Analytical methods of generating multiple sound zones
for open and baﬄed circular loudspeaker arrays
Takuma OKAMOTO, National Institute of Information and Communication Technology, Japan

1. Introduction

3. Proposed method

Personalizing listening areas using multiple loudspeakers

Analytical solution of modeling sound pressure by a rectangular window

Multiple sound zones with a linear array (e.g. T. Okamoto@ICASSP 2014.)
Property: spectral division method based analytical solution
Problem: many loudspeakers and large production space are required
Motivation: Deriving analytical solutions for compact arrays

Spatial Fourier series expansion of sound pressure
Bright zone✓◆P(rnref , φ, 0, ω) = 1
P̊n ( , S , !) = since jn s2⇡ΦSpatial filtering for generating multiple sound zones
φssinc(n /2⇡)e jn sF̊n (r0 , rref , , s , !) =2⇡r0 G̊n (rref > r0 , !)Filter coeﬃcients in temporal frequency domain

Conventional methods with circular arrays and their problems
Least squares (LS) methods (e.g. T. Betlehem et al. 2006.)
Very ill-conditioned and unstable driving signals of loudspeakers
Beamforming methods using a baﬄed circular array (e.g. F. M. Fazi et al. 2014.)
Far field approximation is introduced and near field cannot be controlled

F (r0 ,r = [r, φ, z]T0FD̊n (r0 , !) =Y axis [m]LSY axis [m]-200-1-10-40-10

1X axis [m]z = 0 [m]1.501-20-10

Y axis [m]00.50-20-0.5-30-30-1-400

1-1.5

[dB]

-40-1X axis [m]01[dB]

20X axis [m]z = 0 [m]1.5z = 0 [m]1.501010-20-0.5-10Y axis [m]-100.50.50-20-0.5-30-1-30

15
LS method (open)
LS method (baffled)
Beamforming (open)
Beamforming (baffled)
Proposed (open)
Proposed (baffled)

105-1-40-10open1[dB]-1.5s)

Spatial Nyquist Frequency

25-1-1S)

P|P (r b , !)| / bBDR(!) = 20log10 Pr br d |P (r d , !)| / dwith = ⇡/2, r0 = 0.25 m and rref = 1.0 m

1-0.5-1.5[dB]

0-10

X axis [m]2⇡r0 G̊n (r, r0 , !)

[dB]

0.5

-1.5

P̊n (r, !)1X axis [m]z = 0 [m]-1.5sinc(n /2⇡)ejn( l2⇡j n Jn (kr0 )NX
kr0 Hn0 (kr0 ) sinc(n /2⇡)ejn( l=4j (1 n)
n= N

-20

-0.5
-30

-40

1.5

Y axis [m]Beamforming0)

Driving function of circular monopole sound source distributions
described by 2.5D sound field representation
0 , z = 0, !)G(r, r 0 , !)r0 d 0

0-0.5

0.5NX

n= N

FBF,ba✏ed-10-1G̊ba✏ed,n (r > r0 , !)D(r0 ,10.5-1.5FBF,open =0-30

3D sound field propagated by a baﬄed cylindrical sound source describedby 3D cylindrical harmonics expansion
Z 11(1)X
1Hn (kr r)0)
Gba✏ed (r, r 0 , !) =dkz ejkz (z z0 ) ejn((1)02⇡
1 2⇡kr r0 Hn (kr r0 )n= 1
for r > r0

P (r, !) =1.50-10r

3D sound field propagated by an open cylindricalxsound source described by 3D cylindrical harmonics expansion
Z 11Xejk|r r 0 |1jGopen (r, r 0 , !) ==Jn (kr r0 )Hn(1) (kr r)dkz ejkz (z z0 ) ejn(4⇡|r r 0 | n= 1 2⇡
1 4
for r > r0
G̊open,n (r > r0 , !)Z 2⇡

f = 3000 Hz z = 0 [m]1yφDark zone
P (rref , φ, 0, ω) = 0x

BDR [dB]

D(r 0 , !)G(r, r 0 , !)r0 dz0 d 01z = 0 [m]1.5

Y axis [m]0r0Produced sound pressure level and bright to dark ratio

Proposed2⇡ 1

y

4. Computer simulations

2. 2.5D sound field produced by circular sources
P (r, !) =F̊n (r0 , !)ejn lrref

Discarding evanescent wave component for calculating stable driving signals⇢b(L 1)/2c, for b|kr0 |c b(L 1)/2c
N=b|kr0 |c,for b|kr0 |c < b(L 1)/2c

Analytical solutions for circular monopole sound source distributions
Precise solutions are derived based on 2.5D sound field representation
Bright and dark zones are directly modeled by a rectangular windowzl , !) =Reference circle

n= N

Proposed methods for open and baffled circular loudspeaker arrays3D soundZfield
produced by a cylindrical sound source
ZNX

Circular sound sourcez-40-10X axis [m]

baﬄed

1[dB]

00

1

234

Temporal Frequency [kHz]This study was partly supported by JSPS KAKENHI Grant Number 15K21674.

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