PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Sound field reproduction with exterior field cancellation using
variable-directivity loudspeakers
Bokai DU1; Michael KOHNEN2; Michael VORLÄNDER2；Xiangyang ZENG1
1

School of Marine Science and Technology, Northwestern Polytechnical University, China
2

Institute of Technical Acoustics,RWTH-Aachen University, Germany

ABSTRACTSound field reproduction aims to create or reproduce a certain sound environment where not only the audio
content, but also the spatial property of the sound field has to be preserved. For a real system which isusually placed in a “listening room”, the reproduction performance is affected or even destroyed bysuperimposing of reflections from walls. As a result, it is important to control the sound energy whichradiates to the space outside the loudspeaker array. Compared with omnidirectional loudspeakers, the use ofvariable-directivity loudspeakers can eliminate the exterior field according to the Kirchhoff-Helmholtz
integral equation. This paper investigates the performance of variable-directivity arrays using pressure
matching-acoustic contrast control(PM-ACC) and compares its performance with that of omnidirectionalone.
Keywords: Variable-directivity loudspeakers, Pressure matching, Exterior cancellation

1. INTRODUCTION
Sound field reproduction (SFR) is the process of reproducing a desired sound field to makeperson or people in listening region of loudspeaker array experience a virtual sound environment. In
this virtual environment, not only the content of the sound, but also the spatial property of the soundfield needs to be recreated.
An overview of requirements and technical challenges related to audio immersive systems can befound in [1-2]. Generally speaking, there are three most popular approaches to SFR for modern
reproduction system: Wave field synthesis (WFS) [4], Higher-order Ambisonics (HOA) [5] and thepressure matching method. Huygen's principle and the Kirchhoff-Helmholtz (K-H) boundary surface
integral equation are theory foundations for WFS method. Reproduction using this method is usuallybased on a planar loudspeaker array. HOA method decomposes the primary sound field into sphericalharmonics domain and Mode-matching is often applied to design the loudspeaker signals. For thepressure matching method proper loudspeaker signals are obtained by minimizing the difference
between target sound pressures (sampled by microphone array) and reproduced sound pressuresstirred by a loudspeaker array at sample points. Usually, a least-square (LS) criterion is used todesign the loudspeaker signals and limit the loudspeaker energy.
However, the performance of a reproduction system placed in the ‘listening room’ suffers muchfrom superimposing of reflections from walls. Active methods can be applied to compensate theeffect caused by reflections [5], but this compensation requires measurement of the loudspeaker
response at one or more positions for different ‘listening rooms’. Instead, Poletti proposed to use
fixed or variable directivity loudspeakers in sound field reproduction systems to constrain theexterior sound field energy [6-7]. The loudspeaker signal design was achieved by HOA method. But
the application of HOA method requires a spherical loudspeaker array whereas a more flexible
geometry of loudspeaker array can be taken if we use a pressure matching method approach.
Considering works on multi-zone reproduction techniques [8-10], sound field reproduction withexterior sound field energy constrain method is proposed and the performance of variable-directivity
loudspeaker array is investigated by simulation. At last, a subjective listening experiment isconducted to find the border of the reproduction ‘sweet point’ when primary sources are at different
12

dubokai123@mail.nwpu.edu.cn
mko@akustik.rwth-aachen.de6393

directions. This listening experiment provides some indication on primary sound field sample. The
remaining part of this paper is outlined as follows. The preliminaries and problems are stated in Sec.
2. A proposed reproduction method is described in Sec. 3. Results of the simulation and subjective
experiment are presented in Sec. 4 and conclusions drawn in Sec. 5.

2. Preliminaries and Problem Statements
2.1 Pressure Matching for SFR Problem
For a PM-based reproduction system, loudspeaker elements are at located at rsl , l  1,...L ,where L is the number of loudspeakers. Coordinates of control points of primary sound field are
rbm , m  1,...M , where M is the number of control points. In the frequency domain, the soundpressure at m-th control point stirred by loudspeaker array can be expressed as
the sound pressure stirred by each loudspeaker of the reproduction array::the summation ofLpm   wl ( ) g m ,l ( )

(1)

l 1

where wl is l-th loudspeaker's driving signal and g ml is the acoustic transfer function (ATF)

between the l-th loudspeaker and m-th matching point. For the following part of this paper the
methods are investigated in the frequency domain and the angular frequency  is no longer
indicated.
For the whole reproduction system, the reproduced sound pressure at matching points isexpressed in matrix form:
p rep  Gw
(2)
where p rep is a M by 1 vector of reproduced sound pressure at matching points, w is an L by 1
vector of loudspeaker weights, G is a M by L matrix whose (m,l)-th entry is equal to g m ,l .Now, theLS-based SFR problem goes to solve the an optimization problem: minimizing the squared
approximation error between the target sound pressure p des and reproduced sound pressure p pre , namely:22

w : arg min p rep  Gw   w 2
2

w

(3)

where  is a preselected regularization parameter, it is used to avoid obtaining some physicalunrealizable solution (large magnitude), as a result, the driving signal can easily be obtained as:





1

w  G H G  λI G H p des(4)

2.2 Variable-Directivity LoudspeakerIn this paper, loudspeaker radiation is simulated using a variable-directivity approach. For
example, the variable-directivity loudspeaker can be expressed as a weighted combination ofmonopole and dipole. Weights of monopole and dipole can be manipulated independently, which
results in variable- directivity of a loudspeaker. Green functions of ideal monopole and dipole arereviewed as follows.
The sound field generated by an ideal monopole source in the free field can be expressed as:  jk r  r





s
e
 

p (r , rs )  G (r rs )  4 r  rs(5)

where r  ( x, y , z ) and rs denotes the position of sound source, the wave number k   / c . Fora dipole at position rs and oriented in direction  , sound pressure at any point is expressed as:  jk r  rs
G (r rs )ei p (r , rs )    jk  1    cos  v4 r  rs  k r  rs


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(6)

 



where  is the angle between v and r  rs . And in order to ensure flat responses down to afrequency f l , we must keep a distance from any equalized dipole greater than c /(2f l ) [7].

3. Sound Field Reproduction with Exterior Energy Constraint

Figure 1 illustrates a circular reproduction loudspeaker array. In order to achieve thereproduction and control to energy that radiates outwards the loudspeaker array simultaneously, theregion outside the loudspeaker array should be taken into consideration, additional to the centerregion. These two regions are called dark zone and bright zone, respectively.

Figure 1 – A schematic of the sound field reproduction system with exterior energy constrain(red dots
denote loudspeakers).
For the bright zone S b , which is corresponding to the so called ‘listening area’, reproduced soundfield in this circular area should be the same to the desired sound field. On the other hand, soundfield energy in the dark region should to be as low as possible. In practice, the listening and darkzone can be sampled at discrete points, and sound pressure in these two zones can be expressed as
vectors:







T



Pb ( )  P ( rb(1) ;  ) P ( rb( 2 ) ;  ) ...P ( rb( M b ) ;  )TPd ( )  P ( rd(1) ;  ) P ( rd( 2 ) ;  ) ...P ( rd( M d ) ;  )(7)Combine what was introduced in Sec 2, the sound filed reproduction and the exterior sound fieldenergy control turn to two optimization problems:2

w : arg min p des  G b w 2
w

(8)

2

w : arg min 0  G b w 2
wwhere G b and G d are transfer function matrix between loudspeaker and sample points of brightzone and dark zone. And w denotes the loudspeakers' weights vector.

3.1 Omnidirectional Loudspeaker
Loudspeaker weights can be solved as weighted combination of these two optimization
problems and finally goes to find the minimum J.
22J  (1  a ) Pdes  G b w 2  a 0  G d w 2 w H aG dH G d  (1  a )G bH G b w

HH (1  a ) Pdes
Pdes  Pdes
G b w  w H G bH Pdes(9)where parameter a is the weighting factor that determines the relative extent of less mean squareerror in bright area and less outwards energy. H denotes the conjugate transpose. This optimization isa convex problem and therefore J has global minimum and can be obtained using coordinateddescend algorithm. As a result, the optimal solution is :

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

w  aG dH G d  (1  a )G bH G b

 (1  a)G P
1

H
b

des

(10)

3.2 Variable-Directivity Loudspeaker
As mentioned in Sec. 2.2, variable-directivity loudspeaker is a combination of monopole anddipole loudspeaker where amplitude and phase of these two loudspeakers can be manipulated
independently. As a result, for every variable-directivity loudspeaker, there are two unknown weights
to design. The solution for variable-directivity loudspeaker reproduction system is :
1

H
H
H

G bu  G bu G bu u   G du  G du 
 v   a G  G   (1  a ) G  G   0 I  (1  a ) G  Pdes    dv   dv  bv   bv  bv 

(11)

where u and v are two L  1 vectors for monopole weights and dipole weights, respectively. G bu isthe acoustic transfer function matrix from the monopole to bright zone control points. G bv is theacoustic transfer function from the dipole to bright zone control points. Similarly, G du and G dv aretransfer function matrix from monopole and dipole to dark zone control points. And 0 is theregularization parameter to limit the total energy of loudspeakers.

4. Simulation and Subjective Listening Test
4.1 Objective Evaluation
We now present simulations of sound field reproduction using pressure matching with variable
loudspeakers in free field conditions. We use a circular array of 25 loudspeakers which are uniformly
arranged at a radius 1 m in the horizontal plane. The reproduction will be implemented in the central
area of the array. For comparison, omnidirectional loudspeaker array simulation will also be
investigated.
The desired sound field is radiated by a point source placed on the x-axis at [10,0,0]. Due to theuniformly arrangement of loudspeakers, the error performance was found to be similar for point
source at different angles. Taking the size of the head of a listener, the radius of the bright zone rb isselected to be 0.25 m. The bright and dark zones are sampled with discrete points which distribute
uniformly with distance 6 cm. In order to achieve outwards energy radiation limitation, the dark zone
should be the half space out of the array which is not feasible to sample. In this paper, the dark zone
is only sampled at those points 2.5m  rd  3m . The corresponding Nyquist frequency is more than
2800 Hz which covers our interest frequency range(up to 900 Hz) well.
4.1.1 Reproduction error
To evaluate the reproduction performance of the system in the listening area, the relative
reproduction error of every point in listening area is defined as mean energy error between primarysound field p pre stirred by the point source and reproduced sound field pre . The final relative error isexpressed in dB, and the exterior sound field relative energy(ESL) is also defined in Eq. (13),where p pre 0 is primary sound pressure at the point (0,0).

E (dB )  10 log10

pre  p pre
p pre

2

2
2

(12)

2

2

ESL  10 log10

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pre 2
p pre 02
2

(13)

4.1.2 Simulation Verification
As mentioned before, a circular array with 25 loudspeakers on the horizontal plane is used as the
reproduction system. The minimum distance of adjacent loudspeakers is 0.26 m. The corresponding
Nyquist frequency is 660 Hz. Figure 1 shows the simulation result about reproduction error andrelative exterior sound field energy for variable-directivity loudspeaker array, reproduced soundfields on the horizontal plane, the energy and directivity for each variable-directivity loudspeaker
and relative sound field energy of at 200 Hz. The weighting parameter in Eq. 11 is set to be a=0.5.

(a)

(b)

(c)
(d)
Figure 2 – Simulation for variable-directivity loudspeaker array at 200 Hz.(a) Reproduction error (r<1m)and relative exterior sound field energy(r>1 m), the red line denotes variable-directivity loudspeakerarray and the blue line denotes monopole loudspeaker array.(b) Reproduced sound field(real part) (c)Directivity and energy of variable-directivity loudspeakers.(d) Exterior and interior sound field relative
energy.
As what can be observed in figure 2 (a), compared with monopole array, the variable-directivity
loudspeaker array achieves both sound field reproduction in the bright zone and exterior sound fieldenergy control in the dark zone at 200 Hz,. In the bright zone, the reproduction error is below -30 dB
but the number for monopole one is around -8 dB. And when we move on the exterior sound field
relative energy, the value of variable one goes down to -30 dB in 1 m away from the array but themonopole one decreases to around -20 dB at 1.5 m and goes down continue but slowly. This means,
for the variable-directivity sound field reproduction system, the reproduction performance in thebright zone of the area will be less affected by reflections from the walls (if walls are in sampleddark region)when this array is placed in a ‘listening room’. The energy of the sound field isconstrained to be in the array rather than propagates outwards(refer to figure 2(d)).

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Figure 3 shows the related reproduction performance of the array at 900 Hz which is higher thanthe Nyquist frequency(660 Hz) of the array.

(a)

(b)

(c)

(d)

Figure 3 – Simulation for variable-directivity loudspeaker array at 900 Hz. (a) Reproduction error (r<1m)and relative exterior sound field energy(r>1 m), the red line denotes variable-directivity loudspeakerarray and the blue line denotes monopole loudspeaker one. (b) Reproduced sound field(real part). (c)Directivity and energy of variable-directivity loudspeaker. (d) Exterior and interior sound field relative
energy.
Similar to the result at 200 Hz, variable-directivity loudspeaker array still shows better performance
than the monopole one. Its reproduction error is less than -20 dB in the bright zone and exterior sound field
relative energy is -30 dB at 2.5 m, but the corresponding number for monopole one is -7 dB and -22
dB. But compared with the performance at 200 Hz, this variable-directivity loudspeaker system's
performance at 900 Hz faded. Taking a careful observation to the energy distribution and the
directivity of loudspeakers, at high frequency, this algorithm fails to design a proper loudspeakerdirectivity and make a good energy allocation.
4.2 Subjective Evaluation
Reproduction system is often used for creating a virtual environment, so people's subjective
perception to the reproduced sound field is the most important thing. And for a real system, a larger
effective region(sweet point) means more people can share the best reproduction result. Taking this
into consideration, a subjective experiment was conducted to find the border of the effective regionof reproduced sound field.
In this subjective experiment, 12 virtual variable-directivity loudspeakers are combined together

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following the loudspeaker setup in the VR laboratory at the Institute of Technical Acoustic (ITA),RWTH-Aachen. This virtual array is placed in an anechoic chamber (weighting parameter is set to bea=0) simulated by RAVEN [11]. Sound fields stirred by 5 primary point sources on different
directions are sampled respectively with 72 microphones which uniformly distribute in a 0.25 m
spherical space in the center of loudspeaker array. Primary sources are 7 m away from the center ofthe loudspeaker array. The HRTF (head related transfer function) of a dummy head was used to get
the binaural signals. Directions of primary sources and the dummy head are shown in figure 4. In
this reproduction, the up-limit frequency of reproduction is 900 Hz. Twenty-eight people (8 people
working at ITA, Germany and 20 students at NWPU, China) participated in this listening experiment.
In this subjective experiment, the dummy head moved along the axis towards ‘front virtual source’,
people were asked to try to find the closest position to the coordinate origin where they hear thenoticeable difference compared with sound signal in the array center. In order to improve the speedof this process, the Bayesian adaptive psychometric method was used[12].

Figure 4 –Directions of primary sources and the dummy head used in this paper.
As shown in figure 5 (a), the sweet point border varies with the change of the direction of primarysource. Taking the symmetrical setup of the array into consideration(the physical reproduction
performance should be same), this means that on these five directions people are most sensitive tothe change of the sound field which comes from the front of them, by contrast, they are not so muchsensitive to the change of the sound field from the left. This result provides us with some indication
about the sample of the primary sound field and the design of loudspeaker array. Uniformly
distributed sample points are not a bad option, but it seems that there should be more sample points
when the primary sound field comes front of people or place more loudspeakers in the front ofpeople compared with other directions(a)(b)
Figure 5 –Sweet point border: (a) Different primary sound directions. (b) Primary source on the front
reproduced by monopole array and variable-directivity loudspeaker array.
The figure 5(b) shows the result of sweet point borders with the same primary source direction but
different loudspeakers. As what we expected, variable-directivity loudspeaker shows better
performance than monopole.

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5. CONCLUSIONSIn this paper, the performance of variable-directivity loudspeakers array using pressure matching
method acoustic contrast control has been checked. Compared with the monopole loudspeaker array,variable-directivity loudspeaker system can reproduce the target sound field and constrain the energy
radiating outwards simultaneously, especially when the frequency is lower than Nyquist frequency.
At frequency higher than the Nyquist frequency, this method with variable loudspeakers still shows
better performance than reproduction with omnidirectional ones. On the other hand, a subjective
experiment to find the border of sweet point is conducted. The result shows that people's sensitivity
to sound fields from the different directions is not constant. This provides us with some indication on
decision of sample points distribution and loudspeaker array design.
The result in this paper is based on the simulation and a subjective experiment, and the practical
performance of this kind of loudspeaker array in a virtual or a real listening room has not been
investigated. In the future work, the performance evaluation, both subjective and objective ones, invirtual rooms with different reverberation time and its practical performance in the listening room
will be discussed.

ACKNOWLEDGEMENTS
This work is sponsored by China Scholarship Council. And our work is also partly supported by
National Natural Science Fund of China(11774291) and Aviation Science Fund(2015153021).
Authors thank participants who attended the subjective experiment for their kindly help.

REFERENCES

1.Apostolopoulos J G, Chou P A, Culbertson B, et al. The road to immersive communication[J].
Proceedings of the IEEE, 2012, 100(4): 974-990.
2.Huang Y, Chen J, Benesty J. Immersive audio schemes[J]. IEEE Signal Processing Magazine, 2010, 28(1):20-32.
3.Boone M M, Verheijen E N G, Van Tol P F. Spatial sound-field reproduction by wave-field synthesis[J].
Journal of the Audio Engineering Society, 1995, 43(12): 1003-1012.
4.Poletti M A. Three-dimensional surround sound systems based on spherical harmonics[J]. Journal of the
Audio Engineering Society, 2005, 53(11): 1004-1025.
5.Spors S, Buchner H, Rabenstein R, et al. Active listening room compensation for massive multichannelsound reproduction systems using wave-domain adaptive filtering[J]. The Journal of the AcousticalSociety of America, 2007, 122(1): 354-369.
6.Poletti M, Fazi F M, Nelson P A. Sound-field reproduction systems using fixed-directivity
loudspeakers[J]. The Journal of the Acoustical Society of America, 2010, 127(6): 3590-3601.
7.Poletti M A, Fazi F M, Nelson P A. Sound reproduction systems using variable-directivity
loudspeakers[J]. The Journal of the Acoustical Society of America, 2011, 129(3): 1429-1438.
8.Wu Y J, Abhayapala T D. Spatial multizone soundfield reproduction: Theory and design[J]. IEEE
Transactions on audio, speech, and language processing, 2010, 19(6): 1711-1720.
9.Chang J H, Jacobsen F. Sound field control with a circular double-layer array of loudspeakers[J]. The
Journal of the Acoustical Society of America, 2012, 131(6): 4518-4525.
10.Zhang W, Abhayapala T D, Betlehem T, et al. Analysis and control of multi-zone sound field
reproduction using modal-domain approach[J]. The Journal of the Acoustical Society of America, 2016,
140(3): 2134-2144.
11.Schröder D, Vorländer M. RAVEN: A real-time framework for the auralization of interactive virtualenvironments[C]//Forum Acusticum. Denmark: Aalborg, 2011: 1541-1546.
12.Watson A B, Pelli D G. QUEST: A Bayesian adaptive psychometric method[J]. Perception &psychophysics, 1983, 33(2): 113-120.

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