Analysis and Synthesis of Sound-Radiation with Spherical Arrays

192 Pages·2009·9.15 MB·English

Analysis and Synthesis of Sound-Radiation with
Spherical Arrays
Dissertation by
Franz Zotter
For the Degree
Doktor der Naturwissenschaften
Comittee:
Professor Robert H¨oldrich, Chair (IEM, KU Graz)
Professor David Wessel (CNMAT, UC Berkeley)
Institute of Electronic Music and Acoustics
University of Music and Performing Arts, Austria
September 2009
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ABSTRACT
This work demonstrates a comprehensive methodology for capture, analy-
sis, manipulation, and reproduction of spatial sound-radiation. As the challenge
herein, acoustic events need to be captured and reproduced not only in one but in
a preferably complete multiplicity of directions, instead. The solutions presented
in this work are using the soap-bubble model, a working hypothesis about sound-
radiation, and are based on fundamental mathematical descriptions of spherical
acoustic holography and holophony. These descriptions enable a clear methodic
approach of sound-radiation capture and reproduction. In particular, this work
illustrates the implementation of surrounding spherical microphone arrays for the
capture of sound-radiation, as well as the analysis of sound-radiation with a func-
tional model. Most essential, the thesis shows how to obtain holophonic reproduc-
tion of sound-radiation. For this purpose, a physical model of compact spherical
loudspeaker arrays is established alongside with its electronic control.
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KURZFASSUNG
Diese Arbeit beinhaltet eine umfassende Methodik zur Aufnahme, Ana-
lyse, Manipulation und Wiedergabe von r¨aumlicher Klangabstrahlung. Die neue
Herausforderung liegt darin, akustische Ereignisse nicht nur in einer Richtung,
sondern einer m¨oglichst vollst¨andigen Vielzahl an Richtungen zu erfassen und wie-
derzugeben. Die L¨osungen in dieser Arbeit gehen vom Seifenblasenmodell, einer
Arbeitshypothese u¨ber die Schallabstrahlung, aus und stu¨tzen sich auf mathema-
tische Grundbeschreibungen von kugelf¨ormiger akustischer Holografie und Holo-
phonie. Diese Beschreibungen erm¨oglichen einen klaren methodischen Zugang zu
Abstrahlungsaufnahme und -wiedergabe. Insbesondere wird damit die Umsetzung
von umgebenden kugelf¨ormigen Mikrofonanordnungen zur Abstrahlungsaufnahme
sowie die Auswertung der Abstrahlung anhand eines funktionalen Modells gezeigt.
Als wesentlichsten Beitrag zeigt die Dissertation, wie Abstrahlung holophon wie-
dergegeben werden kann. Dazu wird herausgearbeitet, wie kompakte kugelf¨ormige
Lautsprecheranordnungen physikalisch modelliert und elektronisch gesteuert wer-
den.
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ACKNOWLEDGEMENTS
I am very grateful for the employment I enjoyed during the research for my the-
sis, within the project Virtual Gamelan Graz (VGG) at the Institute of Electronic
MusicandAcousticsinGraz(IEM),onGamelan-music. Thisprojectwasaninter-
disciplinary research platform between the Institute of Ethno-Musicology in Graz
(IME) and IEM due to an initial idea of Gerhard Nierhaus that was further devel-
oped by Professor Gerd Grupe, Alois Sontacchi, and Professor Robert H¨oldrich. I
thank my colleagues at IEM and IME for their teamwork and the Zukunftsfonds
Steiermark for their financial support (Prj. 3027). For great parts of this work, I
have gratefully received support from the Austrian Research Promotion Agency
(FFG), the Styrian government and the Styrian business promotion agency (SFG)
under the COMET program, in the project advanced audio processing (AAP),
sculptural sound.
Special thanks go to Alois Sontacchi for his project management at IEM, Jo-
hannes Zm¨olnig, and Thomas Musil for their PD-software and assistance. I also
want to thank Josef Schalk for the construction of the icosahedral bass loud-
speaker array, and the students of the Musical Acoutics Seminar 2005 for assist-
ing the directional recordings of the Gamelan sounds. Above all, Peter Reiner
and Christian Jochum for their effort in mounting the speakers, cabling, and the
first acoustic and laser-vibrometry measurements, and prototype driving filters
(seminar and student projects). Furthermore, thanks go to Markus Noisternig
for his encouragement to explore the world, his interest and suggestions, Hannes
Pomberger for his excellent work on near- and far-field beamforming with com-
ˇ
pact spherical loudspeaker arrays [Pom08]. Many thanks also go to Nino Skilji´c
for his work on binaural synthesis considering directional sound sources; and to
Margherita Jammer and FabianHohl [Hoh09] for their efforts in spectral modeling
of sounds and directivity (student projects). I thank Professor Helmut Fleischer
(Bundeswehruniversit¨at Mu¨nchen) for his extensive studies on gongs and bells,
and Peter Møller-Juhl (University of South Denmark, Odense) for recommending
the book Fourier Acoustics.
I very deeply thank the Society for the Promotion of Electronic Music and
Acoustics (GesFEMA) and the University of Music and Performing Arts for their
generous support of my research visits at the Center of New Music and Audio
Technology (CNMAT) in Berkeley (CA, USA). In this context, many thanks go
to Professor David Wessel for the overwhelmingly warm welcome at CNMAT and
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advisory, Richard Andrews for his administrative assistance, and Peter Kassakian
for his interesting scientific work. Adrian Freed provided me with thousands of
invaluable suggestions and critical questions and was a great inspiration in my
work. Thanks to Andrew Schmeder for his improvements of the wording and
proof-reading, discussions and interest in the subject, and to Rimas Avizienis for
his electronic hardware prototypes.
For providing me the opportunity to talk at their institutions, I thank Ra-
mani Duraiswami and his group PIRL, UMIACS at the Maryland University,
Jens Meyer and Gary Elko (mhacoustics, Summit, NJ), Gottfried Behler (ITA,
RWTH, Aachen) and Martin Pollow, as well as Sascha Spors and Jens Ahrens
(Deutsche Telekom Labs, Berlin).
Special thanks to all the people at ITA Aachen, also for the important correc-
tionstosomemistakes intheelectroacousticmodelandforprovidingmewiththeir
multi-channel saxophone radiation recordings. Thanks to Zhiyun Li, Svend-Oscar
Petersen, and Boaz Rafaely for their inspiring works on rigid compact spherical
microphone arrays.
Professor Robert H¨oldrich deserves to be mentioned separately. He helped
takingmanystepsthroughthisthesis, spending alotoftimeinfruitfuldiscussions,
encouragement, and important corrections.
Finally,aboveall: ThankyouIngridandWilliforsupportingsomuchacademic
education! Helmuth, thank you very much!
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Contents
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
KURZFASSUNG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 A Soap-Bubble Model of Sound-Radiation . . . . . . . . . . . . . . . . 6
1.2 Musical Instrument Model . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Organization of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . 10
II ACOUSTICS IN SPHERICAL COORDINATES . . . . . . . . . . . 13
2.1 Solving the Helmholtz-Equation in Spherical Coordinates . . . . . . . . 14
2.1.1 Selection of Physical Solutions . . . . . . . . . . . . . . . . . . . 15
2.2 Spherical Base-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Spherical Boundary Value Problems . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Spherical Wave Spectrum . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Spherical Boundary Value/Condition Examples . . . . . . . . . 25
2.4 Spherical Source Distributions . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Spherical Source Distribution Problem . . . . . . . . . . . . . . 30
2.4.2 Expansion of a Point-Source . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Expansion of a Plane-Wave . . . . . . . . . . . . . . . . . . . . 32
2.4.4 Expansion of a Line-Source . . . . . . . . . . . . . . . . . . . . 33
2.5 Acoustic Holography and Holophony with Spherical Arrays . . . . . . . 34
III MANIPULATION OF SPHERICAL BASE SOLUTIONS . . . . . 35
3.1 Coordinate Transforms of Spherical Base Solutions (Addition Theorem
for the Scalar Wave Equation) . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Gradient and its Commutativity with Transforms . . . . . . . . 38
3.1.2 Deriving the Gradient on Spherical Base-Solutions . . . . . . . 40
3.1.3 General Recurrence Relations for Coordinate Transforms . . . . 46
3.1.4 Decomposition of General Transforms into Simpler Steps . . . . 46
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