sound field synthesis of virtual cylindrical waves using...

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...


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


Circular Harmonics Representation

Line sourcemodel




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 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


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


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)


Spherical Harmonics Representation of a Line Source

Line sourcemodel




Smls,n(ω) = −πim−n+1H(2)m (ωc rls)Y

mn (π2 , αls)

∗


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


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


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)


e imα0


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)

∗


e imα0

Line sourcemodel





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

https://github.com/sfstoolbox/sfs

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


)


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


)


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


) f = 500 Hz


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


)


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


)


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


)


Summary and Discussion

Line sourcemodel




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

http://spatialaudio.net

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

sound field synthesis of virtual cylindrical waves using...

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