17
Design of a Highly Directional Endfire Loudspeaker 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 length direction of the array, is discussed. Hence this acoustic array aims at the endfire beamforming direction, resulting in a “spotlight” of sound in a preferred direction. Optimized beamforming techniques are employed, which have been developed earlier for the reciprocal problem of directional microphone arrays. Effects of the design parameters of the loudspeaker array system on its performance have been investigated. It is shown that the stability factor of the optimized beamformer can be a useful parameter to control the directional characteristics of the array. In addition the effect of mutual interaction between individual loudspeaker elements in the array system has been considered to reduce the difference between designed and actual performance of 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

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Page 1: Design of a Highly Directional Endfire Loudspeaker Array*

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

Page 2: Design of a Highly Directional Endfire Loudspeaker Array*

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

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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

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Page 4: Design of a Highly Directional Endfire 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

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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

Page 6: Design of a Highly Directional Endfire 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

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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.

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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.

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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.

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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

[1] B. D. van Veen and K. M. Buckley, “Beamforming:

A Versatile Approach to Spatial Filtering,” IEEE Acoust.,Speech, Signal Process. Mag., vol. 5, pp. 4–24 (1988 Apr.).

[2] J. L. Flanagan, D. A. Berkley, G. W. Elko, and

M. M. Sondhi, “Autodirective Microphone Systems,”

Acoustica, vol. 73, pp. 58–71 (1991).

[3] D. B. Ward, R. A. Kennedy, and R. C. Williamson,

“Theory and Design of Broadband Sensor Arrays with

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[4] W. Soede, A. J. Berkhout, and F. A. Bilsen,

“Development of a Directional Hearing Instrument Based

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[5] W. Soede, F. A. Bilsen, and A. J. Berkhout, “As-

sessment of a Directional Microphone Array for Hearing-

Impaired Listeners,” J. Acoust. Soc. Am., vol. 94, pp. 799–808 (1993).

[6] D. E. Weston, “Jacobi Sensor Arrangement for

Maximum Array Directivity,” J. Acoust. Soc. Am., vol.80, pp. 1170–1181 (1986).

[7] H. Cox, R. M. Zeskind, and T. Koou, “Practical

Supergain,” IEEE Trans. Acoust., Speech, Signal Pro-cess., vol. ASSP-34, pp. 393–398 (1986).

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).

[9] J. M. Kates and M. R. Weiss, “A Comparison of

Hearing-Aid Array-Processing Techniques,” J. Acoust.Soc. Am., vol. 99, pp. 3138–3148 (1996).

[10] I. Merks, “Binaural Application of Microphone

Arrays for Improved Speech Intelligibility in a Noisy

Environment,” Ph.D. thesis, Technical University of

Delft, Delft, The Netherlands (2000).

[11] M. Brandstein and D. Ward, Microphone Arrays.(Springer, New York, 2001), chap. 2.

[12] E. N. Gilbert and S. P. Morgan, “Optimum

Design of Directive Antenna Arrays Subject to Ran-

dom Variations,” Bell Sys. Tech. J., vol. 34, pp. 637–663 (1955).

[13] M. M. Boone, “Directivity Measurements on a

Highly Directive Hearing Aid: The Hearing Glasses,” pre-

sented of the 120th Convention of the Audio Engineering

Society, J. Audio Eng. Soc. (Abstracts), vol. 54, p. 728(2006 July/Aug.), convention paper 2869.

[14] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V.

Sanders, Fundamentals of Acoustics (Wiley, New York,

2000), chap. 7.

[15] D. A. Brandwood, “A Complex Gradient Operator

and Its Application in Adaptive Array Theory,” IEE Proc.,pts. F and H, vol. 130, pp. 11–16 (1983).

[16] A. J. Berkhout, “Pushing the Limits of Seismic

Imaging. Part 1: Prestack Migration in Terms of Double

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(1997).

[17] R. L. Pritchard, “Maximum Directivity Index of a

Linear Point Array,” J. Acoust. Soc. Am., vol. 26, pp.1034–1039 (1954).

[18] E. G. Williams, Fourier Acoustics—Sound Radia-tion and Nearfield Acoustical Holography (Academic

Press, London, 1999), chap. 6.

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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