Sound Propagation Theory for Linear Ray Acoustic Modelling

114 Pages·2011·0.53 MB·English

AB
HELSINKI UNIVERSITY OF TECHNOLOGY
Department of Computer Science and Engineering
Telecommunications Software and Multimedia Laboratory
Sami Kiminki
Sound Propagation Theory for
Linear Ray Acoustic Modelling
Supervisor: ProfessorLauriSavioja
Instructor: TapioLokki,D.Sc. (Tech.)
HELSINKI UNIVERSITY ABSTRACT OF THE
OF TECHNOLOGY MASTER’S THESIS
Author: SamiKiminki
Nameofthethesis: SoundPropagationTheoryforLinearRayAcousticModelling
Date: March7,2005 Numberofpages: 100+14
Department: ComputerScienceandEngineering Professorship: T-111
Supervisor: ProfessorLauriSavioja
Instructor: TapioLokki,D.Sc. (Tech.)
Inthiswork,alinearrayacousticmodellingtheoryisconstructed. Thetheoryformsabase
for linear ray acoustic modellingmethods. As such, the theory can be used to derive and
analyseraymethods. Threeexistingraymodellingmethods(theimagesourcemethod,the
radiosity method, and the ray tracing method) are shown to be derivable from the theory.
Itisalsosuggestedthatthetheorycanbeusedtoderiveacousticcharacteristicsestimators
such as the average reverberation time of a room. To the author’s knowledge, this is the
(cid:2)rstattempttocreate atheoryfor acousticraymodelling.
The theory is divided into two parts: general and acoustic. The general theory consists
of general de(cid:2)nitions, time-dependent energy propagationequations, and detection equa-
tions. The general part yieldstime-independentray modellingtheoryby eliminatingtime
dependency, thus linking the acoustic and the graphic ray modelling. The acoustic part
speci(cid:2)esthegeneralde(cid:2)nitionsasacousticde(cid:2)nitions. Thetheorylackssub-surfacescat-
tering re(cid:3)ection and edge diffraction. A well-de(cid:2)ned extension path for the inclusion is
considered,however.
The general de(cid:2)nitions consist of mathematical and physical de(cid:2)nitions. Energy propa-
gationequationsare constructedindetail, resultingin the re(cid:3)ection-iterativeconstruction
and the acoustic rendering equation. The (cid:2)rst is a straightforward construction, and the
second is a balance equation (cid:151) extension of the Kajiya’s rendering equation. The equa-
tionsevaluateimpulseenergyresponsesandareshowntobeequivalentusinglinearoper-
atoranalysis. Anexamplede(cid:2)nitionforauralizationofenergyresponsesisconstructed.
Keywords: generalmodellingtheory,imagesourcemethod(ISM),radiosity,raytracing,rayacous-
ticmodelling
ii
TEKNILLINEN KORKEAKOULU DIPLOMITY(cid:214)N TIIVISTELM˜
Tekij(cid:228): SamiKiminki
Ty(cid:246)nnimi: ˜(cid:228)nenetenemisteoria lineaarisessa s(cid:228)deakustiikassa
P(cid:228)iv(cid:228)m(cid:228)(cid:228)r(cid:228): 7.3.2005 Sivuja: 100+14
Osasto: Tietotekniikan osasto Professuuri: T-111
Ty(cid:246)nvalvoja: ProfessoriLauriSavioja
Ty(cid:246)nohjaaja: TkTTapioLokki
Ty(cid:246)ss(cid:228) rakennetaan pohjateoria lineaariselle s(cid:228)deakustiselle mallinnukselle. Teoriaa voi-
daank(cid:228)ytt(cid:228)(cid:228)s(cid:228)demenetelmienjohtoonjaanalyysiin.Kolmeolemassaolevaas(cid:228)deakustista
mallinnusmenetelm(cid:228)(cid:228) osoitetaan olevan johdettavissa teoriasta (kuval(cid:228)hde-, radiositeetti-
jas(cid:228)teenseurantamenetelm(cid:228)).Lis(cid:228)ksiehdotetaan,ett(cid:228)teoriaavoitaisiink(cid:228)ytt(cid:228)(cid:228)my(cid:246)sakus-
tisten tunnuslukujen estimointiin, esimerkkin(cid:228) j(cid:228)lkikaiunta-aika. T(cid:228)m(cid:228) on tekij(cid:228)n tiet(cid:228)-
myksenmukaanensimm(cid:228)inenyritysluodakattavas(cid:228)deakustisenmallinnuksenteoria.
Teoriajaetaankahteenosaan,yleiseenjaakustiseen.Yleinenosak(cid:228)sitt(cid:228)(cid:228)yleisetm(cid:228)(cid:228)ritel-
m(cid:228)t,aikariippuvatenergiankulkuyht(cid:228)l(cid:246)tsek(cid:228)havainnointiyht(cid:228)l(cid:246)t.Teorianyleisest(cid:228)osasta
saadaan lis(cid:228)ksi teoria s(cid:228)degra(cid:2)ikalle, kun eliminoidaan aikariippuvuudet. Akustinen osa
spesi(cid:2)oiyleisetm(cid:228)(cid:228)ritelm(cid:228)takustisiksim(cid:228)(cid:228)ritelmiksi.Teoriastapuuttuupinnanalaissiron-
taheijastuksissasek(cid:228)reunadiffraktio.Teorianlaajennettavuusn(cid:228)idenpuutteidenosaltaon
otettuhuomioon.
Yleiset m(cid:228)(cid:228)ritelm(cid:228)t koostuvat matemaattisista ja fysikaalisista m(cid:228)(cid:228)ritelmist(cid:228). Energian-
kulkuyht(cid:228)l(cid:246)t konstruoidaan yksityiskohtaisesti. T(cid:228)m(cid:228) johtaa heijastusiteratiiviseen kons-
truktioonsek(cid:228)akustiseenmallinnusyht(cid:228)l(cid:246)(cid:246)n.Ensimm(cid:228)inenonsuoraviivainenkonstruktio.
J(cid:228)lkimm(cid:228)inen on tasapainoyht(cid:228)l(cid:246), joka on Kajiyan mallinnusyht(cid:228)l(cid:246)n laajennus. Yht(cid:228)l(cid:246)t
tuottavat energiaimpulssivasteita ja konstruktiot osoitetaan yht(cid:228)l(cid:228)isiksi lineaarioperaatto-
rianalyysilla. Ty(cid:246)ss(cid:228) rakennetaan esimerkinomainenm(cid:228)(cid:228)ritys energiavasteiden auralisaa-
tioon.
Avainsanat:yleinenmallinnusteoria,kuval(cid:228)hdemenetelm(cid:228),radiositeetti,s(cid:228)teenseuranta,akustiikan
s(cid:228)demallinnus
iii
Acknowledgments
FirstofallIthankmysupervisingprofessorLauriSaviojaandmyinstructorTapio
Lokki. Theymadeitpossibleformetoconductthisworkandprovidedinvaluable
insightsintothevast(cid:2)eldofacousticmodelling. Ialsothankmyclosestassociate
Heli Nironen for providing lots of background information and reference mate-
rial,andespeciallyforthetirelessconversationaleffortsonvarioussound-related
topics.
I thank professor Timo Eirola for very useful mathematical discussions, and for
allowing me to present the work in his seminar in computational science and en-
gineering. I also thank Jaakko Lehtinen for helping me in radiance calculus, and
JanneKontkanenforvaluableinformationoncomputergraphics.
Additional thanks should go to Vesa Hirvisalo, Jan Jukka Kainulainen, and Juha
Tukkinen for support and otherwise important in(cid:3)uence on my work. Finally, I
thank Maarit Tirri for numerous comments on the textual layout. Any remaining
linguisticanomaliesare duetomypersonalstubbornnessandobsessions.
iv
Contents
Preface xiii
1 Introduction 1
1.1 OrganizationoftheThesis . . . . . . . . . . . . . . . . . . . . . 5
2 Background 7
2.1 SoundandAcoustics . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 PhysicsofSound . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 EnergyofSound . . . . . . . . . . . . . . . . . . . . . . 12
2.2 AcousticModelling . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 ImpulseResponses . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 RayMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 ImageSourceMethod . . . . . . . . . . . . . . . . . . . 17
2.4.2 RadiosityMethod . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 RayTracingMethods . . . . . . . . . . . . . . . . . . . . 21
3 Simpli(cid:2)ed GeneralEnergy PropagationTheory 24
3.1 Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 FundamentalDe(cid:2)nitions . . . . . . . . . . . . . . . . . . 25
3.1.2 PolyhedralEnvironments . . . . . . . . . . . . . . . . . . 29
3.1.3 DiscreteEnvironments . . . . . . . . . . . . . . . . . . . 29
3.1.4 VisibilityComputation . . . . . . . . . . . . . . . . . . . 30
3.2 RadiationandRe(cid:3)ection . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 EnergyFlow . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 TheEnergySource . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 TheObserver . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.4 Radiance . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.5 Re(cid:3)ection . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.6 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 EnergyPropagationEquations . . . . . . . . . . . . . . . . . . . 48
3.3.1 TemporalIntensityAlgebra . . . . . . . . . . . . . . . . 48
3.3.2 GeometryTermandRe(cid:3)ectionKernel . . . . . . . . . . . 52
v
3.3.3 Re(cid:3)ection-iterativeConstruction . . . . . . . . . . . . . . 55
3.3.4 AcousticRenderingEquation . . . . . . . . . . . . . . . 60
3.3.5 EquivalencyofRe(cid:3)ection-iterativeConstructionandARE 62
3.3.6 RemarkonDetection . . . . . . . . . . . . . . . . . . . . 64
3.4 RadiationatVariousFrequencies . . . . . . . . . . . . . . . . . . 64
3.4.1 MathematicalDiscussion . . . . . . . . . . . . . . . . . . 65
3.5 ConsiderationsonExtensions . . . . . . . . . . . . . . . . . . . . 67
3.5.1 ExtendingforEdgeDiffraction . . . . . . . . . . . . . . 67
3.5.2 ExtendingforSub-surface Scattering . . . . . . . . . . . 68
4 AcousticEnergy PropagationTheory 70
4.1 AdaptationoftheGeneralEnergyPropagationTheory . . . . . . 71
4.2 AuralizationofEnergyResponse . . . . . . . . . . . . . . . . . . 72
4.3 SpecializationsoftheTheory . . . . . . . . . . . . . . . . . . . . 73
4.3.1 ImageSourceMethod . . . . . . . . . . . . . . . . . . . 74
4.3.2 RadiosityMethod . . . . . . . . . . . . . . . . . . . . . . 77
4.3.3 RayTracingMethod . . . . . . . . . . . . . . . . . . . . 78
4.4 ConsiderationsonUsingtheTheory . . . . . . . . . . . . . . . . 79
5 Conclusion 81
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 FurtherWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A SomeEssentialMathematics 86
A.1 EuclideanSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.3 LinearOperatorAnalysis . . . . . . . . . . . . . . . . . . . . . . 93
B ABriefNoteon BDRFsand Lambertian DiffuseRe(cid:3)ections 96
Bibliography 98
vi
List of Figures
2.1 A2-dimensionalspring-masssystemwithdisplacement . . . . . . 11
2.2 Direct,(cid:2)rst,andsecondorderimagesources . . . . . . . . . . . . 17
3.1 Possibleraypathsinspecularre(cid:3)ectingenvironment . . . . . . . 32
3.2 Possibleraypathsindiffusere(cid:3)ectingenvironment . . . . . . . . 33
3.3 Incidentenergy fromapointsourcetoa smallsurface patch . . . 34
3.4 Radiantincidentenergytoa smallsurface patch . . . . . . . . . . 35
3.5 Energy(cid:3)owofincidentradiation . . . . . . . . . . . . . . . . . . 40
3.6 Re(cid:3)ectionofplanarbeam . . . . . . . . . . . . . . . . . . . . . . 42
3.7 Intensitymeasurementofplanarpropagatingwavefront . . . . . . 49
3.8 Parametersofthere(cid:3)ectionkernel . . . . . . . . . . . . . . . . . 54
3.9 Diffractionalbendingofrays . . . . . . . . . . . . . . . . . . . . 68
4.1 Re(cid:3)ectingbeam . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.1 Mirrorre(cid:3)ection. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
vii
Notations
Latin alphabets
A variableusedinderivationsandproofs
a m3 pointinsurface geometry
area m2 surfacearea operator(p.77)
(cid:2) (cid:3)
B re(cid:3)ectionoperator(p.62)
(cid:2) (cid:3)
B allre(cid:3)ectionsoperator(p.66)
c m speedofwavefront(p.36)
b s
card cardinality,sizeofset(p.21)
(cid:2) (cid:3)
D delaypattern(p.72)
D linearsystemresponseoperator(p.14)
d totaldetection(p.48,64)
b
representation function of linear system response op-
erator(p.14)
d detectionofdirectsourcetoobserverradiation(p.55)
0
d detectionofradiance,detectionoftotalre(cid:3)ectedradi-
r
ationviaanynumberofre(cid:3)ections(p.57)
det matrixdeterminant
dtf directionaltransferfunctionforobservation(p.39,47)
exitant(pre(cid:2)x)(p.37)
e
F patchemissionvector(p.20,78)
f 1 frequency(p.42)
s
f generalfunction
(cid:2) (cid:3)
viii
f 1 bidirectional re(cid:3)ection distribution function (BRDF)
r sr
(p.41)
(cid:2) (cid:3)
f BRDF ofanidealdiffusere(cid:3)ection(p.45)
r;d
f BRDF ofanidealspecularre(cid:3)ection (p.47)
r;s
G environmentalsurface geometry(p.27)
g Kajiyangeometrytermwithpropagationdelay(p.53)
g(cid:136) Kajiyan geometry term without propagation delay
(p.53)
Ha mediumabsorptionoperator(p.51)
I W intensity, irradiance, energy (cid:3)ow per surface area
m2
h i (p.36)
I identitymatrix,identityoperator
incident(pre(cid:2)x) (p.37)
i
(cid:17)(cid:136)(t) impulseresponse(p.72)
L(W) W radiance(p.40,60)
m2
L (W) hWi primaryradiance (p.60)
0 m2
‘(t) h i time-dependent radiance in non-absorptive medium
(p.52)
totalpropagatedradiance(p.60,61)
‘ primaryradiance (p.54,57(cid:150)58)
0
‘ primaryradiance afterk re(cid:3)ections(p.57,60)
k
‘(cid:136) time-dependent intensity in non-absorptive medium
(p.50)
M mirrorre(cid:3)ection operator(p.45,87)
n surfacenormal(p.27)
O() asymptoticcomplexityclass
P vectorofenergy(cid:3)owsofpatches(p.20,77)
P patch,smallbutnotin(cid:2)nitesimalsurfacearea (p.19)
k
p patternofemittance(p.38)
e
R re(cid:3)ectionkernel(p.53)
r distance,radiusofa sphere
ix
S losslesspropagationoperatorfor distancer (p.50)
r
S(cid:136) propagationoperatorwithmediumabsorptionfordis-
r
tancer. UsedintheconstructionofTIA(p.49)
s noisesignalwithaverageunitintensity(p.72)
T triangle(p.89)
t time
v m 3 velocity(vectorquantity)
s
3
x [m] pointinspace
(cid:2) (cid:3)
x locationoftheobserver(p.39,55)
o
x raypath(p.10)
p
x locationoftheenergy source(p.37,55)
s
Lower-case Greekalphabets
b re(cid:3)ectance factor,totalre(cid:3)ectance (p.42)
g sensitivitypattern(p.72)
d Diracdeltafunctional(p.15,45,91)
q elevation angle (cid:151) angle between W and surface nor-
mal(p.26)
n visibilityfunction(p.27)
n 1 visiblegeometry(p.28)
(cid:0)
n inverseprojection(p.28)
p
r biconicalre(cid:3)ectance factor (p.42)
probabilitydensity(p.92)
s energypro(cid:2)leofsub-band(cid:2)lteredimpulse(p.66,72)
signal(p.14)
t generalscalarorvector
f azimuthangleofW(p.26)
j angle
y waverepresentationofafundamentalparticle(p.13)
w [sr] solidangle,setofdirections(p.26)
x