Reproduction of a 3D Sound Field Using an Array of Loudspeakers

105 Pages·2003·1.68 MB·English

Reproduction of a 3D Sound Field
Using an Array of Loudspeakers
David Excell – 3222940
Supervisor: Dr. Thushara Abhayapala
June 2003
A thesis submitted in part fulfilment of the degree of
Bachelor of Engineering (Honours) at
The Department of Engineering
Australian National University
This thesis contains no material which has been accepted for the award of any other
degree or diploma in any university. To the best of the author’s knowledge and belief,
itcontainsnomaterialpreviouslypublishedorwrittenbyanotherperson,exceptwhere
duereferenceismadeinthetext.
DavidExcell
18June2003
Abstract
In this thesis the fundamental problem in signal processing of controlling a sound field
withinaregionofspaceisstudied. Usingsphericalandcylindricalharmonics,reproduc-
tion systems in both two and three dimensions have been designed. From the design of
thesystems,equationsarederiveddescribingtheperformanceofthesystemforarbitrary
soundfields. Theequationsaredefinedintermsofthenumberofloudspeakers,thesize
ofthelisteningareaandthefrequencycontentofthesoundfields.
Results from numerical verification have been included within this thesis. The im-
plementation of the 2D system is also included in this thesis demonstrating the ability
of the designed system to reproduce ‘real’ sound fields. In the implementation process,
a synthesis system was developed allowing conventionally recorded sound samples to
be placed in 2D space forming a spatial sound field. In the process it is demonstrated
thatthereproductionsystemdoesnotsolelyuseamplitudepanningtechniquesbutalso
considersthephaseanddelayofthesoundfieldcomponentstorecreatethesoundfield.
ii
Acknowledgements
Without support and encouragement from a number of people and organisations the
researchpresentedinthisthesiswouldnothavebeenpossible. Iwouldliketoespecially
thankthefollowing:
•
Dr. Thushara Abhayapala for his excellent supervision, guiding my efforts into
producingapieceofresearchthatIamproudof;
•
TheResearchSchoolofInformationSciencesandEngineering(RSISE)forproviding
mewithaSummerResearchScholarship;
•
Both the Department of Telecommunications (RSISE) and the Department of En-
gineering (Faculty of Engineering and Information Technology) for providing the
facilitiesenablingthecompletionofthisresearchwork;
•
TomandMichaelfortheirproofreadingofthisthesisalongwithprovidingpositive
feedback;
•
GlennDickinsforhissuggestedsolutionstoproblemsencounteredwhileworking
onthisthesis;and
•
My family for their continual support and guidance with everything I set out to
achieve.
Finally, Iwouldliketothankmygirlfriend, Bronnie, whoseunderstandingandsupport
hasbeenmostappreciatedwhileIhavespentcountlesshoursworkingtowardsthisthe-
sis.
iii
Trademarks
AC-3, Dolby, Dolby Digital and Dolby Surround are trademarks of Dolby Laboratories
Licensing Corporation. DTS is a registered trademark for Digital Theater Systems In-
corporated. NXT and SurfaceSound are registered trademarks of NXT Public Limited
Company. THXisaregisteredtrademarkofLucasfilmLimited.
iv
Contents
Glossary vii
ListofFigures ix
1 Introduction 1
1.1 ThesisMotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LiteratureSurvey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 CommercialOpportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 BackgroundTheory 6
2.1 TheoryofAcoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 CoordinateSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 ModalWaveEquationSolution . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 SoundFieldControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 SoundSources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 SoundFieldReconstruction 12
3.1 ThreeDimensionalReconstruction . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 ExactReproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2 ApproximateReproduction . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.3 CalculatingtheLoudspeakerWeights . . . . . . . . . . . . . . . . . 17
3.1.4 LoudspeakerPlacement . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.5 AlternativeLoudspeakerWeightCalculation . . . . . . . . . . . . . 20
3.2 TwoDimensionalReconstruction . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 ExactReproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 ApproximateReproduction . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 CalculatingtheLoudspeakerWeights . . . . . . . . . . . . . . . . . 26
3.2.4 LoudspeakerPlacement . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 NumericalSimulation 27
4.1 ThreeDimensionalSoundFieldReconstruction . . . . . . . . . . . . . . . . 27
4.1.1 ErrorMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 ErrorEquationVerification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 EffectofLoudspeakerWeightEstimation . . . . . . . . . . . . . . . . . . . 31
v
Contents vi
4.4 ComparisonofSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 SummaryofResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 PhysicalImplementation 35
5.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 SoundFieldSynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 LoudspeakerSignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 ComputationalCost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 ReproductionAccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5.1 ModalLimitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5.2 Loudspeakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.6 ListeningTests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.7 SummaryofResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 SoundFieldControl 45
6.1 Non-uniformLoudspeakerPlacement . . . . . . . . . . . . . . . . . . . . . 45
6.2 ListeningAreaRepositioningandDuplication . . . . . . . . . . . . . . . . 46
7 ConclusionsandFurtherWork 49
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 FurtherResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A MathematicalProofs 51
B DetailedImplementationDesignandSetup 64
B.1 DetailedImplementationDesignandSetup . . . . . . . . . . . . . . . . . . 64
B.2 WindowingFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C ListeningTestSummary 75
C.1 SingleStationaryNarrowbandSoundSource . . . . . . . . . . . . . . . . . 75
C.2 SingleRotatingNarrowbandSoundSource . . . . . . . . . . . . . . . . . . 75
C.3 SingleRotatingBroadbandSoundSource(Circle) . . . . . . . . . . . . . . 77
C.4 SingleRotatingBroadbandSoundSource(Square) . . . . . . . . . . . . . . 77
C.5 MultipleStationaryBroadbandSoundSources . . . . . . . . . . . . . . . . 78
D MatLabCodeListing 79
D.1 ThreeDimensionalSimulation . . . . . . . . . . . . . . . . . . . . . . . . . 80
D.2 TwoDimensionalImplementation . . . . . . . . . . . . . . . . . . . . . . . 86
E SummaryofContentonAttachedMedia 91
Bibliography 92
Glossary
Symbol Definition
2D TwoDimensional
3D ThreeDimensional
R
RealNumbers
Z
Integers
al LoudspeakerWeight
c SpeedofSound
c
Radian
DFT DiscreteFourierTransform
DSP DigitalSignalProcessor
E NormalisedExpansionError
f Frequency(Hertz)
(cid:1)
FourierTransform
(cid:1)−1
InverseFourierTransform
h(n1)(·) SphericalHankelFunctionoftheFirstKind
h(n2)(·) SphericalHankelFunctionoftheSecondKind
Hn(1)(·) OrdinaryHankelFunctionoftheFirstKind
√
i ImaginaryPartofaComplexNumber(i = −1)
jn(·) SphericalBesselFunctionoftheFirstKind
Jn(·) OrdinaryBesselFunctionoftheFirstKind
m
Metre
nn(·) SphericalBesselFunctionoftheSecondKind
Nn(·) OrdinaryBesselFunctionoftheSecondKind
L NumberofLoudspeakers
Pnm(·) AssociatedLegendreFunction
r RadialDistancefromtheOrigin
S(x;k) OriginalSoundFieldtobeReproduced
t TimeinSeconds.
T (x;k) ReproducedSoundField
x
MatrixorVector
xˆ x
UnitVectorinthedirectionof
x VectorLength(i.e. x = xxˆ)
x,y,z ComponentsofaVectorinCartesianCoordinates
Ynm(·) SphericalHarmonics
y SoundSourceLocation
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Glossary viii
Symbol Definition
y Loudspeakerposition
l
α SoundFieldCoefficient
δ Kronecker-DeltaFunction
(cid:3) NormalisedTruncationError
γ SoundSurfaceCoefficient
θ Elevation
ϑ Plane-waveElevation
φ Azimuth
ϕ Plane-waveAzimuth
ω RadianFrequency
λ Wavelength
Υ
NormalisedReproductionError
∗
ComplexConjugate
◦
Degrees
|·|
AbsoluteValueorComplexMagnitude
(cid:3)·(cid:3)
EuclidianDistance
(cid:4)·(cid:5)
IntegerCeilingOperator
List of Figures
1.1 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 ThreeDimensionalCoordinateSystem . . . . . . . . . . . . . . . . . . . . . 7
3.1 SphericalBesselFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Normalisedexpansionerrorasplane-wavedirectionisvaried. . . . . . . . 16
3.3 Plots of formulated 3D error equations for the approximate reproduction
solution. (a),(c)and(e)containthemeannormalisederrorobservedfrom
a thousand random fifty plane-wave sound fields, (b), (d) and (f) shows
the standard deviation of the normalised error from these sound fields.
Theplane-wavefrequencyforallsoundfieldswas1kHz.. . . . . . . . . . 18
3.4 Thetwofiguresshowthelayoutoftwenty-fiveandthirty-sixloudspeakers
placedinanuniformpackingaroundaunitsphere. . . . . . . . . . . . . . 20
3.5 NormalisedTruncationErrorfortwoplane-wavesoundfield. . . . . . . . 23
3.6 Plots of formulated 2D error equations for the approximate reproduction
solution. (a),(c)and(e)containthemeanerrorobservedfromathousand
randomfiftyplane-wavesoundfields,(b),(d)and(f)containthestandard
deviationofthenormalisederrorfromthesesoundfields. Theplane-wave
frequencyforallsoundfieldswas1kHz. . . . . . . . . . . . . . . . . . . . 25
4.1 Soundfieldreproductionatheightz = 0.1m . . . . . . . . . . . . . . . . . 28
4.2 Soundfieldreproductionatheightz = 0m . . . . . . . . . . . . . . . . . . 28
4.3 Illustration of the three errors defined in Section 3.1.2. The height of the
soundfieldillustratedisz = 0m,thetwenty-fiveloudspeakerswereplaced
onaradiusofr = 1m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Comparison of the reproduction errors measured via numeric simulation
and using the error formulas presented in Section 3.1.2. The sound field
consistedoffiftyrandomplane-wavesat1kHz. . . . . . . . . . . . . . . . 30
4.5 Effectofvaryingthenumberofloudspeakersforafixednumberofsound
fieldcoefficientsontheestimationoftheloudspeakerweights. Therepro-
duction errors at the four spherical radii were averaged from ten random
fiftyplane-wavesoundfieldsat1kHz. Theloudspeakerradiuswas1m. . 31
4.6 Reproduction error observed at different radii when the reproduction or-
der (N) is varied for a fixed number of loudspeakers. Figure (a) shows
the result for L=25 and Figure (b) shows the result for L=36. The results
shown have been averaged from ten randomly selected sound fields con-
tainingfiftyplane-wavesat1kHz. Theloudspeakerradiuswas1m.. . . . 32
ix