19th Sept. 2018

2.5D localized sound zone generation
with a circular array of fixed-directivity loudspeakers
Takuma OKAMOTO, National Institute of Information and Communications Technology, Japan

1. Introduction

3. Analytical driving function of circular source
Driving function in 2.5D spherical harmonic spectrum domain
Cancelling horizontal sound field produced by D0

Generating personalized sound zones with loudspeakers

Qu et zone

Qu et zone

Sound zone<latexit sha1_base64="p9dyx2KZBi8NtzcEz7TWjqE/ihQ=">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</latexit>

Sound zone

D̊0,1 =

S̊0,1 (rref )=ajkh0 (krref )8⇡ 2 T̊0 (rref , r1 )kh1 (krref )D̊±1,1 == (1 a)2⇡ T̊±1 (rref , r1 )16⇡ 2 T̊±1 (rref , r1 )Localized sound zone in horizontal plane
S0+1 (r, ) = 0 for r = rre
1
X
S0+1 (r, ) =S̊m,0+1 (r)ejm
m= 1
(
)
(
jk
T̊0 (r, r1 )k=ah0 (kr) h0 (krref )(1 a)
h1 (kr)4⇡
4⇡
T̊0 (rref , r1 )<latexit sha1_base64="7n9RbqJYMNWV7NbCQQDIF+3GjWc=">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</latexit>2⇡ T̊0 (rref , r1 )S̊±1,1 (rref )<latexit sha1_base64="AV3WWwyBGxFKf8t/w9kpgBgdLe0=">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</latexit>(a): Contrast control(b): Multizone sound field(c): Localized sound zone

Conventional methods of localized sound zone generation
Evanescent wave production (H. Itou et al., WASPAA 2011, ICASSP 2012)
Circular and linear array combination (T. Okamoto, ICASSP 2015, JIHMSP 2017)
Dimension mismatch between a linear array and a point source (JIHMSP 2017)
Problem: Higher order modal control and many loudspeakers are required
Proposal: Dimension mismatch between a circular array and a point source
2.5D exterior field synthesis with a circular array of fixed-directivity loudspeakers
Introducing fixed-directivity loudspeakers for avoiding forbidden frequencies
Fewer loudspeakers for only controlling 0-th and 1-st order components<latexit sha1_base64="lYE45sti+VdWfy5s5p0N7EU3iv4=">AAACo3ichVFBa9RAFP4arda12q1ehF6C25UVy/IihRZBKXgp9NLudtvCZlmS7LQ7NMmEZHbbGvIH/AMePFkqUvwZXsSreuhPEI8VvHjwbTYgWtQXMvO9b9735psZN/JloonOJoxLlyevXJ26Vro+fePmTHn21laiBrEnWp7yVbzjOonwZShaWmpf7ESxcALXF9vu/tPR+vZQxIlU4aY+i
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1X
jkh0 (krref )kh1 (krref )D1 ( l ) =D̊m,1 ejm l = a+ (1 a)cos
2 T̊ (r , r )2 T̊ (r , r )
8⇡
8⇡
0
ref
1
1
ref
1
m= 1

)

cos

l

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4. Computer simulations

Simulation condition
Circular array of 8 loudspeakers with a = 0.5 (cardioid), rre = 0.5 m and r1 = 0.1 m
Results

2. Proposed analytical formulation
Exterior sound field produced by circular fixed-directivity sound source
Z 2⇡
FS̊m,1 (r) = 2⇡ D̊m,1 T̊m (r, r1 )S1 (r, ✓, ) =D1 ( 1 )T (r, r 1 )d 1

h1 (krref )T̊±1 (r, r1 )hn (kr)jn,1 (a, r1 )n=|m|<latexit sha1_base64="FhYX3+E3Iu+OHQqC9Fmq5x7gBNs=">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</latexit><latexit sha1_base64="bWbsPhYr0Umt0oLsCjl0BO7b9o8=">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</latexit>

jn,1 (a, r1 ) = ajn (kr1 ) + j(12n + 1 (n |m|)! |m| 2
P (0)4⇡ (n + |m|)! na)jn0 (kr1 )Exterior sound field produced by fixed-directivity sound source at array center
S0 (r, ✓, ) = D0 T (r, r 0 )D0 = 1 for simplicity and r0 = 0 centered origin
⇢

y
jkr
e
j
S0 (r, ✓, ) =a + (1 a) 1 +
cos ↵
4⇡r
kr
r
jk
k
D
= a h0 (kr) (1 a) h1 (kr) cos ↵4⇡4⇡
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S0 (r, )S1 (r, )S0+1 (r, )Monopo eT̊m (r> , r1 ) = jk

1
X

f = 1000 Hz<latexit sha1_base64="LoJphsA71gQpkNK4mCqJ2MXJUZY=">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</latexit>

0<latexit sha1_base64="Ro6cb5EpdF/0Vn/QvdgUyED9ldw=">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</latexit><latexit sha1_base64="OqNi9v58fRmAV5WzL8+ZkAt49wM=">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</latexit>

20 log10 |S0+1 (r, )|Fjk
h0 (kr)4⇡
kS̊±1,0 (r) = (1 a) h1 (kr)8⇡r

S̊0,0 (r) = a

Attenuat on property (<latexit sha1_base64="oyNkMMk6xXYMqJIDsuQ4vP1KW9c=">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</latexit><latexit sha1_base64="swcO9HYIHq+7kyeu5IcQ8IzL7Zo=">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</latexit>

Card o d<latexit sha1_base64="DSNdAMFJKNPOksQobnHNuegkjDM=">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</latexit>

Qu

on

=0 )

Frequency response

Forb dden frequenc es

This study was partly supported by JSPS KAKENHI Grant Number JP18K11387.