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Design of a Highly Directional EndfireLoudspeaker Array*
MARINUS M. BOONE, AES Fellow([email protected])
Laboratory of Acoustical Imaging and Sound Control,Delft University of Technology, 2600 GA, Delft, The Netherlands
AND
WAN-HO CHO AND JEONG-GUON IH, AES Member([email protected]) ([email protected])
Center for Noise and Vibration Control, Department of Mechanical Engineering,Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
The applicability of a loudspeaker line array, of which the main directivity is in the lengthdirection of the array, is discussed. Hence this acoustic array aims at the endfire beamformingdirection, resulting in a “spotlight” of sound in a preferred direction. Optimized beamformingtechniques are employed, which have been developed earlier for the reciprocal problem ofdirectional microphone arrays. Effects of the design parameters of the loudspeaker array systemon its performance have been investigated. It is shown that the stability factor of the optimizedbeamformer can be a useful parameter to control the directional characteristics of the array. Inaddition the effect of mutual interaction between individual loudspeaker elements in the arraysystem has been considered to reduce the difference between designed and actual performanceof the array system. A prototype constant-beamwidth array system has been tested by simula-tion and measurement and the results support findings in a parametric analysis.
0 INTRODUCTION
Directional loudspeaker systems have been studied in-
tensively by many researchers because of their useful
application, such as a column array that addresses sound
information in the plane of the listeners’ ears. For a single
loudspeaker unit the directional characteristics depend on
the Helmholtz number, which is related to the size of the
radiating membrane and the wavelength. When multiple-
loudspeaker units are concerned, forming a so-called
loudspeaker array, the directional characteristics depend
on the placement of the loudspeaker units within the array
and on the filtering of the audio signals that are provided
to the loudspeakers.
A lot of research work on the characteristics of transducer
array systems has already been conducted in the field of
antennas, which are for electromagnetic use, and for micro-
phone systems. Various spatial filtering methods have been
devised from this previous work, and the method is in
general called beamforming [1]. For microphone arrays,
directivity control has been one of the major research
topics, in particular on the design of a constant beamwidth
over a broad frequency range [2], [3]. Some array designs
have been applied to hearing aids [4], [5] to obtain a highly
directive characteristic. The representative methods to ob-
tain highly directive beam patterns can be summarized by
three methods—the delay and sum technique [1], the gradi-
ent method (such as Jacobi arrays) [6], and optimal beam-
forming [7], [8]. Among these, the optimal beamforming
method is known to deliver a relatively high directivity as
compared to other methods [9], [10]. The solution for opti-
mal beamforming was suggested in the middle of the 20th
century. However, at that time it was only considered to be
of academic interest because of noise problems associated
* This work was presented in part at the 122nd Convention of
the Audio Engineering Society, Vienna, Austria, 2007 May 5–8.
Manuscript received 2008 February 22; revised 2009 February
11 and April 6.
J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May 309
PAPERS
with the available equipment [11], but also because the
implementation of the required filters was not possible with
the analog equipment at that time. A constrained solution to
solve the noise problem was suggested by Gilbert and
Morgan [12], and with the advent of modern digital signal
processing equipment, this technique has been applied to
many practical situations. One of these applications is the
optimized beamforming that has been implemented in the
hearing glasses [13]. These are highly directive hearing aids
mounted in the arms of a pair of spectacles, with four
microphones at each side.
Theoretically, based on the reciprocity principle of the
acoustic field, the theory for a microphone array system
can also be applied to a loudspeaker array system. In this
study the theory to optimize the directivity of microphone
arrays is reviewed and then applied to a loudspeaker array
system. Moreover, a modification process has been
adapted to overcome the problems induced by the differ-
ence between microphones and loudspeakers. The major
concern in this investigation is the optimization of the
design parameters of the optimal beamformer. These are
directivity index and noise sensitivity for microphone
arrays and directivity index and power index for loud-
speaker arrays. In these optimizations the stability factor
that will be introduced in Section 1.2 will play an impor-
tant role.
1 BASIC THEORY
1.1 Basics of the Array System
Fig. 1 shows the configurations of typical microphone
and loudspeaker arrays. Here Fn(o), (n ¼ 1, 2, . . ., N),denotes the filters that control the input and output, and
which are connected to each acoustic transducer (micro-
phone or loudspeaker). They can be written in vector
form as
Fð!Þ ¼ F1ð!Þ F2ð!Þ � � � FNð!Þ½ �T (1)
where o is the angular frequency and the superscript T
denotes the transpose of the vector. In general, the acous-
tic transducers are distributed in an orderly manner in a
real implementation. Most representative configurations
are broadside arrays, in which the transducers are aligned
perpendicular to the direction of sound propagation, and
endfire arrays, in which the transducers are aligned along
the direction of sound propagation. For a broadside loud-
speaker array the vector of transfer functions to a far-field
immission point consists of the directivities of the indi-
vidual transducers and their relative propagation delays,
as described by the vector equation [10]
Wð�; �; !Þ ¼�1 �; �; !ð Þej!cx1 sin � cos��2 �; �; !ð Þej!cx2 sin � cos�
..
.
�N �; �; !ð Þej!cxN sin � cos�
26664
37775: (2)
Here y is the azimuthal angle, f the zenithal angle (see
Fig. 2), c the speed of sound, xi the transverse position
of the ith transducer, and Gn, n ¼ 1, 2, . . ., N, denotesthe directional factor of each transducer. In the case of
an endfire array the propagation vector W is described
by [10]
Wð�; �; !Þ ¼�1ð�; �; !Þej!cz1 cos ��2ð�; �; !Þej!cz2 cos �
..
.
�Nð�; �; !Þej!czN cos �
26664
37775 (3)
where zi is the longitudinal position of the ith transducer.
A broadside array concentrates the acoustic information
onto a two-dimensional plane. On the other hand, an end-
fire array focuses the sound on a one-dimensional line.
For that reason an endfire array is more suitable for the
objective of this study than a broadside array to obtain a
highly directive beam pattern. The geometry of an endfire
array is shown in Fig. 2.
To compare the acoustic performances of array sys-
tems, many evaluation parameters have been suggested,
of which the directivity factor is the most important. For
loudspeaker systems the directivity factor is defined by
the ratio of the acoustic intensity at a far-field point in
Fig. 1. Typical basic configurations of acoustic array systems.(a) Microphone array. (b) Loudspeaker array.
310 J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May
BOONE ET AL. PAPERS
a preferred direction and the intensity obtained at the
same point with a monopole source that radiates the
same acoustic power as the loudspeaker system [14]. This
measure implies how much available acoustic power is
concentrated in the preferred direction by the designed
system as compared to the total radiated power. In matrix
notation this definition is expressed by [10]
Qð!Þ¼max�;�fFHð!ÞW�ð�;�;!ÞWHð�;�;!ÞFð!ÞgFHð!ÞSTzzð!ÞFð!Þ
(4)
where* denotes the conjugate operator, H denotes the
Hermitian transpose, and Szz(o) is defined by its elements,
Szzð!Þ¼½Smn�
¼ 1
4�
R 2�
0
R �
0Wmð�;�;!Þ�W�
nð�;�;!Þsin�d�d�(5)
Here the subscripts m and n denote the indexes of the
loudspeaker elements.
For microphone array systems the same expressions
can be used. Here the directivity factor expresses the ratio
between the sensitivity in a main direction and the sensi-
tivity in a diffuse or isotropic sound field. The matrix
Szz(o) is commonly called the coherence matrix of the
array. For an endfire array the elements of the coherence
matrix can be written as sinc functions [10], [11],
Smn ¼ sin½kðzm � znÞ�kðzm � znÞ : (6)
Here zm and zn are the positions of the transducer elements
and k denotes the wavenumber. Usually the directivity
index (DI), which is the logarithmic value in dB of the
directivity factor, is used.
Another important evaluation parameter for micro-
phone arrays is the noise sensitivity (NS). This quantity
shows the amplification ratio of uncorrelated noise, so-
called internal noise, to the signal and is given by [10]
�ð!Þ ¼ FHð!ÞFð!ÞFHð!ÞW�ð!ÞWTð!ÞFð!Þ : (7)
The noise sensitivity C(o) is the reciprocal of the white
noise gain [8]. It is not only a measure for the sensitivity
of the microphone array to internal microphone noise
but also a measure of the robustness of the array to
errors in signal processing and deviations of the transdu-
cers from their ideal gains (in amplitude and phase).
Eq. (7) is also a useful performance measure for loud-
speaker arrays. Here the denominator relates to the
sound intensity in the target direction, whereas the nom-
inator, which is the sum of the squared filter values, is
proportional to the electric input power of the loud-
speaker array. Hence we designate C(o) for loudspeakerarrays the power factor (PF) or, when expressed on a dB
scale, the power index (PI).
With the notation described in this section the di-
rectivity pattern of the discrete array system can be
expressed as
�ð�; �; !Þ ¼ FTð!ÞWð�; �; !Þ: (8)
1.2 Optimal Beamformer
As a starting point let us take a look at a simple
delay-and-sum microphone array beamformer in endfire
configuration. In this case the transducer elements are
only filtered with delays that compensate for the travel
times of a wave from the target direction to the different
microphones. The consequence is that this gives a high
output from the target direction because all contributions
of the desired wave are completely in phase. This results
in a high ratio between the target response and the
internal noise response. Hence the noise sensitivity of
such a system is low. The directivity of the array is a
result of the fact that from directions other than the
target direction the wavefield contributions will add up
more or less destructively. However, at low frequencies
the phase differences between those nontarget waves are
small and hence the directivity index is small at low
frequencies.
The optimal beamformer tries to solve this problem by
not maximizing the output of the array in the target direc-
tion, but by maximizing the ratio between the response
from the target direction and the average uncorrelated
response from all other directions.
This optimization problem of an array system can
be defined by the following minimization expression
[8], [10]:
minFð!Þ
FHð!ÞSTzzð!ÞFð!Þ subject to FTð!ÞWð!Þ¼1: (9)
The real-valued cost function derived by the Lagrange
method is given by
minFð!Þ
JðFÞ¼FHSTzzFþ�ðFTW�1Þþ��ðFTW��1Þ (10)Fig. 2. Configuration of endfire loudspeaker array and direc-tional definition of radiated sound field.
J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May 311
PAPERS HIGHLY DIRECTIONAL LOUDSPEAKER ARRAY
where m is a Lagrange multiplier. The optimal value of
F is obtained at the stationary point where the differen-
tial of J(F) is zero. The optimal solution Fopt is esti-
mated as
rFJðFÞ ¼ SzzF�opt þ �W ¼ 0 (11a)
or
FTopt ¼ ���WHS�1
zz : (11b)
Substituting the constraint into Eq. (11b), the Lagrange
multiplier is expressed as
� ¼ �1
WHS�1zz W
: (12)
Then the optimal solution of Eq. (9) is given by [15]
FToptð!Þ ¼
WHð!ÞS�1zz ð!Þ
WHð!ÞS�1zz ð!ÞWð!Þ : (13)
The optimized beamformer aims at maximum directivity
and, at the same time, frequency-independent sensitivity
in the target direction. This gives a principle problem at
low frequencies, because the magnitudes of the individual
transducer filters will increase considerably.
This characteristic can be demonstrated with a simple
example. Fig. 3 shows the filter coefficients estimated by
the optimal beamforming method with N ¼ 2, d ¼ 0.15 m
as a function of frequency. We see that the magnitudes of
the filter coefficients increase drastically in the low-fre-
quency range. Also notice an opposite phase relationship
between both filters. Due to this characteristic, the self-
noise of the microphones is amplified greatly. It also
means that the system should have a high precision to
obtain the desired target response. For loudspeaker arrays
this means that high input signals are needed and also that
the output control must be very precise for a correct re-
sponse in the target direction. Therefore in the past the
optimal beamformer could not be implemented easily by
means of conventional analog circuits.
To solve this problem, Gilbert and Morgan [12] sug-
gested a method adding a specific value b to the diagonal
of the coherence matrix. Cox et al. [8] suggested a
generalized form of the optimization problem. To restrict
the noise sensitivity or power index, an additional con-
straint is required, such as
FHð!ÞFð!Þ ¼ �2�max: (14)
Using the Lagrange multiplier b, the real-valued cost
function can be written as [9],[10]
minFð!Þ
JðFÞ ¼ FHðSzz þ �IÞTFþ �ðFTW � 1Þ
þ ��ðFTW� � 1Þ � ��2(15)
where I is an identity matrix of size N. The solution of
Eq. (15) is obtained as [9], [10]
FToptð!Þ ¼
WHð!Þ Szz þ �ð!ÞI½ ��1
WHð!Þ Szz þ �ð!ÞI½ ��1Wð!Þ : (16)
The role of b is to obtain a behavior of the array that is
between the maximized directivity result of Eq. (13) and
the delay-and-sum result. We will call the coefficient
b(o) the stability factor, because it controls the robustness
of the array.
2 OPTIMAL DESIGN OF AN ARRAY SYSTEM
2.1 Design Parameters
The directional characteristics of the loudspeaker array
system depend on several design parameters—the number
of transducers, their spacing and distribution pattern, the
directional characteristics of the individual loudspeakers,
and the applied beamforming filters. For the optimal
beamformer the filter shape of the array system is deter-
mined by Eq. (16). Therefore the parameter to be opti-
mized is the stability factor b(o).To investigate the effect of each design parameter, a
parametric study was conducted with Eqs. (4) and (7).
Each loudspeaker element is assumed to be a monopole
and the effects of reflection and scattering are ignored.
With uniform spacing d and the same number of transdu-
cers, it is observed that the same directional characteris-
tics apply if we normalize the frequencies according to
the high-frequency limit (in Hz) given by
fh ¼ c
2d: (17)
Fig. 4 shows the directivity index and power index of an
array with four equally spaced loudspeakers. The stability
factor b was set at 0.01. Also, using the delay-and-sum
beamformer, it was compared with the array system
having the same configuration. The filter derived by the
delay-and-sum beamformer is given by [16]
Fdsð!Þ ¼ W�ð!ÞWHð!ÞWð!Þ : (18)
Fig. 3. Filter coefficients estimated using optimal beamformingmethod with N ¼ 2. Both filters have the same magnitude. —phase of first filter; – – phase of second filter.
312 J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May
BOONE ET AL. PAPERS
For the case of an endfire array, Eq. (18) can be rewrit-
ten as
Fds;endfireð!Þ ¼ 1
N1 e�jkðz2�z1Þ . . . e�jkðzN�z1Þh iT
: (19)
The result shows that the directivity index is higher for
the optimal beamformer than for the delay-and-sum
beamformer in the frequency range below f/fh � 0.9. This
is consistent with previous work, which mentioned that
the optimal beamformer converges to the delay-and-sum
beamformer at fh [10], [11].The number of loudspeaker elements determines the
maximum value of the directivity index. For an endfire
array system, the maximum directivity index (in dB) is
determined by [17]
DImax ¼ 20 logN (20)
where N denotes the number of transducers. Fig. 5 depicts
the results of a parametric study varying the number of
transducers (N ¼ 4–8) with b ¼ 0.01. The increase in DI
follows the increase of N over the whole frequency range
lower than fh. The frequency with the maximum DI value
also increases, but it remains below fh. The power index
shows a decreasing tendency with the increase in fre-
quency, reaching a minimum value at f ¼ fh. These resultsare in agreement with the aforementioned theory.
Fig. 6 shows the changes in directional characteristics
with varying stability factors (b ¼ 100–10�6). Here the
number of equally spaced transducers was eight. With
increasing b the DI and PI values decrease up to the
frequency of DImax. At frequencies higher than fh DI andPI are no longer controllable by b.
2.2 Design Procedure
Fig. 7 illustrates the design steps of the optimal beam-
former for a loudspeaker array system. First the target
values of the evaluation parameters DI and PI and the
high-frequency limit should be selected. Considering these
target values, the basic design parameters, i.e. the number
of transducers and spacing, are determined using Eqs. (17)
and (20). The second step is the selection of b as suggested
to solve the power index problem of the equipment.
However, it can also be applied to control the directional
Fig. 4. (a) Directivity index and (b) power index of array withfour uniformly spaced loudspeakers. — optimal beamformer;– – delay-and-sum beamformer.
Fig. 5. Changes in (a) directivity index and (b) power indexfor various numbers of loudspeakers. — N ¼ 4; – – N ¼ 6; • • •N ¼ 8.
J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May 313
PAPERS HIGHLY DIRECTIONAL LOUDSPEAKER ARRAY
characteristics of the array system without changing the
array configuration. Consequently this parameter can be
determined after selecting the other parameters. Finally
the optimal filter is derived by Eq. (16).
The optimal value of b, considering both DI and PI,
cannot be obtained directly. For this reason several itera-
tive methods were suggested to obtain the optimal value
[10]. It is thought that, as an alternative method, the
plot of PI versus DI can be used in the selection of b.Consider the array system with N ¼ 8, which was used
in the previous section. The range of b was varied from
10�6 to 100. Fig. 8 shows the DI–PI plots for various
normalized frequencies (f/fh ¼ 0.08–0.75). It can be
Fig. 6. Change in (a) directivity index and (b) power indexfor various stability factors b. — b ¼ 10�6; – – b ¼ 10�5; • • •b ¼ 10�4; – � – b ¼ 10�3; – �� – b ¼ 10�2, - - - b ¼ 10�1, ������b ¼ 100. Fig. 7. Design steps for optimal beamformer.
Fig. 8. Directivity index versus power index for array system with N ¼ 8 for different values of b and f / fh.
314 J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May
BOONE ET AL. PAPERS
seen that, with the increase of b, both DI and PI decrease
for a certain frequency or, conversely, the range of varia-
tions of DI and PI decreases with an increase in fre-
quency for a fixed value of b. This fact is related to the
result of the previous section, namely, that the directional
characteristics are no longer controllable at frequen-
cies higher than fh. If the target performance of the array
system is given by a specific range of DI and PI, the
value of the stability factor can be selected in such a
DI–PI plot.
3 CONSIDERATIONS FOR A PRACTICALSYSTEM
3.1 General Directivity Pattern
In the foregoing sections the effect of reflection and
scattering induced by the loudspeaker enclosures has been
ignored, that is, Gn ¼ 1, n ¼ 1, 2, . . ., N. In the case of a
microphone array system, the size of the transducers is
usually sufficiently small compared to the smallest wave-
length of interest. However, for loudspeaker arrays it is
recommended that the loudspeaker units be of a larger
size to obtain a sufficiently large radiation power. In prac-
tice the loudspeaker is not a simple source, but its radia-
tion pattern is a function of frequency or the Helmholtz
number, which is the ratio of diaphragm size to wave-
length. Therefore the Gn value in Eq. (3) cannot be con-
sidered unity, and the coherence function of the array
system described by Eq. (6) may not be valid. This fact
will affect the design of the optimal filters, especially in
the selection of the stability factor to obtain the target
directivity index value, because Eq. (4) is a function of
the coherence matrix Szz (o).Two methods are applicable to solve this problem.
The first is to return to Eq. (5) as a more general
definition of the coherence matrix. The other method is
using the general definition of directivity, which is
given for a source with only radiation dependence as a
function of y,
Qð!Þ ¼ 2=R �
0�2ð�Þ sin � d� (21)
where it is assumed that G2 (ytarget) ¼ 1 in the target
direction.
Eq. (6) can be used as one of the evaluation parameters,
but it can also be employed in the derivation of the filter
coefficient, as in Eqs. (13) and (16). Therefore it is
expected that a direct calculation using Eq. (5) to design
the optimal filter can yield a further optimized solution
than the use of Eq. (6).
3.2 Mutual Interaction between Transducers
Scattering effects induced by objects increase accord-
ing to the size of the objects and decrease according to the
distance compared to the wavelength [18]. In the case of a
microphone system, the size of the microphones is usually
small enough to ignore the scattering effect; hence the
equations described in Section 2 can be used with the
assumption of omnidirectionality. In the case of a loud-
speaker array, however, the size of the loudspeakers is
much larger than that of the microphones to obtain
enough output power. Therefore not only the directivity
of the single transducer itself related to its own geometry
should be considered, but also the scattering from the
other transducers.
Usually the scattering effect is considered as being
induced by an incident field. The total field is described
by a summation of incident and scattered sound fields.
The directional pattern of the individual unit can be
obtained by summing the direct field from the transducer
itself and the scattering field induced by the other units.
The analytical solution for the scattered field can be found
under some special conditions [18]. However, the direc-
tional pattern of a general array system is not easy to
obtain theoretically, because the scattering field of each
loudspeaker also becomes the incident field to the other
loudspeakers, recursively. Therefore numerical methods
or measurements would be useful to obtain the resultant
sound field.
3.3 Modified Design Procedure
Fig. 9 explains the modified design procedure for the
design of an optimal beamformer. Considering the acous-
tical facts as outlined in the previous sections, an addi-
tional step to obtain the directional pattern of each
transducer in the array system is required. In this step the
directional pattern must contain the directional pattern of
Fig. 9. Modified design steps for optimal beamformer consider-ing directivity pattern and mutual interaction between loud-speaker elements.
J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May 315
PAPERS HIGHLY DIRECTIONAL LOUDSPEAKER ARRAY
each transducer as well as the scattering effect induced by
every other transducer. Fig. 10 depicts the geometrical
definition of the observation plane to obtain the direc-
tional pattern of the total sound field for each transducer
in the array system. Values of Gn, n ¼ 1, 2, . . ., N, can be
obtained by either calculation or measurement of the
sound pressure over the selected plane. When the direc-
tional pattern of the first unit was calculated, only the
diaphragm center of the first loudspeaker was activated
and the other units were inactive.
4 NUMERICAL SIMULATION
4.1 Modeling of the Loudspeaker Array System
As a design example, a loudspeaker array system was
chosen that consisted of eight loudspeakers at regular
intervals of 0.15 m. The size of each loudspeaker box
was 0.11 (W) � 0.16 (H) � 0.13 (D) m, and the diam-
eter of the loudspeaker diaphragm was 0.075 m. The
boundary element method (BEM) was used to calculate
the directional pattern of each transducer in the given
array configuration. Each loudspeaker was modeled by
106 linear triangular elements, as illustrated in Fig. 11.
The characteristic length of the elements was 0.057 m,
which gives a high-frequency limit of 1 kHz based on
the l/6 criterion (fh of the array system was 1.1 kHz).
All elements except the center of the diaphragm were
modeled as rigid boundaries. In order to obtain the di-
rectional pattern of each loudspeaker in the array sys-
tem, the calculation was carried out by activating the
units one by one as part of the complete system. The
observation plane was selected as a circle, as shown in
Fig. 10, which centered at the active node of each loud-
speaker with a wavelength of the lowest frequency of
interest as the radius (3.43 m).
4.2 Example 1—Maximization of the DirectivityIndex
The target performance of the array system was the
maximization of the directivity index within a range of
the power index that does not exceed 20 dB. Optimal
filters were calculated by two methods. In the first
method we assumed that every loudspeaker unit would
behave as a monopole and the scattering effect was
ignored. In the second method the directional pattern of
each unit and the effect of scattering were taken into
account in the design procedure. The acoustical perfor-
mance of the designed filters was tested by numerical
simulation using BEM.
Fig. 12 shows the power index of the designed filters
calculated by Eq. (7), and Fig. 13 shows the directivity
index of the designed loudspeaker system using the opti-
mal beamforming method to maximize the directivity
index. One can observe that, compared to the result using
the simple source assumption, values of the directivity
index considering the scattering effect are very similar to
the predicted values. This fact is also confirmed in the
directional pattern, as shown in Fig. 14.
4.3 Example 2—Constant-Beamwidth Array
As a second example we considered a constant-
beamwidth array (CBA) system. The simplest concept to
design a CBA is using the different array sets, as com-
puted for different values of the Helmholtz number kd.With this method, however, we need redundant acoustic
devices. In a specific array system one can say that the
same value of DI means the same beamwidth. Therefore
the CBA system can be designed by selecting the fre-
quency-dependent factor b(o) that gives a constant DI
over the entire target frequency range.
The same array system as described in the previous
example was used for the test. Values of DI and PI of this
system as a function of b are as shown in Fig. 8. The
target frequency range was 0.1–1 kHz and the target value
of DI was 12 dB, which is the highest value in Fig. 8 for
the condition of PI < 20 dB. To satisfy these conditions,
the b values along the DI line of 12 dB were selected from
Fig. 8.
Figs. 15 and 16 show the directivity index and pattern
of the designed loudspeaker system. One can observe
that every case satisfies the target value within �2 dB.
As in the previous test example, the filter designed
considering the directional pattern of each loudspeaker
in the array system, as shown in Fig. 15(b), is very
much like the result obtained using the simple source
assumption.
Another factor that should be considered is the power
index of the designed filters. Although it is possible to
obtain a constant directivity index as a function of
frequency, this may lead to high power index values
with consequently a low array output and also the pos-
sibility of large deviations from the desired pattern in
real situations that do not strictly follow the model
Fig. 10. Observation planes to obtain directional pattern of totalsound field for each loudspeaker element in system.
316 J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May
BOONE ET AL. PAPERS
Fig. 11. Boundary element model. (a) One loudspeaker. (b) Entire loudspeaker array.
Fig. 12. Power index of filters. ▪ filter designed under simple source assumptions; ○ filter designed considering directional pattern ofeach loudspeaker in system.
J. Audio Eng. Soc., Vol. 57, No. 5, 2009 May 317
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Fig. 13. Directivity index of loudspeaker system designed using optimal beamforming method to maximize directivity index. ▪Eq. (16); ○ BEM. (a) Filter designed under simple source assumptions. (b) Filter designed considering directional pattern of eachloudspeaker in system.
Fig. 14. Comparison of directivity patterns of loudspeaker system designed using optimal beamformer to maximize directivity index.(a) Predicted by Eq. (16) using filter designed under simple source assumptions. (b) Predicted by BEM using filter designed undersimple source assumptions. (c) Predicted by Eq. (16) using filter designed considering directional pattern of each loudspeaker insystem, (d) Predicted by BEM using filter designed considering directional pattern of each loudspeaker in system.
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Fig. 15. Comparison of directivity patterns of loudspeaker system designed using optimal beamforming method to obtain constantbeamwidth. ▪ Eq. (16); ○ BEM. (a) Filter designed under simple source assumptions. (b) Filter designed considering directionalpattern of each loudspeaker in system.
Fig. 16. Comparison of directivity patterns of loudspeaker system designed using optimal beamformer to obtain constant beamwidth.(a) Predicted by Eq. (16) using filter designed under simple source assumptions. (b) Predicted by BEM using filter designed undersimple source assumptions. (c) Predicted by Eq. (16) using filter designed considering directional pattern of each loudspeaker insystem, (d) Predicted by BEM using filter designed considering directional pattern of each loudspeaker in system.
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used. Fig. 17 shows an example of such a case. It is
observed that the difference between predicted and
simulated results increases at the points of high power
index (400 Hz). From this result one can conclude that
it is helpful to maintain a low power index for precise
predictions.
5 MEASUREMENTS
5.1 System Configuration
To see the performance of the designed filters in a real
situation, measurements were conducted in an anechoic
condition. Fig. 18 is a schematic of the loudspeaker array
and measurement system setup. Fig. 19 shows an endfire
array system. The loudspeaker sizes and all the mea-
surements involved were the same as in Section 4. The
loudspeaker array was mounted on a turntable. In the
measurement system the distance between the micro-
phone and the frontal surface of the loudspeaker array
was 2.5 m when the angle was 0. It is noted that the
geometric center of the array system was also the rotating
center of the turntable. Because of the limited distance
between loudspeaker array and measurement microphone
some near-field effects may occur. One effect is that in
the endfire orientation, level differences occur because of
distance changes. For the outer loudspeakers of the array
this results in a level change at the microphone location of
� 1.5 dB. Another effect occurs in broadside orientation.
Here the outer loudspeakers produce signals with in-
correct phases at the microphone location of about 5 at
100 Hz and 50 at 1000 Hz.
Measurements were conducted in steps of 10 from 0 to
180, and symmetrical data are assumed in the other half-
circle. The sound pressure was measured using a sound
level meter (B&K 2239) and the signal was transferred to
the control computer.
Signal processing was done using the MATLAB soft-
ware for both signal generation and analysis of the
recorded signal. A swept-sine signal was adopted in the
test, and the filter designed for the optimal beamformer
was applied to the swept signal. For multichannel sound
Fig. 17. Effect of power index. ▪ PI limited to 20 dB; ○ PIlimited to 50 dB. (a) Applied power index. (b) Resulting direc-tivity index.
Fig. 18. Schematic diagram of loudspeaker array and measurement system setup.
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playback the filtered signal was sent to an ADAT audio
interface (RME ADI-648) using a sound card (RME
HDSP MADI), and then this signal was amplified by a
multichannel amplifier (Sonic Emotion M3S Amp).
In all measurements two different filter design methods
were used as discussed in the previous section—the filter
design under simple source assumptions and the filter
design including the directional characteristics of each
loudspeaker for the total radiation field.
5.2 Test—Constant b
As a first example we considered the acoustic perfor-
mance of the optimal beamformer with constant b for all
frequencies. The b value was set to 0.01, which is usually
a satisfactory value for optimized beamformers. A com-
parison of the directivity index of the designed and
measured loudspeaker system is given in Fig. 20. Similar
to the foregoing simulation results, the filters designed
considering the directional characteristics of the loudspea-
kers were closer to the predicted values than the filters
using the uniform pattern. Moreover a far higher directiv-
ity index value was obtained by the former filter than the
latter, which did not consider loudspeaker directivity.
Fig. 21 compares simulated and measured directional
patterns of the designed loudspeaker system. In the
measured result the beamwidth of the main lobe is nearly
the same or somewhat narrower in the low-frequency
range, but with higher sidelobes, than the predicted value.
This may be caused by the near-field effects in the mea-
surements, as mentioned in Section 5.1.
5.3 Test 2—Constant-Beamwidth Array
In a second example the filter of the constant-
beamwidth array, which was introduced in Section 4.3,
was implemented. The target value of the directivity in-
dex was chosen to be 12 dB. Fig. 22 compares the target
values and the results of two designed filters. The filter
designed considering the directional characteristics of
the total radiation field of each loudspeaker shows values
that are higher and closer to the target than the filter
using a uniform pattern. However, it is noted that the
designed filter has a still higher sound level in the off-axis
directions than expected from theoretical prediction.
Fig. 23 shows the measured directional patterns of the
array system, which supports the aforementioned findings
with regard to the acoustic performance of the designed
filters.
Fig. 19. Endfire array system.
Fig. 20. Comparison of directivity patterns of loudspeaker sys-tem designed using optimal beamforming method with b ¼ 0.01.▪ Eq.(16), ○ measured. (a) Filter designed under simple sourceassumptions. (b) Filter designed considering directional patternof each loudspeaker in system.
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6 CONCLUSIONS
In this study the basic theory of an endfire loudspeaker
array system was investigated with a parametric study.
The design parameters included the number of loud-
speaker units, their spacing, the array length, and the effect
of the stability factor of the optimal beamformer. Two
types of filter design procedures were tested—one method
was to design a filter under a simple source assumption
and another was to design a filter by additionally consider-
ing the directional pattern of each loudspeaker in the array
system. Measurements on an array of eight loudspeakers
revealed that inclusion of the directivity patterns of the
loudspeakers resulted in an increase in the directivity
by 2–3 dB as compared with a design based on a mono-
pole assumption. Moreover it was observed that the
difference between expected and actual results of the
directivity index also changed from 3–6 dB to 1–3 dB
Fig. 21. Comparison of directivity patterns of loudspeaker system designed using optimal beamformer with b ¼ 0.01. (a) Predicted byEq. (16) using filter designed under simple source assumptions. (b) Measured result using filter designed under simple sourceassumptions. (c) Predicted by Eq. (16) using filter designed considering directional pattern of each loudspeaker in system.(d) Measured result using filter designed considering directional pattern of each loudspeaker in system.
Fig. 22. Comparison of the directivity patterns of loudspeakersystem designed using optimal beamforming method to obtain con-stant beamwidth with DI ¼ 12 dB. ▪ target value; ○ simple sourceassumptions,△ realistic source model with loudspeaker directivity.
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when considering a realistic radiation condition without
too much effort.
7 ACKNOWLEDGMENT
This work was supported in part by the Korea Research
Foundation under grant KRF-2006-612-D00004, the BK21
Project, and NRL (M10500000112-05J0000-11210).
8 REFERENCES
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[7] H. Cox, R. M. Zeskind, and T. Koou, “Practical
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Fig. 23. Comparison of directivity patterns of the loudspeaker system designed using optimal beamformer to obtain constantbeamwidth with DI ¼ 12 dB. (a) Predicted by Eq. (16) using filter designed under simple source assumptions. (b) Measured resultusing filter designed under simple source assumptions. (c) Predicted by Eq. (16) using filter designed considering directional patternof each loudspeaker in system, (d) Measured result using filter designed considering directional pattern of each loudspeaker in system.
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[8] H. Cox, R. M. Zeskind, and M. M. Owen, “Robust
Adaptive Beamforming,” IEEE Trans. Acoust.,Speech, Signal Process., vol. ASSP-35, pp. 1365–1376(1987).
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dom Variations,” Bell Sys. Tech. J., vol. 34, pp. 637–663 (1955).
[13] M. M. Boone, “Directivity Measurements on a
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[14] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V.
Sanders, Fundamentals of Acoustics (Wiley, New York,
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W.-H. Cho
THE AUTHORS
Marinus M. Boone was born in 1947. He received
an M.Sc. degree in subjective acoustics, with the
topic of loudness perception, from the Delft Univer-
sity of Technology, Delft, The Netherlands.
Since 1974 he has been with the Laboratory of
Acoustical Imaging and Sound Control at Delft Uni-
versity of Technology, where he received a Ph.D. de-
gree on the development of a 32-channel microphone
system for directional outdoor noise measurements.
Later on he designed several microphone systems,
based on the same technology, for the measurement of
traffic and aircraft noise as well as microphone arrays
for directional hearing aids, leading to the design of the
so-called hearing glasses. At present his interests are in
audio transducers and multichannel sound reproduc-
tion for applications with wave field synthesis.
Dr. Boone is a fellow of the Audio Engineering
Society, and from 1996 to 2006 he was secretary of
the Netherlands Section of the AES.
Wan-Ho Cho was born in Seoul, Korea, in 1980.
He received a B.S. degree in mechanical engineer-
ing and M.S. and Ph.D. degrees in acoustics from
the Korean Advanced Institute of Science and
Technology (KAIST) in 2002, 2004, and 2008, re-
spectively. He is now a postdoctoral fellow in the
Acoustics Laboratory at KAIST.
During his doctoral studies he joined the Labora-
tory of Acoustical Imaging and Sound Countrol at
Delft University of Technology, Delft, The Nether-
lands, as a guest student researcher from 2006 to
2007, doing research on the directional loudspeaker
array system. His doctoral thesis was about sound
field control based on acoustical holography using
loudspeaker array systems. His research interests are
in the area of audio engineering, especially sound
field control and reproduction by multichannel au-
dio systems. He is also involved in product sound
quality (PSQ) and noise control.
M. M. Boone J.-G. IH
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Jeong-Guon Ih received a B.S. degree in mechani-
cal engineering in 1979 from the Seoul National Uni-
versity and M.S. and Ph.D. degrees in Acoustics/
mechanical engineering in 1981 and 1985 from
KAIST. He was with Loughborough University (UK)
in 1999, with Seikei University (Japan) in 2005,
and with Canterbury University (New Zealand) in
2006 as a visiting professor, either doing research or
lecturing.
He joined the Department of Mechanical Engi-
neering of the Korean Advanced Institute of Science
and Technology (KAIST) in Daejeon, Korea, as an
assistant professor in 1990, teaching courses in
acoustics and vibrations, and is currently a professor
there. Before joining the faculty at KAIST he was
with the DaewooMotor Company (now GMDaewoo
Auto Company) in Inchon, Korea, from 1979 to 1990.
From 1985 to 1990 he was in charge of the Noise,
Vibration and Harshness (NVH) Group in the Tech
Center. In 1987 he spent a year at ISVR, Southampton
University (UK) as a postdoctoral researcher.
Dr. Ih received domestic and international awards,
including academic awards from the Acoustical So-
ciety of Korea (ASK) and the Korean Society for
Noise and Vibration Engineering (KSNVE). He was
Secretary General of Inter-Noise 2003 in Jeju,
Korea. Currently he is a vice president of ASK and
formerly he served as editor in chief of the Journal ofthe Acoustical Society of Korea. He is a member of
the editorial board of Applied Acoustics.
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