Joeri_Labeeuw_199809

OPTIMIZATION OF

AMPLITUDE DISTRIBUTIONS

IN WAVE FIELD SYNTHESISAPPLIED TO EARLY REFLECTIONS

AND VIRTUAL SOURCES

J.M.J. Labeeuw

M.Sc. Thesis

Supervisor: Prof. dr. ir. A.J. Berkhout

Mentors: dr. ir. D. de Vries

ir. J.J. Sonke

Laboratory of Seismics and Acoustics

Department of Applied Physics

Delft University of Technology

Delft, September 1998

ii

Afstudeercommissie: prof. dr. ir. S.W de Leeuw

Computational Physics

Technische Natuurkunde, TU Delft

dr. ir. D. de Vries

Akoestische en Seismische Technieken


dr. ir. M.M. Boone



dr. ir. J. Raatgever

Akoestische Perceptie


dr. ir. E. Start

Lichtveld Buis & Partners B.V.

ir. J.J. Sonke



iii

iv

Abstract

This thesis describes how the sound �eld of a virtual hall can be synthesized by making use of rows

of loudspeakers; \loudspeaker arrays". The acoustics of a hall is built up by direct sound, early

re ections and reverberation. The sound �eld of the early re ections is assumed to come from virtual

sources according to the mirror image source model. Presuming that the room where the sound �eld

is synthesized has a negligible response, the acoustics of the virtual hall can partly be constructed by

generating the sound �eld of the mirror image sources with the loudspeaker arrays.

Using loudspeaker line arrays, the sound �eld is synthesized in an accurate way in a large receiver area

with respect to its phase. The amplitude of the sound �eld, however, is only synthesized correct on a

curve. The optimal curve will be found, whereby the sound �eld in a speci�c receiver area is synthesized

as well as possible. To do this, an expression will be developed which describes the amplitude behavior

of the sound �eld in a manageable way.

The amplitude behavior of the sound �eld coming from speci�c planar arrays of loudspeakers, whereby

each loudspeaker on a vertical line emits the same signal, is also considered. Since also in this con-

�guration, the amplitude of the sound �eld can only be synthesized in a correct way on a curve, the

optimal curve will be found in the same way as done for the line arrays.

The amplitude deviations of a loudspeaker line array and the speci�c loudspeaker planar array, syn-

thesizing the sound �eld of sources, will be compared. Here, one will see that when the sources to be

synthesized have a larger distance to the array than the receiver, the planar array is best used to obtain

the minimal amplitude deviation. For small source-to-array distances, the line array is preferred.

The incorrect amplitude distribution is evaluated with perceptual experiments. From these experi-

ments, it appears that the amplitude deviations are audible when the receiver is beyond a certain

distance from the curve where the amplitude of the sound �eld is synthesized correctly.

v

vi

Samenvatting

Deze thesis beschrijft hoe met behulp van luidsprekerreeksen het geluidsveld van een virtuele zaal

gesynthetiseerd kan worden. De akoestiek van een zaal bestaat uit direct geluid, vroege re ecties en

nagalm. Er wordt aangenomen dat het geluidsveld van de vroege re ecties afkomstig is van virtuele

bronnen, waarbij de virtuele bronnen verkregen zijn met behulp van het spiegel-bronnen model. Als

verondersteld wordt dat de kamer waarin het geluidsveld gesynthetiseerd wordt een verwaarloosbare

responsie heeft, dan kan de akoestiek van de virtuele zaal gedeeltelijk opgewekt worden door het

geluidsveld van de spiegel-bronnen te synthetiseren met luidsprekerreeksen.

Het geluidsveld kan met behulp van luidsprekerreeksen op een lijn met betrekking tot de fase nauwkeu-

rig gesynthetiseerd worden in een groot ontvangers gebied, daarentegen wordt de amplitude van het

geluidsveld alleen correct gesynthetiseerd op een kromme. Er wordt een optimale kromme gevonden,

waarbij het geluidsveld in een bepaald gebied zo goed mogelijk gesynthetiseerd wordt. Om dit te

kunnen doen zal er eerst een functie afgeleid worden die het amplitude-gedrag van het geluidsveld op

een gemakkelijk hanteerbare wijze beschrijft.

Ook zal het amplitude-gedrag van het geluidsveld dat afkomstig is van speci�eke luidsprekerreeksen

in vlakken bekeken worden. Hierbij zenden alle luidsprekers op een verticale lijn hetzelfde signaal uit.

Aangezien bij deze con�guratie de amplitude ook alleen maar op een kromme goed gesynthetiseerd

kan worden, zal er voor deze situatie ook een optimale kromme gevonden worden.

De amplitude-fouten van een luidsprekerreeks op een lijn en een luidsprekerreeks in een vlak, die het

geluidsveld van bronnen synthetiseren, zullen met elkaar vergeleken worden. Hierbij zal blijken dat

wanneer de te synthetiseren bronnen verder weg van de luidsprekerreeks staan dan de waarnemer, de

luidsprekerreeks in een vlak het best gebruikt kan worden om een minimale amplitude-afwijking te

verkrijgen. Als de bron dichtbij staat, kan de luidsprekerreeks op een lijn het best gebruikt worden.

De verkeerde amplitude-verdeling is ge�evalueerd met perceptie experimenten. Uit deze experimenten

blijkt dat de amplitude-verschillen te horen zijn als de waarnemer zich te ver van de kromme bevindt

waar het geluid goed gesynthetiseerd wordt.

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viii

Contents

Abstract v

Samenvatting vii

1 Introduction 1

1.1 Synthesizing the wave �eld of virtual sources . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Background of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Construction of the wave �eld of a virtual hall 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The components of the wave �eld in a hall . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 The direct sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 The early re ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 The reverberation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Models of wave �eld extrapolation in halls . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The Mirror Image Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Generating the sound �eld of mirror image sources . . . . . . . . . . . . . . . . . . . . 11

3 Synthesizing a wave �elds with loudspeaker arrays 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Basic acoustic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Kirchho�-Helmholtz integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 The 3D Rayleigh-I integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 2 12D Rayleigh-I integral for a line array . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 2 12D Rayleigh-I integral for a distribution of vertical line sources . . . . . . . . . . . . 22

3.7 Point solution versus line solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 The amplitude function for a line array and for a DVLS 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 The amplitude function for a line array . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 The amplitude function for a DVLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Optimizing the amplitude factor 37

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Optimizing the amplitude factor gcm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Optimizing the amplitude factor gcl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Simulations of amplitude distributions 43

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

ix

6.2 Simulations using a line array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.3 Simulations using a DVLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.4 Simulations of a virtual hall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 A line array versus a DVLS 55

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2 The gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.3 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4 Truncating the DVLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Wave �eld synthesis of virtual halls in practice 63

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.2 Hardware con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.3 Software used for wave �eld synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.4 The amplitude factor of the driving signal . . . . . . . . . . . . . . . . . . . . . . . . . 66

9 Perceptual measurements 71

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2 Method of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.1 Moving the optimal position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.2 The two alternative forced choice . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.3 Fitting the psychometric curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.3.1 Presentation of the processed data . . . . . . . . . . . . . . . . . . . . . . . . . 73

9.3.2 The weighting factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9.3.3 The chi-square test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.4 Adjusting the amplitude factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

10 Conclusions and recommendations 81

10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

10.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

A Con�gurations used for simulations and measurements 83

A.1 The con�guration used for simulations without a virtual hall . . . . . . . . . . . . . . 83

A.2 The room used for simulations and measurements . . . . . . . . . . . . . . . . . . . . . 84

A.3 The con�guration of the virtual hall used for simulations . . . . . . . . . . . . . . . . . 84

A.4 The con�guration used for perceptual measurements . . . . . . . . . . . . . . . . . . . 85

B Numerical results of perceptual measurements 87

Bibliography 90

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Chapter 1

Introduction

1.1 Synthesizing the wave �eld of virtual sources

Today, when one wants to see a movie, a play or listen to music, he prefers to go to a cinema, a theater

or a concert hall respectively. A reason one goes out for this, while it is easier to sit back at home

looking at television or listening to the stereo system, is that one wants to be merged in the total. An

important aspect for this is that the acoustics is much better in real situation than at home, but an

attentive listener notices a few shortcomings.

In the cinema, one cannot exactly hear from which direction the sound is coming and how far away the

sources are located. In a theater, the sound is often ampli�ed. When the sound enhancement system

of today is used, it can happen that one sees the actor in the middle of the stage while one hears him

from a direction of one of the loudspeakers at the side of the stage. Another example is related to

the acoustics of concert halls. Many halls that do not ful�ll the requirement of a good concert hall,

mainly concerning the early re ections and the reverberation time. Finally, multipurpose rooms or

halls are needed, where one day one can listen to a concert while another day one can listen to a

speech. For each situation, di�erent acoustic properties of the room or the hall are needed. Take for

example reverberation. While listening to classical music, one needs a much longer reverberation than

when listening to a speaker who is hardly intelligible when the reverberation is too long.

All these shortcomings can be solved with the aid of loudspeaker arrays. With loudspeaker line arrays,

which are distributions of loudspeakers on lines, or loudspeaker plane arrays, which are distributions

of loudspeakers in planes, a sound or wave �eld is synthesized very well. By using such arrays to

synthesize the wave �eld of sources, a listener is able hear exactly from which directions the sound of

the sources is coming and is even able to estimate their relative distance.

In this thesis, the way to construct the wave �eld of a virtual hall by synthesizing the wave �eld of

early re ections is described. The early re ections are that part of the wave �eld that is re ected some

times against the walls and other objects and will be treated as the wave �eld coming from mirror

image sources. This �eld is supposed to be synthesized by line arrays or by distributions of vertical

line sources (speci�c plane arrays). The line arrays are driven by a signal derived from the Rayleigh

2 12D integral. The distributions of vertical line sources are driven by a signal derived from an adapted

version of the same integral. The wave �eld synthesized by a line array and a distribution of vertical

line sources (DVLS) will be compared to decide which array is best used to synthesize the wave �eld

of a virtual hall. The room in which the wave �eld is synthesized is assumed to have a negligible

1

2 Chapter 1. Introduction

response. See [Larsen, 1998] how to include the acoustics of the room.

Especially, attention is paid to the amplitude of the synthesized wave �eld and its error with respect

to the amplitude of a real source's wave �eld because line arrays and distributions of vertical line

sources are able to synthesize the wave �eld correctly concerning its phase, but not concerning its

amplitude. No substantial problem occurs due to this error if the sources to be synthesized lie in

a relative small source area, which is the case by direct sound enhancement. If the sources may lie

everywhere in space, this error could result in a perceptual problem such as an incorrect relative

distance estimation, because the error leads to a supplementary amplitude di�erence between the

sources. Therefore, a model shall be developed to describe the amplitude of the wave �eld of the

sources, synthesized by di�erent types of arrays. With this model, the driving signal for the arrays

will be optimized such that the amplitude error is minimal in a speci�ed receiver area.

1.2 Outline of this thesis

In chapter 2, some models that describe the wave �eld in an enclosed space are treated and a short

introduction is given about synthesizing the wave �eld of virtual sources with loudspeaker arrays.

Thereafter, chapter 3 deals with the theory of how to synthesize wave �elds with arrays, using the

Raleigh operators to obtain their driving signals. From the Raleigh 3D integral, the Rayleigh 2 12D

integral for a line array and for a DVLS will be derived. In this chapter, it is also shown that amplitude

deviations occur when using an array for synthesizing a wave �eld. In chapter 4, an amplitude model

will be developed which describes the amplitude distribution of the synthesized wave �eld by a line

array or a DVLS. In chapter 5, the previous named model will be used to calculate the driving signals

of the arrays such that the amplitude deviation is minimal in a speci�c receiver area. Results of

some amplitude simulations are found in chapter 6. In chapter 7, the amplitude errors of the wave

�eld synthesized by the di�erent kinds of arrays are compared, to examine which array results to

the smallest amplitude deviation of the synthesized wave �eld, given a source con�guration and a

receiver area. Chapter 8 gives a global description of the hardware and software setup that is used to

synthesize the wave �eld of a virtual hall in practice. Perceptual evaluations of amplitude deviations

are described in chapter 9. Chapter 10 contains conclusions of this research and suggestions for future

research. At the end of this thesis, two appendices are found. Appendix A contains a description of the

virtual hall, the loudspeaker arrays used for the measurements and simulations and a description of

the reproduction room in which measurements were done. Appendix B contains tables with numerical

results of perceptual measurements.

1.3 Background of this thesis

In 1993, [Vogel, 1993] described a way of applying direct sound enhancement with a loudspeaker line

array in a dissertation. The concept was invented by [Berkhout, 1988]. The impractical loudspeaker

plane that is theoretically described by the Rayleigh 3D operator was replaced by a line array driven

with the Rayleigh 2 12D operator. Listening tests in an anechoic room showed that this system synthe-

sizes the wave �eld of virtual sources with a well de�ned source localization in a horizontal listeners

plane. To record the direct sound, the source area was divided into sub-areas. Each sub-area was

covered by a microphone with a special directivity property.

Hereafter, [Start, 1997] continued the project. The e�ects on the wave �eld due to discretization and

truncation of the line array were examined. Truncation e�ects, due to the �nite length of a line array,

could be reduced by tapering the far ends of the line array. To reduce spatial aliasing, the discrete

1.3. Background of this thesis 3

wave �eld synthesis process is divided into four steps:

1. Spatial anti-aliasing �ltering

2. Spatial sampling

3. Spatial synthesis

4. Spatial reconstruction �ltering

Psycho-acoustical measurements lead to the conclusion that if the sound �eld is synthesized correctly

up to a frequency of at least 1.5 kHz, the localization accuracy equals the accuracy in real sound �elds.

Due to spatial aliasing, the spaciousness of the synthesized sound �eld is larger than for real sound

�elds. Distortion of the temporal spectrum due to spatial aliasing results in a source and receiver

position-dependent coloration of synthesized signals. Coloration e�ects due to di�raction at the end

of the line array are negligible for practically realizable loudspeaker arrays.

4 Chapter 1. Introduction

Chapter 2

Construction of the wave �eld of a

virtual hall

2.1 Introduction

To synthesize the wave �eld of a virtual hall[1], it is necessary to know how the wave �eld of a hall is

built up. Then, models can be made which are used to simulate each component of the wave �eld.

2.2 The components of the wave �eld in a hall

The wave �eld in a hall can be divided in three components namely:

� (Pseudo) direct sound

� Early re ections

� Reverberation

A schematic representation of an impulse response of a hall is shown in �gure 2.1. In this �gure, the

three components are visualized. The next subsections describe each of these three components of the

wave �eld.

2.2.1 The direct sound

After a sound source has emitted a signal, the �rst sound arriving at the receiver is called the direct

sound. The direct sound is nearly identical to the emitted signal. Only the amplitude of the sound

�eld has decreased by propagating away from the source. The human ear integrates the direct sound

and its re ections over 20 ms, i.e., one only hears one signal while the information is spread over a

longer time interval. Therefore, these very early re ections are often called the pseudo direct sound.

In a hall, damping and dispersion due to inhomogeneities of the medium and due to temperature

gradients may be neglected.

[1]In this thesis the word \hall" is used for an arbitrary enclosed space.

5

6 Chapter 2. Construction of the wave �eld of a virtual hall

(Pseudo) direct sound

Statistic

Early reflections

Reverberation

Deterministic

Am

plitu

de

time (s)

Figure 2.1: A schematic representation of an impulse response for a �ctitious enclosure.

2.2.2 The early re ections

Re ections arriving between 20 and 100 ms are called the early re ections. The density is so low that

there is no signi�cant overlap and they may be treated deterministically. Models to describe the early

re ections will be discussed in a later part of this chapter.

When a sound �eld meets a boundary, a part is absorbed by the wall. The absorbed sound is transferred

into energy heating up the wall or propagates through the wall. This way the energy of the sound �eld

decreases. This decrease is not the same for each frequency and strongly depends on the properties

of the walls, so that the models that describes the sound �eld of the early re ections become more

complicated. A second complication occurs because a part of the re ected wave is scattered in a

non-specular way as is seen in �gure 2.2.

incidentwavefield

specularreflection

non-specularreflections

Figure 2.2: A part of the wave �eld is scattered in a non-specular way.

Re ections can be calculated using Snell's Law, which says that the angle of the incident wave equals

the angle of the re ected wave. By taking non-specular scattering into account, the sound �eld cannot

be calculated using Snell's Law only. Hence, one needs an extension of this model or has to use other

2.3. Models of wave �eld extrapolation in halls 7

models. Finally, di�raction e�ects due to di�ractors appear. These di�ractors are discontinuities, such

as objects and corners, in the hall or on the walls. When a sound �eld meets such a discontinuity, the

discontinuity acts as a sound source with a speci�c direction characteristic. However, by neglecting

the contributions such as frequency dependent re ections, non-specular scattering and di�raction, the

sound �eld of the early re ections are described with not too complicated models as like the sound

�eld of the direct sound. Therefore, the sound �eld is said to be deterministic.

The presence of early re ections is important for some impressions. According to [Barron and Mar-

shall., 1981], the early lateral re ections are the primary cause for the feeling of spaciousness, i.e., the

early re ections coming from the sides of the hall give information about the size of the hall. Some

acoustic parameters that describe these impressions are the Lateral e�ciency and the Lateral energy

fraction. Together with the direct sound, the early re ections have an important contribution to the

speech intelligibility. The larger the energy of the direct sound and the early re ections compared

with the energy of the reverberation, the better intelligible a voice is. Some parameters describing this

\early-to-late energy ratio" are \De�nition", \Clarity" or \Center Time" (see [Boone et al., 1994]).

2.2.3 The reverberation

Finally, the wave �eld that has re ected many times against the walls arrives to the receiver. Due

to the many re ections the sound �eld has undergone, where for each frequency depended re ection

di�raction and scattering occurs, the sound �eld cannot be speci�ed by a deterministic model. Some

statistic models have been derived from the assumed property that the reverberant sound �eld is

di�usive. The way to obtain information about the reverberant wave �eld will not be treated in this

thesis. For some methods, one can read [Boone et al., 1994].

The reverberation contributes for instance to \fullness of tone". Reverberation in low frequency bands

contributes to perception of \warmth" and reverberation in high frequency bands to the perception

of `brilliance".

2.3 Models of wave �eld extrapolation in halls

In the past, some methods were developed to calculate the wave �eld in a hall as a function of time.

Until now, there is no model that describes the wave �eld perfectly. Each model has its advantages

and disadvantages taking into account di�erent parts of in uences on the wave �eld as described in

the previous section, or by di�erences in simplicity of implementation. A few models will be mentioned

here: the Mirror Image Source Model (MISM), the Ray Tracing Model (RTM) and the WRW model.

The MISM is discussed in the next section.

The RTM divides the source �eld into a distribution of rays. These rays can be interpreted as uncor-

related sound beams with in�nitely high directivity and plane wave characteristics. Absorption and

scattering can be included easily in this model. More about this Ray Tracing Model is found in [Boone

et al., 1994].

The WRW model describes the wave �eld with the aid of matrices. For the description of propaga-

tion of the wave �eld from boundary to boundary, the W-matrix is used. For the description of the

re ections, the R-matrix is used. By combining these matrices, the wave �eld can be predicted. The

WRW model is able to include di�raction and scattering. A wide treatment of this model is found in

[van den Oetelaar, 1997].


2.4 The Mirror Image Source Model

In this thesis, the 2D Mirror Image Source Model is used to calculate and synthesize the wave �eld

of a hall in two dimensions. Because the wave �eld will be synthesized with loudspeaker arrays so

that many real-time signals have to be processed, complicated models cannot easy be used because

o� lack of computational capacity. The 2D Mirror Image Source Model describes the wave �eld in a

virtual hall in two dimensions as if it was coming from multiple virtual sources. The driving signals

for loudspeaker arrays that have to synthesize the wave �eld of virtual sources, are calculated easily

and fast. This section contains only a brief explanation of the Mirror Image Source Model. Interested

readers are referred to [Kuttru�, 1979].

First, consider one source and one wall. The source is emitting a sound signal in all directions. When

the wave �eld of the signal arrives at the wall, it is re ected with the same angle as the angle of

incidence as described in Snell's Law. If non-specular re ections are supposed to be negligible, the

re ected wave �eld may be expected to come from a mirror image source which position is obtained

by mirroring the primary source in the wall, as seen in �gure 2.3. This operation can be done for

enclosures with more than one boundary. This way, one obtains the �rst order mirror image sources.

Figure 2.3: The re ected sound �eld can be seen as coming from a mirror image source.

Hereafter, the hall can also be mirrored in each wall so that one obtains mirror image halls, as shown in

�gure 2.4. Each mirror image source can again be mirrored, but now in each wall of its corresponding

mirror image hall, to obtain the second order mirror image sources. This can be repeated until enough

mirror sources are available. This way, one obtains a distribution of many mirror image sources due to

the re ections of the wave �eld against the walls. To calculate the source strength of a mirror image

source, the source signal has to be multiplied with the re ection coe�cients of the walls in which the

mirror image source is re ected. The re ection coe�cients are a function of the absorption coe�cients.

As read in [Borish, 1984], not all mirror image sources are audible at every position in the hall, but

only those which are \seen" through all the walls in which they are re ected. Consider �gure 2.5. A

straight line is drawn between a source and the receiver. Thereafter, the line can be projected back

through the walls in which the source was re ected. The line that is obtained represents the path the

sound �eld has traveled. A continuous path is obtained if the straight line intersects all the walls in

2.4. The Mirror Image Source Model 9

hallprimary source

1st order MI source

1st order MI hall

Figure 2.4: The mirror image sources are obtained by mirroring recursively the hall and the sourcein each wall.

which the source is re ected and the source is said to be visible. For an invisible source, the path

shows discontinuities such that the sound never arrives at the receiver. All the mirror image sources

have to undergo separately the \visibility" test. Figure 2.6 shows for a given hall and a given source

position the mirror sources up to the 8th order. The mirror image sources that are visible in the center

are encircled.

Invisible

*Primarysource

Receiver

Ψ

Soundpath ofΨ

Figure 2.5: Only the \visible" mirror image sources are audible at the receiver position. A mirrorimage source is visible if the line that connects the mirror image source with thereceiver intersects all walls in which the source is mirrored.


Figure 2.6: Mirror image sources calculated until the 8th order, shown as dots. The mirror imagesources that are visible in the center of the hall (*) are encircled (167 of 696 mirrorimages sources are visible).

2.5. Generating the sound �eld of mirror image sources 11

2.5 Generating the sound �eld of mirror image sources

In the previous section, it is seen how to locate the visible mirror image sources with the Mirror Image

Source Model. To synthesize the sound �eld of these mirror image sources one can place transducers

in the open �eld at every position where a source has to be located, which is of course an impractical

solution. A more realistic solution is to make use of loudspeaker arrays to synthesize the sound �eld

of the virtual sources.

By placing one array in a hall, one is able to synthesize the sound �eld of virtual sources placed behind

the array, or even lying in front of the array This con�guration can also be used to amplify the sound

�eld of a real source, then the virtual source coincides with the real source, as seen in �gure 2.7. The

synthesized sound �eld forms an ampli�ed replica of the primary �eld [Start, 1997].

Stage

Figure 2.7: An array can synthesize and amplify the sound �eld of a source.

It is easy to expand this system such that the sound �eld of multiple sources everywhere space is

synthesized by placing loudspeaker arrays all around the hall as shown in �gure 2.8.

Stage*

Figure 2.8: Arrays can be used to synthesize the sound �eld of multiple virtual sources from dif-ferent directions such as mirror image sources.


In this way, a "virtual reality" is created where the sound �eld of the virtual sources is perceived as

coming from positions lying inside or even outside the enclosure. Synthesizing acoustics of a hall is

done by synthesizing the sound �eld of the mirror image sources and reverberation with the arrays.

Hereby, the real source may supplementary be ampli�ed as described above. The next chapter, a

description is given how to use an array to synthesize the sound �eld of a source.

Chapter 3

Synthesizing a wave �elds with

loudspeaker arrays

3.1 Introduction

This chapter gives an introduction to the underlying basics of wave �eld synthesis in homogeneous

media and shows how a given wave �eld in a closed volume is synthesized by a distribution of secondary

monopole and/or dipole sources. First, the basics of wave �elds are treated. Thereafter the Kirchho�-

Helmholtz operator will be derived. The special case of synthesizing the wave �eld with a planar

array or a line array will be described when the Rayleigh integrals are treated. Interested readers

are referred[Berkhout, 1987]. Finally, the last section of this chapter is dedicated to the amplitude

deviation that occurs when a sound �eld is synthesized by means of the Rayleigh 2 12D operator.

A remark has to be made about the notation of the vectors in this thesis. The vector ~r is the position

in Cartesian coordinates (~r = (x; y; z)T ). The vector from a primary source[1] to a receiver R is

denoted by ~̂r. The vector from a primary source to a secondary monopole/dipole distribution will be

written as ~r0 and the vector from a secondary monopole/dipole distribution to the receiver is indicated

by ~�r. When a subscript is used, the subscript stands for the object where the vector is pointing to,

for example, ~rS is a vector to the surface S and ~rR is a vector to the receiver R. Finally, when the

length of the vector is meant, the arrow above the variable and its \bold face" are removed, i.e.,�� ~�r�� = �r.

3.2 Basic acoustic equations

In this section, the basic acoustic equations are summarized to use them for derivations in the suc-

ceeding sections. Consider a homogeneous, isotropic and sourceless medium with negligible viscosity

losses, thermal conductivity and molecular relaxation. With the aid of the equation of motion, New-

ton's second Law, the �rst basic equation is derived which gives the relation between the temporal

[1]In this section, a \primary source" stands for an arbitrary source to be synthesized.

13

14 Chapter 3. Synthesizing a wave �elds with loudspeaker arrays

derivative of the particle velocity v(~r; t) and the gradient of the pressure p(~r; t):

�0@

@tv(~r; t) +rp(~r; t) = 0; (3.1)

where �0 is the density of the medium and t indicates the time. Another basic equation is found by

using the law of conservation of mass that quanti�es the relation between the divergence of the particle

velocity and the temporal derivative of the pressure:

1

c2@

@tp(~r; t) + �0r � v(~r; t) = 0: (3.2)

Here, c represents the propagation velocity of acoustical compressional waves, the sound speed. Com-

bining equation (3.1) and equation (3.2) results in the wave equation of compressional acoustic waves:

r2p(~r; t)� 1

c2@2

@t2p(~r; t) = 0: (3.3)

This equation is only valid if there are no sources in the medium. If there are sources in the medium,

equation (3.3) changes to

r2p(~r; t)� 1

c2@2

@t2p(~r; t) = r � f(~r; t)� �0

@

@tq(~r; t): (3.4)

Here, f represents an external force per unit volume and q represents the source strength density.

In the previous equations, the acoustic pressure and velocity are represented in the space-time domain,

i.e., p(~r; t) and v(~r; t). They can also be represented in the space-frequency domain by using the Fourier

transform. Then, the acoustic wave equation (3.3) is written as

r2P (~r; !)� k2P (~r; !) = 0: (3.5)

This is the so-called homogeneous Helmholtz equation, where ! represents the angular frequency and

k the wave number:

k =!

c: (3.6)

Note that in this thesis, any variable as a function of space and time is represented with a small char-

acter. The Fourier transform of this variable in the space-frequency domain is indicated by the capital

of the same character. Henceforward, the space-frequency domain will be used. After transforming

equation (3.4) to the space-frequency domain one obtains:

r2P (~r; !) + k2P (~r; !) = r � F(~r; t)� j!�0Q(~r; t): (3.7)

This is the so-called inhomogeneous Helmholtz equation. To solve equation (3.7), some extra infor-

mation is necessary such as boundary and initial conditions. For an in�nite homogeneous medium,

Sommerfeld's radiation condition holds, which requires that only outgoing waves are allowed. The

outward propagation is characterized by the two criteria:

limr!1

�r

�@

@rP (~r; !) + jkP (~r; !)

��= 0;

P (~r; !) = O

�1

r

�:

(3.8)

3.3. Kirchho�-Helmholtz integrals 15

One wave �eld that satis�es these conditions is the wave �eld of a monopole. The source strength Q

of the monopole placed at position ~r0 is given by

Qm(~r; !) = U(!)�(~r �~r0); (3.9)

where U(!) represents the volume velocity. With equation (3.9) the inhomogeneous Helmholtz equa-

tion (3.7) changes to

r2Pm(~r; !) + k2Pm(~r; !) = 4�S(!)�(~r�~r0): (3.10)

Note that there are no external forces F in this case. The solution of equation (3.10) is:

Pm(~r�~r0; !) = S(!)e�jkj~r�~r0j

j~r�~r0j ; (3.11)

giving the pressure �eld of a monopole. Here the source spectrum S(!) is given by

S(!) = j!�0U(!)

4�: (3.12)

The particle velocity of a monopole located at the origin is will be given here:

Vm(~r) =S(!)

�0c

(1 + jkr)

jkr

e�jkr

r

~r

r: (3.13)

3.3 Kirchho�-Helmholtz integrals

In this section, an expression will be derived for a wave �eld inside a volume V caused by a secondary

source distribution on the surface S of that volume. The secondary source distribution synthesizes

the wave �eld of a source outside the volume. The expression is known as the Kirchho�-Helmholtz

integral.

Consider a source which has a spectrum S(!) and lies outside a volume V. The volume is enclosed

by a surface S as shown in �gure 3.1. The wave �eld inside the volume is given by the homogeneous

Helmholtz equation (3.5). Outside the volume, the wave �eld obeys the inhomogeneous Helmholtz

equation (3.7). Next, locate a monopole source R with spectrum S(!) = 1 inside the volume V at ~rR,

generating a pressure �eld:

G(~r �~rR; !) = e�jkj~r�~rRj

j~r�~rRj : (3.14)

This function is a Green's function and is a solution of the inhomogeneous Helmholtz equation (3.7):

r2G(~r�~rR; !) + k2G(~r �~rR; !) = �4��(~r�~rR): (3.15)

Henceforth, the indication of the angular frequency ! is omitted for notational convenience. Multipli-

cation of equation (3.5) by G(~r � ~rR) and equation (3.15) by P (~r) and subtracting the results, one

obtains after integration over volume V:

P (~rR) = � 1

4�

ZV

�P (~r)r2G(~r �~rR) � G(~r �~rR)r2P (~r)

�dV: (3.16)


R

~n

V

�

S

~r0

~�r

~̂r

Figure 3.1: A surface S that encloses a volume V and contains an in�nite distribution of monopoleand dipole sources to synthesize the wave �eld of source

It should be noted that this integral is only de�ned in volume V. Outside the volume the integral

equals zero, on the surface S the integral equals 12P (~rR).

A property of the Green's function (3.14) is that it is a reciprocal function: it remains unchanged if

the positions of the source and the receiver are reversed (G(~r � ~rR) = G(~rR � ~r)). Hence, the unit

monopole source may be placed on the surface whereas the receiver position becomes ~rR. Applying

Green's second theorem:

ZV

�Pr2G�Gr2P

�dV = �

IS

[PrG�GrP ] � ~n dS (3.17)

(with ~n the inward pointing vector perpendicular on the surface S) and the reciprocal property of

Green's function on equation (3.16), one obtains

P (~rR) =1

4�

IS

hP (~rS)rG( ~�r)�G( ~�r)rP (~rS)

i� ~n dS: (3.18)

This is the Kirchho�-Helmholtz integral where ~�r stands for the vector from the surface to the receiver

R, i.e., ~�r = ~rR � ~rS. Finally, substituting the Green's function (3.14) in the Kirchho�-Helmholtz

integral and using equation (3.2) , one obtains:

P (~rR) =1

4�

IS

j!�0V~n(~rS)e�jk�r

�rdS +

1

4�

IS

P (~rS)1 + jk�r

�rcos�

e�jk�r

�rdS:

(3.19)

This is called the Kirchho� representation integral for homogeneous media. The angle � is the angle

between ~�r and ~n. The factor V~n is the particle velocity of the incident wave �eld, perpendicular to

the surface S.

The Kirchho� representation integral can be interpreted as follows: The wave �eld of a source outside

a volume V is synthesized within that volume by an in�nite monopole (�rst term of equation (3.19))

3.4. The 3D Rayleigh-I integral 17

and dipole (second term of equation (3.19)) distribution lying on the reproduction surface S of volume

V. The source strength of each monopole is proportional to the normal component of the local particle

velocity V~n(~rS) of the incident wave �eld. The source strength of each dipole is proportional to the

local pressure P (~rS) of the incident wave �eld.

It is possible to inverse the con�guration, i.e., to place the source in volume V and the receiver R

outside the volume. Equation (3.19) does not change in this case.

3.4 The 3D Rayleigh-I integral

In the previous section the Kirchho�-Helmholtz integral was introduced which describes how a wave

�eld is synthesized by a distribution of monopoles and dipoles. The Rayleigh integral is a special case

of the Kirchho�-Helmholtz integral.

Consider the con�guration shown in �gure 3.2. A wave �eld P (~r) is generated by a source in the

half space z < zS0 . The volume V is enclosed by the plane S0 at z = zS0 and the hemisphere S1with radius r, which contain a distribution of monopoles and dipoles. By letting the radius tend to

in�nity (r ! 1) and applying the Sommerfeld's radiation condition (equation (3.8)) to boundary

Figure 3.2: An outline of the volume V enclosed by the surfaces S0 at z = zS0 and S1 with radiusr that contains an in�nite monopole/dipole distribution. Outside the volume, a source is placed.


S1, the Kirchho�-Helmholtz integral over S1 results into a zero contribution to the pressure at R (see

[Wapenaar and Berkhout, 1989]).

In order to simplify the Kirchho�-Helmholtz integral,

G�(~r) =e�jk�r

�r+ �(~r); (3.20)

is token as Green's function, which obeys the inhomogeneous Helmholtz equation (3.15) and the

condition:

rG�(~r) =@

@~n

�e�jk�r

�r+ �(~r)

�= 0 (3.21)

for every point on the reproduction surface S0. This condition is chosen such that the second term

(the dipole term) of the Kirchho� representation integral (3.19) vanishes. Hence, only monopoles have

to be used to synthesize the wave �eld. To ful�ll the condition (3.21), �(~r) obeys:

�(~r) =e�jk�r

0

�r0; (3.22)

in which �r0

=��~r 0

R �~rS0�� and ~r 0

R(x; y; z) = ~rR(x; y;�z), or the receiver mirrored in the plane S0.

This means that the wave �eld gets an extra contribution of a monopole source located at the receiver

position mirrored in the surface S0, being in phase with G( ~�r). Now, the Kirchho�-Helmholtz integral

equation (3.18) can be written as

P (~rR) =1

2�

ZS0

j!�0V~n(~rS0)e�jk�r

�rdS0; for ~rR 2 V; (3.23)

which is called the Rayleigh-I representation integral. It states that any wave �eld generated by the

sources in the half space z < zS0 can be constructed in the half space V (z > zS0) by means of a

continuous distribution of monopoles at the reproduction surface S0. Note that such a distribution of

monopoles will radiate a \mirror" wave �eld into the half space z < zS0 , since there are no dipoles to

cancel this wave �eld.

Rewriting the Rayleigh-I integral (3.23) to

P (~rR) =

ZS0

Q(~rS0)W (�r) dS0; (3.24)

the integral is divided in two factors. The �rst factor Q(~rS0) is the driving signal for each secondary

monopole on the reconstruction plane S0. It is proportional to the normal component of the particle

velocity V~n(~rS0) of the incident wave �eld, measured at the position of each monopole:

Q(~rS0) =1

2�j!�0V~n(~rS0): (3.25)

The second factor, W (�r), describes the propagation of the wave �eld generated by the secondary

monopole distribution on the reconstruction plane S0:

W (�r) =e�jk�r

�r: (3.26)

3.5. 2 12D Rayleigh-I integral for a line array 19

3.5 212D Rayleigh-I integral for a line array

In the previous section, the Rayleigh-I integral was presented, which describes the wave �eld synthesis

for a source by a planar distribution of monopoles. When someone wants to synthesize a wave �eld

in practice with a plane consisting of a secondary monopole source distribution, the monopoles, which

use the driving signals given in equation (3.25), have to be realized by loudspeakers. Theoretically,

an in�nite planar array of loudspeakers should be needed. Di�raction e�ects occur since the array

has to be truncated to �nite dimensions. In addition, the distance between the loudspeakers must

be small enough to prevent spatial aliasing, which in practice is always impossible above a certain

frequency (see [Start, 1997]). Supposing that di�raction and aliasing are to overcome, there still exist

some problems. Apart from the fact that, in some applications, a planar array may unacceptably a�ect

the visibility of actors, singers and scenery pieces, the major problem that occurs is the huge demand

on hardware and computational power since the driving signals has to be calculated and ampli�ed for

each loudspeaker separately. These problems can be solved using line arrays. In this section, the 2 12D

Rayleigh operator for a line array is derived, which describes 3D wave �eld synthesis with a line array

of secondary monopole sources.

To start, consider �gure 3.3. An expression will be found for the contribution to the wave �eld in

~r0~r0~r0~r0~r0~r0

R

z

y

x

~�r

~n

~̂r

S

C

~r00

~�r0

L

Figure 3.3: A diagram of a primary source in the plane y = 0 and an in�nite secondarymonopole distribution on the reconstruction plane S at z = zS. The contribution ofthe wave �eld in R generated by the sources on column C can be approximated by acontribution of the point at ~r00 only.

receiver point R, due to the source distribution on column C of the reconstruction plane S. All wave

�eld contributions due to the sources on column C are assigned to one point, namely ~r00. This point

is the intersection of the surface y = 0 and column C. Hence, the Rayleigh-I integral (3.23) is written

as an integral over all column contributions:

P (~rR) =1

2�

1Z�1

PC(~rR) dxL: (3.27)


PC(~rR) is the pressure at the receiver position due to the sources on a column C. L stands for the

line where the columns intersect the plane y = 0. If the sound �eld of a monopole source is supposed

to be synthesized, the velocity of the wave �eld due to a monopole, equation (3.13), is used for the

velocity appearing in Rayleigh-I integral (3.23). Then, the contribution of a column C to the pressure

at the receiver R is expressed as

PC(~rR) =

1Z�1

S(!)

�1 + jkr0

r0

�cos�inc

e�jkr0

r0e�jk�r

�rdyC: (3.28)

In this integral, r0 is the distance from the source to a point on the column: j~rC �~rj, �r the distancefrom that point on the column to the receiver R: j~rR �~rCj and �inc the angle between ~r0 and the

normal vector on the reconstruction plane S.

The points around the plane y = 0 of the column C (indicated width ~r00) give the major contribution

to the wave �eld at receiver point R. Then, according to [Bleistein, 1984], the contribution to the wave

�eld in the receiver point generated by the column monopoles can be given by the \stationary phase

approximation". It states that the contribution of all monopoles on C to the wave �eld at R can be

approximated by one contribution of a monopole at ~r00 only. In section 3.7, the concept \stationary

phase" is explained in more detail. See [Start, 1997] for a complete derivation of the stationary phase

approximation in this speci�c case. The result of this approximation on equation (3.28) is stated here:

PC(~rR) = S(!)p2�jk

s�r0

r0

0 +�r0cos�inc

e�jkr0

0pr

0

0

e�jk�r0

�r0; for kr

0

0 � 1: (3.29)

Since equation (3.29) is a far �eld or high frequency approximation, the last condition has to be

satis�ed. Henceforth, the indication \0" of r0

0 and �r0 are omitted since always the plane y = 0 is

considered.

Combining equations (3.27) and (3.29) results into the equation that describes the 3D wave �eld

synthesis of a primary monopole by a distribution of secondary monopole sources on a line, as seen

in �gure 3.4:

PL(~rR) =

1Z�1

gm(r0;�r) S(!)

rjk

2�cos�inc

e�jkr0

pr0

e�jk�r

�rdxL; for kr0 � 1: (3.30)

Equation (3.30) is called: the Rayleigh 2 12D operator[2]. To distinguish this operator and an operator

derived in the next section, the operator will be called: the Rayleigh 2 12D operator for a line array.

The subscript L means that the pressure is synthesized by secondary monopole source distribution on

a line L. In equation (3.30), the amplitude factor gm(r0;�r) for each secondary monopole m on the

line L appears. This factor is given by

gm(r0;�r) =

r�r

r0 +�r: (3.31)

It corrects the amplitude of each secondary monopole source such that the wave �eld has the right

amplitude at position ~rR.

[2]Equation equation (3.30) is the general Rayleigh 2 12D operator, as stated by [Vogel, 1993]. After applying the

stationary phase approximation described in section 3.7, and choosing a straight line parallel on the line array asreceiver curve, the speci�c case for the Rayleigh 2 1

2D operator as described in [Start, 1997] and [Verheijen, 1997] is

obtained.

3.5. 2 12D Rayleigh-I integral for a line array 21

~r0

R

z

y

x ~�r~n

~̂rL

Figure 3.4: A diagram of the Rayleigh 2 12D setup for a line array.

The far �eld approximation of the normal component of the particle velocity V line~n on the line L, due

to a line source at ~r, is approximated by ([Berkhout, 1987]):

V line~n (~r0) = S(!)1

�0c

r2�

jkcos�inc

e�jkr0

pr0

; for r0 � 1: (3.32)

Substituting equation (3.32) in the 2 12D Rayleigh integral for a line array (3.30), one obtains:

PL(~rR) =1

2�

1Z�1

gm(r0;�r) j!�0V

line~n (~r0)

e�jk�r

�rdxL; for kr0 � 1: (3.33)

This equation describes how a wave �eld can be synthesized, by an in�nite secondary monopole

distribution on a line, driven by a signal due to a line source at ~r, as shown in �gure 3.4. Hence,

equation (3.33) can be split up in driving signals and propagation factors, as done in equation (3.24):

PL(~rR) =

1Z�1

Qm(~r0)Wm(�r) dxL; for kr0 � 1; (3.34)

and it is seen that the driving signals of the secondary monopole sources Qm are the product of the

amplitude factors gm, the normal components of the particle velocity at ~r0 due to a line source located

at ~r and an imaginary frequency dependent factor:

Qm(~r0) = gm(r0;�r)S(!)

rjk

2�cos�inc

e�jkr0

pr0

: (3.35)

It has to be noticed that the driving signals can be adapted for one receiver position only. This receiver

position is fully determined by the amplitude factors gm(r0;�r). The propagation factorWm describes

the wave �eld synthesized by the secondary monopole sources.

Wm =e�jk�r

�r: (3.36)


3.6 212D Rayleigh-I integral for a distribution of vertical line

sources

Another way to synthesize a wave �eld is to make use of a secondary source distribution where all

the monopole sources on a vertical line have the same phase and amplitude. The reason why such a

horizontal array of vertical line sources is introduced will become clear in the next chapters. In this

thesis, this speci�c array will be indicated as a Distribution of Vertical Line Sources, abbreviated as

\DVLS". The con�guration is shown in �gure 3.5. Note that all positions in �gure 3.5 lie in the plane

y = 0.

~r0

z

y

x ~�r~n

~̂r

L

R

~�r~n~�r~n

S

Figure 3.5: A diagram drawing of the DVLS. The secondary source distribution consists of verticallines.

Equation (3.30) describes how a wave �eld is synthesized by a secondary monopole distribution on a

line, with the driving signals derived from a line source at ~r. Here, the same equation is interpreted

in another way to derive the Rayleigh 2 12D operator for a DVLS :

The far �eld approximation of the normal component of the particle velocity V point~n due to a monopole

point source at the source position ~r is approximated by ([Berkhout, 1987])

V point~n (~r0) =1

�0cS(!) cos�inc

e�jkr0

r0; for kr0 � 1: (3.37)

3.7. Point solution versus line solution 23

The pressure �eld of a vertical line source with a source spectrum S(!) = 1 is given by

Wl(�r) =

r2�

jk

e�jk�rp�r

: (3.38)

Hence, using equation (3.37) and equation (3.38), equation (3.30) is expressed as:

PS(~rR) =1

2�

1Z�1

gl(r0;�r) j!�0V

point~n (~r0) Wl(�r) dxL; for kr0 � 1: (3.39)

This is the Rayleigh 2 12D operator for a DVLS. The subscript S means that the pressure �eld is

synthesized by a distribution of vertical line sources placed on the reproduction surface S, intersecting

the plane y = 0 on the line L parallel to the x-axis. The amplitude factor gl for each vertical line

source on the line L is given by

gl(r0;�r) =

rr0

r0 +�r: (3.40)

Reformulating equation (3.39) in the same way as equation (3.24):

PS(~rR) =

1Z�1

Ql(~r0)Wl(�r) dxL: for kr0 � 1; (3.41)

one sees that the driving signals Ql for each vertical line source are the product of the amplitude

factor gl, the normal component of the particle velocity at ~r0 due to a point source located at ~r and

an imaginary frequency dependent constant:

Ql(~r0; !) = gl(r0;�r)S(!)

jk

2�cos�inc

e�jkr0

r0: (3.42)

The propagation factor Wl, given in equation (3.38), describes the wave �eld synthesized by the

secondary line sources. Note that also here, the driving signals can be adapted for one receiver position

only. This receiver position is fully determined by the amplitude factors gl(r0;�r).

3.7 Point solution versus line solution

In the previous sections, it is seen how the wave �eld of a primary source is synthesized by a line array

or a DVLS. Considering the Rayleigh 2 12D integral for a line array or for a DVLS, equation (3.33)

and equation (3.39) respectively, one can see that the amplitude of the synthesized pressure �eld only

matches the amplitude of the primary monopole's pressure �eld at the receiver position R, because

the amplitude factor gm, being part of the driving signal, can be adapted for one position R only[3].

Therefore, this is called a \point solution". However, it is possible to match the wave �eld on a curve

and obtain a \line solution". This will be explained using �gure 3.6.

[3]The phase is synthesized accurate at each receiver position.


RC

Rstat

~�rstat

~�r

AR

~r0stat

~r0

L

z

x

stat

~̂r

Lstat

Figure 3.6: A representation of the stationary phase source stat, the stationary phase receiverRstat and the stationary phase line Lstat.

To start, the \stationary phase line" Lstat is introduced as the line that goes through the primary

source and the receiver Rstat, which will be called the \stationary phase receiver". The secondary

monopole source that lies on the intersection point of the secondary source distribution L and the

stationary phase line Lstat will be called the \stationary phase source" stat. The vector from the

primary source to the stationary phase source stat will be denoted by ~r0stat

and the vector from the

stationary phase source to the stationary phase receiver Rstat will be denoted by ~�rstat

.

The stationary phase approximation, as mentioned in section 3.5, states that the wave �eld on the

stationary phase line Lstat is highly dominated by the contributions of the secondary sources in the

neighborhood of the stationary phase source stat because they are in stationary phase for this line:

�0 stat (r0stat;�rstat) = 0; �00 stat(r

0stat;�rstat) 6= 0; 8 (r0stat

;�rstat) 2 Lstat: (3.43)

The phase function � of the integrand in equation (3.30) and equation (3.39) is determined by

�(r0;�r) = �k(r0 +�r): (3.44)

Other sources are not in phase for positions on the line Lstat and, therefore, their contribution to the

wave �eld partially or nearly totally annihilates at these positions due to destructive interference. This

means that if the amplitude of the stationary phase sources around stat is changed, only the wave

�eld in the neighborhood of the stationary phase line Lstat changes, while the wave �eld on all other

places does not change much. The amplitude of each secondary source can be changed by adjusting its

3.7. Point solution versus line solution 25

amplitude factor, attributing for each stat a unique stationary phase receiver Rstat on its stationary

phase line Lstat. Hence, the amplitude factors g(r0stat

;�rstat) of each stationary phase source stat

may be unlinked, instead of using g(r0;�r) which are all linked to one receiver point R only.

All amplitude factors g(r0stat

;�rstat) together de�ne a receiver curve RC in the receiver area. Note

that the curve RC must be a smooth curve. A discontinuitie violates the assumption that the wave

�eld at a Rstat is approximately only in uenced by the amplitude at stat. If a straight line is chosen

for the receiver curve, the particular case of the Rayleigh 2 12D operator described by [Vogel, 1993],

[Start, 1997] and [Verheijen, 1997] is obtained.

To make distinction between the general amplitude factor g(r0;�r) and the amplitude factor for a

stationary phase source g(r0stat

;�rstat), the notation gp and gc will be used respectively where gp

indicates that the receiver position is de�ned in a point R, whereas gc indicates that the receiver

positions are de�ned on a receiver curve RC by making use of the stationary phase approximation.

For notational convenience, the superscript \stat." will be dropped.

For so far, a line array is taken under consideration. For the DVLS, the same as above can be said

by replacing \stationary phase source" and \secondary source" by \stationary phase line source" and

\secondary line source" respectively.

To �nish the chapter, the equations for the di�erent amplitude factors are summarized:

The amplitude factors where the receiver position is only de�ned in a point R, the amplitude factors

gpm(r0;�r) =

r�r

r0 +�r; (3.45)

gpl (r0;�r) =

rr0

r0 +�r(3.46)

are used for a line array and a DVLS respectively.

The amplitude factors for a stationary phase source-receiver pair, where the receiver position is de�ned

on a receiver curve RC, the amplitude factors

gcm(r0;�r) =

r�r

r0 +�r; r̂ = r0 +�r; (3.47)

gcl (r0;�r) =

rr0

r0 +�r; r̂ = r0 +�r (3.48)

are used for a line array and a DVLS respectively.

For the vector from the source to the receiver R: ~̂r = ~r0 + ~�r may be used in all cases, while

only for the stationary phase (line) source-receiver pair: r̂ = r0 +�r is valid because in that case,

~�r lies in the prolonging of ~r0.


Chapter 4

The amplitude function for a line

array and for a DVLS

4.1 Introduction

In the previous chapter, the theory to synthesize a source with a line array or planar arrays is described.

With an in�nite planar array driven by signals obtained from the Rayleigh 3D operator, the sound

could be synthesized perfectly as long as the distance between the loudspeakers is small enough to

prevent spatial aliasing. The result is called a \volume solution", i.e., the sound �eld is synthesized

correctly in every point in volume. When using a line array or a DVLS, the sound �eld is synthesized

in an accurate way with respect to its phase. The amplitude of the sound �eld is only synthesized

correctly on a receiver curve RC in a receiver area, as can be read in section 3.7. The result is called

a \line solution" If the area where the virtual sources may be placed is relative small, no substantial

problem occurs due the amplitude deviation because the amplitude due to each source does not di�er

much and remains nearly the same in a receiver area when the sources are moving. When synthesizing

the �eld of multiple sources, which may lie everywhere in space, the amplitude contribution due to

each source has to be correct at all the receiver positions. This is a necessity to be able to estimate

the relative distance between sources or, in our case, to realize a good acoustic balance between the

wide spatial distribution of mirror image sources.

To clarify this, �gure 4.1 contains two simulated impulse responses of a virtual hall described in section

A.3 of appendix A. The primary source is placed at (x; z) = (0;�4) (m). The Mirror Image Source

Model (see chapter 2) is used to calculate the positions of the mirror image sources until the �fth

order (56 \visible" mirror image sources are found). How the amplitude is calculated will become clear

later in this chapter. Figure 4.1(a) shows the correct impulse response, �gure 4.1(b) shows the impulse

response synthesized by four line arrays and thus has amplitude deviations. The impulse responses

are \measured" at (x; z) = (1; 1) (m). The receiver curve RC for the primary source and each mirror

image source is the straight line though the center of the reproduction room, perpendicular to the line

from that mirror image source to the center, as is seen in �gure 4.2. From �gure 4.1, it is seen that

27

28 Chapter 4. The amplitude function for a line array and for a DVLS

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.05

0.1

0.15

0.2

time (s) �!

Amplitude�!

(a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.05

0.1

0.15

0.2

time (s) �!

Amplitude�!

(b)

Figure 4.1: Two impulse responses of the same hall simulated at (x; z) = (1; 1) (m).(a): Simulation of the correct impulse response.(b): Simulation of an impulse response synthesized with four line arrays while the

sound �eld is optimized at (x; z) = (0; 0) (m).

4.2. The amplitude function for a line array 29

2

1

RC1

RC2

R

AR

Figure 4.2: The choice of the receiver curve: The line through the center of the room, perpendicularto the line that connects the center with the source.

a volume solution is needed for synthesizing the wave �eld of a virtual hall, which apparently cannot

be reached using line arrays. However, the amplitude factors gcm of equation (3.30) for each secondary

phase source can be adjusted such that the amplitude error of the synthesized pressure �eld is minimal

in a given receiver area AR. To do this, easy implementable expressions are needed for the amplitude

of the pressure �eld synthesized by a line array or by a DVLS. These expressions will be derived in

this chapter. In the next chapters, these expressions will be used for optimizing the amplitude of the

synthesized pressure �eld and for all other amplitude calculations.

4.2 The amplitude function for a line array

Since the phase properties of the synthesized wave �eld are correct, only to the amplitude behavior is

considered. Therefore, the scaled amplitude will be de�ned as

A(r̂; !) =

��P (r̂; !)S(!)

�� ; (4.1)

to discard the phase properties and the source spectrum S(!). Furthermore, the wave �eld is considered

along a stationary phase line Lstat (see �gure 3.6). Substitution equation (3.11), the expression for the

pressure �eld of a monopole, into equation (4.1), the scaled amplitude of a monopole pressure �eld is

obtained:

Am(r̂) =

��Pm(r̂; !)S(!)

�� =1

r̂: (4.2)


At a stationary phase receiver position Rstat, the scaled amplitude of the pressure �eld synthesized

by a line array equals

AL(r0;�r; !) =

��PL(r̂; !)S(!)

�� = gcm(r0;�r)�L(r

0;�r; !); (4.3)

where PL is given by the Rayleigh 2 12D integral for a line array: equation (3.30). Since the amplitude

factors gcm are unlinked, they may be put before the integral of the integral sign of the Rayleigh 2 12D

integral. Hence, �L(r0;�r) contains all amplitude functions of the line array other than the factor gcm.

According to equation (3.47) this amplitude factor is de�ned as

gcm(r0;�r) =

r�r

r0 +�r; r̂ = r0 +�r: (4.4)

The scaled amplitude AL equals the scaled amplitude of a monopole at the stationary phase receiver,

given by �r:

AL(r0;�r) = Am(r̂): (4.5)

Hence, combining equations (4.2) to (4.5) and noting that r̂ = r0 +�r, �L(r0;�r) is given by:

�L(r0;�r) =

1r0+�r

gcm(r0;�r)

=

s1

(r0 +�r)�r: (4.6)

Notice that the attenuation �L(r0;�r) as de�ned here is independent of the angular frequency !.

According to equation (4.4), for a given source position, gcm should be consistent for all receiver

positions on the line Lstat. In practice, however, this is impossible. Being part of the driving signal of

the considered stationary phase source, gcm can be adjusted for one stationary receiver position �r0

only as illustrated in �gure 4.3. This means that the scaled amplitude, which is correctly synthesized

in �r0, is given by

AL(r0;�r;�r0) = gcm(r

0;�r0)�L(r0;�r) = gcm(r

0;�r0)

s1

(r0 +�r)�r; (4.7)

and is independent of the angular frequency !. Figure 4.3 shows two curves. One represents the

amplitude of the pressure �eld due to a monopole source (Am), located at one meter to the left of the

line array that is placed on �r = 0 (m). The other line is the amplitude of the monopole, synthesized

by the line array (AL), seen along a stationary phase line. By changing the constant amplitude factor

gc(r0;�r0) of the secondary source, the curve that represents the amplitude of the synthesized sound

�eld shifts up or down to match the two curves at the position �r0. Note that the vertical axis is

expressed in dB.

Equation (4.7) will be called: \the amplitude function for a line array". This function is an easy

expression for calculating the amplitude in a given receiver area without using the Rayleigh 2 12D

integral. Figure 4.4 shows the relative error of the amplitude obtained using the Rayleigh 2 12D integral

for a line array (3.30) with a �ne sampled monopole distribution and the amplitude calculated with

4.2. The amplitude function for a line array 31

0 0.5 1 1.5 2 2.5 3−15

−10

−5

0

5

10

�r (m) �!

Amplitude[dB]�!

AL

Am

�r0

Figure 4.3: Two amplitude attenuation curves. AL is the amplitude of the pressure �eld of a linearray along a stationary phase line as a function of the distance from the array. Theline array aims at synthesizing the pressure �eld of a monopole located at 1 (m) tothe left of the array. Am is the true amplitude of the monopole's pressure �eld.

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7 0

0.005

0.01

0.015

0.02

0.025

x (m)

z(m)

Relativeerror

Figure 4.4: The relative error of the amplitude calculated with the Rayleigh 2 12D integral and the

amplitude calculated with the amplitude function for a line array. In this setup, thesource is located at (x; z) = (0;�13)(m) and the array at z = 0, �4(m) � x � 4(m).The di�raction e�ects are due to the limitations of the array in the x-direction.


the amplitude function for a line array (4.7). The array and receiver area setup is found in section

A.1 of appendix A. From the bar at the right side of the �gure, one can read that the relative error

is negligible small. The most important factor of the di�erence is caused by the absence of di�raction

e�ects. These di�raction e�ects have a \high" contribution at the positions with a high aperture as

can be read in [Start, 1997]. One also sees a higher error in the center of the receiver area. It is possible

that di�raction interferes at these positions.

From equation (4.7), one can derive the next limits:

AL �

8>>><>>>:

1

�r� 1

r̂for

�r

r0� 1;

1p�r

� 1p�r

for�r

r0� 1:

(4.8)

From these limits one can see, looking at the amplitude in the receiver area speci�ed by �r while

r0 is taken constant, that if the source-to-array distance is much smaller than the array-to-receiver

distance (�rr0 � 1, �gure 4.5(a)), the amplitude of the pressure �eld is the same for a line array and

for a monopole. If the source is placed far away from the line array in comparison with the array-to-

receiver distance (�rr0 � 1, �gure 4.5(b)), the line array has a pressure �eld amplitude that equals

that of a line source, with a fall o� of 1=p�r.

R

r̂

�rr0

(a)

R

r̂

r0�r

(b)

Figure 4.5: Two possible con�gurations. In �gure 4.5(a) the source is placed relative close to thearray (�r

r0� 1). In �gure 4.5(b) the source is placed relative far away from the array

(�rr0� 1).

To conclude, an easy way to calculate the amplitude of the pressure �eld of a line array is found.

Although this method cannot be used to get frequency depended information, it is very practical in

use because it does not contain an in�nite integral. By this, also di�raction and spatial aliasing e�ects

(see [Start, 1997]) vanishes from the calculated results. To realize the same results with the normal

Rayleigh 2 12D integral, one has to integrate over a long array with a �ne sampled secondary source

distribution.

4.3 The amplitude function for a DVLS

It has been made clear that one cannot realize a volume solution by using a line array. Therefore, the

Rayleigh 2 12D operator for a DVLS was developed. The advantage of using this operator compared

4.3. The amplitude function for a DVLS 33

with the Rayleigh 3D operator is that the driving signal Ql has only to be calculated on a horizontal

line (as is done by the Rayleigh 2 12D operator for a line array) because all vertical line sources have

the same phase and amplitude. Using this setup, the phase is synthesized again correctly everywhere

in the receiver area, assuming that spatial aliasing e�ects and di�raction are negligible. The amplitude

is only synthesized correctly on a curve. In this section, the amplitude function for a DVLS will be

derived and the properties of the pressure �eld synthesized by this array is discussed.

Just as the amplitude factor gcm is used in the previous section as a measure of the amplitude atten-

uation, the amplitude factor gcl :

gcl (r0;�r) =

rr0

r0 +�r; r̂ = r0 +�r: (4.9)

will be used in this section to obtain the amplitude function for a DVLS. Both amplitude factors

have the same meaning, only the mathematical formulation is di�erent. After substitution gcl in

equation (4.6), the amplitude attenuation of the pressure �eld of the DVLS is reads

�L(r0;�r) =

1r0+�r

gcl (r0;�r)

=

s1

(r0 +�r)r0; (4.10)

so that the amplitude of the pressure �eld synthesized by a DVLS, seen along a stationary phase line

is expressed as:

AS(r0;�r;�r0) = gcl (r

0;�r0)�L(r0;�r) = gcl (r

0;�r0)

s1

(r0 +�r)r0: (4.11)

Here, �r0 is again the distance from the secondary source to the receiver position where the pressure

�eld's amplitude of the real source has to match that one of the to be synthesized source. Equation

(4.11) will be called: \the amplitude function for the DVLS".

Figure 4.6 shows the amplitude of the pressure �eld due to a monopole and synthesized by the DVLS.

The amplitude of the pressure �eld of the DVLS equals the amplitude of a monopole's pressure �eld

at �r0. This position can only be changed by changing the amplitude factor gcm(r0;�r0).

The relative error of the amplitude of the pressure �eld synthesized by a DVLS obtained using the

Rayleigh 2 12D integral for a DVLS (3.39) and calculated from the amplitude function for the DVLS is

shown in a �gure 4.7. The array and receiver area con�guration is found in section A.1 of appendix A.

Also here, the major error exists due to the absence of di�raction e�ects in the amplitude function.

The error is in the same order of magnitude as that in �gure 4.4.

Looking at the limits for the source placed close to, or far away from the DVLS, one obtains:

AS �

8>>><>>>:

1p�r

� 1pr̂

for�r

r0� 1;

1

r0� 1

r̂for

�r

r0� 1:

(4.12)

This means that the amplitude of the synthesized pressure �eld in the receiver area, speci�ed by �r,

equals that of a line source if the distance from the source to the DVLS is much smaller than the


0 0.5 1 1.5 2 2.5 3−14

−12

−10

−8

−6

−4

−2

0

�r (m) �!

Amplitude[dB]�!

Am

AS

�r0

Figure 4.6: Two amplitude attenuation curves. AS is the amplitude of the pressure �eld of a DVLSalong a stationary phase line as a function of the distance from the array. The DVLSaims at synthesizing the pressure �eld of the monopole located at 1 (m) to the left ofthe array. Am is the true amplitude of the monopole's pressure �eld.

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

70.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

x (m)

z(m)

Relativeerror

Figure 4.7: Relative error of the amplitude calculated with the Rayleigh 2 12D integral for a DVLS

and the amplitude calculated from the amplitude function for the DVLS in a speci�creceiver area. In this setup, the source is located at (x; z) = (0;�13)(m) and the arrayat z = 0 (m),�4(m) � x � 4(m). The di�raction e�ects are due to the limitations ofthe array in the x-direction.

4.3. The amplitude function for a DVLS 35

distance between the DVLS and the receiver, i.e., �rr0 � 1 (see �gure 4.5(a)). Making the distance

between the source and the DVLS much larger than the distance between the DVLS and the receiver

(�rr0 � 1, �gure 4.5(b)), the pressure �eld in the receiver area attenuates the same as the pressure

�eld of a monopole. This is the opposite of what happens with the line array.

From �gure 4.6 and the limits in equation (4.12), one can see also that the Rayleigh 2 12D operator

for a DVLS does not give a volume solution. Whereas the amplitude attenuation of the line array's

pressure �eld is larger than that of a monopole's pressure �eld, the amplitude attenuation of the

DVLS's pressure �eld is smaller than that of a monopole's pressure �eld, which may sometimes be a

bene�t.

Concluding: A line array is needed to synthesize the pressure �eld of sources when they lie close to

the array compared with the array-to-receiver distance, while sources remote from the array have to

be synthesized with a DVLS. Note that if the monopole's distance is in�nite, the DVLS synthesizes

plane waves. If reverberation is synthesized by plane waves ([Sonke and de Vries, 1997]) the sound �eld

due to the DVLS results to no amplitude deviation. In chapter 7, the array to be used for di�erent

con�gurations will be given.


Chapter 5

Optimizing the amplitude factor

5.1 Introduction

In chapter 3, it has been made clear that it is possible to synthesize a wave �eld correct on a curve RC.

In the past, often a straight line was token for this curve. This is not a trivial choice as can be read in

section 3.7. In this chapter, the optimal receiver curve is found for which the amplitude error between

the synthesized pressure �eld and that of a monopole is minimal in a speci�ed receiver area. Since the

phase of the synthesized wave �eld is correct in each receiver position, only the amplitude is considered

and the amplitude functions derived in the previous chapter are used. First, the amplitude error in a

receiver area due to a line array is minimized by optimizing the amplitude factor gcm. Thereafter, the

same is done for the DVLS with the amplitude factor gcl . To signify the amplitude error, the amplitude

ratio level is de�ned. This level is derived from the pressure ratio level LR:

LR(r̂; !) = 10 log

�� P (r̂; !)Pd(r̂; !)

��2!

[dB]; (5.1)

where Pd is the \desired" pressure. By making use of equation (4.2) and equation (4.3), the amplitude

ratio level is written as:

LRA(r̂) = 20 log

�A(r̂)

Am(r̂)

�[dB]: (5.2)

This amplitude ratio level describes the relation between the desired amplitude of a monopole's pres-

sure �eld Am and the amplitude of the pressure �eld synthesized by a line array (AL) or a DVLS

(AS). The amplitude ratio level is independent of the angular frequency !.

5.2 Optimizing the amplitude factor gcm

In this section, the amplitude distribution of the pressure �eld synthesized by a line array will be

optimized. Therefore, the amplitude ratio level for AL will be minimized in a receiver area AR (as

37

38 Chapter 5. Optimizing the amplitude factor

shown in �gure 5.1) by solving the least squares problem:

�maxZ�min

r̂max(�)Zr̂min(�)

�20 log

�AL(r̂)

Am(r̂)

��2r̂dr̂ d� = minimal (5.3)

r̂min

d�

r0

r̂max

r̂maxr̂min

AR

Figure 5.1: A room with four line arrays. To calculate the optimal amplitude distribution in thereceiver area AR, one needs to know the integration boundaries r̂min; r̂max.

As seen in the previous chapters, a change in the amplitude factor of a secondary source, only gives

rise to the amplitude change of the wave �eld around its stationary phase line Lstat (as shown in

�gure 5.2):

@

@gcm(r0;�r0)

�maxZ�min

r̂maxZr̂min

�20 log

�AL(r̂)

Am(r̂)

��2r̂ dr̂ d� �

@

@gcm(r0;�r0)

r̂maxZr̂min

�20 log

�AL(r

0;�r;�r0)

Am(r0 +�r)

��2r̂ dr̂ d� = 0:

(5.4)

5.2. Optimizing the amplitude factor gcm 39

r̂

dr̂ d�

Astatd�

dr̂

Lstat

�r

r0

stat

Figure 5.2: The area covered by the stationary phase source.

By substituting equations (4.3), (4.5) and (4.7) in (5.4), one obtains:

@

@gcm(r0;�r0)

r̂maxZr̂min

�20 log

�gcm(r

0;�r0)�Lgcm(r

0;�r)�L

��2r̂ dr̂ d� =

800

ln 10

r̂maxZr̂min

�ln(gcm(r

0;�r0))� ln(gcm(r0;�r0))

gcm(r0;�r)

�r̂ dr̂ d� = 0:

(5.5)

Note that r̂ = r0 + �r, while for a given source position, r0 is a constant in equation (5.4) and

equation (5.5). Therefore, gcm(r0;�r0) is only a function of �r0, whereas g

cm(r

0;�r) and r̂ are a function

of �r. If gcm does not equal in�nity, the optimal amplitude factor is obtained from equation (5.5):

gc;optm (r0;�r0) = exp

8>>><>>>:

r̂maxRr̂min

ln(gcm(r0;�r)) r̂ dr̂

r̂maxRr̂min

r̂ dr̂

9>>>=>>>;

= exp

8>>><>>>:

r̂maxRr̂min

ln(q

r̂�r0

r̂ ) r̂ dr̂

r̂maxRr̂min

r̂ dr̂

9>>>=>>>;: (5.6)

Solving (5.6), the optimal amplitude factors for the monopole sources on a line array are given by:

gc;optm (r0;�r0) =

exp

8<:�12�r

2(ln(�r) � 12 ) + r0�r(�r � 1)

��rmax

�rmin� �

12 r̂

2(ln(r̂)� 12 )�r̂max

r̂min

[r̂2]r̂max

r̂min

9=; :

(5.7)

Giving each secondary source an optimal amplitude factor, the optimized pressure �eld is determined

by the Rayleigh 2 12 integral for a line array, equation (3.33). Using them in the amplitude function for

a line array (4.7), the optimized amplitude is found. To determine the optimal receiver curve RoptC ,

equation (3.47) is expressed as

�ropt0 =r0

( 1gc;optm

)2 � 1: (5.8)


Figure 5.3 shows for three di�erent sources their optimal receiver curves RoptC , being the curves where

the amplitude of the pressure �eld generated by the monopole source equals that of the line arrays.

The straight part of the optimal curve is shifted in the direction of the source. Near the edges, the

optimal curve bends towards one of the corners. This is because the integration boundaries r̂min and

r̂max along the stationary line Lstat belonging to the optimal stationary phase receiver, approach each

other, as seen in �gure 5.1. The optimal receiver position always lies between r̂min and r̂max. In the

next chapter, more will be said about the shape of the optimal receiver curves.

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

1

2

3

2

3

1

x (m)

z(m)

Figure 5.3: Three optimal receiver curves for three di�erent sources, using three line arrays tosynthesize the sound �eld.

5.3 Optimizing the amplitude factor gcl

In this section, the amplitude error of the pressure �eld of a monopole and a DVLS will be minimized.

Since both amplitude factors gcl and gcm mean the same, the same shall be done as in the previous

section, to retrieve the optimal amplitude factor gc;optl . If the amplitude factor gcm is replaced by gcl ,

equation (5.6) changes to:

gc;optl (r0;�r0) = exp

8>>><>>>:

r̂maxRr̂min

ln(gcl (r0;�r)) r̂ dr̂

r̂maxRr̂min

r̂ dr̂

9>>>=>>>;

= exp

8>>><>>>:

r̂maxRr̂min

ln(q

r0

r̂ ) r̂ dr̂

r̂maxRr̂min

r̂ dr̂

9>>>=>>>;: (5.9)

5.3. Optimizing the amplitude factor gcl 41

The solution of the integrals, yielding the optimal amplitude factors gc;optl for the vertical line sources

is:

gc;optl (r0;�r0) = exp

8><>:h12 r̂

2�ln�r0

r̂

�+ 1

2

�ir̂max

r̂min

[r̂2]r̂max

r̂min

9>=>; : (5.10)

Using for each secondary line source its optimized amplitude factor, the optimal pressure �eld dis-

tribution is calculated with the Rayleigh 2 12D integral for a DVLS, equation (3.39). If one is only

interested in the amplitude, the amplitude function for a DVLS, equation (4.11), can be used. The op-

timal receiver position, i.e., the position where the synthesized pressure �eld equals that of a monopole

source, is obtained by rewriting equation (3.48):

�ropt0 = r0

1

(gc;optl )2� 1

!: (5.11)

Figure 5.4 shows for three sources their optimal receiver curve RoptC , using three vertical line source

arrays to synthesize the pressure �eld of a monopole. The curves have the same shape as when the

sound �eld is synthesized by line arrays. Note that the straight part of the optimal curve is shifted

away from the source, whereas it shifts towards the source when line arrays are used.

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

1

2

3

2

3

1

x (m)

z(m)

Figure 5.4: Three optimal receiver curves for three di�erent sources, using three vertical linesource arrays to synthesize the sound �eld.


Chapter 6

Simulations of amplitude

distributions

6.1 Introduction

In this chapter some simulations of the amplitude distribution of a wave �eld synthesized by line

arrays and vertical line source arrays will be shown. See [Start, 1997] for information on the phase

properties of the wave �eld synthesized by line arrays. The con�guration of the arrays and the virtual

hall are given in appendix A.

6.2 Simulations using a line array

In �gure 6.1, one can see three amplitude distributions in a speci�c receiver area given in section

A.1. Figure 6.1(a) shows the amplitude distribution of a monopole's pressure �eld calculated with

equation (3.11) and equation (4.1). The source is located at (x; z) = (0;�1) (m). Figure 6.1(b) and

�gure 6.1(c) shows the amplitude distributions of the pressure �eld synthesized by a line array, aiming

to synthesize the pressure �eld of the monopole. The amplitude is calculated with the amplitude

function (4.7). The line array, located at z = 0 (m), contains gcm or gc;optm in its driving signal so

that the amplitude is synthesized correctly on a line (z = 4 (m)) or the optimal receiver curve RoptC

respectively. The amplitude scale is chosen such that the amplitude equals 0 [dB] in the center of

the receiver area. From �gure 6.1 it is again clear that a line array cannot synthesize the amplitude

correctly at every position. Although all the distributions have nearly the same shape, those shown in

�gures 6.1(b) and (c) have a larger attenuation than that one shown in �gure 6.1(a). The amplitude

is in the neighborhood of the array at x = 0 (m) 2 [dB] louder in �gures 6.1(b) and (c). The error

is much smaller at x = 6 (m). The optimal amplitude distribution equals the real distribution in the

corners of the receiver area at z = 1 (m).

43

44 Chapter 6. Simulations of amplitude distributions

−2

0

22

4

6

−2

0

2

4

6

8

10

x (m)z (m)

Amplitude[dB]

(a)

−2

0

22

4

6

−2

0

2

4

6

8

10

x (m)z (m)

Amplitude[dB]

(b)

−2

0

22

4

6

−2

0

2

4

6

8

10

x (m)z (m)

Amplitude[dB]

(c)

Figure 6.1: Amplitude distributions of the pressure �eld of a source, simulated in a rectangularreceiver area. The source is located at (x; z) = (0;�1) (m). The pressure �eld is(a): that of a real source,(b): synthesized by a line array, optimized on the line z = 4 (m);(c): synthesized by a line array, optimized on the optimal receiver curve.

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

z(m)

LRA

(a)

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x (m)

z(m)

LRA

(b)

Figure 6.2: The absolute amplitude ratio level of the amplitude distributions shown in�gures 6.1(b) and 6.1(c). The source is located at (x; z) = (0;�1) (m).

6.3. Simulations using a DVLS 45

The absolute value of the amplitude ratio levels LRA of the amplitude distributions shown in �g-

ures 6.1(b) and 6.1(c) are shown in �gures 6.2(a) and 6.2(b) respectively. In these �gures, the receiver

curves RC where the synthesized pressure �eld matches the real pressure �eld, are clearly seen. It

is seen that the two amplitude distributions have nearly the same error around x = 0 (m) that ap-

proximates 2 [dB]. In �gure 6.2(a), this error remains constant along the x-axis, whereas it decreases

towards the positive and negative x-axis in �gure 6.2(b). The edges of the optimal receiver curve bend

strongly towards the array because the ratio between the minimal and maximal integration radius of

equation (5.6) changes much for the stationary phase lines through this area, as can be read in the

previous chapter. It should be noted that the optimal amplitude distribution in �gure 6.1(c) lies a bit

lower than the distribution in �gure 6.1(b) so that the optimal receiver curve shifts from the centerline

towards the array, as also can be seen in �gure 5.3. For the explanation of this, consider the limits

given equation (4.8). At positions nearby the array, �rr0 << 1) is valid, so that the amplitude behavior

approximates the amplitude behavior of the line array's pressure �eld. Moving away from the array,

the amplitude behavior changes towards that of the real pressure �eld. Therefore, the positions nearby

of the array will have a larger weight in the optimization than other positions. Hence, the amplitude

lowers and, as can be seen in �gure 4.3, the optimal position shifts towards the array.

If the source-to-array distance increases, the amplitude error also increases. In �gures 6.3 and 6.4 the

same functions as in �gures 6.1 and 6.2 respectively are shown, but with a source placed seven meter

behind the array, at (x; z) = (0;�7) (m). From 6.4 one sees that the maximal amplitude error equals

approximately 4 12 [dB], two times larger than the previous case. The straight part of the optimal

receiver curve is longer than in the previous case. The optimal receiver curve is shifted more towards

the array, because �rr0 becomes smaller, giving positions in the near of the array an even larger weight

in the optimization than in the previous case.

From these simulations, it is seen that when using a line array to synthesize the sound �eld of a

monopole source, the amplitude deviations grow when the source moves away from the array. The

optimal amplitude distribution has a smaller amplitude deviation in the receiver area than when the

sound �eld is optimized on a straight line in the center. Nevertheless, the amplitude deviation remains

large.

6.3 Simulations using a DVLS

In the same way as in the previous section, the amplitude distribution of a DVLS in a particular receiver

area is investigated. Two di�erent source positions are considered. The DVLS uses the amplitude

factors gcl or gc;optl in its driving signal such that the amplitude of the pressure �eld matches that of

the monopole on a line (z = 4 (m)) or the optimal curve RoptC respectively.

In �gure 6.5 the source is placed at (x; z) = (0;�1) (m). The amplitude distributions shown in

�gure 6.5(b) and �gure 6.5(c) are calculated with equation (4.11). It is seen that in this particular

case, the amplitude attenuation of the pressure �eld synthesized by the DVLS is much smaller than

when a line array is used, even too small in comparison with the attenuation of the real pressure �eld:

the amplitude attenuates from 5 to �1 [dB] for a DVLS, from 10 to �3 [dB] for a line array (see


−2

0

22

4

6

−2

0

2

4

6

x (m)z (m)

Amplitude[dB]

(a)

−2

0

22

4

6

−2

0

2

4

6

x (m)z (m)

Amplitude[dB]

(b)

−2

0

22

4

6

−2

0

2

4

6

x (m)z (m)

Amplitude[dB]

(c)

Figure 6.3: Amplitude distributions of the pressure �eld of a source, simulated in a rectangularreceiver area. The source is located at (x; z) = (0;�7) (m). The pressure �eld is(a): that of a real source,(b): synthesized by a line array, optimized on the line z = 4 (m);(c): synthesized by a line array, optimized on the optimal receiver curve.

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x (m)

z(m)

LRA

(a)

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0.5

1

1.5

2

2.5

3

3.5

4

x (m)

z(m)

LRA

(b)

Figure 6.4: The absolute amplitude ratio levels of the amplitude distributions shown in�gures 6.3(b) and 6.3(c). The source is located at (x; z) = (0;�7) (m).

6.3. Simulations using a DVLS 47

−2

0

22

4

6

−2

0

2

4

6

8

x (m)z (m)

Amplitude[dB]

(a)

−2

0

22

4

6

−2

0

2

4

6

8

x (m)z (m)

Amplitude[dB]

(b)

−2

0

22

4

6

−2

0

2

4

6

8

x (m)z (m)

Amplitude[dB]

(c)

Figure 6.5: Amplitude distributions of the pressure �eld of a source, simulated in a rectangularreceiver area. The source is located at (x; z) = (0;�1) (m). The pressure �eld is(a): that of a real source,(b): synthesized by a DVLS, optimized on the line z = 4 (m);(c): synthesized by a DVLS, optimized on the optimal receiver curve.

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0

0.5

1

1.5

2

2.5

3

3.5

x (m)

z(m)

LRA

(a)

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0.5

1

1.5

2

2.5

3

3.5

4

x (m)

z(m)

LRA

(b)

Figure 6.6: The absolute amplitude ratio level of the amplitude distributions shown in�gures 6.5(b) and 6.5(c). The source is located at (x; z) = (0;�1) (m).


�gure 6.1) and from 9 to �3 [dB] for a monopole source. Whereas the optimized amplitude factor

for a line array gc;optm produces a sound �eld with a smaller amplitude on the edges of the array, the

amplitude factor of the DVLS gc;optl makes the amplitude increase at these places.

The absolute amplitude ratio levels LRA of the amplitude distributions shown in �gures 6.5(b) and

(c) are represented in �gure 6.6. When using the optimized amplitude factor, the maximum amplitude

deviation is larger but the area with the large deviation is much smaller. The error has the same order

of magnitude as in the case using the line array with the source at a distance of seven meter from the

array (around 4 [dB] nearby the array). Note that also here the amplitude distribution shifts down

due to the optimization. In this situation, this leads to an optimal receiver curve lying at a z-position

larger than the center of the receiver area.

It is seen from the simulations in �gure 6.7 and �gure 6.8, where the source is placed at (x; z) =

(0;�7) (m), that the amplitude error becomes much smaller. This is opposite to what happens for a

−2

0

22

4

6

−2

−1

0

1

2

x (m)z (m)

Amplitude[dB]

(a)

−2

0

22

4

6

−2

−1

0

1

2

x (m)z (m)

Amplitude[dB]

(b)

−2

0

22

4

6

−2

−1

0

1

2

x (m)z (m)

Amplitude[dB]

(c)

Figure 6.7: Amplitude distributions of the pressure �eld of a source, simulated in a rectangularreceiver area. The source is located at (x; z) = (0;�7) (m). The pressure �eld is(a): that of a real source,(b): synthesized by a DVLS, optimized on the line z = 4 (m);(c): synthesized by a DVLS, optimized on the optimal receiver curve.

line array. The amplitude attenuates from 1 12 to �1 [dB] for a DVLS, from 7 to �3 [dB] for a line

array (see �gure 6.3) and from 2 12 to �2 [dB] for a monopole source. Also when using a DVLS, the

total amplitude error is reduced by using of the optimal amplitude factor, but still remains too large.

6.4. Simulations of a virtual hall 49

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0

0.2

0.4

0.6

0.8

1

1.2

x (m)

z(m)

LRA

(a)

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

7

0.2

0.4

0.6

0.8

1

1.2

1.4

x (m)

z(m)

LRA

(b)

Figure 6.8: The absolute amplitude ratio levels of the amplitude distributions shown in�gures 6.7(b) and 6.7(c). The source is located at (x; z) = (0;�7) (m).

The conclusion is that when a source is remote from a line array or a DVLS, the DVLS is used best

to synthesize the wave �eld. For sources in the neighborhood of a line array or a DVLS, the line array

is used best. In the next chapter, one can read which array has to be used for an optimal amplitude

distribution in connection with a particular source distance and a speci�c receiver area.

6.4 Simulations of a virtual hall

So far, only the amplitude error of one single source is taken under consideration. In this section,

the amplitude errors that appear when synthesizing multiple sources, will be discussed. Those sources

are the mirror image sources of a primary source placed at (x; z) = (0;�6) (m) in the virtual hall

described in section A.3.

First, the sound �eld will be synthesized such that it is synthesized correct on a receiver point R,

giving a point solution. Thereafter, the con�guration is considered where the amplitude of the sound

�eld is synthesized correctly for each source on a straight line RC, as shown in �gure 4.2 of chapter

4. This con�guration is considered, because it is often used in the past. Finally, the sound �eld will

be synthesized correctly for each source on the optimal curve RoptC To calculate the amplitude, the

amplitude function for a line array, equation (4.6), is which the amplitude factors gpm, gcm and gc;optm

are substituted respectively.

Figure 6.9 shows the mean deviation when four line arrays are used to synthesize the wave �eld.

Hereby, the absolute value of the amplitude ratio level LRA is considered for each source individually.

In real, interaction occurs between the pressure �elds due to the di�erent sources. In table 6.1 the

mean (over the receiver are area), the maximal and the minimal amplitude ratio level are shown.


(a)

−1 0 1

−2

−1

0

1

2

(b)

−1 0 1

−2

−1

0

1

2

(c)

−1 0 1

−2

−1

0

1

2

0 1 2 3 4 5 6 7 8 9 10

x (m)

z(m)

x (m)

z(m)

x (m)

z(m)

LRA

Figure 6.9: The mean absolute amplitude ratio level of a primary source and 56 image mirrorimage sources. The wave �eld is synthesized using four line arrays. It matches thedesired sound �eld of each (mirror image) source(a) in a point,(b) on straight lines,(c) on the optimal receiver curves.

Table 6.1: The mean, the maximal and the minimal amplitude ratio levels LRA of the synthesizedsound �eld by line arrays in the receiver area.

point solution line solution optimal curve solution

mean error 2.9 dB { 2.5 dBmaximal error 10.7 dB { 6.7 dBminimal error 0 dB 0 dB 0.8 dB


It is clear that using the optimized amplitude factor gc;optm , the mean error is minimal over the whole

receiver area, but nowhere becomes zero. When gpm and gcm are used, the error is zero in the center of

the receiver area.

The maximal and the mean error of the line solution are not shown because the simulated wave

�eld equals zero at some receiver position when the amplitude function for a line array (4.7) is used.

Hence, the amplitude ratio level (equation (5.2)) becomes in�nite, shown as dark areas in the corners

of �gure 6.9(b). To clarify why the amplitude becomes zero at some positions, consider �gure 6.10.

}}

~�r

~r0

~r0

AR

gcm not de�ned

~�r~r0

AR

gcm not de�ned

R

RC

Figure 6.10: The choice of the line RC to obtain a line solution. The amplitude factors for thespeakers in the left corner are not de�ned.

From this �gure, one sees some problems occur calculating the amplitude factor. Because some sec-

ondary sources lie on the wrong side of the receiver curve RC (�r is negative), the amplitude factor

gcm becomes imaginary, as seen in equation (3.47). If only the real part of the amplitude factor is used

in the amplitude function (4.7), the amplitude will be zero along the stationary phase lines of these

secondary sources. If in practice also the real part of the amplitude factor is used, then the pressure

�eld is highly dominated by di�raction contributions at these positions.

Also for a DVLS, three mean absolute amplitude ratio levels are considered as has been done above

for the line arrays. The amplitude of the synthesized sound �eld is calculated with equation (4.11).

It is seen from the �gures and the tables that the mean and the maximal errors are much smaller


(a)

−1 0 1

−2

−1

0

1

2

(b)

−1 0 1

−2

−1

0

1

2

(c)

−1 0 1

−2

−1

0

1

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

x (m)

z(m)

x (m)

z(m)

x (m)

z(m)

LRA

Figure 6.11: The mean amplitude ratio level of a primary source and 56 image mirror imagesources. The wave �eld is synthesized using four distributions of vertical line sources.It matches the desired sound �eld of each (mirror image) source(a) in a point,(b) on straight lines,(c) on the optimal receiver curves.

Table 6.2: The mean, the maximal and the minimal amplitude ratio levels LRA of the synthesizedsound �eld by a DVLS in the receiver area.

point solution line solution optimal curve solution

mean error 0.20 dB 0.18 dB 0.17 dBmaximal error 0.46 dB 0.37 dB 0.28 dBminimal error 0 dB 0 dB � 0 dB


when the DVLS is used. This is because the mirror image sources lie far away from the arrays, while,

as read in the previous sections, a DVLS can synthesize the sound �eld of remote sources much better

than line arrays. The mean and the maximal amplitude ratio levels are also smaller in the case the

optimal driving signals are used. Attention is drawn to the fact that in the case the sound �eld is

optimized for each source on a straight line, the error is de�ned at each receiver position, while the

receiver curve still lies at the wrong side of the line array for some secondary sources. The amplitude

factors for these sources exceed unity. In practice, the amplitude factors are given the restriction to

be maximally unity, resulting to a right amplitude distribution on the position of these sources.


Chapter 7

A line array versus a DVLS

7.1 Introduction

In the previous chapters, two ways to synthesize a sound �eld with a secondary monopole or line

source distribution are treated. One using a line array of monopole sources, each driven by a signal

derived from the Rayleigh 2 12D integral for a line array, the other using an array with a vertical line

source distribution (DVLS), each driven by a signal derived from the Rayleigh 2 12D integral for a

DVLS. The amplitude of the sound �eld synthesized by the line array approximates best the desired

sound �eld if the virtual sources lie in the neighborhood of the array, whereas the sound �eld of the

DVLS has the smallest error for sources lying far away from the array. In this chapter, one �nds which

array is best used given a particular receiver area.

7.2 The gradient method

In this section, the gradients of the amplitude attenuation of the pressure �eld synthesized by a line

array and a DVLS are compared with that of a monopole's pressure �eld. By this, some insight is

obtained about the behavior of the wave �eld synthesized by the arrays as a function of the source

and receiver distance.

Consider �gure 7.1 where the amplitude function of the pressure �eld for a line array, a DVLS and a

monopole are shown as a function of the array-to-receiver distance �r in a double logarithmic scale.

The limits given in equation (4.8) and equation (4.12) are shown as asymptotes. The shape of the

curves remains the same by changing the source-to-array distance r0, only the amplitude values along

the vertical axis change. The slope of a curve is the power of �r. For example, if �r approaches

zero, the slope of AL approaches � 12 . Therefore, the amplitude for the line array attenuates at this

distance with �r(�1

2). To get some insight of the attenuation behavior of the arrays, the gradient of the

amplitude functions (4.7) and (4.11), which for a given source-to-array distance r0 is fully determined

by the array-to-receiver distance �r, are logarithmically expressed by:

55

56 Chapter 7. A line array versus a DVLS

10−2

10−1

100

101

102

10−3

10−2

10−1

100

101

102

AL

ASAm

�r �!

Amplitude�!

Figure 7.1: Amplitude functions of the pressure �eld of a line array, a DVLS and a monopole.

A0

L =@

@log�r

(20 log

gcm(r

0;�r0)

s1

(r0 +�r)�r

!)= �10

�rr0

1 + �rr0

+ 1

!(7.1)

for the line array and

A0

S =@

@log�r

(20 log

gcl (r

0;�r0)

s1

(r0 +�r)r0

!)= �10

�rr0

1 + �rr0

!(7.2)

for the DVLS. Similarly, for the amplitude of the pressure �eld of a monopole:

A0

m =@

@log�r

�20 log

�1

r0 +�r

��= �20

�rr0

1 + �rr0

!: (7.3)

Notice that the gradients are independent of the amplitude factors. The gradients are shown in �g-

ure 7.2(a), the errors of the gradients of AL and AS with respect to Am are shown in �gure 7.2(b)

7.2. The gradient method 57

10−2

10−1

100

101

102

−1

−0.8

−0.6

−0.4

−0.2

0

�rr0 �!

gradient�!

A0

S

A0

m

A0

L

(a)

10−2

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

�L �S

�rr0 �!

Error�!

(b)

Figure 7.2: (a): The gradients of the amplitude of the pressure �eld, synthesized by a line array,a DVLS and a monopole.

(b): The gradient errors of A0

L and A0

S with A0

m.

and are given by

�L =��A0

L �A0

m

�� ;�S =

��A0

S �A0

m

�� :(7.4)

One can say that if the gradient error of the line array is smaller than that of the DVLS in a particular

receiver area, or vice versa, there always exist an amplitude factor that makes the amplitude error

due to the �rst array in this receiver area smaller than the amplitude error due to the second array.

By equating the gradient errors �L and �S, the ratio for which both arrays give an equal gradient error


is obtained (note that A0

L is always larger than A0

m and A0

S always smaller than A0

m):

A0

L �A0

m = �(A0

S �A0

m) ()�r

r0= 1 () �r = r0: (7.5)

Hence, if all the positions in the receiver area have a distance to the array larger than the source-to-

array distance, i.e., �r > r0 8 �r as is seen in �gure 7.3(a), the gradient error of the line array is

smaller than that of the DVLS (�L < �S). In this case, a line array could best be used to obtain the

smallest amplitude deviation in the whole receiver area.

If all the positions in the receiver area have a distance to the array smaller than the source-to-array

distance, as seen in �gure 7.3(b)), the gradient error of the DVLS is always smaller than that of the

line array so that the DVLS could best be used.

~r0

L

AR~�r

(a)

~r0AR

~�r

S

(b)

Figure 7.3: (a): The distance from all receiver positions to the array are larger than the distancefrom the source to the array.

(b): The distance from all receiver positions to the array are smaller than the distancefrom the source to the array.

In all other cases, where the at some array-to-receiver distances the gradient error of the line array

is larger, while at other distances that one of the DVLS is larger, or vice versa, the gradient method

does not give a solution that shows which array to use. In this case, the amplitude error of one array

is at some positions larger and at other positions smaller than the amplitude error of the other array.

Figure 7.4 shows which array to use for combinations of r0 and �r. Here, the receiver area given in

section A.1 is considered whereby only the depth is varied.

When considering early re ections and making the virtual hall at least twice as large as the receiver

area, such that the mirror-sources-to-array distances exceeds all array-to-receiver distances, the DVLS

is always best used.

7.2. The gradient method 59

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

receiver area depth (m) �!

r0(m)�!

DVLS

Line array

?

Figure 7.4: A diagram showing which array to use for di�erent depths of the receiver area.


7.3 Numerical approximation

In the previous section, a method to determine the array to be used given a source position and a

speci�ed receiver area is shown. Because this method does not give a solution in all cases, this gap

will be �lled up by a numerical solution in this section. The source-to-array distance r0 for which in

a speci�c receiver area AR the mean amplitude ratio level of the line array equals that of a DVLS is

found. Here, the mean amplitude ratio level in a receiver area AR is expressed by:

LRA =1

AR

ZAR

20 log

��A(r0; r̂; r̂0)

Am(r̂)

��dAR: (7.6)

LRA depends on the de�nition of the receiver area AR and the method of optimizing the sound �eld.

Figure 7.5(a) shows the mean amplitude ratio levels for the �elds synthesized by a line array and a

DVLS as function of the source-to-array distance. The �elds are optimized in the center of the receiver

area. Figure 7.5(b) shows the same for the sound �eld optimized on the optimal receiver curve RoptC .

1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

r0 (m) �!

LRA

�!

Line array

DVLS

(a)

1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

r0 (m) �!

LRA

�!

Line array

DVLS

(b)

Figure 7.5: Mean amplitude ratio levels in a receiver area for the DVLS and the line array, asfunction of the source-to-array distance.(a): The amplitude is optimized in the center of the receiver area.(b): The amplitude is optimized on the optimal receiver curve.

The point of intersection of the two curves represents the source-to-array distance for which both

arrays have the same amplitude deviation. This distance will be called the transition distance r0t.

For smaller source-to-array distances than the transition distance, the line array works better, for

larger source-to-array distances the DVLS. The mean amplitude ratio levels shown in �gure 7.5 are

calculated numerically by replacing the integral in equation (7.6) by a summation. One can see that

for the receiver area considered, this transition distance is 3:45 (m) when the wave �eld is optimized

in the center and 3:36 (m) when the wave �eld is optimized on the optimal curve.

7.3. Numerical approximation 61

Figure 7.6 shows the source-to-array transition distance as function of the receiver area depth, just

as done in the previous section. The three curves shown represent the transition distance for the

amplitude optimized in the center of the receiver area, optimized on a line through the center and

optimized on the optimal curve. When combinations of r0 and �r lie below a line in �gure 7.6 a line

array has to be used, otherwise the DVLS give the best results. Because optimization in the center

result to approximately the same amplitude factors as by optimization on a line through the center,

the transition distances for these methods are as good as equal. In �gure 7.6, it is seen that if the

source-to-array distance is larger than a third to a half times the receiver area length, the DVLS gives

always the best results.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Optimized in Center Optimized on line Optimized on optimal curve

receiver area depth (m) �!

r0(m)�!

DVLS

Line array

Figure 7.6: The transition distance for three di�erent optimization methods as function of thedepth of the receiver area, starting at 1 meter to the right of the array,


7.4 Truncating the DVLS

Until now, the assumption was made that the DVLS is in�nite extended in the vertical direction.

However, in practice this is impossible. One should truncate the in�nite vertical line sources to �nite

dimensions[1], which changes the properties of the DVLS. In this section, only some qualitative remarks

will be given.

When the array is truncated, the pressure �eld has an amplitude behavior between that of an in�nite

DVLS and a line array. Hence, the truncated DVLS may already be better used for smaller source-to-

array distances than given in the previous sections. However, for sources remote, its pressure �eld has

a larger amplitude deviation than the in�nite extended DVLS. In addition, di�raction e�ects occur

which are larger than the di�raction e�ects in the horizontal direction. This is because all sources in

vertical direction have the same phase, so that the aperture decreases and a larger receiver area is

in uenced by the di�raction e�ects ([Start, 1997]). The di�raction e�ects can be reduced by using a

taper in the vertical direction. This gives an extra reduction to the vertical extension.

[1]In practice, the line source has also to be sampled in the vertical direction. In this direction, a larger sample distancethan in horizontal direction is allowed in order to ful�ll the anti aliasing condition ([Start, 1997]).

Chapter 8

Wave �eld synthesis of virtual halls

in practice

8.1 Introduction

This chapter describes the hardware and the software used for the practical implementation of syn-

thesizing the wave �eld of a virtual hall in a smaller reproduction room. The software implements the

theory derived in the previous chapters. The con�gurations of the reproduction room and the virtual

hall are described in section A.2 and A.3 of appendix A respectively.

8.2 Hardware con�guration

To synthesize the wave �eld of a virtual hall with loudspeaker arrays, many speakers and powerful DSP-

boards to process lots of signals in real-time are needed. Figure 8.1 shows the scheme of the processing.

The primary sound, which can be a recorded signal, is o�ered to Analog-to-Digital (A/D) converters.

Thereafter, the digital signal is sent to 17 \Digital Signal Processors" (DSPs) that keep the signal in

a bu�er. The DSPs calculate for each loudspeaker the driving signal Q, obtained from weighting and

delay tables calculated by the software program. This driving signal is sent to the Digital-to-Analog

(D/A) converters. Finally, the analog signals are ampli�ed and sent to each loudspeaker. An in-depth

study how to use the hardware to synthesize the sound �eld of virtual sources is found in [Verheijen,

1997].

8.3 Software used for wave �eld synthesis

In order to let the DSPs emit the correct signal, some values have to be calculated. First, as read in

chapter 2, the position of the mirror image sources are calculated, given a primary source position and

a virtual hall. The source strength of each virtual source is dependent on the absorption coe�cients

63

64 Chapter 8. Wave �eld synthesis of virtual halls in practice

Computer

Digital Signal Processors

Software

D/AConvertor

Amplifier

Speakers

Sound

A/DConvertor

Tables

Virtual hall

Figure 8.1: A block diagram of the process to synthesize the wave �eld of a virtual hall

8.3. Software used for wave �eld synthesis 65

of each wall in which it is re ected:

Si(!) =Yj

S(!)p1� �j : (8.1)

Here, S(!) is the source spectrum of the primary source (the recorded signal), Si(!) the source

spectrum of the mirror image source i and �j the absorption coe�cient of the wall j. Note that in our

case the absorption coe�cient �j is independent of the frequency !. Frequency dependent absorption

could be included but is not easy in practice because a �lter should be implemented for each mirror

image source.

The DSPs calculate driving signals from weighting and delay tables, such that each loudspeaker[1]

emits for each source a weighted source signal after it has undergone a delay:

ql(t) =Xi

Wl;i s(t�Dl;i): (8.2)

Here, ql is the driving signal for loudspeaker l in the time domain, s the source signal, W the weighting

table and D the delay table. The strength of each mirror image source is included in the weighting

factors of the weighting table. Transforming the expressions for the driving signal (equation (3.35)

and equation (3.42)) to the time domain, one obtains for a single loudspeaker l and a single virtual

source i:

qml;i(~r0l;i; t) = gml;i

si(t) cos�incl;i

rjk

2�

�(t� r0l;ic )p

r0l;i�x (8.3)

for a single loudspeaker l on a line array and

qll;i(~r0l;i; t) = gll;i si(t) cos�incl;i

jk

2�

�(t� r0l;ic )

r0l;i�x (8.4)

for the loudspeakers l on a vertical line of the DVLS. Here, �x is the sample distance between the

loudspeaker l and its neighboring loudspeakers l�1 and l+1. From equation (8.3) and equation (8.4),

the delay entry for the loudspeaker l and the virtual source i is seen:

Dl;i =r0l;ic: (8.5)

Using line arrays to synthesize the sound �eld, the frequency dependent factorq

jk2� (see equation (8.3))

occurs. An in�nite impulse response �lter (IIR �lter) is used taking care of the factorpk and the

loudspeaker characteristics so that a at spectrum is realized[2]. The other part under the root sign

is a phase shift of 45� and an ampli�cation factor. If this part is ignored for all loudspeakers when

synthesizing all sources, no problem occurs. When only the early re ections are synthesized, without

enhancing the primary source, the ampli�cation factor should be considered in the weighting factor to

[1]The word \loudspeaker" is used for a loudspeaker of the line arrays and for the loudspeakers on a vertical line ofthe DVLS[2]By lack of computational power, the factor

pk may be ignored in our case without distorting the spectrum to much,

because it is compensated more or less by the loudspeaker characteristics.


realize a good acoustic balance. Using a DVLS to synthesize the sound �eld, also an IIR �lter has to

be used[3], to take care of the term jk2� of equation (8.4). The phase shift of 90� and the ampli�cation

may be ignored like by the line arrays. The remaining part of the driving signal, containing the

amplitude factor g, cos�inc, the sample distance and the amplitude decrease due to re ections (see

equation (8.1)), are stored in the weighting table W.

8.4 The amplitude factor of the driving signal

As read in the previous chapters, there are several ways to calculate the amplitude factor gm. Three

ways were used:

� optimizing the sound �eld in a point, using gp

� optimizing the sound �eld on a line, using gc

� optimizing the sound �eld on the optimal curve, using gc;opt.

Hereafter, these subjects are described individually. All simulations are done with the setup of the

loudspeaker arrays in the reproduction room described in section A.2.

Optimizing in a point

In order to obtain the correct amplitude in one point, the equation of the amplitude factor gp for a

�xed point is used. Suppose that a source is located at (x; z) = (0;�4) (m) and an optimal sound �eld

is desired in the center of the reproduction room (at (x; z) = (0; 0) (m)). Then, using equation (3.45)

(or equation (3.46) for a DVLS) one obtains an amplitude factor for each loudspeaker of the nearest

array as shown in �gure 8.2. For the line array, the amplitude factor decreases in the directions of the

edges of the array. This is the case when the source-to-array distance (r0) is smaller than the distance

from the array to the optimal receiver position �r0. If r0 > �r0, then the amplitude factor increases

in the direction of the edges of the array. For the DVLS, the opposite happens. To reduce di�raction

e�ects, the driving signals at the outer limits of the arrays have to decrease to zero (see [Start, 1997]).

Therefore, a taper window has to be used.

Although the sound �eld is optimized in a point, there are more places in the receiver area where

amplitude deviation is minimal. One obtains the receiver curve RC, the curve where the amplitude

of the synthesize sound �eld matches the sound �eld of the synthesized monopole source, from the

amplitude factor of each secondary source, just as done in chapter 5. Figure 8.3 shows receiver curves

for three di�erent sources, calculated with equation (5.8) where gpl is substituted for of gc;optl . Here,

it is seen that a point solution is properly also a line solution. The receiver curves are the same for a

line array and a DVLS.

[3]The loudspeakers do not compensate the factor k, so that an IIR �lter may not be omitted.

8.4. The amplitude factor of the driving signal 67

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(a)

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(b)

Figure 8.2: The amplitude factors gpm (a) and gpl (b) which results to a point solution.

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

1

2

3

2

3

1

x (m)

z(m)

Figure 8.3: The receiver curves where the amplitude of the synthesized sound �eld equals theamplitude of the real sound �eld in the case gp is used in the driving signal Q, forthree di�erent sources.


Optimizing on a straight line

Another way to optimize the sound �eld is using the amplitude factors gc, such that the synthesized

wave �eld matches the real wave �eld on a straight receiver line. The receiver line is chosen perpen-

dicular to the line that connects the source with a speci�c point in the receiver area, as shown in

�gure 4.2. If the receiver curve is parallel with the loudspeaker array, which is the case when the

source is placed at (x; z) = (0;�4) (m), the amplitude factor is a constant for all loudspeakers (see

�gure 8.4).

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(a)

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(b)

Figure 8.4: The amplitude factors gcm (a) and gcl (b) which results to a (straight) line solution.

Optimizing on the optimal curve

Finally, �gure 8.5 shows some optimal amplitude factors gc;opt, for a source located at (x; z) =

(0;�4)(m). Note that for the line array, the amplitude factors become smaller towards the edges

so that no taper window is needed to reduce di�raction e�ects. Looking at the amplitude factors of

the DVLS, it is clearly seen that they become larger towards the edges. This increases di�raction

e�ects and a taper window should not be omitted. Note that this taper a�ects the optimal ampli-

tude distribution so that, if a large receiver area is de�ned in comparison to the area between the

loudspeaker arrays, the amplitude deviation does not approach zero in the corners of the receiver

area.

8.4. The amplitude factor of the driving signal 69

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(a)

−2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (m) �!

g�!

(b)

Figure 8.5: The amplitude factors gc;optm (a) and gc;opt

l (b) which results to the optimal curvesolution.


Chapter 9

Perceptual measurements

9.1 Introduction

In previous chapters, it is seen that the synthesized pressure �eld's amplitude matches the real pressure

�eld's amplitude of a monopole on a curve. On all other places, the amplitude distribution is incorrect.

In this chapter, the extend in which the incorrect amplitude distribution is perceptual is investigated.

Therefore, perceptual experiments were done in a reproduction room where the sound �eld of the

virtual hall described in section A.4. was synthesized using four line arrays. Especially attention was

paid to the sound �eld of the early re ections, assuming that the direct sound is not ampli�ed and

the arrays are only used to synthesize the acoustics of the virtual hall. The goal was to �nd how far

away along the x-axis a subject can move from a point where the signal is optimized, without hearing

an amplitude di�erence. This distance is indicated as �x̂0.

9.2 Method of measurement

9.2.1 Moving the optimal position

A subject may not move to another location during a measurement, otherwise he should have too

much information if he is located on the position where the sound �eld is optimized or not. Therefore,

the optimization position �x is changed while the subject stayed on the same position. However,

this is not reciprocal. A short investigation showed that the amplitude deviations are larger when the

subject moves to another position while the optimization position is kept constant, than when the

optimization position is changed while the subject stays at the same position. By using the second

method, an upper limit �x0 for �x̂0 is found.

71

72 Chapter 9. Perceptual measurements

9.2.2 The two alternative forced choice

To �nd the distance �x0 where a subject hears a di�erence between the correct optimized and the

incorrect optimized signal, a \two alternative forced choice" (2 AFC) was used. Here, the subject has

to judge stimuli by choosing between two alternatives of which one meets a certain requirement. If the

di�erence between the stimuli is large enough, the subject gives the correct answer with a probability

that approximates one. If the di�erence between the stimuli is small, he cannot detect it so that if

enough stimuli are o�ered, the number of times the incorrect answer is given approximates the number

of times the correct answer is given (the probability on a correct answer approximates 0.5). When the

probability exceeds 0.75, the di�erence is said to be noticeable.

The two alternatives a subject could choose from were two pairs of signals. One pair existed of a signal

optimized at the position where the subject was located (�x = 0) and a signal optimized at another

position, �x apart from the subject. The second pair existed of two identical signals optimized at

the subject's position. The two pairs were at random o�ered to the subject, while the optimization

position ~�x of the signal changed in each series of pairs.

To shorten the length of the experiments an adapted 2 AFC method was used. Instead of two pairs of

signals, only three signals were presented. Each triple consisted of one incorrect signal and two cor-

rect signals: incorrect-correct-correct, or two correct signals and one incorrect signal: correct-correct-

incorrect. The combinations correct-incorrect-correct or incorrect-correct-incorrect did not occur. The

subject was asked the question which pair of signals, (1 & 2 or 2 & 3) were the same. This adaption

of the 2AFC method does not a�ect the qualities of the normal 2 AFC method.

9.3 Fitting the psychometric curve

In order to determine the distance �x0 from the subject's position where the sound �eld must be

optimized so that he hears a di�erence between the sound �elds, i.e., where the probability of correct

answers exceeds 0.75, a psychometric curve was �tted on the answers. The psychometric curve is an

approximation of the integrated normal distributed Gauss probability curve and is given by

p(�A;�A0; s) =1

2+1

2

1

1 + exp �(�A��A0)s

!(9.1)

for the two alternative forced choice method [Raatgever, 1991]. In this equation, p stands for the

probability of detecting a correct signal, �A for the variable in question (a perceived amplitude

change due to a change of the optimization position), �A0 for the value where the Gauss-curve has

a maximum (or where the psychometric curve equals 0.75) and s is related to the standard deviation

of the subject.

If the number of correct answers for optimization distance �x is indicated as g�x, and the number of

o�ered triplets for the same distance as N�x, the estimation of the measured probability reads

~p�x =g�x

N�x: (9.2)

9.3. Fitting the psychometric curve 73

To �t the psychometric curve through the probabilities, the software package Matlab was used to solve

the non-linear least squares problem:

minimize

(X�x

f(p(�A;�A0; s)� ~p�x)W�xg2): (9.3)

In this equation,W�x stands for a weighting factor that is described in section 9.3.2. �A is a function

of �x as clari�ed in the next section.

According to the NAG manual [N.A.G., 1995], the standard deviations of the estimated parameters

~�A0 and ~s are written as

� ~�A0=

�2S

m� nH ~�A0; ~�A0

�2and �~s =

�2S

m� nH~s;~s

�2: (9.4)

Here, m is the number of points where the curve has to be �t through, n the number of parameters

of the psychometric curve to be estimated, S is the sum of squares and H is the Hessian calculated

with:

H = (2JTJ)�1: (9.5)

The Jacobian J is given as a result from the least squares minimization by Matlab. The con�dence

interval on �A0 is given by

~�A0 � � ~�A0t(�=2;m�n) < �A0 < ~�A0 + � ~�A0

t(�=2;m�n); (9.6)

where t(�=2;m�n) is the Student-distribution with � = (m�n) degrees of freedom and � the inreliability

threshold. This distribution is found in a t�(�) table ([van Soest, 1992]).

9.3.1 Presentation of the processed data

To �t a psychometric curve through the data, one has to ensure that a variation in the optimization

distance �x is linear with a change in the perception of the sound �eld di�erence. The di�erence

(or the cue) in our measurement is the amplitude di�erence of the sound �eld, which is not linear

with a variation of �x. Hence, the amplitude di�erence has to be set on the horizontal axis of the

curve �tting to obtain a linear scale. Because the human hearing is logarithmic sensitive for amplitude

changes, the logarithmic amplitude change of the sound �eld is considered. In the experiments, the

sound �eld of multiple sources were synthesized (the primary source and the mirror images sources),

each having a contribution to the perceived amplitude di�erence while their sound �eld is coming from

di�erent directions. Most subjects stated that a shift in the optimization position was perceived as a

decrease in the amplitude or a shift of the source position in the same direction. In addition, a timbre

change could be perceived. Therefore, a scale for �A was used depending on the di�erence between

the total amplitude of the pressure �eld at the subjects' position due to all mirror image sources at


his left-hand side and all mirror image sources on his right-hand side[1]:

�A = 20 log

0@Xi;left

ALi(r0;�r;�r0)

1A � 20 log

0@ Xi;right

ALi(r0;�r;�r0)

1A : (9.7)

In this equation, ALi is the amplitude of source number i synthesized by the line arrays. Note that

~�r0 is the position on the receiver curve RC belonging to the stationary phase line of the stationary

phase receiver (the subject). This receiver curve is the straight line through ~�x perpendicular to the

line from this position to the primary source (or a mirror image source) as shown in �gure 9.1.

�x�r0

RC

R

Lstat

Figure 9.1: The optimization position for the stationary phase source belonging to the stationaryphase receiver for a sound �eld optimized �x (m) from the receiver.

Table 9.1 shows the values of the optimization distances and the values of �A calculated with equa-

tion (9.7) for the speci�c con�guration given in section A.4.

Table 9.1: The optimization distance �x and the amplitude di�erence of the sound �eld betweenthis position and the position of optimization.

�x (m) �A (dB)0 012 1.05671 2.1602

1 12 3.37092 4.7880

2 12 6.63123 9.8903

After �tting the psychometric curve, the distance ~�x0 and its con�dence interval are calculated from

the found threshold ~�A0 and the con�dence interval given in equation (9.6) respectively. This is

[1]This does not include the perception of timbre deviations.

9.3. Fitting the psychometric curve 75

done by linear interpolation of the values calculated with equation (9.7), and shown in table 9.1 and

�gure 9.2.

0 0.5 1 1.5 2 2.5 30

1.1

2.2

3.4

4.8

6.6

9.9

�x �!

�A�!

~�x0

~�x0+�~�x0

~�x0 ��~�x0

~�A0

~�A0 + � ~�A0

~�A0 � � ~�A0

Figure 9.2: The linear interpolation used to derive the threshold optimization distance �x and itscon�dence interval from the threshold �A and its con�dence interval.

9.3.2 The weighting factors

To �t the psychometric curve through the data, the data is weighted with a weighting factor W�x:

W�x =1

�2~p�x(9.8)

where �~p�x is the standard deviation of a binomial distribution belonging to the estimated probability

~p�x. This standard deviation is given by

�~p�x =

s~p�x(1� ~p�x)

N�x: (9.9)

In the case all answers of a given optimization position are all correct or all incorrect, ~p�x or (1�~p�x)

equals zero, making the weighting factor W�x in�nite large. To prevent this an alternative deviation

is used ([Maat, 1997]):

�~p�x =

vuut�g�x+ 1

2

N�x+1

��1� g�x+

1

2

N�x+1

�N�x + 1

: (9.10)

This formula says that an imaginary next answer is given that is half correct, half incorrect.


9.3.3 The chi-square test

To test the goodness of �t, the chi-square (�2) test was used. Chi-square is given by ([Taylor, 1982]):

�2 =X�x

~p�x � p(�A; ~�A0;~s)

�~p�x

!2

: (9.11)

The chi-square test rejects the results if

�2 > �2�(m� n); (9.12)

where �2�(m � n) is the chi-square value with an unreliability � in the case of (m � n) degrees of

freedom. This value is found in a �2�(�) table ([van Soest, 1992]). The unreliability � means that if

the results are rejected, one has a probability of � they were truthful and thus incorrect rejected.

9.4 Adjusting the amplitude factors

After pilot experiments, it seemed that subjects were able to recognize the correct signal, i.e., the signal

that was optimized at the position of the subject. This way, he is able to know which combination

was o�ered after only the �rst of the three signals was o�ered (if the subject recognizes the �rst signal

as the correct one, he knows that the combination should be correct-correct-incorrect, otherwise, is

should be incorrect-correct-correct). This had to be eliminated because this recognition may be due

signal characteristics other than the amplitude properties. Hence, the primary source and the mirror

image sources were shifted randomly along the x-direction after each triple of signals such that each

new triple was perceived coming from a di�erent distribution of directions and having di�erent spectral

properties. Because a shift of the sources results in an amplitude change in the whole receiver area,

the amplitude factors were adjusted such that the distribution equals the amplitude distribution due

to the non-shifted sources.

As told in the introduction, only the amplitude di�erences were concerned by changing the opti-

mization positions of the early re ections. Hence, the amplitude factor of each loudspeaker was kept

constant in each triple for the primary source, giving a correct amplitude at the subject's position.

This way, the primary source produced a constant masking of the perception of the amplitude dif-

ferences, caused by the variation of the optimization position of the sound �eld of the mirror image

sources.

9.5 Results

Figure 9.3 to 9.11 show the results of the psychometric curve �tting through the measurements for 9

di�erent subjects. The sound �eld was optimized on 13 di�erent positions along the x-axis, 6 to the

left-hand side of the subject, 6 to the right-hand side of the subject, and one, the correct signal, at the

subject's position. Because optimizing a distance �x towards the left-hand side gives the same result

9.5. Results 77

as optimizing a distance �x towards the right-hand side, the results are taken together resulting in 6

optimization distances �x giving an incorrect amplitude distribution. The �gures show the fraction of

correct answers ~p�x and their standard deviance �~p�x . All numerical values are shown in appendix B.

Through the values, the psychometric curve is �tted. Because there are 6 �t-points and two parameters

to estimate for the curve �t, the number of degrees of freedom (�) equals 4. The results of the �t

together with the con�dential interval are shown in the text-box in the lower right corner. The rescaled

threshold optimization distance ~�x0 and the con�dential interval of �x0 is shown in the upper right

text-box. The middle curve belongs to the estimated parameters ( ~�A0 and ~s) for the psychometric

curve. The other two curves are the outer limits of the maxima and minima due to the deviance of

the variables ~�A0 and ~s. Figure 9.12 shows the results of all the experimental subjects together.

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1

= 1.0165 ± 0.57302 = 5.0799 ± 1.0368 = 2.3676

= 2.0792, 1.7372< <2.3605

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.3 Results of DV, N�x = 30

0 0.5 1 1.5 2 2.5 30.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

= 0.12788 ± Inf = 9.9422 ± Inf = 1.2592

~p�!

�x �!

~�A0~s

�2

Figure: 9.4 Results of EL, N�x = 30

0 0.5 1 1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 3.197 ± 4.8119 = 5.7711 ± 4.7277 = 14.0867

= 2.2667, 0.49369< <NaN

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.5 Results of JB, N�x = 30

0 0.5 1 1.5 2 2.5 30.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

= 1.3595 ± 0.84476 = 3.2447 ± 1.3576 = 2.9527

= 1.4479, 0.87628< <1.9345

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.6 Results of JG, N�x = 40


0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1

= 1.7747 ± 0.51257 = 3.5458 ± 0.9745 = 3.4763

= 1.5617, 1.1698< <1.9055

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.7 Results of JL, N�x = 80

0 0.5 1 1.5 2 2.5 30.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

= 1.9558 ± 1.4517 = 3.1171 ± 1.8829 = 5.4528

= 1.3952, 0.58044< <2.0575

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.8 Results of JS, N�x = 40

0 0.5 1 1.5 2 2.5 30.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 1.4044 ± 2.2208 = 5.6133 ± 2.1759 = 7.9971

= 2.2239, 1.5235< <2.6777

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.9 Results of PS, N�x = 30

0 0.5 1 1.5 2 2.5 30.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 3.1479 ± 3.772 = 5.3308 ± 3.5401 = 9.3242

= 2.1472, 0.83261< <2.8436

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.10 Results of RR, N�x = 30

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1

= 1.9096 ± 1.1563 = 4.5206 ± 1.5896 = 3.7514

= 1.9056, 1.3183< <2.3587

~p�!

�x �!

~�A0~s

�2

~�x0 �x0

Figure: 9.11 Results of TP, N�x = 30

9.5. Results 79

DV EL JB JG JL JS PS RR TP0

0.5

1

1.5

2

2.5

3

3.5

4

← −

←

← +

�x�!

Experimental subject

�x0 � ~�x0

�x0

�x0 � ~�x0

Figure 9.12: The results of all subjects.

From the �gures is seen that some subjects could perceive an optimization shift much better than

others. The results of subject EL do even not exceed a probability of 0.75. Note that the optimization

distance threshold and its con�dential intervals are not calculated for EL because ~�A0 and ~�A0 �� ~�A0

t(�=2;m�n) fall outside the intervals of table 9.1 and, therefore, they cannot be interpolated back.

The same happens for the maximum of the con�dential interval for JB.

Table 9.2 shows the �2 values calculated with equation (9.11) and the values for �2�(4) found in a

table, for 4 degrees of freedom and di�erent improbabilities �.

Table 9.2: Goodness of �t using the �2 test.

Subject �2 � = 0:25 � = 0:10 � = 0:05

DV 2.37 5.39 7.78 9.49EL 1.26 5.39 7.78 9.49JB 14.09 5.39|| 7.78|| 9.49||JG 2.96 5.39 7.78 9.49JL 2.95 5.39 7.78 9.49JS 5.45 5.39|| 7.78 9.49PS 8.00 5.39|| 7.78|| 9.49RR 9.32 5.39|| 7.78|| 9.49TP 3.75 5.39 7.78 9.49

From the table it is seen that the measurements of JB may be rejected with only 5 % probability of


incorrect rejection. The �2 value of EL is very low, giving a high probability that the measurements

may not be rejected. But note that the �2 test only tells in what extent the data distribution can be

described by the psychometric curve.

It is possible that the measurements are in uenced by other cues than amplitude di�erences. DV

told a di�erence in timbre could be perceived. A short investigation is done to see if this e�ect

was large enough to be an undesired dominant cue. The conclusion is that the timbre di�erences

change in a totally di�erent way than the amplitude di�erences, i.e., the change of the spectrum does

not change monotonically with a change in optimization distance. In addition, a spectrum variation

occurs by shifting all the sources to omit source recognition. However, the spectrum did not change

randomly and therefore it may in uence the measurements, but if this variation was a dominant cue,

the psychometric curve �t should be much worse. Maybe this is the explanation why JB and RR have

a high probability of detection when the sound �eld is already optimized on �x = 1 (m) in comparison

with the surrounding optimization distances. For the measurements of these subjects, the high values

of �2 give a low probability of incorrect rejection of the model based on amplitude di�erences. It is

remarkable the �2 test for the measurements of DV who heard the di�erences in timbre shows a low

value of �2.

Although, the perception may be in uenced by other factors, one may conclude that the incorrect

amplitude distribution is audible. For the subjects, the estimated upper limit �x0 for �x̂0 varies

between 1:4 (m) to 2:2 (m), which is roughly 14 to 1

3 of the total room length. Note that these values

hold for the con�guration used in our measurements. For other con�gurations the following counts:

� For a larger virtual hall, the mirror image source-to-array distance grows. Hence, the amplitude

error increases and �x0 should decrease.

� For a larger array, while the virtual hall remains the same, the mirror image source-to-array

distance decreases. Hence, the amplitude error decreases and �x0 should increase.

� If the size of the array and the virtual hall have the same proportional increase, the threshold

�x0 increases also with this proportion.

The last point is explained using the amplitude function for a line array, equation (4.7). If the con�g-

uration grows with a fraction n, the amplitude due to a particular source reads:

AL(nr0; n�r; n�r0) =

rn�r0

nr0 + n�r0

s1

(nr0 + n�r)n�r=

AL(r0;�r;�r0)

n(9.13)

Note that an increase of �r0 by the factor n results into an increase of �x0 by the same factor

n as seen in �gure 9.1. Considering the logarithm of equation (9.13), it is seen that everywhere in

the receiver area, the amplitude changes for every source with � log(n)[dB]. Hence, the amplitude

di�erences between the sources remain the same if the con�guration and the threshold distance �x0

grow with factor n.

Using a DVLS to synthesize the wave �eld of the early re ections, the amplitude errors are much

smaller. Hence, the perception of the amplitude errors should be less or even inaudible.

Chapter 10

Conclusions and recommendations

10.1 Conclusions

The goal of this research was to synthesize as well as possible the wave �eld of early re ections of a

�ctive hall with a line array. The amplitude distribution of the synthesized wave �eld is chosen to be

optimized because a line array can synthesize a sound �eld correctly up to a certain frequency with

respect to its phase, but the amplitude is only synthesized in a correct way on a single curve. Perceptual

measurements showed that an incorrect amplitude distribution can be perceived at positions where

the sound �eld is not optimized.

To examine the amplitude distribution, the \amplitude function" has been derived. The assumption for

this derivation is that the amplitude along a stationary phase line is only in uenced by the amplitude

of its stationary phase source. This function has proved to be very convenient because of its easy

implementation and its simple form compared with the existing wave �eld extrapolation methods.

With the amplitude function, the optimal driving signal for secondary sources of the line array is found,

making the amplitude deviation minimal in a given receiver area. Hereby, the amplitude deviation is

especially reduced at positions for which the receiver area addressed by a stationary phase source is

relatively small. Since the amplitude attenuation cannot be changed along a stationary phase line, the

optimization does not lead to a tremendous improvement.

To reduce the amplitude signi�cantly, the distribution of vertical line sources (DVLS) is introduced.

For this array, the same computational e�ort is needed as for the line array. Also a DVLS does not

synthesize the amplitude of the pressure �eld correctly in the whole receiver area, but only on a

contour. The amplitude deviation of the pressure �eld generated by a DVLS is reduced by minimizing

the error in a receiver area like it has been done for the line array. Also here, the improvement is not

very large.

Synthesizing the wave �eld of a source lying close to the array, the DVLS has a larger amplitude

deviation than the line array. If the source-to-array distance is larger than a third to a half times the

receiver area length, the DVLS has to be used to obtain the smallest deviation.

81

82 Chapter 10. Conclusions and recommendations

When synthesizing the sound �eld of a primary source in a virtual hall the following can be stated:

� The primary source signal should be synthesized by a line array to obtain the minimal amplitude

deviation, only if the source-to-array distance is smaller than a third to a half times the receiver

area length.

� The mirror image sources, representing the early re ections of a �ctive hall, lie far away from

the arrays so that using a DVLS to synthesize this wave �eld leads to a large improvement.

� The reverberation, represented as plane waves, has a negligible amplitude deviation when a

DVLS is used to synthesize the wave �eld.

� The optimized driving signal has to be used for reducing the remaining amplitude deviation.

10.2 Recommendations for future research

It has been made clear that the DVLS is best used for synthesizing the wave �eld of early re ections

and reverberation of virtual halls and therefore, it is highly recommendable to realize this array to do

further research. In this thesis, a DVLS that is in�nitely extended in the vertical direction is considered

by simulation. This is impossible in practice. Hence, one should investigate the e�ects of truncating

the DVLS in vertical direction and of spatial sampling with a �nite distance.

By truncating the array in the vertical direction, the properties of the array will change. The pressure

�eld will have an amplitude behavior that lies between an in�nite DVLS and a line array and on

the edges di�raction occurs, which are larger than the di�raction e�ects in the horizontal direction.

The di�raction e�ects can be reduced by using a taper in the vertical direction, but this reduces the

extension in vertical direction even more. One could investigate how the reduction of the vertical

length in uences the amplitude behavior of the DVLS. The transition distance, for which the line

array and the shortened DVLS have the same amplitude error, could move so close to the array that

the shortened DVLS may even be better used for synthesizing the sound �eld of the direct sound.

For a DVLS, the maximal aperture is smaller in vertical direction than in horizontal direction, so that

the DVLS may be sampled with a larger distance in the vertical direction to ful�ll the anti-aliasing

condition.

Finally, it may be possible to synthesize the di�raction of the walls of the virtual hall by a line array

or a DVLS, by accessing only parts of the array. The di�raction of the array should coincide with the

di�ractions of the virtual walls. The active parts of the array can be determined with the visibility

test.

Appendix A

Con�gurations used for simulations

and measurements

A.1 The con�guration used for simulations without a virtual

hall

For the simulations in chapters 4, 6 and 7 the con�guration shown in �gure A.1 is used. The loudspeaker

array is located on the line z = 0 (m) and supposed to be in�nite along the x-axis. The listening area,

where the sound �eld's amplitude is calculated, is a rectangular area. The corners are positioned at

(x; z) = (�3; 1), (x; z) = (3; 1), (x; z) = (3; 7) and (x; z) = (�3; 7) (m). The position of the source

di�ers for each simulation.

Receiver area

Loudspeaker array1

2

7

3

8

4

5

6

0-5 -4 -3 -2 -1 1 2 3 4 5

-1

-2

-3

0

Source area

x (m)

z (m

)

Figure A.1: The con�guration used for simulations.

83

84 Appendix A. Con�gurations used for simulations and measurements

A.2 The room used for simulations and measurements

The reproduction room, used for synthesizing the sound �eld of a virtual hall is shown in �gure A.2.

The length of the room is 7:5 (m) whereas the width measures 5:9 (m). The height of the room is

2:4 (m). In this room, a rectangular speaker array is suspended existing of ten loudspeaker bars. Each

loudspeaker bar exists of 16 loudspeakers that are driven individually. The rectangular loudspeaker

array dimensions are 4:06 (m) by 6:10 (m). These dimensions are also used for some simulations.

Figure A.2: The reproduction room in which the sound �eld of a virtual hall is synthesized.

A.3 The con�guration of the virtual hall used for simulations

The virtual hall used for the simulations is shown in �gure A.3. The corners are de�ned by (x; y) =

(�4;�6), (x; y) = (�8; 7), (x; y) = (8; 7) and (x; y) = (4;�6) (m). All the walls are assumed acousti-

cally hard with an absorption coe�cient of 0.05. Figure A.3 also shows the walls of the reproduction

room, the loudspeaker arrays and the listening area AR, which was taken to be 90% of the area

enclosed by the loudspeaker arrays.

A.4. The con�guration used for perceptual measurements 85

0

0

-8 -6 -4 -2 2 4 6 8

-2

-8

-4

-6

2

4

6

Receiverarea

Virtual room

Reproduction room Loudspeaker array

x (m)

z (m

)

Figure A.3: The virtual hall used for simulations.

A.4 The con�guration used for perceptual measurements

For the perceptual measurements, another virtual hall con�guration as for the simulations is used.

Here, the corners are de�ned by (x; y) = (�4;�5), (x; y) = (�9; 9), (x; y) = (9; 9) and (x; y) =

(4;�5) (m). The absorption coe�cients for all walls are still 0.05. The primary source is positioned at

(x; y) = (0;�6) (m). Mirror image sources are calculated with the 2-dimensional mirror image source

model until the 3th order. Only the primary source and 25 mirror image sources were synthesized. The

DSPs can process this maximum in our con�guration. The subject's position is (x; z) = (0; 0)(m), in

the center of the reproduction room. The virtual hall, the mirror sources and their placement with

respect to the reproduction room is shown in �gure A.4.

-40 -30 -20 -10 0 10 20 30 40

-40

-30

-20

-10

0

10

20

30

virtual room

reproduction room

loudspeaker array

subject primary source

x (m)

z (m

)

Figure A.4: The con�guration used for the perceptual measurements

86 Appendix A. Con�gurations used for simulations and measurements

Appendix B

Numerical results of perceptual

measurements

This appendix contains the numerical results of the perceptual measurements. For each subject a

table is shown. The �rst rows of a table give the number of correct answers g ~�x for the optimization

position ~�x and for di�erent sessions. For each session, �ve triple of signals per optimization position

were o�ered. The estimated probability of correct answers and its standard deviation, calculated with

equation (9.2) and equation (9.10) respectively, are shown in the last two rows. Note that these values

are calculated for the absolute value of ~�x, given in percent. Table B.10 contains the results of JL

when the primary source was optimized together with all mirror image sources at ~�x, while for all

other results, the primary source optimization position was taken at �x = 0 (m) for each signal.

Table B.1: Results of DV

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

g ~�x session 1 5 5 4 4 3 3 2 2 3 3 3 5g ~�x session 2 5 5 5 3 2 3 4 4 3 4 5 5g ~�x session 3 5 5 1 2 3 2 2 5 4 3 4 5

~p�x 53 63 63 67 90 100�~p�x 9 8.7 8.7 8.5 5.7 2.3

Table B.2: Results of EL

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3


~p�x 50 50 43 57 47 70�~p�x 9 9 8.9 8.9 9 8.3

87

88 Appendix B. Numerical results of perceptual measurements

Table B.3: Results of JB

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3


~p�x 37 80 60 63 77 90�~p�x 8.7 7.3 8.8 8.7 7.7 5.7

Table B.4: Results of JG

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

g ~�x session 1 5 5 5 2 2 4 1 3 3 4 5 5g ~�x session 2 5 5 4 4 5 3 5 4 3 5 2 5g ~�x session 3 5 5 4 3 3 1 4 2 5 5 4 5g ~�x session 4 5 5 4 4 3 4 3 5 4 5 5 5

~p�x 63 68 70 90 90 100�~p�x 7.6 7.3 7.2 4.9 4.9 1.7

Table B.5: Results of JL

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

g ~�x session 1 5 5 5 4 3 3 3 3 5 5 5 5g ~�x session 2 5 4 3 3 3 4 3 3 4 4 5 5g ~�x session 3 5 5 3 3 5 5 2 4 5 4 4 5g ~�x session 4 5 5 4 5 2 3 1 3 2 4 5 5g ~�x session 5 5 4 4 4 4 4 2 3 5 5 5 5g ~�x session 6 5 5 4 4 3 2 1 3 4 5 3 5g ~�x session 7 5 4 5 5 4 1 3 3 2 3 4 4g ~�x session 8 5 5 4 5 2 3 4 2 4 3 5 5

~p�x 55 63 80 81 91 99�~p�x 5.5 5.4 4.5 4.4 3.2 1.5

Table B.6: Results of JS

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

g ~�x session 1 5 4 5 4 3 2 2 5 4 5 4 5g ~�x session 2 5 4 5 4 2 2 1 3 4 3 4 5g ~�x session 3 5 5 5 4 3 3 1 4 1 4 5 5g ~�x session 4 4 5 5 5 4 3 5 3 5 4 5 5

~p�x 48 68 78 90 90 98�~p�x 7.8 7.3 6.6 4.9 4.9 2.9

89

Table B.7: Results of PS

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3


~p�x 53 40 47 73 87 87�~p�x 9 8.8 9 8 6.3 6.3

Table B.8: Results of RR

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3


~p�x 43 77 53 77 80 90�~p�x 8.9 7.7 9 7.7 7.3 5.7

Table B.9: Results of TP

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3


~p�x 53 63 53 83 87 97�~p�x 9 8.7 9 6.9 6.3 3.9

Table B.10: Results of JL,primarysourcevariable.

~�x (m) -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3

g ~�x session 1 5 5 3 5 2 3 1 4 4 5 5 5g ~�x session 2 5 5 5 4 4 3 4 2 5 4 5 5g ~�x session 3 5 4 4 3 4 3 4 3 4 5 5 5g ~�x session 4 5 5 5 2 3 2 3 4 5 4 5 5g ~�x session 5 5 5 5 4 3 3 4 3 4 5 5 5

~p�x 60 64 80 90 98 100�~p�x 6.9 6.7 5.7 4.3 2.4 1.4

90 Appendix B. Numerical results of perceptual measurements

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