Is Sound Field Control Determined at all Frequencies? How is it Related to Numerical Acoustics? Franz Zotter 1 and Sascha Spors 2 1University of Music and Performing Arts, Institute of Electronic Music and Acoustics 2Universität Rostock, Institute of Communications Engineering AES 52nd International Conference Guildford, UK
Sound Field Synthesis (SFS) Physical reconstruction of a sound field is assumed to result in high perceptual quality pin(r) ∂V V virtual source This contribution yet another review of the underlying problem focusing on uniqueness issues links to established theories in numerical acoustics Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Introduction 1
The Kirchhoff-Helmholtz Integral The Kirchhoff-Helmholtz integral provides the solution of the interior problem for a bounded source free region V with smooth boundary ∂V s∈∂V p(s) ∂G(r − s) ∂n(s) − G(r − s) ∂p(s) ∂n(s) dS(s) = −p(r), r ∈ V −p(r)/2, r ∈ ∂V 0, r / ∈ V Sound Field Synthesis V V n(s) n(s) p(s), ∂p(s) ∂n(s) ∂V ∂V p(r) −p(r) −p(r)/2 0 Green’s function represents secondary monopole and dipole sources no issues with non-uniqueness [Copley 1968] Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Background 2
Synthesis with Secondary Monopoles V (s) ∂V p(r) µ p(r) = ∂V µ(s) G(r − s) dS(s) Goal: Synthesized sound field p(r) should match desired pin(r) for r ∈ V Single layer solutions 1. explicit solution of the single layer potential [Daniel, Fazi, Ahrens, ...] 2. equivalent scattering approach [Fazi, ...] 3. high-frequency approximation of the Kirchhoff-Helmholtz integral [Herrin, Spors, ...] Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Background 3
Equivalent Scattering Approach Scattering of the desired field pin at the scatterer Vsc ∂Vin −p(r) −p(r)/2 0 Vin 0 ∂V sc V sc −pin (r) pin (r) −p sc (r) n(s) − ∂Vsc ∂pin (s) ∂n(s) + ∂psc (s) ∂n(s) −µ(s) G(r − s) dS(s) = pin (r), r ∈ Vsc pin (r), r ∈ ∂Vsc −psc (r), r / ∈ Vsc no acoustic interaction between Vin and Vsc assumed total sound field p = pin + psc sound soft boundary condition p(s) = 0 is applied theoretical basis of the Boundary Element Method (BEM) [Juhl, 1993] Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Equivalent Scattering 4
Uniqueness of the Simple Source Solution Superposition of non-trivial solutions pl (s) of the sound soft boundary condition ∂V ∂pl (s) ∂n(s) G(r − s) dS(s) = 0, for r ∈ ∂Vsc changes the synthesized sound field − ∂V ∂pin (s) ∂n(s) + ∂psc (s) ∂n(s) + l αl ∂pl (s) ∂n(s) G(r − s) dS(s) = = pin (r), r ∈ Vsc = pin (r), r ∈ ∂Vsc driving function µ(s) is not uniquely determined solutions pl (s) exist at specific frequencies (resonances) Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Uniqueness 5
Methods to Cope for Non-Uniqueness Various methods have been published to cope for uniqueness, e.g. 1. CHIEF point method [Schenk 1968] field of non-trivial solutions pl (s) is not zero inside Vsc null-field constraints in Vsc rule out non-uniqueness additional constraints by CHIEF-points placed inside Vsc 2. Burton-Miller method [Burton et al. 1971] solutions pl (s) cause an erroneous value of the boundary integral for p(r) ∈ Vsc additional derived integral constraint on ∂Vsc removes non-uniqueness Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Uniqueness 6
Example – Higher-Order Ambisonics (HOA) spherical secondary source distribution with radius R representation of sound fields in spherical harmonics Single layer boundary integral equation for psc without further constraints yields −i(ω c R)2 bnm j′ n (ω c R) + cnm h′ n (ω c R) jn(ω c r) hn(ω c R) = bnm jn(ω c r) Non-unique for r = R whenever jn(ω c R) = 0 Unique solution can be achieved by solving 1. for r < R (CHIEF) [Ahrens et al. 2008] , since always a jn(ω c r) = 0 exists 2. equation derived with respect to r (Burton-Miller), since whenever jn(ω c R) = 0, j′ n (ω c R) = 0 for R > 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | HOA 7
Illustration – Higher-Order Ambisonics (HOA) Scattered sound field psc(r) x / m y / m −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 3D ESA, f = 500 Hz, virtual point source at s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | HOA 8
Illustration – Higher-Order Ambisonics (HOA) Scattered and incident sound field psc(r) + pin(r) x / m y / m −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 3D ESA, f = 500 Hz, virtual point source at s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | HOA 8
Illustration – Higher-Order Ambisonics (HOA) Synthesized sound field p(r) x / m y / m −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 3D ESA, f = 500 Hz, virtual point source at s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | HOA 8
High-Frequency Approximation introducing the 3D free-field Green’s function into the Kirchhoff-Helmholtz integral stationary phase approximation 1. (r − s)||n(s) ⇒ cos φ = 1 2. ω c r ≫ 1 ⇒ 1+i ω c r r → i ω c approximation is known as high-frequency BEM [Herrin et al. 2003] ∂V p(s) cos φ 1 + iω c r r + ∂p(s) ∂n(s) G(r − s) dS(s) = p(r) for r ∈ V 0 pin (r) ∂V V virtual source n r s r p(r) φ r = r − s φ = ∠(n, (r − s)) Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | High-Frequency Approximation 9
High-Frequency Approximation introducing the 3D free-field Green’s function into the Kirchhoff-Helmholtz integral stationary phase approximation 1. (r − s)||n(s) ⇒ cos φ = 1 2. ω c r ≫ 1 ⇒ 1+i ω c r r → i ω c approximation is known as high-frequency BEM [Herrin et al. 2003] ∂V −iω c p(s) + ∂p(s) ∂n(s) G(r − s) dS(s) ≈ p(r) for r ∈ V 0 pin (r) ∂V V virtual source n r s r p(r) φ r = r − s φ = ∠(n, (r − s)) Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | High-Frequency Approximation 9
Example – Wave Field Synthesis synthesis of a virtual point source stationary-phase approximation w.r.t virtual source − ∂V iω c + cos φ0 1 + iω c r0 r0 G(r0 ) G(r − s) dS(s) ≈ pin (r) ∂V V φ0 s s0 r0 point source r0 = s − s0 φ0 = ∠(n(s), (s − s0 )) Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | WFS 10
Example – Wave Field Synthesis synthesis of a virtual point source stationary-phase approximation w.r.t virtual source − ∂V 2iω c max nT(s−s0) s−s0 , 0 G(r0 ) µ(s) G(r − s) dS(s) ≈ pin (r) ∂V V φ0 s s0 r0 point source r0 = s − s0 φ0 = ∠(n(s), (s − s0 )) equal to driving function of 3D WFS [Spors et al. 2008] WFS constitutes a high-frequency approximation of the BEM Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | WFS 10
Illustration – Comparison of WFS and HOA Continuous distribution of secondary sources Wave Field Synthesis x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Higher-Order Ambisonics x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5D WFS/HOA, s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | WFS 11
Illustration – Comparison of WFS and HOA Spatially discrete distribution of 56 secondary sources Wave Field Synthesis x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Higher-Order Ambisonics x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5D WFS/HOA, s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | WFS 11
Illustration – Comparison of WFS and HOA Spatially discrete distribution and spatially bandlimited (N = 28) HOA Wave Field Synthesis x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Higher-Order Ambisonics x / m y / m −2 −1 0 1 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5D WFS/HOA, s0 = [0 5 0] m, R = 1.5 m, z = 0 Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | WFS 11
Practical Constraints The synthesized sound field is affected by spatial sampling spatial bandlimitation directivity of loudspeakers frequency response of loudspeakers diffraction/reflections by loudspeakers listening room acoustics artifacts of 2.5-dimensional synthesis These practical artifacts have a major impact on the perceived quality! Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Conclusion 12
Conclusions theory of SFS is well covered by numerical acoustics uniqueness of equivalent scattering approach can be ensured WFS is a reasonable approximation with benefits (uniqueness, geometry) influence of practical constraints not well researched Thanks for your attention! www.spatialaudio.net Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | Conclusion 13
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