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Sound Field Synthesis of VirtualCylindrical Waves using Circular andSpherical Loudspeaker Arrays
Nara Hahn and Sascha SporsUniversity of Rostock, Institute of Communications Engineering
138th AES ConventionWarsaw, 10. May 2015
Sound Field Synthesis
aims at the physical reconstruction of a desired sound field S(x, ω) within a target regionusing a large number of secondary sources driven by individual signalsD(x0, ω)
S(x, ω)S(x, ω)
D(x0, ω)
Analytic methods
Wave Field Synthesis (WFS)
Near-field compensated higher-orderAmbisonics (NFC-HOA)
Spectral division method (SDM)
. . .
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Introduction 1 / 18
Analytic Source Models
Various analytic source models are used
Closed-form driving functions are known for
NFC-HOA WFSplane wave 3 3
line source 7 3
point source 3 3
focused source 3 3...
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Introduction 2 / 18
Outline
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
1. spherical harmonics representation of the sound field of a line source
2. analytic NFC-HOA driving function
3. evaluation of the synthesized sound field
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Introduction 3 / 18
Circular Harmonics Representation
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
circular harmonics expansion of a two-dimensional sound field (independent to the z -axis)
S(x, ω) =
∞∑m=−∞
Sm(ω)Jm(ωc r sinβ)e imα
α: azimuth angle, β: colatitude angle
Sm(ω): expansion coefficient
Jm(·): m-th Bessel function of the first kind
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Modal representation 4 / 18
Circular Harmonics Representation
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
circular harmonics expansion of the sound field of a line source
− i4H
(2)
0 (ωc ‖x−xls‖) =
∞∑m=−∞
− i4H
(2)m (ωc rls)e
−imαls︸ ︷︷ ︸Sls,m(ω)
Jm(ωc r sinβ)e imα.
xls = (rls, αls,π2
)
H(2)m (·): m-th Hankel function of the second kind
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Modal representation 4 / 18
Spherical Harmonics Representation
S(x, ω) =
∞∑n=0
n∑m=−n
Smn (ω)jn(ωc r)Y mn (β,α)
Smn (ω): expansion coefficient
jn(·): n-th spherical Bessel function of the first kind
Y mn (β,α) =√
2n+14π
(n−m)!(n+m)!
Pmn (cosβ)e imα : spherical harmonics
Pmn (·): associated Legendre function
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Modal representation 5 / 18
Spherical Harmonics Representation
S(x, ω) =
∞∑n=0
n∑m=−n
Smn (ω)jn(ωc r)Y mn (β,α)
=
∞∑m=−∞
e imα∞∑
n=|m|
Smn (ω)jn(ωc r)Y mn (β, 0)
︸ ︷︷ ︸=Sm(ω)Jm(
ωc r sinβ)
Smn (ω) = 4πim−nY mn (π2 , 0)∗Sm(ω)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Modal representation 5 / 18
Spherical Harmonics Representation of a Line Source
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
Smls,n(ω) = −πim−n+1H(2)m (ωc rls)Y
mn (π2 , αls)
∗
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Modal representation 6 / 18
Near-field Compensated Higher-order Ambisonics
G(x−x0 , ω)
D(x0, ω)
S(x, ω)∂V0 V0
explicit solution of the continuous synthesis equation
S(x, ω) =
∮∂V0
D(x0, ω)G(x− x0, ω)dA0
based on the spherical harmonics expansion
Smn (ω), Gmn (ω)
considers radially symmetric secondary sourcedistribution
3D: spherical distribution of point sources2D: circular distribution of line sources
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 7 / 18
Near-field Compensated Higher-order Ambisonics
G(x−x0 , ω)
D(x0, ω)
S(x, ω)∂V0 V0
explicit solution of the continuous synthesis equation
S(x, ω) =
∮∂V0
D(x0, ω)G(x− x0, ω)dA0
based on the spherical harmonics expansion
Smn (ω), Gmn (ω)
considers radially symmetric secondary sourcedistribution
3D: spherical distribution of point sources2D: circular distribution of line sources
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 7 / 18
Near-field Compensated Higher-order Ambisonics
G(x−x0 , ω)
D(x0, ω)
S(x, ω)∂V0 V0
explicit solution of the continuous synthesis equation
S(x, ω) =
∮∂V0
D(x0, ω)G(x− x0, ω)dA0
based on the spherical harmonics expansion
Smn (ω), Gmn (ω)
considers radially symmetric secondary sourcedistribution
3D: spherical distribution of point sources2D: circular distribution of line sources2.5D: circular distribution of point sources
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 7 / 18
Driving Functions
3D NFC-HOA
D3D(α0, β0, ω) =
∞∑n=0
n∑m=−n
1
r20
Smn (ω)
G0n(ω)︸ ︷︷ ︸
Dmn (ω)
Y mn (β0, α0)
2.5D NFC-HOA
D2.5D(α0, ω) =
∞∑m=−∞
1
2πr0
Sm|m|(ω)
Gm|m|(ω)︸ ︷︷ ︸Dm(ω)
e imα0
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 8 / 18
Driving Functions
3D NFC-HOA (2D sound field)
D3D(α0, β0, ω) =
∞∑n=0
n∑m=−n
1
r20
4πim−nY mn (π2 , 0)∗Sm(ω)
G0n(ω)︸ ︷︷ ︸
Dmn (ω)
Y mn (β0, α0)
2.5D NFC-HOA (2D sound field)
D2.5D(α0, ω) =
∞∑m=−∞
1
2πr0
4πim−|m|Y m|m|(π2 , 0)∗Sm(ω)
Gm|m|(ω)︸ ︷︷ ︸Dm(ω)
e imα0
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 8 / 18
Driving Functions
3D NFC-HOA (line source)
D3D(α0, β0, ω) =
∞∑n=0
n∑m=−n
1
r20
−πim−n+1H(2)m (ωc rls)Y
mn (π2 , αls)
∗
G0n(ω)︸ ︷︷ ︸
Dmn (ω)
Y mn (β0, α0)
2.5D NFC-HOA (line source)
D2.5D(α0, ω) =
∞∑m=−∞
1
2πr0
−πim−|m|+1H(2)m (ωc rls)Y
m|m|(
π2 , αls)
∗
Gm|m|(ω)︸ ︷︷ ︸Dm(ω)
e imα0
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | NFC-HOA 8 / 18
Numerical Simulation
3D NFC-HOA 2.5D NFC-HOAr0 1.5 m 1.5 mNloudspeaker 4841 64maximum order 21 31fartifact-free 764 Hz 1128 Hz
−1
0
1
−1
0
1
−1.5
−1
−0.5
0
0.5
1
1.5
xy
z
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
x
y
Sound Field Synthesis toolbox (https://github.com/sfstoolbox/sfs)
secondary monopole sources Gmn (ω) = −i ωch
(2)n (ω
cr)Y mn (β0, α0)∗
1Riesz s-energy approachN.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 9 / 18
3D Synthesis500 Hz
xy -plane
x / m
y/m
−2 −1 0 1 2
−2
−1
0
1
2
yz -plane
y / m
z/m
−2 −1 0 1 2
−2
−1
0
1
2
r0 = 1.5 m,Nloudspeaker = 484,M = 21, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 10 / 18
3D Synthesis1500 Hz
xy -plane
x / m
y/m
−2 −1 0 1 2
−2
−1
0
1
2
yz -plane
y / m
z/m
−2 −1 0 1 2
−2
−1
0
1
2
r0 = 1.5 m,Nloudspeaker = 484,M = 21, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 11 / 18
3D SynthesisAmplitude Decay for Fixed Source Position
NFC-HOA
−2 −1 0 1 2
−12
−9
−6
−3
0
3
6
2 kHz
1 kHz
500 Hz
250 Hz
y / m
|S(x,ω
)|/dB
real line source
WFS
−2 −1 0 1 2
−12
−9
−6
−3
0
3
6
2 kHz
1 kHz
500 Hz
250 Hz
y / m
|S(x,ω
)|/dB
real line source
r0 = 1.5 m,Nloudspeaker = 484,M = 21, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 12 / 18
3D SynthesisAmplitude Decay for Varying Source Position
NFC-HOA
0 2 4 6 8
−6
−3
0
3
6
9
rls / m
|S(0,ω
)|/dB
real line sourcez = 0.0z = 0.5z = 1.0
WFS
0 2 4 6 8
−6
−3
0
3
6
9
rls / m
|S(0,ω
)|/dB
real line sourcez = 0.0z = 0.5z = 1.0
r0 = 1.5 m,Nloudspeaker = 484,M = 21, xls = (rls, αls,π2
) f = 500 Hz
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 13 / 18
2.5D SynthesisNFC-HOA
1000 Hz
x / m
y/m
−2 −1 0 1 2
−2
−1
0
1
2
2500 Hz
x / m
y/m
−2 −1 0 1 2
−2
−1
0
1
2
r0 = 1.5 m,Nloudspeaker = 64,M = 31, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 14 / 18
2.5D SynthesisAmplitude Decay
NFC-HOA
−2 −1 0 1 2
−12
−9
−6
−3
0
3
6
2 kHz
1 kHz
500 Hz
250 Hz
y / m
|S(x,ω
)|/dB
real line source
WFS
−2 −1 0 1 2
−12
−9
−6
−3
0
3
6
2 kHz
1 kHz
500 Hz
250 Hz
y / m
|S(x,ω
)|/dB
real line source
r0 = 1.5 m,Nloudspeaker = 64,M = 31, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 15 / 18
2.5D SynthesisAmplitude Decay
Fixed source position
−2 −1 0 1 2
−6
−3
0
3
6
9
12
y / m
|S(x,ω
)|/dB
real line sourcevirtual point sourcevirtual line sourcevirtual plane wave
Varying source position
0 2 4 6 8
−6
−3
0
3
6
9
yls / m
|S(0,ω
)|/dB
real line sourcevirtual point sourcevirtual line sourcevirtual plane wave
r0 = 1.5 m,Nloudspeaker = 64,M = 31, xls = (rls, αls,π2
)
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Evaluation 16 / 18
Summary and Discussion
Line sourcemodel
Circular harmonicsexpansion coefficients
Spherical harmonicsexpansion coefficients
NFC-HOAdriving function
sound field with a mild amplitude decay
compensation of the low-pass characteristic (FEQ(ω) =√i ωc )
efficient realization of driving function required
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Conclusion 17 / 18
THANK YOU!
http://spatialaudio.net
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Conclusion 18 / 18
A1. Converting Sm(ω) to Smn (ω)
i−mJm(z) =
∞∑n=|m|
4πi−njn(z)Y mn (π2 , 0)∗Y mn (β, 0)
LHS
m
z
−20 0 200
10
20
30
40
50
dB−60
−50
−40
−30
−20
−10
0
RHS (M=30)
m
z
−20 0 200
10
20
30
40
50
dB−60
−50
−40
−30
−20
−10
0
error
m
z
−20 0 200
10
20
30
40
50
dB−60
−50
−40
−30
−20
−10
0
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Appendix 19 / 18
A2. Green’s Function
Spherical harmonics expansion ofG(x− x0, ω)
3D NFC-HOA: x0 = (r0, 0, 0)
2.5D NFC-HOA: x0 = (r0, 0, π2 )
Free-field Green’s function
Gmn (ω) = −i ωc h(2)n (ωc r0)Y mn (β0, α0)∗
N.Hahn and S.Spors | 10. May. 2015 | Virtual Cylindrical Waves in NFC-HOA | Appendix 20 / 18
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