G(x x 0,ω) P(x,ω) x V 1 ds 0 = 2P(x,ω) x 0 x V n G(x x 0, ω) V P(x,ω) V x 0 x 0 ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 3
Fundamentals of Sound Field Reproduction The field of a primary source S(x,ω) within the region V is given by its pressure and pressure gradient on the boundary S(x,ω) = ( G(x x 0,ω) n S(x 0,ω) S(x 0,ω) ) n G(x x 0, ω) ds 0 x V primary source n G(x x 0, ω) V P(x,ω) V S(x, ω) x 0 x 0 ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 3 Fundamentals of Sound Field Reproduction The Green s function and its gradient can be interpreted as sources (secondary sources) that generate the desired virtual wave field S(x,ω) inside the listening area V S(x,ω) = ( G(x x 0,ω) n S(x 0,ω) S(x 0,ω) ) n G(x x 0, ω) ds 0 x V virtual source n G(x x 0, ω) V P(x,ω) V S(x, ω) x 0 x 0 ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 3
Fundamentals of Sound Field Reproduction The theoretical basis of sound field reproduction is given by the Kirchhoff-Helmholtz integral S(x,ω) = ( G(x x 0,ω) n S(x 0, ω) S(x 0, ω) ) n G(x x 0, ω) ds 0 x V The explicit form of the Green s function depends on the dimensionality of the problem: Reproduction in a volume (3D) G 0,3D (x x 0,ω) = 1 e jω c x x 0 4π x x 0 secondary point sources Reproduction in a plane (2D) G 0,2D (x x 0,ω) = j 4 H(2) 0 (ω c x x 0 ) secondary line sources G(x x 0, ω) monopole secondary source n G(x x 0, ω) dipole secondary source ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 3 Elimination of Dipole Secondary Sources There exist various approaches to eliminate the dipole secondary sources, e.g. 1. Modification of Green s function used in the Kirchhoff-Helmholtz integral assumption of a Neumann Green s function secondary sources may be hard to realize for complex geometries basis of Wave Field Synthesis (WFS) 2. The Simple Source Approach provides formulation for monopole only reproduction driving function is given by considering a disjunct interior/exterior problem solution known to be not unique [Copley, 1967] 3. Assume monopole only reproduction ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 4
Reproduction Equation The problem of sound field reproduction can be formulated by a boundary integral equation for x S(x,ω) = D(x 0,ω) G(x x 0, ω) ds 0 that has to be solved w.r.t. the driving function D(x,ω). Analogy to acoustic scattering (Dirichlet boundary condition) incidence wave field equals wave field of virtual source scattered wave field equals wave field in V ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 5 Theory of Fredholm Operators The boundary integral equation can be interpreted as an operator (AD)(x) = D(x 0,ω) G(x x 0, ω) ds 0 A is a Fredholm operator (acting on a Sobolev space) if its range is closed kernel is finite dimensional cokernel is finite dimensional From the equivalent scattering problem it is known [Giroire, 1982] that A is a Fredholm operator of zero index is an isomorphism iff ω 0 is not an eigenvalue of the interior Dirichlet problem A is a compact operator! ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 6
Solution by Expansion of Operator Each compact (Fredholm) operator can be expanded into a series (A D)(x) = N n=1 G(n, ω) ψ n (x),d(x,ω) ψ n (x) with 1 N. G(n, ω) eigenvalues of operator ψ n is a orthogonal and normalized set in that fulfils the wave equation ψ n (x 0 ) ψ m (x 0 )ds 0 = δ nm n ψ n (x)ψ n (x 0 ) = δ(x x 0 ) typical Green s functions are symmetric, hence ψ n (x) = ψ n(x) ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 7 Solution by Expansion of Operator II Projection of the driving signal onto basis functions D(n, ω) = ψ n (x),d(x,ω) = D(x,ω) ψ n (x) ds Solution is derived by expanding the desired virtual wave field S(x,ω) w.r.t. the ψ n (x) comparison of coefficients Resulting loudspeaker driving function with the expansion coefficients N D(x,ω) = n=1 D(n, ω) ψ n (x) D(n, ω) = S(n, ω) G(n, ω) ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 8
Interpretation of Solution basis functions ψ n (x) are typically termed as acoustic modes non-uniqueness problem at eigenfrequencies of interior Dirichlet problem solutions given by null-space of operator A reported as less severe problem [Copley,1967], [Giroire,1982] ill-conditioning problem small eigenvalues values G(n, ω) may give rise to ill-conditioning modes with zero eigenvalues cannot be controlled solution: apply regularization or discard uncontrollable modes interpretation of transformation diagonalization of operator by expansion into acoustic modes basis of Wave-Domain Adaptive Filtering [Spors et al.,2004], [Buchner et al.,2004] Problem: Derivation of basis functions for a given geometry ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 9 Basis Functions for Regular Geometries Suitable basis functions are known for a variety of regular geometries e.g. [Morse,Feshbach,1953]. Examples two dimensional reproduction circular boundary exponential functions (Fourier series) 2D higher order Ambisonics elliptical boundary Mathieu functions three dimensional reproduction spherical boundary spherical harmonics oblate/prolate spheroidal boundary oblate/prolate spheroidal harmonics 3D higher order Ambisonics ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 10
Example: Two-Dimensional Higher Order Ambisonics expansion of wave fields on in terms of Fourier series using shift-theorem of Hankel functions and Jacobi-Anger expansion f pw = 1000 Hz f pw = 2000 Hz [R = 1.50 m, N = 56, θ pw = 90 o ] ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 11 Spatial Sampling In practice, the secondary source distribution is spatially sampled virtual source S(x, ω) V sampling has large impact for reproduction with audio-signal bandwidth on typical systems sampling may degrade the reproduction of the desired wave field sampling has to be considered in a generalized theory ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 12
Spatial Sampling of Driving Function Typical approach: model sampling of secondary sources by sampling of driving function D(x 0,ω) D S (x 0,ω) P S (x,ω) G 0 (x,ω) n x spatial sampling Sampling for linear/circular boundaries [Spors et al.,2006] repetitions (and overlap) of spatial spectrum of driving function band-limitation of spatial spectrum improves sampling artifacts Open questions for a generalized sampling theory applicability of existing approaches (scattering, boundary element method)? does sampling influence the non-uniqueness problem? formulation as discrete system more promising? ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 13 2 1 /2D Reproduction Typical systems aim at correct reproduction in a plane using secondary point sources Mismatch of secondary source types! The large-argument (far-field) approximation of the Hankel function G 0,2D (x x 0,ω) G 0,2 1/2D(x x 0, ω) = 2π jk x x0 e jk x x 0 4π x x 0 states that secondary line sources can be exchanged by point sources when applying a geometry independent spectral correction, and an amplitude correction with respect to a reference position/line Application to sound field reproduction: expansion of 2 1 /2D Green s function required, e.g. [Ahrens et al.,2008] ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 14
Conclusions This talk summarized the physical foundations of sound field reproduction and presented the outline of a generalized theory based on functional analysis. Benefits closed form solution for driving function provides insights into the underlying physics Drawbacks derivation of suitable basis functions for a given geometry non-uniqueness and ill-conditioning of the problem very limited description due to secondary source sampling Open Question: Is physically motivated sound field reproduction by the explicit solution of the (continuous) reproduction equation a promising concept? ===!" Laboratories Spors et al.: Arbitrarily Shaped Sound Field Reproduction Page 15