Invited lecture in the 178th Meeting of the Acoustical Society of America@San Diego 2nd Dec. 2019

1. Spatial Fourier transform-based
localized sound zone generation methods
with loudspeaker arrays
Takuma Okamoto
National Institute of Information and Communications Technology (NICT), Japan
1
2nd Dec. 2019, 178th Meeting of the Acoustical Society of America@San Diego

2. Introduction: Generating localized sound zone with loudspeakers
Previous methods and their problems
Proposed spatial Fourier transform-based localized sound zone
generation methods
Acoustically bright and dark zones with linear and circular arrays
Localized sound zones near loudspeakers with linear and circular arrays
Challenges and prospects
Conclusions
Outline
2

3. Generating localized sound zones
with loudspeakers
Acoustically bright and dark zones
Generating listening and quiet zones
Multiple sound zones
Superposition of bright and dark zones
Localized sound zones near loudspeakers
Listening zone is only near loudspeakers
Used in reverberant environments
Multizone sound field synthesis
Multiple sound fields are synthesized
in multiple zones
Not only sound pressures but also wavefronts
Introduction
3Ket point: How to realize “quiet zones”!!
Bright zone
(Listening area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Dark zone
(Quiet area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
こんにちは
Hello
你好
Japanese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
English
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Chinese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Brightzone
(Listeningarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
Darkzone
(Quietarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>

4. Discretized multiple point control: Kirkeby93, Choi02
Principle
Numerical calculation of inverse
of spatial (correlation) matrix
Problems
Quite unstable (ill-conditioned)
Numerical solutions and NOT based on wave equation
Spatial Fourier transform-based analytical sound field synthesis
Higher-order Ambisonics (HOA): Ahrens08
Spherical / circular harmonic analysis-based method for spherical / circular arrays
Spectral division method (SDM): Ahrens10
Angular spectrum-based method for planar / linear arrays
Previous methods and their problems
4
P (!) = G(!)D(!)
D(!) = G+
(!)P (!)
5
P (!) = G(!)D(!)
↓
D(!) = G+
(!)P (!)
例：BoSCシステム
Pressure matching-based
sound field control
...
...
...
...
G(!)
P (!)D(!)
Loudspeakers<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
Controlpoints<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
Stable driving function without evanescent component
Physically derived solutions based on wave equation

5. Generating localized sound zones
with loudspeakers
Acoustically bright and dark zones
Generating listening and quiet zones
Multiple sound zones
Superposition of bright and dark zones
Localized sound zones near loudspeakers
Listening zone is only near loudspeakers
Used in reverberant environments
Multizone sound field synthesis
Multiple sound fields are synthesized
in multiple zones
Not only sound pressures but also wavefronts
Introduction
5Ket point: How to realize “quiet zones”!!
Bright zone
(Listening area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Dark zone
(Quiet area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
こんにちは
Hello
你好
Japanese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
English
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Chinese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Brightzone
(Listeningarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
Darkzone
(Quietarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>

6. Analytically derived spatial filters in angular spectrum domain
Spatial filters in angular spectrum domain
Shift theorem introduced to shift rectangular window along with -axis
Spatial filters for generating bright and dark zones
x
y
y = yb
Secondary source
Bright
DarkDark
P(x, yb) = 1
P(x, yb) = 0
lb
xb
Shift
˜F(kx, !) =
˜P(kx, yref , !)
˜G(kx, yref , !)
P(x, yref , !) = 1
P(x, yref , !) = 0
Modeled by
Rectangular window
P(x, yb) = ⇧
✓
x
lb
◆
=
⇢
1, for |x|  lb/2
0, elsewhere
˜P(kx) = lb sinc
✓
kxlb
2⇡
◆
Fx
˜F(kx, !) =
lb sinc (kxlb/2⇡) exp(jkxxb)
˜G(kx, yb, !)
˜Pshift(kx) = ˜P(kx) exp(jkxxb) = lb sinc
✓
kxlb
2⇡
◆
exp(jkxxb)
x
Arbitrary length :
Arbitrary position : [xb, yb]T
lb
Proposed method with a linear array
6
ICASSP14

7. Demo with an actually implemented array
7
ICASSP14

8. 8

9. Comparison with 5 methods
DS: Delay-and-sum bermforming
CM: Contrast maximization: Choi02
LS: Pressure-matching: Kirkeby93
SFT: Proposed spatial Fourier transform
rect: Rectangular window-based filter
Hann: Hann window-based filter
Results
9
DS CM LS SFT (rect.) SFT (Hann)
z = 0 [m]
[dB]1.5 2 2.5
y [m]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x[m]
-30
-25
-20
-15
-10
-5
0
z = 0 [m]
[dB]1.5 2 2.5
y [m]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x[m]
-30
-25
-20
-15
-10
-5
0
z = 0 [m]
[dB]1.5 2 2.5
y [m]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x[m]
-30
-25
-20
-15
-10
-5
0
z = 0 [m]
[dB]1.5 2 2.5
y [m]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x[m]
-30
-25
-20
-15
-10
-5
0
z = 0 [m]
[dB]1.5 2 2.5
y [m]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x[m]
-30
-25
-20
-15
-10
-5
0
0 1 2 3 4 5
Temporal frequency [kHz]
0
5
10
15
20
25
30
35
40
Brighttodarkratio[dB]
DS CM LS SFT (rect.) SFT (Hann)
JASA17

10. Compaison between rectangular and Hann windows
Fourier transform of rectangular window = sinc function
Fourier transform of Hann window = superposition of sinc functions
Comparison of calculated results on control line
Fewer sidelobes in Hann window
-> Fewer plane waves orthogonal to linear array
-> Still higher accuracy except for control line
Property of SFT-based methods
Synthesized accuracy can be physically interpreted
-40 -30 -20 -10 0 10 20 30 40
kx
0
5
10
15
20
25
30
35
40
45
|˜F(kx,k)|
SFT (rect.) SFT (Hann)
Z 1
1
Prect (x, lb) ejkxx
dx =
Z lb/2
lb/2
ejkxx
dx =
lb sin (kxlb/2)
(kxlb/2)
= lb sinc
✓
kxlb
2⇡
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Z 1
1
PHann (x, lb) ejkxx
dx =
Z lb/2
lb/2
cos (⇡x/lb) ejkxx
dx =
Z lb/2
lb/2
ej⇡x/lb
+ e j⇡x/lb
2
ejkxx
dx
=
lb
2
sinc
✓
1
2
+
kxlb
2⇡
◆
+
lb
2
sinc
✓
1
2
kxlb
2⇡
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Result analysis
JASA17
10

11. Spatial filters in 2D cylindrical harmonic spectrum domain
Proposed method with a circular array
11
WASPAA15
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
z = 0 [m]
[dB]-1 0 1
X axis [m]
-1.5
-1
-0.5
0
0.5
1
1.5
Yaxis[m]
-40
-30
-20
-10
0
LSBeamformingSFT
f=3000Hz
Open Baffled
0 1 2 3 4 5
Temporal Frequency [kHz]
0
5
10
15
20
25
BDR[dB]
Spatial Nyquist Frequency
LS method (open)
LS method (baffled)
Beamforming (open)
Beamforming (baffled)
Proposed (open)
Proposed (baffled)
y
x
z
r0
rref
Φ
φs
P(rref , φ, 0, ω) = 1
P(rref , φ, 0, ω) = 0
Dark zone
Bright zone
Circular sound source
Reference circle

12. Generating localized sound zones
with loudspeakers
Acoustically bright and dark zones
Generating listening and quiet zones
Multiple sound zones
Superposition of bright and dark zones
Localized sound zones near loudspeakers
Listening zone is only near loudspeakers
Used in reverberant environments
Multizone sound field synthesis
Multiple sound fields are synthesized
in multiple zones
Not only sound pressures but also wavefronts
Introduction
12Ket point: How to realize “quiet zones”!!
Bright zone
(Listening area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Dark zone
(Quiet area)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
こんにちは
Hello
你好
Japanese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
English
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Chinese
area<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Brightzone
(Listeningarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
Darkzone
(Quietarea)<latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit><latexitsha1_base64="(null)">(null)</latexit>
NOT realized by beamforming and parametric arrays

13. Proposed localized sound zone generation based on residual sound
field between two types of sound sources
A circular and linear array combination
ICASSP15, JIHMSP17b
A linear array and an additional loudspeaker
JIHMSP17a
A circular array and an additional loudspeaker
IWAENC18
Problems
Undesired sound pressures propagate vertical directions: JIHMSP17a, IWAENC18
Complicated arrangement of loudspeakers are required: ICASSP15, JIHMSP17b
Proposed methods and their problems
13
Quiet zone<latexit sha1_base64="5Qy+Y6CSS/IpelXJ98RZ0gZQckw=">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</latexit><latexit sha1_base64="5Qy+Y6CSS/IpelXJ98RZ0gZQckw=">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</latexit><latexit sha1_base64="5Qy+Y6CSS/IpelXJ98RZ0gZQckw=">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</latexit><latexit sha1_base64="5Qy+Y6CSS/IpelXJ98RZ0gZQckw=">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</latexit>
Listening
zone
<latexit
sha1_base64="50UZq2Mr5xzL42T5YG9Qp8GpHGk=">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</latexit>
<latexit
sha1_base64="50UZq2Mr5xzL42T5YG9Qp8GpHGk=">AAACcXichVHLSsNAFD2N7/po1Y3iprQoglBu3CiuBDcuXNhqtVCrJHGsQ9NJSNKCFn/AH1BwpSAifoYbf8CFnyAuK7hx4W0aEBX1hsycOXPPvWdmTNeWfkD0FNO6unt6+/oH4oNDwyOJ5OjYlu/UPUsULMd2vKJp+MKWShQCGdii6HrCqJm22DarK+397YbwfOmozeDIFeWaUVHyQFpGwNTuGncQSqpK6thRYi+ZoSyFkfoJ9AhkEMW6k7zBDvbhwEIdNQgoBIxtGPD5K0EHwWWujCZzHiMZ7gucIM7aOmcJzjCYrfJY4VUpYhWv2zX9UG1xF5t/j5UpTNMj3VKLHuiOnun911rNsEbbyxHPZkcr3L3E6cTG27+qGs8BDj9Vf3oOcIDF0Ktk727ItE9hdfSN47PWxlJ+ujlDV/TC/i/pie75BKrxal3nRP4CcX4A/ft1/wRb81mdsnpuPrO8GD1FP6aQxizf9wKWsYp1FLivh3Nc4irW0ia1lJbupGqxSDOOL6HNfQAf149/</latexit>
<latexit
sha1_base64="50UZq2Mr5xzL42T5YG9Qp8GpHGk=">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</latexit>
<latexit
sha1_base64="50UZq2Mr5xzL42T5YG9Qp8GpHGk=">AAACcXichVHLSsNAFD2N7/po1Y3iprQoglBu3CiuBDcuXNhqtVCrJHGsQ9NJSNKCFn/AH1BwpSAifoYbf8CFnyAuK7hx4W0aEBX1hsycOXPPvWdmTNeWfkD0FNO6unt6+/oH4oNDwyOJ5OjYlu/UPUsULMd2vKJp+MKWShQCGdii6HrCqJm22DarK+397YbwfOmozeDIFeWaUVHyQFpGwNTuGncQSqpK6thRYi+ZoSyFkfoJ9AhkEMW6k7zBDvbhwEIdNQgoBIxtGPD5K0EHwWWujCZzHiMZ7gucIM7aOmcJzjCYrfJY4VUpYhWv2zX9UG1xF5t/j5UpTNMj3VKLHuiOnun911rNsEbbyxHPZkcr3L3E6cTG27+qGs8BDj9Vf3oOcIDF0Ktk727ItE9hdfSN47PWxlJ+ujlDV/TC/i/pie75BKrxal3nRP4CcX4A/ft1/wRb81mdsnpuPrO8GD1FP6aQxizf9wKWsYp1FLivh3Nc4irW0ia1lJbupGqxSDOOL6HNfQAf149/</latexit>
rref<latexit sha1_base64="CFUkeuTib2k20J6+6s9lyqoKtc0=">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</latexit><latexit sha1_base64="CFUkeuTib2k20J6+6s9lyqoKtc0=">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</latexit><latexit sha1_base64="CFUkeuTib2k20J6+6s9lyqoKtc0=">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</latexit><latexit sha1_base64="CFUkeuTib2k20J6+6s9lyqoKtc0=">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</latexit>
r1<latexit sha1_base64="kECeVdONZkREA05auDq+YDTMi20=">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</latexit><latexit sha1_base64="kECeVdONZkREA05auDq+YDTMi20=">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</latexit><latexit sha1_base64="kECeVdONZkREA05auDq+YDTMi20=">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</latexit><latexit sha1_base64="kECeVdONZkREA05auDq+YDTMi20=">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</latexit>
D0<latexit sha1_base64="akObRUbfr1lPOulNpHk6Sr3hGBY=">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</latexit><latexit sha1_base64="akObRUbfr1lPOulNpHk6Sr3hGBY=">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</latexit><latexit sha1_base64="akObRUbfr1lPOulNpHk6Sr3hGBY=">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</latexit><latexit sha1_base64="akObRUbfr1lPOulNpHk6Sr3hGBY=">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</latexit>
D1<latexit sha1_base64="0jEde/BFIENs7k9a32fBI5gsmnU=">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</latexit><latexit sha1_base64="0jEde/BFIENs7k9a32fBI5gsmnU=">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</latexit><latexit sha1_base64="0jEde/BFIENs7k9a32fBI5gsmnU=">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</latexit><latexit sha1_base64="0jEde/BFIENs7k9a32fBI5gsmnU=">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</latexit>
x<latexit sha1_base64="sJ3Yc9jxYbeQynDF3dTMwgVRBRY=">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</latexit><latexit sha1_base64="sJ3Yc9jxYbeQynDF3dTMwgVRBRY=">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</latexit><latexit sha1_base64="sJ3Yc9jxYbeQynDF3dTMwgVRBRY=">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</latexit><latexit sha1_base64="sJ3Yc9jxYbeQynDF3dTMwgVRBRY=">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</latexit>
y<latexit sha1_base64="rf5WtD2FO1bFDQSrh96tL06skOQ=">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</latexit><latexit sha1_base64="rf5WtD2FO1bFDQSrh96tL06skOQ=">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</latexit><latexit sha1_base64="rf5WtD2FO1bFDQSrh96tL06skOQ=">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</latexit><latexit sha1_base64="rf5WtD2FO1bFDQSrh96tL06skOQ=">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</latexit>
y
x
z
r0
D
L(x,!)
˜P
C
+
L(r>
r0,kx,!)=
0
˜P
C
+
L(r<
r0,kx,!)6=
0
D
C(r0,!)
Y axis [m]
Xaxis[m]
z = 0 [m]
[dB]
2 1 0 1 2
2
1.5
1
0.5
0
0.5
1
1.5
2
40
35
30
25
20
15
10
5
0
1076 T. Okamoto
z = 0 [m]
[dB]-1 0 1 2 3 4
y [m]
-2
-1
0
1
2
x[m]
-40
-35
-30
-25
-20
-15
-10
-5
0
(a) x-y plane (f = 500 Hz)
x = 0 [m]
[dB]-1 0 1 2 3 4
y [m]
-2
-1
0
1
2
z[m]
-40
-35
-30
-25
-20
-15
-10
-5
0
(b) y-z plane (f = 500 Hz)
Figure 6. Results of sound pressure level produced
using linear array of 64 loudspeakers with x =0.05
Produced sound pressure level at x = [0, 0.1, 0]T
i
circles are loudspeakers.
[d0 0.5 1 1.5 2 2.5 3
Distance [m]
0.5
1
1.5
2
2.5
3
3.5
4
Temporalfrequency[kHz]
Figure 7. Results of distance amplitude properties
1072 T. Okamoto
(a) Producing monopole source!
using a point source A(1)
y = yrefy
Point source
x
y = yd
(b) Producing virtual monopole source!
using the linear sound source A(2)
y = yrefy
Virtual point source
x
Linear sound source
y = yd
(c) Proposed method!
Producing (a) and (b) simultaneously
y = yrefy
Point source
x
Linear sound source
y = ydDA(1) = S(!)
PA(1)(x, yref , 0, !) PA(2)(x, yref , 0, !)
DA(2) DA(2)
DA(1) = S(!)
PA(x, yref , 0, !) = 0
Figure 2. Arrangements of sound sources and scheme of proposed method.
DA(1) and DA(2) are respective driving functions of point source A(1) given
as equation (7) and linear sound source A(2) given as equation (11). PA(1)
and PA(2) are sound pressures produced by sound sources A(1) and A(2)
defined in equations (9) and (12). PA is total produced sound pressure in
proposed method given in equation (13).
in the spectral division method (SDM) [28]. From equations (3) and (5), the produced
sound pressure is then represented:
˜P(kx, y, 0, !) =
˜P(kx, yref, 0, !)
H
(2)
0 (k⇢yref)
H
(2)
0 (k⇢y) . (6)
In the SDM using a linear sound source (2.5-dimensional SDM), equation (6) shows
that the produced sound pressure is correct only at reference receiver line y = yref because

14. 3D localized sound zone generation with multiple circular arrays and
an additional loudspeaker
Principle
Sound field produced by a center loudspeaker is cancelled by using multiple circular
arrays
Property
Sound field produced by only has 0th order spherical harmonic spectrum
Alternative proposed method
14
(b) Exterior sound field synthesis
+
(c) Proposed method
D D
D0 D0
− ˇB0
0 = −
jk
√
4π
for r > rQ
rQ
ˇB0
0 =
jk
√
4π
for r > 0
(a) Monopole sound field
Residual sound field
for r < rQ
WASPAA19
RODUCED BY POINT
TAL PLANE
lative to Cartesian coordi-
und field for regions exte-
n(kr)Y m
n (✓, ), (1)
onic spectra of the exterior
Hankel function of the first
ber [22];
!
!
P|m|
n (cos ✓)
}
ejm
(2)
|
is the associated Legen-
assumption, the transfer
= [r, ✓0, 0]T
to a point
ld Green’s function [22]:
m
(✓ , )⇤
Y m
(✓, ), (3)
n + |m| even because P
|m|
n (0) = 0 when n + |m| is odd [27–30].
Therefore, to synthesize a complete 3D exterior sound field using
loudspeakers on the horizontal plane, both monopole and vertical
dipole components are required. These vertical dipole components
include the vertical derivatives of P
|m|
n (0) and can control the odd
spectra. To approximately implement the vertical dipole compo-
nents using actual monopole loudspeakers, monopole pairs are in-
troduced [27, 28]. Therefore, the conventional approach requires
double the number of loudspeakers.
In the proposed approach, the sound field produced by a point
source located at the origin only including the 0-th order spherical
harmonic spectrum component (Fig. 1(a)), is cancelled using multi-
ple co-centered circular loudspeaker arrays on the horizontal plane
(Fig. 1(b)). As a result, the proposed method can be simply real-
ized using a practical planar omni-directional array with half the
number of loudspeakers required for the conventional method using
monopole pairs [27,28].
3. PROPOSED 3D LOCALIZED SOUND ZONE
GENERATION
From (3), the sound field produced by a monopole loudspeaker with
driving function D0 located at the origin (Fig. 1(a)) is given as:
S0(r, ✓, ) = D0G(r, r0 = 0)
= D0
0X
n=0
nX
m= n
jk
p
4⇡
hn(kr)Y m
n (✓, ) = D0
ejkr
4⇡r
, (5)

15. Sound pressure Level: 500 Hz
Sound pressure level on -axis
Results of alternative proposed method
15(d) 20 log10 |S0+1(x, y = 0, z)|0)|
(b) S0+1(x, y = 0, z) (d) 20 log10 |S0+1(x, y = 0, z)|(c) 20 log10 |S0+1(x, y, z = 0)|
(a) S1(x, y = 0, z) (b) S0+1(x, y = 0, z)
再生音圧レベル：500 Hz
z軸上の音圧レベル：∼ 800 Hz
(a) Point source (b) Method with monopole pairs (c) Proposed method
Simulation results
(b) 20 log10 |S0+1(x, y = 0, z)|(a) S0+1(x, y = 0, z)
従来法(モノポール対) 提案法(無指向性)
提案法の有効性を確認
(d) 20 log10 |S0+1(x, y = 0, z)|(c) 20 log10 |S0+1(x, y, z = 0)|
(a) S1(x, y = 0, z) (b) S0+1(x, y = 0, z) (d) 20 log10 |S0+1(x, y =(c) 20 log10 |S0+1(x, y, z = 0)|
(a) S1(x, y = 0, z) (b) S0+1(x, y = 0, z)
Previous problems can be successfully solved
音圧レベル：500 Hz
の音圧レベル：∼ 800 Hz
(a) Point source (b) Method with monopole pairs (c) Proposed method
Simulation results
10
(b) 20 log10 |S0+1(x, y = 0, z)|S0+1(x, y = 0, z)
従来法(モノポール対) 提案法(無指向性)
提案法の有効性を確認
(d) 20 log10 |S0+1(x, y = 0, z)|(c) 20 log10 |S0+1(x, y, z = 0)|
(a) S1(x, y = 0, z) (b) S0+1(x, y = 0, z) (d) 20 log10 |S0+1(x, y = 0, z)|(c) 20 log10 |S0+1(x, y, z = 0)|
(a) S1(x, y = 0, z) (b) S0+1(x, y = 0, z)
従来法(モノポール対)：(5 + 7 + 9) x 2 + 1 = 43チャネル
Δz = 0.05 m
点音源位置：[0, 0, ­0.1]T
提案法(無指向性)：5 + 7 + 9 + 1 = 22チャネル
(a) Method with monopole pairs
∆z
(b) Proposed method12+1=22 loudspeakers
WASPAA19
opole pairs, and the proposed method.
y Beven
0 and Bodd
0 components and the
entional method, Deven+odd
, was also
mum order N in (12) was set at 100.
ts for both the conventional and pro-
n Fig. 2.
ds S1(x, y, z) and S0+1(x, y, z), and
0 log10 |S0+1(x, y, z)|, produced us-
h with monopole pairs and using the
nar omni-directional array at a fre-
shown in Figs. 3 and 4, respectively.
he sound pressure levels on the z-axis
Hz  f  800 Hz produced using a

16. Applied to flexible array configuration
Spatial Fourier transform: Arrangements of arrays MUST be planar / linear or circular /
spherical
Arrangements and applications are limited
-> Investigating other arrangements and their formulations
3D localized sound zone near loudspeakers
Implementing a system with actual loudspeakers
Investigating distance attenuation control
Challenges and prospects
16

17. Spatial Fourier transform-based localized sound zone generation
Multiple sound zones with linear and circular arrays
Localized sound zone near loudspeakers with circular and linear arrays
Challenges and prospects
Acknowledgements
These studies were partly supported by JSPS KAKENHI grand numbers JP25871208,
JP15K21674, and JP18K11387
JP25871208: Apl. 2013 to Mar. 2015
ICASSP14, ICASSP15
JP15K21674: Apl. 2015 to Mar. 2018
WASPAA15, ICASSP16, JASA17, ICASSP17, JIHMSP17a, WASPAA17,
JIHMSP17b
JP18K11387: Apl. 2018 to Mar. 2021
Appl. Sci18, IWAENC18, ICASSP19, WASPAA19
Conclusions
17

18. Deep neural network-based text-to-speech synthesis
T. Okamoto, T. Toda, Y. Shiga, and H.Kawai,“Tacotron-based acoustic model using
phoneme alignment for practical neural text-to-speech systems,” in Proc. ASRU, Dec.
2019. (to appear)
ASRU 2019: IEEE Automatic speech recognition and understanding workshop,
14–18, December, 2019, Sentosa, Singapore
Announcement
18

19. Sound field control with least square-based Pressure-matching
Kirkeby93: O. Kirkeby and P.Nelson, “Reproduction of plane wave sound fields,” J. Acoust.
Soc. Am., vol. 94, no. 5, pp. 2992–3000, Nov. 1993.
Acoustic contrast control
Choi02: J.-W. Choi and Y.-H. Kim, “Generation of an acoustically bright zone with an
illuminated region using multiple sources,” J. Acoust. Soc. Am., vol. 111, no. 4, pp. 1695–
1700, Apr. 2002.
Spectral division method
Ahrens08: J. Ahrens and S. Spors, “An analytical approach to sound field reproduction
using circular and spherical loudspeaker distributions,” Acta Acust. Acust., vol. 94, no. 6,
pp. 988–999, Nov. 2008.
Spectral division method
Ahrens10: J. Ahrens and S. Spors, “Sound field reproduction using planar and linear arrays
of loudspeakers,” IEEE Trans. Audio, Speech, Lang. Process., vol. 18, no. 8, pp. 2038–
2050, Nov. 2010.
Related works
19

20. ICASSP14: T. Okamoto, “Generation of multiple sound zones by spatial filtering in
wavenumber domain using a linear array of loudspeakers,” in Proc. ICASSP, May 2014, pp.
4733–4737.
ICASSP15: T. Okamoto, “Near-field sound propagation based on a circular and linear array
combination,” in Proc. ICASSP, Apr. 2015, pp. 624–628.
WASPAA15: T. Okamoto, “Analytical methods of generating multiple sound zones for open
and baffled circular loudspeaker arrays,” in Proc. WASPAA, Oct. 2015.
JASA17: T. Okamoto and A. Sakaguchi, “Experimental validation of spatial Fourier transform-
based multiple sound zone generation with a linear loudspeaker array,” J. Acoust. Soc. Am.,
vol. 141, no. 3, pp. 1769–1780, Mar. 2017.
JIHMSP17a: T. Okamoto, “Localized sound zone generation based on external radiation
canceller,” J. Inf. Hiding Multimed. Signal Pro- cess., vol. 8, no. 6, pp. 1335–1351, Nov.
2017.
Publications: Localized sound zone (1)
20

21. JIHMSP17b: T. Okamoto, “Horizontal local sound field propagation based on sound source
dimension mismatch,” J. Inf. Hiding Multimed. Signal Process., vol. 8, no. 5, pp. 1069–1081,
Sept. 2017.
IWAENC18: T. Okamoto, “2.5D localized sound zone generation with a circular array of fixed-
directivity loudspeakers,” in Proc. IWAENC, Sept. 2018, pp. 321–325.
WASPAA19: T. Okamoto, “3D localized sound zone generation with a planar omni-directional
loudspeaker array,” in Proc. WASPAA, Oct. 2019, pp. 370–374.
Publications: Localized sound zone (2)
21

22. ICASSP16: T. Okamoto, “2.5D higher-order Ambisonics for a sound field described by
angular spectrum coefficients,” in Proc. ICASSP, Mar. 2016, pp. 326–330.
ICASSP17: T. Okamoto, “Analytical approach to 2.5D sound field control using a circular
double-layer array of fixed-directivity loudspeakers,” in Proc. ICASSP, Mar. 2017, pp. 91–95.
WASPAA17: T. Okamoto, “Angular spectrum decomposition-based 2.5D higher-order
spherical harmonic sound field synthesis with a linear loudspeaker array,” in Proc. WASPAA,
Oct. 2017, pp. 180–184.
Appl. Sci.18: T.Okamoto,“Mode-matching-based sound field recording and synthesis with
circular double-layer arrays,” Appl. Sci., vol. 8, no. 7, 1048, July 2018.
ICASSP19: T. Okamoto, “Horizontal 3D sound field recording and 2.5D synthesis with omni-
directional circular arrays,” in Proc. ICASSP, May 2019, pp. 960–964.
Publications: Sound field synthesis
22