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Is Sound Field Control Determined at all Frequencies? How is it Related to Numerical Acoustics?

Sascha Spors
September 02, 2013

Is Sound Field Control Determined at all Frequencies? How is it Related to Numerical Acoustics?

Talk given at the 52nd International Conference of the Audio Engineering Society in Guildford, UK

Sascha Spors

September 02, 2013
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  1. 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

    View Slide

  2. 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

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  3. 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

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  4. 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

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  5. 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)
    −pin
    (r) +
    Vsc
    p(s)
    ∂G(r − s)
    ∂n(s)
    − G(r − s)
    ∂p(s)
    ∂n(s)
    dS(s) =





    0, r ∈ Vsc
    −p(r)/2, r ∈ ∂Vsc
    −p(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

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  6. 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

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  7. 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

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  8. 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

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  9. 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

    c
    R) + cnm
    h′
    n

    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

    c
    R) = 0 for R > 0
    Zotter, Spors | 2.9.2013 | Is Sound Field Control Determined at all Frequencies? | HOA 7

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  10. 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

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  11. 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

    View Slide

  12. 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

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  13. 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

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  14. 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

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  15. Example – Wave Field Synthesis
    synthesis of a virtual point source
    stationary-phase approximation w.r.t virtual source

    ∂V

    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

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  16. 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

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  17. 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

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  18. 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

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  19. 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

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  20. 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

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  21. 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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