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The single-layer potential approach applied to sound field synthesis including cases of non-enclosing distributions of secondary sources [Elektronische Ressource] / vorgelegt von Jens Ahrens

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The Single-layer Potential Approach Applied toSound Field Synthesis Including Cases ofNon-enclosing Distributions of Secondary Sourcesvorgelegt vonDiplom-IngenieurJens Ahrensaus FriedbergVon der Fakult¨at IV - Elektrotechnik und Informatikder Technischen Universit¨at Berlinzur Erlangung des akademischen GradesDoktor der IngenieurwissenschaftenDr.-Ing.genehmigte DissertationPromotionsausschuss:Vorsitzender: Prof. Dr.-Ing. Alexander RaakeBerichter: Prof. Dr.-Ing. Sebastian M¨ollerBerichter: Dr.-Ing. Dr. Tech. h. c. Jens BlauertTag der wissenschaftlichen Aussprache: 04.10.2010Berlin 2010D 83AbstractThe present dissertation treats the topic of sound field synthesis. The focus liesthereby on serving human listeners although the results can be also exploited inother applications such as underwater acoustics or ultrasonics. A fundamental for-mulation of the problem is derived based on standard integral equations and thesingle-layer potential approach is identified as a useful tool in order to derive ageneral solution. An explicit solution is derived exemplarily for inward-radiatingspherical distributions of secondary sources.The drawback of the single-layer potential approach is the fact that it requiressecondary source distributions which enclose the receiver volume.
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The Single-layer Potential Approach Applied to
Sound Field Synthesis Including Cases of
Non-enclosing Distributions of Secondary Sources
vorgelegt von
Diplom-Ingenieur
Jens Ahrens
aus Friedberg
Von der Fakult¨at IV - Elektrotechnik und Informatik
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
Dr.-Ing.
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Alexander Raake
Berichter: Prof. Dr.-Ing. Sebastian M¨oller
Berichter: Dr.-Ing. Dr. Tech. h. c. Jens Blauert
Tag der wissenschaftlichen Aussprache: 04.10.2010
Berlin 2010
D 83Abstract
The present dissertation treats the topic of sound field synthesis. The focus lies
thereby on serving human listeners although the results can be also exploited in
other applications such as underwater acoustics or ultrasonics. A fundamental for-
mulation of the problem is derived based on standard integral equations and the
single-layer potential approach is identified as a useful tool in order to derive a
general solution. An explicit solution is derived exemplarily for inward-radiating
spherical distributions of secondary sources.
The drawback of the single-layer potential approach is the fact that it requires
secondary source distributions which enclose the receiver volume. Extensions to
the single-layer potential approach are proposed which allow for a derivation of ex-
plicit solutions for circular, planar, and linear distributions of secondary sources.
Based on above described formulation it is shown that the two established analyt-
ical approaches of Wave Field Synthesis and Near-field Compensated Higher Order
Ambisonics constitute specific solutions to the general problem which are covered
by the single-layer potential solution and its extensions.
The physical theory of the single-layer potential approach requires that the em-
ployed distributions of secondary sources are continuous. Such continuous distribu-
tions can not be implemented in practice with today’s available loudspeaker tech-
nologybut discrete distributions ofloudspeakers have tobeused. The consequences
of this spatial discretization of the secondary source distribution are analyzed in
detail for all above mentioned geometries in different spatial frequency domains, in
temporal frequency domain, and in time domain. Two fundamental results are de-
rived: Firstly, the discretization leads to repetitions of the secondary source driving
function in a spatial frequency domain which is determined by the geometry of the
secondary source distribution under consideration. And secondly, the bandwidth of
the driving function with respect to the according spatial frequency domain has es-
sential influence on the properties of the synthesized sound field. As a consequence,
the concept of categorizing sound field synthesis approaches according to the band-
width of the driving function into narrowband, wideband, and fullband approaches
is proposed.
It is finally shown how different types of spatial bandwidth limitation can be
employed in order to locally increase the accuracy of the synthesized sound field.
This concept is termed local sound field synthesis.
This thesis presents an instrumentalized analysis of the fundamental physical
properties of the problem. Although the presented work aims at audio presentation
to human listeners, perception can only be marginally be considered. However, care
was taken that the results are presented such that they can be directly used as a
basis for experimental perceptual evaluation.Zusammenfassung
Dievorliegende Dissertationbehandelt dasThema derSchallfeldsynthese. Im Fokus
steht dabei die Darbietung von Audiosignalen. Die vorgestellten Ergebnisse lassen
sich jedoch in anderen Gebieten wie z.B. der Unterwasserakustik oder Ultraschall-
technik anwenden. Eine grundlegende Formulierung des Problems auf Basis von
etablierten Integralgleichungen wird erarbeitet. Die Verwendung der Methode des
Einschichtpotentials alszweckdienliche allgemeineL¨osungsmethodewirdvorgeschla-
gen. Eine explizite L¨osung wird beispielhaft fu¨r einw¨arts strahlende kugelf¨ormige
Sekund¨arquellenverteilungen erarbeitet.
Den Nachteil der Einschichtpotentialmethode stellt der Umstand dar, dass
die behandelten Sekund¨arquellenverteilungen das Zielvolumen einschließen mu¨ssen.
Es werden Erweiterungen der Einschichtpotentialmethode vorgeschlagen, welche
L¨osungen fu¨r kreisf¨ormige, ebene und zeilenf¨ormige Sekund¨arquellenverteilungen
erm¨oglichen. Anhand dieser Ergebnisse wird gezeigt, dass die beiden etablierten
analytischen Methoden der Wellenfeldsynthese und des Near-Field-Compensated-
Higher-Order-Ambisonics Spezialf¨alle der allgemeinen L¨osung darstellen und in der
L¨osung u¨ber die Einschichtpotentialmethode enthalten sind.
Die physikalischen Grundlagen der Einschichtpotentialmethode erfordern, dass
diebetrachtetenSekund¨arquellenverteilungen kontinuierlichsind. Solchekontinuier-
lichen Verteilungen lassen sich mit der zur Verfu¨gung stehenden Lautsprechertech-
nologie praktisch nicht umsetzen. Es mu¨ssen diskrete Lautsprecherverteilungen
verwendet werden. Die Auswirkungen dieser r¨aumlichen Abtastung der Sekund¨ar-
quellenverteilung wird detailliert fu¨r alle oben genannten Geometrien im Raumfre-
quenzbereich, im Zeitfrequenzbereich sowie im Zeitbereich untersucht. Zwei grund-
legende Ergebnisse werden erarbeitet: Erstens, die Abtastung fu¨hrt zu Wiederho-
lungen der Sekund¨arquellenansteuerungsfunktion in einem Raumfrequenzbereich,
der durch die Geometrie der betrachteten Sekund¨arquellenverteilung bestimmt ist.
Und zweitens, die r¨aumliche Bandbreite der Sekund¨arquellenansteuerungsfunktion
hat grundlegenden Einfluss auf die Eigenschaften des synthestisierten Schallfeldes.
Deshalb wird vorgeschlagen, die verschiedenden Methoden der Schallfeldsynthe-
se anhand der r¨aumlichen Bandbreite der Ansteuerungsfunktion in schmalbandige,
breitbandige und vollbandige Methoden einzuordnen.
Letztlich wird gezeigt, wie verschiedene Arten der r¨aumlichen Bandbreitenein-
schr¨ankungangewendetwerdenk¨onnen,umlokaldieGenauigkeitdessynthetisierten
Schallfeldes zu erh¨ohen. Dieses Konzept wird lokale Schallfeldsynthese getauft.
Die vorliegende Dissertation stellt eine instrumentalisierte Analyse der grundle-
genden Eigenschaften des Problems dar. Obwohl die behandelten Methoden prim¨ar
auf den Menschen als Empf¨anger abzielen, kann die menschliche Wahrnehmung
der synthetisierten Schallfelder nur marginal beru¨cksichtigt werden. Die Ergebnisse
wurden jedoch derart aufgearbeitet, dass sie direkt als Basis fu¨reine weiterfu¨hrende
experimentelle Evaluierung der Wahrnehmung dienen k¨onnen.Acknowledgments
My special thanks go to Sebastian M¨oller for putting immeasurable efforts in pro-
viding perfect working conditions and for giving me the freedom to work on the
topic of sound field synthesis. And, of course, I thank him for reviewing the present
dissertation. Irene Hube-Achter’s efforts have also contributed to a considerable
extent to the pleasantness of my working conditions which I am also very thankful
for.
Jens Blauert deserves general acknowledgements for exciting and inspiring dis-
cussions over the years; and he deserves special acknowledgements for reviewing the
present dissertation and for giving valuable comments and suggestions.
I wish to thank all of my colleagues at Quality and Usability Lab, most notably
MatthiasGeier, KarimHelwani andHagenWierstorfoftheaudiotechnology group,
Warcel W¨altermann and Alexander Raake, and I wish to thank the management
ofDeutscheTelekomLaboratoriesfortheirsupportandenthusiasmforspatialaudio.
The last and thus most important paragraph is dedicated to Sascha Spors who
deserves mostpronouncedacknowledgments forvariouseffortsincludingintroducing
me to the topic of sound field synthesis, guiding me through all these years that I
have spent at Quality and Usability Lab and Deutsche Telekom Laboratories, and
also for organizing my employment after a single phone call. And finally, I am
especially thankful for the fact that we have shared and do still share so many of
our interests and for the coincidence that brought us together.Contents
1 Introduction 1
1.1 The History of Audio Presentation . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation of the Presented Research . . . . . . . . . . . . . . . . . . 2
1.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Physical Fundamentals of Sound Fields 7
2.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Solutions in Cartesian Coordinates . . . . . . . . . . . . . . . 8
2.1.3 Solutions in Spherical Coordinates . . . . . . . . . . . . . . . 9
2.2 Representations of Sound Fields . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Representation ofSound Fields asSeries ofSpherical Harmonics 12
2.2.2 Selected Properties of Bandlimited Spherical Harmonics Series 15
2.2.3 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Far-Field Radiation . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.5 The Wavenumber Domain . . . . . . . . . . . . . . . . . . . . 20
2.2.6 The Angular Spectrum Representation . . . . . . . . . . . . . 21
2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . 23
2.3.2 Neumann Boundary Condition. . . . . . . . . . . . . . . . . . 23
2.3.3 Sommerfeld Radiation Condition . . . . . . . . . . . . . . . . 24
2.4 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 The Kirchhoff-Helmholtz Integral . . . . . . . . . . . . . . . . . . . . 25
2.6 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.1 Wave Field Synthesis . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.2 Simple Source Formulation and Equivalent Scattering Problem 33
2.6.3 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.4 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Continuous Secondary Source Distributions 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Spherical Secondary Source Distributions . . . . . . . . . . . . . . . . 39
3.2.1 Derivation of the Driving Function . . . . . . . . . . . . . . . 40
3.2.2 Synthesized Sound Field . . . . . . . . . . . . . . . . . . . . . 42
3.2.3 Incorporation of Secondary Sources With Complex Radiation
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
III Contents
3.2.4 Near-Field Compensated Higher Order Ambisonics . . . . . . 47
3.3 Circular Secondary Source Distributions . . . . . . . . . . . . . . . . 48
3.3.1 Derivation of the Driving Function . . . . . . . . . . . . . . . 49
3.3.2 Synthesized Sound Field . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Incorporation of Secondary Sources With Complex Radiation
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Planar Secondary Source Distributions . . . . . . . . . . . . . . . . . 55
3.4.1 Derivation of the Driving Function . . . . . . . . . . . . . . . 55
3.4.2 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . 57
3.4.3 Synthesized Sound Field And Example Driving Function . . . 58
3.5 Linear Secondary Source Distributions . . . . . . . . . . . . . . . . . 59
3.5.1 Derivation of the Driving Function . . . . . . . . . . . . . . . 59
3.5.2 Synthesized Sound Field And Example Driving Function . . . 60
3.5.3 Incorporation of Secondary Sources With Complex Radiation
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.4 A Note on Wave Field Synthesis Employing Linear Secondary
Source Distributions . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.5 Truncated Linear Secondary Source Distributions . . . . . . . 65
3.6 Approximate Solution for Arbitrary Convex Secondary Source Dis-
tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Discrete Secondary Source Distributions 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Excursion: Discretization of Time-Domain Signals . . . . . . . . . . . 69
4.3 Spherical Secondary Source Distributions . . . . . . . . . . . . . . . . 73
4.3.1 Discretization of the Sphere . . . . . . . . . . . . . . . . . . . 73
4.3.2 Discretization of the Driving Function . . . . . . . . . . . . . 74
4.3.3 Properties of the Synthesized Sound Field in Time-Frequency
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Circular Secondary Source Distributions . . . . . . . . . . . . . . . . 81
4.4.1 Discretization of the Driving Function . . . . . . . . . . . . . 81
4.4.2 On the Spatial Bandwidth of Wave Field Synthesis With Cir-
cular Secondary Source Distributions . . . . . . . . . . . . . . 84
4.4.3 Properties of the Synthesized Sound Field in Time-Frequency
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.4 Properties of the Synthesized Sound Field in Time Domain . . 89
4.4.5 Optimizing the Synthesis with Respect to a Given Receiver
Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5 Planar Secondary Source Distributions . . . . . . . . . . . . . . . . . 99
4.6 Linear Secondary Source Distributions . . . . . . . . . . . . . . . . . 102
4.6.1 Discretization of the Driving Function . . . . . . . . . . . . . 102
4.6.2 Properties of the Synthesized Sound Field in Time-Frequency
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6.3 Properties of the Synthesized Sound Field in Time Domain . . 107
4.6.4 SpatialDiscretizationinWaveFieldSynthesisEmployingLin-
ear Secondary Source Distributions . . . . . . . . . . . . . . . 109