F2016-NVH
SOUND FIELD CONTROL IN THE INTERIOR AND EXTERIOR OF
ELECTRIC VEHICLES
Choi, Jung-Woo *
1
School of Electrical Engineering, Korea Advanced Institute of Science and Technology,
Korea
1
KEYWORDS – personal sound system, directional pedestrian warning, loudspeaker array,
acoustic contrast control, acoustically bright and dark zones
ABSTRACT –
With rapid pervasion of electric vehicles, sound field control in the exterior and interior of
automotive vehicles are facing new challenges. In this talk, two specific applications of sound field
control systems for electronic vehicles are introduced: a directional pedestrian warning system for
exterior sound fields and a personal sound system for interior sound fields. Recently, the directional
pedestrian warning system has received much attention for its ability of delivering alert sounds to
pedestrians with minimal increase of environmental noises. There have been number of studies on
sound beam generation to a pedestrian direction by controlling loudspeaker arrays installed on
vehicles. Although the generation of sound beams is not a new topic in the field of acoustic array
signal processing, special cares must be taken in regard to reflections from an automobile chassis
and ground surface, as well as to the minimization of computational complexity for fast control of
sound beams.
The second topic of sound field control, not limited to electric vehicles, is the realization of personal
sound systems for automotive vehicles. The personal sound system for a car cabin aims at
reproducing independent audio content per seat without disturbing other passengers in different
seats. This simultaneous but independent reproduction of audio contents can provide completely
new experience to the passengers, e.g., by delivering navigation or driver-assistant messages only
to a driver, while playing multimedia contents to other seats. The hands-free communication in
vehicles can be managed in a more private way through the focusing of a received speech signal to
a specific seat. The development of such systems requires comprehensive understanding on sound
fields produced in a car cabin, which exhibit strong reflections and scattering waves that are
completely different from those in a normal room or free-field.
In this talk, it is explained how conventional sound field control strategies should be modified for
automotive vehicles, especially under the context of acoustic contrast control. The acoustic contrast
control, originally proposed for focusing sound fields of arbitrary shapes, has evolved into diverse
forms to meet various kinds of goals and constraints. Through experiments with practical prototype
arrays, it is demonstrated that unique sound propagation in the inside and outside of cars can be
controlled to realize directional warning and personal sound systems.
INTRODUCTION
As noise and sound control technologies evolve, the simultaneous reproduction and suppression of
sound fields have gained a lot of attention. This is possible by controlling array of loudspeaker such
that waves from multiple loudspeakers make constructive and destructive interferences in space.
The concept – personal sound(1) and acoustically bright and dark zones(2)(3) is introduced for this
purpose. The personal sound zone refers to the zone of focused sound energy in which only the
listener can hear loud sound. The purpose of generating a personal sound zone can be either for
minimizing the distraction of other listeners or for constructing personal sound systems that allow
listeners located in different regions to enjoy different sound content at the same time. The great
benefit of producing a personal sound zone is that the user does not need to wear or bring a headset
for enjoying personal sound.
Upon this premise, several types of personal sound systems based on the conventional loudspeaker
array have been developed (Fig. 1) for PC monitors(4), mobile phones(5,6), stereo systems(7).
Among them, two particularly interesting examples can be found from automotive applications –
(A) a directional warning sound system(Fig. 3(a)) and (B) a personal sound system for
passengers(Fig. 3(b)). A directional warning sound system studied under the European project
eVADER (Electric Vehicle Alert for Detection and Emergency Response)(8). Researches have
covered variety of issues on the pedestrian safety against electric vehicles(9)(10), including
subjective test for synthesizing proper warning sound, and a system to deliver artificial warning
sound from electric vehicles to pedestrians. The latter study has to do with the synthesis of sound
beam steered to the pedestrian’s direction, which is for delivering warning sound without increasing
the noise pollution.
On the other hand, the personal sound system for car interior sound(Fig. 3(b) or (11)) is also being
actively developed by many researchers. The primary purpose is to enjoy independent audio
programs at different passenger seats, e.g., enjoying music in a rear passenger seat while listening
voice navigation guide in the driver’s seat. Other scenarios such as securing privacy during the
hands-free communication are promising applications of the personal audio system for automotive
vehicles.
The synthesis of a sound beam or personal sound zone, however, is not a whole new story. For
simple loudspeaker geometries such as linear or circular arrays, a basic method one can utilize is
the beamforming technique. For example, Druyvestyn and Garas(1) proposed the use of active
noise control, beamforming, and loudspeaker directivity, depending on the frequency range of the
input signal. By forming a sound beam directed at the listener, a personal sound zone can be
generated along a line of sound-beam propagation. In many practical problems, however, such
simple array geometries are not possible due to the limitations on the possible loudspeaker
positions. Moreover, reflections, diffraction, and scattering of wavefronts in the listening
environment can seriously degrade the sound focusing performance of the beamforming system.
A generalized theory that can account for these complex acoustic propagations has been in
development since 2002, under the concept of acoustically bright and dark zones(2,3). This concept,
in essence, is simply the extension of the zone of quiet in active noise control (ANC), for multiple
zones with different characteristics. In the bright zone, the spatial average of acoustic potential
energy is controlled to be maximal, while the energy is attenuated within the dark zone. The energy
ratio between two different zones, termed acoustic contrast, is maximized through several
optimization techniques(Fig. 3).
Figure 1: Various loudspeaker arrays for personal sound systems: (a) TV (b) Monitor (c) Mobile.
(a)
(b)
Figure 2: Two examples of personal audio system for automotive applications (a) directional pedestrian warning
(b) playback of independent audio program
Figure 3: Generation of acoustically bright and dark zones using acoustic contrast optimization(28)
(17 monopole sources, L0=5λ, r0=20 λ, λ : wavelength)
The optimization technique itself is not especially different from the directivity or array gain
optimization techniques in beamforming(e.g., (12,13,21)). It should be noted, however, that the
“zone control” and “optimization with transfer functions of loudspeakers with arbitrary geometries”
are two distinct factors discriminating acoustic contrast control from its predecessors(14).
Albeit of its long history, application of acoustic contrast control to automotive vehicle is quite new
and noteworthy. In this article, theories behind the personal sound system are briefly reviewed, and
more importantly, experimental studies on the application of personal sound systems are introduced.
The practical difficulties and ways to overcome them are explained with practical demonstration
systems.
THEORIES ON THE SYNTHESIS OF A PERSONAL SOUND ZONE
Figure 4: Structure of a sound field control system
For a personal sound system, we have multiple loudspeakers, such as shown in Fig. 4, driven by
multichannel control signals, i.e., a source signal filtered by multichannel filters. Sound fields
produced by multiple loudspeakers interfere in space and time to form a certain shape of resultant
sound field. The spatial distribution of a sound field can be controlled over a finite zone of interest(
V ) through the tuning of relative magnitudes and weights of multichannel filters.
For ease of description, assume that a sound field is generated from a source signal of unit amplitude
and single frequency  . Multiple loudspeakers are positioned at rs( n ) ( n  1, , N ), and their
sound fields are measured at discrete locations r ( m ) ( m  1,
, M ) sampling the zone of
interest(Fig. 4). By denoting the transfer function of each loudspeaker measured at r ( m ) as
h(r ( m ) | rs( n ) ) , we can construct a M  N matrix H ( [H]( m,n)  h(r ( m) | rs( n) ) / M ) relating
acoustic responses between multiple loudspeakers and multiple microphones. Note that, for brevity,
the transfer function matrix is normalized by the square root of the number of microphones, and its
frequency dependency is omitted. Then, the sound pressure field vector, also normalized by the
square root of M ( p  [ p(r (1) ), , p(r ( M ) )]T / M ), can be described in terms of the matrix H
and multichannel filter coefficients q  [q(rs(1) ), , q(rs( N ) )]T .
p  Hq
(1)
For a sound field control problem, the final goal is to find multichannel filter coefficients q that
produce a desired pressure field p . The acoustic contrast problem, however, considers acoustic
potential energy of two different zones VB and VD . The sound fields in these zones can be denoted
by the vectors p B  [ p(rB(1) ),
, p(rB( M B ) )]T / M B and p D  [ p(rD(1) ),
, p(rD( M D ) )]T / M D ,
where M B and M D denote the number of measurement positions in VB and VD , respectively. By
using the multichannel filter coefficients q  [q (1) , , q ( N ) ]T and denoting transfer functions for VB
and VD as H B and H D , respectively, the pressure vectors can be expanded to
p B  H B q , p D  H Dq .
(2)
Acoustic contrast control pursues the maximization of the energy ratio between two different zones
VB and VD . To incorporate the acoustic potential energy, acoustic contrast control employs the
spatial average of acoustic potential energy as a measure representing the energy of a zone. Two
energy measures for the bright and dark zones can be described as
eB  p B
2
 q H R B q,
eD  p D
2
 q H R Dq ,
(3)
where each element of the spatial correlation matrix R  H H expresses the correlation of transfer
functions of different loudspeakers. Then the acoustic contrast is defined as a ratio of these two
energies. That is,
H

pB
pD
2
2

q H R Bq
.
q H R Dq
(4)
It is well-known that the maximization of this Rayleigh quotient form shares a solution with the
generalized eigenvalue problem R Bq   R Dq , which can be solved by iterative techniques such
as the QZ factorization algorithm(15). The acoustic contrast can alternatively be defined as

pB
pT
2
2
q H R Bq
,
 H
q RT q
(5)
where q H RT q  pTH p is the energy over the total zone of interest calculated from the pressure
vector pT  [ M B pTB
optimal solution(2).
M D pTD ]T / M B  M D . It has been shown that both forms have the same
Depending on the constraints of a given system, the acoustic contrast problem can be modified
into various forms. For example, when the singularity of matrix R D is of concern, the acoustic
contrast problem can be modified into a regularized form:
q H R Bq
qH R q
(6)
 H
 H B .
q (R D   I )q q R Dr q
The regularization parameter  prevents the abrupt increase of the acoustic contrast near the null
space of R D , by incorporating the total power of the multichannel filter coefficient q H q . This
concept stems from the input power penalty of the beamforming problem and denoted as an acoustic
brightness penalty in the zone control problem. Similarly to the acoustic contrast of Eq. (4), the
acoustic brightness is defined as a ratio of the acoustic potential energy of the bright zone to the
total power of the filters:
qH R q
 HB
q q .
(7)
Therefore, the modified form of Eq. (6) can be viewed as a hybrid form of the acoustic contrast and
brightness problem. The regularization parameter  plays a similar role as in the regularized least
squares problem, and the optimal parameter  is determined from the acoustic brightness-tocontrast curve (Fig. 5(c)). By calculating and collecting brightness and contrast of solutions with
different  , one can select the point of the best compromise between two different measures.
(a)
(b)
(c)
Figure 5: Acoustic contrast optimization for the sound field manipulation in a car cabin.
(a) Transfer function measurement using a microphone array (b) Sound field focused for the driver’s seat
(c) Brightness-Contrast curve (at 500Hz)
To illustrate the mathematical similarity of this technique to the well-known least mean squares
problem, let us reformulate the general acoustic contrast problem as the following optimization
problem: we want to find an excitation signal q opt that maximizes the acoustic potential energy of
the bright zone ( VB ) while that of the dark zone ( VD ) is constrained to a constant value eD . This
statement can be written in a mathematical form as
qopt  arg max L (q),
q
where L = p B
2
  ( p D  eD )   q .
2
2
(8)
Now the acoustic contrast maximization problem is converted to the energy difference
2
2
2
p B   ( p D  eD ) maximization with negative input power penalty   q .
The variable  is a Lagrange multiplier to describe the dark zone constraint as a penalty function,
2
and  is a kind of tuning parameter to prevent the divergence of input power q . By taking the
derivative of L with respect to q and  , the optimal solution can be found at
L
L
 2  R Bq   (R D   I)q   0,
= q H R Dq  eD  0,
q

(9)
where    /  . Accordingly, the problem defined in Eq. (8) is equivalent to the eigenvalue
problem given by
R Bq   (R D   I)q
subject to q H R D q  eD .
(10)
From Eq. (10), it can be seen that the optimal solution q is one of the eigenvectors of the
generalized eigenvalue problem. The resultant value of the objective function L=
  eD is
maximized when the eigenvalue  is maximum, so q opt is the eigenvector corresponding to the
maximum eigenvalue, scaled such that q H R D q  eD . In practical applications, the optimal solution
can be scaled differently to yield a constant output energy in the bright zone or constant pressure at
a reference position. Otherwise, one can take  as the Lagrangian and tune the variable  to a
fixed value. That is,
2
2
2
(11)
qopt  arg max  p B   p D   ( J  q )  ,


q
which is called the energy difference maximization(16). The solutions of these two methods are not
different, in that they span the same acoustic brightness-to-contrast curve and their objective
functions are only different in the expression of the tuning factor.
Although acoustic contrast problems can be approached by simple eigenvalue analysis, it is the
transfer functions of given acoustic environment that determines the maximum achievable acoustic
contrast. The following example describes the synthesis of a directional sound beam for a hybrid
vehicle. In this experiment, six loudspeakers were installed in the front grill, and 31 microphones
arranged at 10 m distance with 6 degree angular interval measured the impulse responses of
individual loudspeakers (Fig. 6). The height of microphones was set at 60 cm and 1.6 m, to inspect
the influence of ground reflection to the beam pattern.
Before the experiment, basic beam patterns can be simulated from a point source assumption for
each loudspeaker. This assumption combined with a free-field propagation model would greatly
simplify the given problem into a classical beam pattern design. Lots of classical theories, such as
the logarithmic array design with denser loudspeaker spacing near the center, or directivity control
algorithms can be incorporated to design and predict a beam shape. In practice, however, the selfscattering from the car body structure and ground reflections make the transfer functions far from
the idealistic condition. The impulse response (IR) actually measured from a real vehicle in an
outdoor condition (Fig. 7) shows very early reflections from the ground, as well as complex
scattering waves right after the direct wave. Most of broad alternations in the IR is due to the
dynamics of loudspeaker itself, but the influence of ground reflection and scattering cannot be
neglected for precise control of sound beams.
Using the measured IRs, multichannel filters were designed, and their beam responses were
measured by the same microphone set-up. Figure 8 shows the responses of sound beams designed
with different brightness constraint (   -1, -3, -6, -12, -100 dB from the maximum brightness) in
frequency domain. Bright and dark zones were configured as angular areas that varies with respect
to the frequency. For example, the bright zone has wider angle in the low frequency region than in
high frequency, considering the beamwidth inversely proportional to the frequency. Above 2 kHz,
spatial aliasing due to discrete loudspeaker arrangement can produce grating lobes. These grating
lobes are physical consequences of spatial sampling and cannot be suppressed by signal processing
techniques. With this regard, only the angular region free from aliased components was considered
as dark zones. It can be seen that beams are sharpened as brightness decreases, but there is a certain
limit of this trend. If brightness is reduced to -100 dB, the beam is no longer in control and shows
different behavior from simulations. This is because of insufficient accuracy of measured IRs and
too small radiation efficiency of the array leading to the nonlinear distortion of loudspeakers.
Figure 6: Measurement of impulse responses in an open space. Total 31 microphones are distributed at 10 m distance
to span the frontal 180 degree with 6 degree interspacing. Six loudspeakers installed behind the front grill are excited
by log-chirp signals between 100 Hz and 22 kHz.
Figure 7: Impulse response from the 4th speaker from left, measured by the center microphone at 10 m distance.
Figure 8: Beam responses of multichannel filters designed with different brightness constraints (   -1, -3, -6, -12, 100 dB).
INTERIOR FIELD CONTROL AND FILTER DESIGN ISSUES
A personal sound system is implemented by filtering a source input signal using multichannel
filters, of which outputs drive multiple loudspeakers or a loudspeaker array such that acoustic
potential energy can be focused only to a designated region while suppressing the energy in other
area. For designing multichannel filters, frequency domain approaches are often preferred because
of its simplicity and relatively less computational effort. In the frequency domain approach(2,3),
optimizations techniques such as acoustic contrast maximization or energy difference method are
carried out at discrete frequencies of concern. The optimal coefficients calculated at discrete
frequencies are then inverse-Fourier transformed to determine the coefficients of FIR filters. The
frequency domain design, however, suffers from several artifacts.
A representative example is the causality problem. The inverse Fourier transform of the filter
coefficients designed at discrete frequencies work properly only with cyclic convolution, which
yields the causality issue in the real systems producing sound through the linear convolution.
Basically, relative weights between filter coefficients of different frequencies need to be aligned in
both magnitude and phase, to ensure the minimal distortion of temporal pressure signals at multiple
listening positions. However, spatial optimization techniques, such as acoustic contrast
optimization, predetermine the spatial distribution of sound, so the modification of temporal
characteristics over multiple positions is inevitably limited. For the alignment of multiple frequency
filter coefficients, Choi and Kim(17) used a single-channel post-filter to recover impulse invariance
at a single listener position within the bright zone. Although it can reduce temporal artifacts to some
extent by rearranging multichannel filters designed in frequency domain, the post-filter cannot fully
resolve the causality problem in principle. A time domain optimization technique(18) proposed by
Cheer and Elliott controls the time-average of the acoustic potential energy and hence can avoid the
causality issue. Nevertheless, owing to the fact that the time averaged potential energy has no
consideration of its spectral distribution, time domain optimization often leads to non-equalized
responses in frequency domain. To minimize the pressure variation over the bright zone, Cai(19)
introduced a differential constraint to the frequency domain formula. For the same objective,
Choi(20) used the subband averaging technique that takes the inter-frequency components into
consideration during the optimization stage. Although none of these techniques can perfectly
control both the temporal and spatial distribution of the resultant sound field, this can be understood
as the downside of the sound field control approach, which enhances some acoustic quantities with
less constraints on the magnitude and phase distribution in space and time.
As a representative example of circular convolution artifacts, the acoustic contrast control using
transfer functions measured in a real car cabin is presented here. The primary objective of this
experiment was to investigate the acoustic contrast that can be achieved only using loudspeakers
pre-installed in the commercial vehicle. The car was a large-sized sedan with 12 independent
loudspeaker channels(Fig. 9(a)). Although the exact dimension of the car cabin is not included, its
size can be inferred from the distance between left and right bass speakers (1.53 m) and the distance
from the front center to surround center loudspeaker (2.59 m). Each of the mid-woofer and tweeter
pair was combined by the passive crossover network, and surround and center speakers were
connected to independent channels.
To measure the transfer function, a rectangular microphone array with 30 microphones positioned
over a 5  6 grid of 4 cm spacing was built (Fig. 9(b)). Transfer functions between 12 loudspeakers
and four listener zones configured at four seat locations were measured by log-chirp excitations
signals of 1s duration within the frequency range of 100 Hz~ 22 kHz. The sampling rate f s was
set to 51.2 kHz, and all measured transfer functions were truncated to 8192 samples and then padded
with extra zeros of 8192 samples (16k samples in total). From the impulse responses shown in Fig.
10(a), it can be seen that the most of impulse responses are bounded within 40 ms (2048 samples)
from the arrival of direct waves. All responses show strong early reflections, which are great
differences compared to the free-field or semi-anechoic case.
(a)
(b)
Figure 9: Experiment configurations (a) loudspeaker positions (b) microphone array
(a)
(b)
Figure 10: (a) impulse responses, and (b) frequency responses of 12 loudspeaker channels measured at the center
microphone of the microphone array positioned in the driver’s seat (front left seat).
Using the measured and zero-padded transfer functions (Fig. 10(b)), the multichannel filter
coefficients were calculated at design frequencies of 12.5 Hz interval, which gives the filter length
of 4096 samples. The frequency spacing of the transfer function was 3.125 Hz. The interpolation
function was truncated to 33 samples in the interpolated frequency axis. The regularized acoustic
contrast maximization was carried out for each design frequencies, and the regularization parameter
 was determined such that the acoustic brightness of Eq. (7) normalized by the maximum
eigenvalue of R b equals to -10 dB. This is for maximizing acoustic contrast under the efficiency
loss constraint. The optimized filter coefficients were then convolved with transfer functions in time
domain. Responses after the convolution were transformed again to frequency domain, in order to
exclude the circular convolution artifact in the simulation. Final acoustic contrast curves with and
without the subband averaging are presented in Fig. 11.
The first graph of Fig. 11 shows the acoustic contrast variation across frequency when the bright
zone ( VB ) was configured at the driver’s seat position (front left) and the dark zone was at the rear
right seat. The striking difference between the conventional and subband averaging can be found
across entire frequency region below 1 kHz. The conventional optimization at discrete frequencies
shows a lot of peaks across design frequencies, but the performance is significantly lowered in
between, which may lead to the significant leakage of sound energy in the dark zone. By contrast,
the subband averaging yields relatively smooth and flat acoustic contrast change, and its value is in
the middle of peaks and troughs of the acoustic contrast curve obtained from the conventional
acoustic contrast maximization.
(a)
(b)
Figure 11: Acoustic contrast (blue) without subband averaging (red) with subband averaging. (a) bright zone: driver’s
seat, dark zone: rear right seat (b) bright zone: driver’s seat, dark zone: all other seats.
The second graph(Fig. 11(b)) is for the case when the energy is focused over the driver’s seat, while
the energy of other three seats are suppressed simultaneously. The similar behavior to the previous
case is also observed in the frequency range below 300 Hz. As frequency goes up, the result with
subband averaging follows the lower bound of the conventional optimization. This tendency
implies that the subband averaging at high frequencies cannot enhance the performance because
the transfer function change in this region is so rapid that a filter coefficient of single frequency
cannot control the entire subband. In the frequency region beyond 700Hz, two curves have little
difference. The acoustic contrast change shown in this example tells us that there is a certain
frequency region where the subband averaging can be effective as compared to the conventional
design at discrete frequencies. For the performance enhancement over the entire frequency range,
however, increase of the filter length or the time domain optimization should be the right answer.
Nevertheless, results show that the computational cost and complexity can be reduced even with
the moderate acoustic contrast in low frequency regions, by considering subband spreading or
leakage.
SUMMARY
Two personal sound systems developed for automotive applications are presented. Both the
directional pedestrian warning for the exterior sound field and individual sound zone system for
interior field exploit the similar principles and control strategies based on the acoustic contrast
control. The scattering waves and early reflections of interior/exterior sound fields, however, are
very different and require judicious modifications of cost functions. Variants of the acoustic contrast
control have been proposed for the improved control efficiency and robustness, and two particular
techniques, brightness and contrast hybrid/subband optimization techniques are explained with
practical prototypes.
ACKNOWLEDGEMENT
This work was supported by supported by the Hyundai Motor Company, and the experiment on the
directional pedestrian was done in collaboration with ARE, Korea.
REFERENCES
[1] Druyvesteyn WF, Garas J. Personal Sound. J Audio Eng Soc. Audio Engineering Society;
1997 Sep 1;45(9):685–701.
[2] Choi J-W, Kim Y-H. Generation of an acoustically bright zone with an illuminated region
using multiple sources. J Acoust Soc Am. 2002 Apr;111(4):1695–700.
[3] Choi J-W, Kim Y-H. Generation of acoustically bright and dark zone using multiple sources.
Proc. Inter-Noise 2002. Dearbone, MI, USA; p. N322.
[4] Lee C-H, Chang J-H, Park JY, Kim Y-H. Personal sound system design for mobile phone,
monitor, and television set; Feasibility study. J Acoust Soc Am. Acoustical Society of
America; 2007 Nov 1;122(5):3053.
[5] Elliott SJ, Cheer J, Choi J-W, Kim Y. Robustness and regularization of personal audio
systems. IEEE Trans Audio, Speech Lang Process. 2012 Sep;20(7):2123–33.
[6] Cheer J, Elliott SJ, Kim Y, Choi J-W. Practical Implementation of Personal Audio in a
Mobile Device. J Audio Eng Soc. Audio Engineering Society; 2013 Jun 7;61(5):290–300.
[7] Park J-Y, Chang J-H, Kim Y-H. Generation of Independent Bright Zones for a Two-Channel
Private Audio System. J Audio Eng Soc. Audio Engineering Society; 2010 Jun
8;58(5):382–93.
[8] Quinn D, Mitchell J, Clark P, and Garcia JJ, eVADER – Development and initial evaluation
of a next generation pedestrian alert solution for quiet electric vehicles. Proc. of FISITA
2014, Maastricht, Netherlands, 2-6th June 2014, paper no. F2014-NVH-080.
[9] Vansant GK, and Berkhoff AP, On the design of a (H)EV steerable warning device using
acoustic beam forming and advanced numerical acoustic simulation, Proc. FISITA 2014,
Maastricht, Netherlands, 2-6th June 2014, paper no. F2014-NVH-069.
[10] Berkhoff AP, Genechten BV, Source arrays for directional and non-directional sound
generation. Proc. ISMA 2014.
[11] Cheer J, Elliott S. Design and implementation of a personal audio system in a car cabin.
Proc. of Meetings on Acoustics. Acoustical Society of America; 2013. p.055009.
[12] Dawoud M, Anderson A. Design of superdirective arrays with high radiation efficiency.
IEEE Trans Antennas Propag. 1978 Nov;26(6):819–23.
[13] Cox H, Zeskind R, Kooij T. Practical supergain. IEEE Trans Acoust. 1986 Jun;34(3):393–
8.
[14] Choi J-W, Kim Y, Ko S, Kim J. Super-directive loudspeaker array for the generation of
personal sound zone. Proc. 125th Audio Engineering Society Convention. 2008.
[15] Golub GH, Loan CF Van. Matrix Computations. 3rd edition. JHU Press; 2012.
[16] Shin M, Lee SQ, Fazi FM, Nelson PA, Kim D, Wang S, et al. Maximization of acoustic
energy difference between two spaces. J Acoust Soc Am. Acoustical Society of America;
2010 Jul 16;128(1):121–31.
[17] Choi J-W, Kim Y-H. Active Control for the Enhancement of Sound Field. Proc Active 04.
Williamsburg, Virginia;
[18] Elliott SJ, Cheer J. Regularization and robustness of personal audio systems [Internet].
ISVR Technical Memorandum 995. 2011 [cited 2014 Aug 15]. Available from:
http://eprints.soton.ac.uk/207989/1/Pub12685.pdf
[19] Cai Y, Wu M, Yang J. Sound reproduction in personal audio systems using the least-squares
approach with acoustic contrast control constraint. J Acoust Soc Am, Acoustical Society of
America; 2014 Feb 1;135(2):734–41.
[20] Choi JW, Subband optimization for acoustic contrast control, Proc. ICSV 22, Held in
Florence, Italy, July 2015.
[21] Kim Y-H, Choi J-W. Sound Visualization and Manipulation. John Wiley & Sons, 2013.