Talk I gave at the EAA Winter School 2013 in Meran, Italy on the theoretical basis of Sound Field Synthesis techniques like higher-order Ambisonics (HOA) and Wave Field Synthesis (WFS).
Zotter 2 1Universität Rostock, Institute of Communications Engineering 2University of Music and Performing Arts Graz, Institute of Electronic Music and Acoustics
within the listening area V by loudspeakers (secondary sources) placed on the border ∂V s(x, t) ∂V V virtual source • assumed to result in authentic reproduction of original scene • frequency dependent weighting of secondary sources ⇒ driving signal Spors, Zotter | Sound Field Synthesis | Overview 1
• thorough review of the underlying physical problem • Higher-Order Ambisonics and Wave Field Synthesis Spors, Zotter | Sound Field Synthesis | Overview 2
∞ −∞ s(t, x)e−iωt dt Fourier-Transformation with respect to position vector x = [x y z]T ˜ S(ω, k) = ∞ −∞ S(ω, x)e+i k,x dxdydz Example: 3D free-ﬁeld Green’s function (acoustic point source) g0,3D (x|x0 , t) = 1 4π δ(t − x−x0 c ) x − x0 G0,3D (x|x0 , ω) = 1 4π e−i ω c x−x0 x − x0 ˜ G0,3D (k|x0 , ω) = 1 k 2 − (ω c )2 ei k,x0 Caution: Different deﬁnitions of the spatio-temporal Fourier-Transformation in the literature Spors, Zotter | Sound Field Synthesis | Overview 3
the interior problem for a bounded source free region V with smooth boundary ∂V ∂V P(x0 , ω) ∂G(x|x0 , ω) ∂(n, x0 ) − G(x|x0 , ω) ∂P(x, ω) ∂(n, x) x=x0 dS0 = P(x, ω) x ∈ V 1 2 P(x, ω) x ∈ ∂V 0 x ∈ ¯ V 0 ∂V V ¯ V n x x0 G(x|x0 , ω) P(x, ω) Spors, Zotter | Sound Field Synthesis | Theoretical Basis 4
the interior problem for a bounded source free region V with smooth boundary ∂V ∂V P(x0 , ω) ∂G(x|x0 , ω) ∂(n, x0 ) − G(x|x0 , ω) ∂P(x, ω) ∂(n, x) x=x0 dS0 = P(x, ω) x ∈ V 1 2 P(x, ω) x ∈ ∂V 0 x ∈ ¯ V Deﬁnition of the directional derivative ∂P(x, ω) ∂(n, x) x=x0 = ∇x P(x, ω), n(x) x=x0 External problem • exchange results for V and ¯ V on right hand side • reverse normal vector n or reverse signs on the right hand side Spors, Zotter | Sound Field Synthesis | Theoretical Basis 4
ω) within the region V is given by its pressure and pressure gradient on the boundary ∂V ∂V S(x0 , ω) Ddipole(x0,ω) ∂G(x|x0 , ω) ∂(n, x0 ) − G(x|x0 , ω) ∂S(x, ω) ∂(n, x) x=x0 Dmonopole(x0,ω) dS0 = S(x, ω) x ∈ V 0 S(x, ω) ∂V V ¯ V virtual source n x x0 G(x|x0 , ω) P(x, ω) Spors, Zotter | Sound Field Synthesis | Theoretical Basis 5
ω)G(x|x0 , ω)dS0 • aim P(x, ω) = S(x, ω) for x ∈ V • solution of integral equation with respect to driving signal D(x0 , ω) Theretical basis • acoustic scattering and radiation • Boundary Element Method (BEM) • Sound Field Synthesis (SFS) Spors, Zotter | Sound Field Synthesis | Theoretical Basis 6
explicit solution of the single layer potential 3. high-frequency approximation of the Kirchhoff-Helmholtz integral Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 7
at a scatterer with boundary ∂V 0 S(x, ω) ∂V V ¯ V virtual source n x x0 G(x|x0 , ω) P(x, ω) • incident ﬁeld (virtual source) is not affected by the presence of the scatterer • scattered sound ﬁeld Psc (x, ω) • total sound ﬁeld Pt (x, ω) = S(x, ω) + Psc (x, ω) Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 8
, αpw ) as virtual source Spw (x, ω) = 4π ∞ n=0 n m=−n (−i)njn ( ω c r)Ym n (βpw , αpw )∗Ym n (β, α) Scattering of plane wave at sphere with pressure release boundaries [Gumerov et al. 2004] Psc,pw (x, ω) = −4π ∞ n=0 n m=−n (−i)n jn (ω c R) h(2) n (ω c R) h(2) n ( ω c r)Ym n (βpw , αpw )∗Ym n (β, α) Calculation of driving function • superposition of virtual source and scattered sound ﬁeld • directional derivative with inward pointing radial unit vector ⇒ Analytic (Near-Field Compensated) Higher-Order Ambisonics [Ahrens et al. 2008] Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 10
a trivial problem if S(x, ω) = 0 for x ∈ ∂V ∂V D(x0 , ω)G(x|x0 , ω)dS0 = 0 Uniqueness of solution can be ensured by additional constraints • CHIEF point method [Schenk 1968] • Burton-Miller method [Burton et al. 1971] • deﬁne pressure on symmetry axis of rotationally symmetric problems e.g. analytic NFC-HOA [Ahrens et al. 2008] Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 12
explicit solution of the single layer potential 3. high-frequency approximation of the Kirchhoff-Helmholtz integral Spors, Zotter | Sound Field Synthesis | Explicit Solution 13
explicit solution of the single layer potential 3. high-frequency approximation of the Kirchhoff-Helmholtz integral Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 15
directional derivative into the Kirchhoff-Helmholtz integral results in ∂V S(x0 , ω) · 1 + j ω c r r · cos φ − ∂S(x, ω) ∂(n, x) x=x0 G(x|x0 , ω) dS0 = S(x, ω) for x ∈ V 0 S(x, ω) ∂V V ¯ V virtual source n x x0 r P(x, ω) φ r = x − x0 φ = ∠(n, (x − x0 )) Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 16
directional derivative into the Kirchhoff-Helmholtz integral results in ∂V S(x0 , ω) · 1 + j ω c r r · cos φ − ∂S(x, ω) ∂(n, x) x=x0 G(x|x0 , ω) dS0 = S(x, ω) for x ∈ V Stationary phase approximation 1. (x − x0 )||n ⇒ cos φ = 1 2. ω c r ≫ 1 ⇒ 1+j ω c r r → j ω c ∂V j ω c S(x0 , ω) − ∂S(x, ω) ∂(n, x) x=x0 D(x0,ω) G(x|x0 , ω) dS0 ≈ S(x, ω) Approximation is known as the high-frequency BEM [Herrin et al. 2003] Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 16
(x, ω) = e−j ω c npw,x Driving function for the synthesis of a plane wave Dpw (x0 , ω) = 1 + npw , n · j ω c · e−j ω c npw,x0 Selection of secondary sources 1 + npw , n ≈ 2 npw , n for npw , n > 0 0 for npw , n < 0 ∂V V n x0 npw plane wave Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 17
(x, ω) = e−j ω c npw,x Driving function for the synthesis of a plane wave Dpw (x0 , ω) = 1 + npw , n · j ω c · e−j ω c npw,x0 Selection of secondary sources 1 + npw , n ≈ 2 npw , n for npw , n > 0 0 for npw , n < 0 Driving function of 3D WFS [Spors et al. 2008] Dpw,3D (x, ω) = apw (x0 ) · 2 npw , n · j ω c · e−j ω c npw,x • same procedure for a virtual point source • WFS constitutes a high-frequency approximation of the BEM Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 17
global dependency of driving function • analytic driving functions for regular geometries only • exact reproduction within entire listening area possible Approximation of Kirchhoff-Helmholtz Integral (e.g. Wave Field Synthesis) • local dependency of driving function • analytic driving functions for arbitrary (convex) geometries • exact reproduction only for planar systems, minor degradations for curved systems Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 19
on the dimensionality of the problem: Reproduction in a volume (3D) G0,3D (x|x0 , ω) = 1 4π e−j ω c |x−x0 | |x−x0 | ⇒ secondary point sources Reproduction in a plane (2D) G0,2D (x|x0 , ω) = − j 4 H(2) 0 (ω c |x − x0 |) ⇒ secondary line sources 2.5-Dimensional Synthesis • secondary point sources are used for the synthesis in a plane • amplitude deviations in the listening area V • minor spectral deviations Spors, Zotter | Sound Field Synthesis | Dimensionality 20
sampled in practice S(x, ω) ∂V V virtual source • high impact on synthesis with full audio-signal bandwidth • may (perceptually) degrade the synthesis of the virtual source ⇒ sampling has to be considered in the evaluation of novel approaches Spors, Zotter | Sound Field Synthesis | Spatial Sampling 21
D(x0 , ω) DS (x0 , ω) PS (x, ω) n ∆x spatial sampling • sampling of driving function, interpolation by secondary sources • description by spatio-temporal spectra of driving function/secondary sources • intuitive anti-aliasing condition ∆x < λ 2 does not hold in general Sampling for linear/circular/spherical boundaries [Spors et al. 2006, Ahrens et al. 2012] • repetitions (and overlap) of spatial spectrum of driving function • band-limitation of spatial spectrum improves sampling artifacts (e.g. HOA) Spors, Zotter | Sound Field Synthesis | Spatial Sampling 22
• spatial sampling reduces achievable accuracy considerably • monochromatic sound ﬁelds are not sufﬁcient for evaluation • perception of synthetic sound ﬁelds plays an inportant role Reproducible Reserach • software implementations have high impact on results • Sound Field Synthesis (SFS) Toolbox http://github.com/sfstoolbox • SoundScape Renderer (SSR) http://www.spatialaudio.net/ssr Spors, Zotter | Sound Field Synthesis | Conclusions 25