Distance Attenuation Control of Spherical
Loudspeaker Array
Shigeki Miyabe∗ , Takaya Hayashi† , Takeshi Yamada‡ and Shoji Makino§Life Science Center of Tsukuba Advanced Research Alliance, University of Tsukuba, Tsukuba, Japan
E-mail: {∗ miyabe, § maki}@tara.tsukuba.ac.jp, † hayashi@mmlab.cs.tsukuba.ac.jp, ‡ takeshi@cs.tsukuba.ac.jp
Tel: +81-29-853-6566 Fax: +81-29-853-7387

Abstract—This paper describes control of distance attenuation
using spherical loudspeaker array. Fisher et al. proposed radialfiltering with spherical microphone to control the sensitivity to
distance from a sound source by modeling the propagation ofwaves in spherical harmonic domain. Since transfer functions
are not changed by swapping their inputs and outputs, we canuse the same theory of radial filtering for microphone arrays tothe filter design of distance attenuation control with loudspeakerarrays. Experimental results confirmed that the proposed method
is effective in low frequencies.

(a) Configuration of sphericalmicrophone array.

(b) Block diagram of sphericalmicrophone array.

(c) Configuration of spherical
loudspeaker array.

(d) Block diagram of spherical
loudspeaker array.

I. I NTRODUCTION
Although loudspeaker playback does not constrain bodies oflisteners unlike headphone playback, it suffers from leakageof sound. Thus limiting the area of the the sound of the loudspeaker playback is an important issue for acoustic insulationand speech privacy. The most popular approach to the area
limitation is directivity control of loudspeakers. For example,
various plane wave transducer devices are developed, e.g.,[1]. Loudspeaker array is another solution for the directivity
control [2]. Also, there has been a rapid progress in parametricspeakers to form sharp directivity using ultrasonic waves, e.g.,[3]. However, the magnitude behind the listeners cannot be
maintained by directive loudspeakers, and the area limitationis also degraded by reflection on walls.
In this paper we propose sound area limitation by thecontrol of distance attenuation using a spherical loudspeakerarray so that we can limit the area of the sound with highmagnitude is limited only around the loudspeaker array. For
this purpose, we employ a theory to control sensitivity ofspherical microphone arrays to distance from sound sources[4] proposed by Fisher et al. Utilizing a nature of transfer
functions which are not changed by spwapping their inputs
and outputs, the Fisher’s method is used for design of filters
to control disttance attenuation of a spherical loudspeakerarray assuming a virtually inverted wave propagation model
as if sound is emitted by a microphone and reaches at theloudspeaker array. As a result of a simulation in an anechoic
environment, effectiveness of the proposed method in low
frequencies is confirmed.
II. P ROPOSED M ETHOD
A. Strategy
Here we review the outline of the Fisher’s method to control
the distance sensitivity of spherical microphone arrays, andFig. 1. Configuration and block diagram of spherical microphone array andspherical speaker array.

discuss how to apply it to distance attenuation control of
spherical loudspeaker arrays.
The radial filtering proposed in [4] controls sensitivity ofspherical microphone arrays to distance from the sources. As
shown in Fig. 1 (a), assuming a spherical wave is radiatedfrom a point source and reaches a microphone array with M
microphone elements, the relation between the magnitude ateach microphone element and the distance is modeled in thespherical harmonic domain. Using the modeled relation, filters
to approximate arbitrary distance characteristics is designed.
The observed signal at each element is filtered separately andsummed to generate the output signal as shown in Fig. 1 (b).
The transfer function qmic (k) between the source and the array
output is given by
qmic (k) =M∑w∗ (k, Ωj ) c (k, Ωj ) ,

(1)

j=1
∗

where {·} denotes complex conjugate, Ωj (j = 1, . . . , M )is the angle of the j-th element in the three-dimensionalpolar coordinate with its origin at the center of the sphericalmicrophone array, w∗ (k, Ωj ) is the filter for the observed
signal at the j-th element, c (k, Ωj ) is the transfer function
between the source and the j-th element, and k = f /v is thewave number with the frequency f and the velocity v.
The problem we discuss in this paper is the playback by thespherical loudspeaker array as shown in Fig. 1 (c). Suppose
the M loudspeaker positions in Fig. 1 (c) is the same as
the ones of microphones in Fig. 1 (a). Also, the source in
Fig. 1 (a) is replaced by the microphone at the same positionin Fig. 1 (c). Since the source is multiple loudspeakers on
the surface of the sphere, the wave front is not a simple
spherical wave but the more complicated superimposition of
the waves from the multiple loudspeaker elements. However,
the transfer functions between the spherical loudspeakers andthe microphone in Fig. 1 (c) is the same as the ones between
the source and the microphones, e.g., c (k, Ω1 ) , . . . , c (k, ΩM )because transfer functions are not changed by swapping their
inputs and outputs. Thus, by outputting the signal filteredby the same filters w∗ (k, Ωj ) , j = 1, . . . , M , the transfer
function qspk (k, Ωj ) from the loudspeaker array input to themicrophone is given by
qspk (k) =M∑c (k, Ωj ) w∗ (k, Ωj )j=1

= qmic (k) ,(2)wave number kmax to control has to satisfy
N
,
(5)
a
where a is the radius of the array sphere. The theoreticalestimate of the error caused by the truncation is discussed
in [6], and in this paper we evaluate the error through theexperiments.
kmax ≪

C. Derivation of Radial Filter
The gain y (k) of the array output of the microphone arrayto observe the sound field assuming near field is expressed asy (k) =

Nn∑∑∗(k) pnm (k) ,
wnm

(6)

n=0 m=−nwhere pnm (k) is the spherical harmonic coefficients of thesound field p (k, r, Ω) with the near-field assumption, whose
detail is described in Appendix B, is given by∫pnm (k, r) =p (k, r, Ω) Ynm∗ (Ω) dΩΩ∈S 2

results as the equivalent one as the transfer function qmic (k) of
the spherical microphone array. Thus the distance characteristics of the loudspeaker output in Fig. 1 (c) is the same as that of
the sensitivity of the microphone array in Fig. 1 (a). Therefore,
we can design the distance characteristics of the spherical
loudspeaker array by the Fisher’s method using the inverted
virtual microphone array model, where a single microphoneemits the spherical wave and the elements of the loudspeakerarray observes the wave emitted from the microphone.
B. Signal Expression in Spherical Harmonic Domain
Arbitrary sound pressure g (k, Ω) of the wave number k at apoint of the angle Ω on a sphere is expressed by the following
linear combination using the spherical harmonic Ynm (Ω) as theorthonormal bases.
∞ ∑n∑g (k, Ω) =gnm (k) Ynm (Ω) ,(3)
n=0 m=−nwhere gnm (k) is the coefficient corresponding to the basis
Ynm (Ω), n is the order of harmonic, m is an integer to satisfy−n ≤ m ≤ n, and the detail of the spherical harmonicYnm (ω) is described in Appendix A. Equation (3) is known as
inverse spherical Fourier transform. The coefficient gnm (k) is
obtained by the following integral on a 2-sphere S 2 , referred
to as spherical Fourier transform:∫gnm (k) =g (k, Ω) Ynm∗ (Ω) dΩ.
(4)
Ω∈S 2

As can be seen in Eq. (3), infinite order of the sphericalharmonic is required to express the pressure g (k, Ω) accurately. However, as discussed in [5], the order n to beapproximated is bounded by the number M of the array2elements as M ≥ (n + 1) . Thus
√ we have to truncate theorder by the maximum N ≤ M − 1, and the maximum= bsn (k, r, rs ) Ynm∗ (Ωs ) ,(7)

where rs and Ωs are respectively the distance and the directionof the source, and bsn (k, r, rs ) is the near-field intensity givenby(8)
bsn (k, r, rs ) = i−n+1 kbn (kr) hn (krs ) ,where hn (·) describes the spherical Hankel function whose
details are described in Appendix C, and bn (·) is the farfield intensity described in Appendix D. Here we consider∗
the following filter coefficient wnm(k) to cancel the terms
unrelated to the distance rs :∗=wnm

dn (k)Y m (Ωl ) ,−n+1i
kbn (ka) n(9)√where i = −1 and Ωl is the look direction of the sourceand dn (k) is the coefficient to design. Its array output gainy (k, rs , Ωl ) is given byy (k, rs , Ωl ) =
=

N
n
∑
∑

∗
wnm
(k) pnm (k, a) Ynm (Ωl )n=0 m=−nN∑

2n + 1dn (k) hn (krs ) Pn (cos Θ) , (10)4πn=0using the spherical harmonics addition theorem [7]:
n
∑
m=−n

Ynm∗ (Ωs ) Ynm (Ωl ) =2n + 1Pn (cos Θ) ,
4π

(11)

where Θ is the angle between Ωl and Ωs , and Pn (·) is
the Legendre function. It can be seen in Eq. (10) that thedirectivity of the control is affected only by the Legendre
function Pn (cos θ), whose plots with low orders are shownin Fig. 2. With any order Pn (cos θ) is one if Θ = 0, π andthe directivity is not so strong for other directions with the
low orders. Thus the radial filtering by Eq. (10) does not have

0
Beampatterm [dB]TABLE IPARAMETERS OF SIMULATION

n=0
n=1
n=2
n=3

−10

Number L of sampled distances
Maximum order N
Number M of loudspeaker elements
Radius a of spherical loudspeaker array
Maximum controlled distance rsupE
Listening distance rlis

−20

−30

−40
−200

−100

0
Angle [deg]

100

200

†

Fig. 2. Legendre function Pn (cos θ) with orders n = 0, 1, 2, 3.

strong directivity.
D. Filter Design by Least Squares Estimate
Suppose the listener is at the distance of rlis from thecenter of the sphere, referred to as listening distance, and wecontrol the gain y (k, rs , Ωl ) to be close to the desired gain
yd (k, rs , Ωl ), which equals to 1 at rs = rlis and equals to zeroat the distance farther than rsup as{
1 (rs = rlis ) ,y (k, rs , Ωl ) ≈ yd (k, rs , Ωl ) =(12)0 (rs ≥ rsup ) .
To design the filter coefficients dn (k) to satisfy Eq. (12), formulate the least squares problem with the sampled distances.
We sample the (L − 1) distances from rsup to rsupE (> rsup ),and we define the L sample distances rl , l = 1, . . . , L together
with rlis as
r1 = rlis , r2 = rsup , rL = rsupE , rl−1 < rl < rl+1 .

(13)Note that L > N . Next, we define the L × (N + 1) matrix
H (k) composed of the right term of Eq. (10) except thecoefficient dn (k) and Legendre functions as[]
2n + 1
H (k) =
hm−1 (krl )
,
(14)
4π
lm
where [x]lm denotes the matrix which has the entry x in the lth row and m-th column. Also with the vector d (k) composedof the coefficients dn (k), given byTd (k) = [d0 (k) , . . . , dN (k)] ,

(15)

T

where {·} denotes the matrix transposition, the Ldimensional vector form y (k) of the series of the array output
gain y (k, rl , Ωl ) is expressed asTy (k) = [y (k, r1 , Ωl ) , . . . , y (k, rL , Ωl )]= H (k) d (k) .

(16)

Also the L-dimensional vector form yd (k) of the desired gain
yd (k, rl , Ωl ) is given byT

yd (k) = [yd (k, r1 , Ωl ) , . . . , yd (k, rL , Ωl )] .

(17)

Thus the least squares solution of the coefficient dn (k) to2minimize ∥yd (k) − y (k)∥ is obtained as†

d (k) = H (k) yd (k) ,

100
2
12
0.05 m5m
0.1 m, 0.2 m(18)

where {·} denotes the pseudo inverse matrix. However, thedesigned distance characteristics y (k, rs , Ωl ) is just a theoretical curve only within the truncated order n ≤ N is considered.
the performance degraded by the truncation is evaluated in thesimulation of the following section.
By substituting the designed coefficient dn (k) in Eq. (9) andapplying the inverse spherical Fourier transform to wmn (k),the frequency characteristics of the array filter w∗ (k, Ωj ) for
j = 1, . . . , M is obtained asw∗ (k, Ωj ) =

N
n
∑
∑

dn (k)Y m (Ωl ) Ynm (Ωj ) .
−n+1 kb (ka) ninn=0 m=−n(19)With the inverse discrete Fourier transform of w∗ (k, Ωj ), we
can obtain the array filter for the l-th element in the form offinite impulse response.
III. S IMULATION
A. Experimental Conditions
We simulate the distance attenuation control in anechoic
environment comparing the performance in different configurations. We compared the simulated responses of the spherical
loudspeaker array of the designed filter given by Eq. (19) withtheoretical values given by Eq. (6) to evaluete the error causedby the truncation of the order, and with the simulated responses
of a single loudspeaker output to evaluate the effectiveness ofthe proposed method. The used spherical loudspeaker arrayhas 12 elements positioned at the vertexes of the icosahedron
inscribed within the open sphere. The simulation parameters
are listed on Table I.
B. Results
The results of the simulation are shown in Fig. 3. Comparing(a)–(c) or (d)–(f), the control in the low frequency is successful, but in the high frequencies the curve of the processed
gain is far worse than the theoretical curve, the gain in thespherical harmonic domain, and approaches to the curve of
the unprocessed signal. For example, in 148 Hz shown in (a)the suppression of about 40 dB is obtained at 1 m comparedwith the unprocessed output, but it degrades to worse than
10 dB in 1000 Hz as in (c). Thus the control of the highfrequency is hard because of the condition in Eq. (5) is hardly
satisfied.
Comparing (a) and (d), (b) and (e) or (c) and (f) reveals
that the contol is easy when the listening distance is close to
the array surface.

(a) f = 148 Hz, rs = 0.1 m

Radial Filter Responce(dB)

Radial Filter Responce(dB)0Theoretical
Simulated
Unprocessed

−20
−40
−60
−80
−100
0

1

2
3
Distance(m)

4

−40

−80

2
3
Distance(m)450Theoretical
Simulated
Unprocessed−20

Radial Filter Responce(dB)

Radial Filter Responce(dB)

1(d) f = 148 Hz, rs = 0.2 m

−40
−60
−80

1

2
3
Distance(m)4Theoretical
Simulated
Unprocessed

−20

−80

1

2
3
Distance(m)45Radial Filter Responce(dB)

Theoretical
Simulated
Unprocessed

−40
−60
−80

1

2
3
Distance(m)45Theoretical
Simulated
Unprocessed

−20
−40
−60
−80
−100
0

n
∞ ∑
∑

bsn (k, r, rs ) Ynm∗ (Ωs ) Ynm (Ωs ) , (24)1

23Distance(m)4whose double sum is canceled by the orthonormal property ofYnm (Ω) and reduces to Eq. (7).
C. Spherical Hankel and Bessel functions(f) f = 1000 Hz, rs = 0.2 m
0

−20

(23)

n=0 m=−n−600e−2πik |r − rs |,|r − rs |where r = (r, Ω) and rs = (rs , Ωs ) and |r − rs | denotes the
Euclidean distance between the vectors r and rs . Applying thespherical Fourier transform, the sound field p (k, r, Ω) in thespherical harmonic domain is given asp (k, r, Ω) =

−40

−100
0

5

(e) f = 383 Hz, rs = 0.2 m
Radial Filter Responce(dB)p (k, r) =

−60

0

−100
0

The sound filed with the near-field assumption is given as
follows in the frequency domain.

Theoretical
Simulated
Unprocessed

−20

−100
0

5

(c) f = 1000 Hz, rs = 0.1 m−100
0B. Sound Field in Spherical Harmonic Domain(b) f = 383 Hz, rs = 0.1 m

05Fig. 3. Experimental results.

IV. C ONCLUSIONSThe spherical Hankel and Bessel functions hn and jn andthe derivatives of them h′n and jn′ are given by√π(1)hn (kr) =H 1 (kr)(25)
2kr n+ 2√π(26)
jn (kr) =J 1 (kr)2kr n+ 2nh′ (kr) =hn (kr) − hn+1 (kr)
(27)
kr
n
jn (kr) − jn+1 (kr) ,(28)
jn′ (kr) =kr(1)This paper described the control of the area where loudspeaker output is audible using loudspeaker array. The filter
coefficients are designed in the spherical harmonic domain
according to the inverted model of the wave propagation which
is treated as if a microphone emits the spherical wave and theloudspeaker array observes it. Effectiveness of the proposed
method in the low frequency is ascertained in the simulation.
ACKNOWLEDGMENT
This work was supported by KAKENHI 23760325.

A. Spherical Harmonic
The spherical harmonic is the solution of three-dimensional
Laplace equation in spherical coordinate, given by√
2n + 1 (n − m)! m
Ynm (Ω) =P (cos θ) eimϕ ,(20)
4π (n + m)! n
where Pnm (·) is the associated Legendre function given bym()m/2 dmPn (x) ,dxm
with the Legendre function given by
[ n
]
)n
d ( 2
1
.
Pn (x) = n
x −1
2 n! dxn
1 − x2

D. Intensity Function
The intensity bn (kr) of the far-field assumption is given foropen and rigid spheres as{
jn (kr)(open sphere),n′bn (kr) = 4πijn(ka)jn (kr) − h′ (kr) hn (kr) (rigid sphere).
n(29)
R EFERENCESA PPENDIX

Pnm (x) = (−1)

where Hn+ 1 , Jn+ 12 and Yn+ 21 are the Hankel function of the2
first kind, the Bessel function of the first kind and the Bessel
function of the second kind, respectively.

(21)

(22)

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