Directivity Patterns for Room Acoustical Measurements and Simulations

188 Pages·2015·18.45 MB·English

by  Martin Pollow

MartinPollow
Directivity Patterns for Room Acoustical
Measurements and Simulations
LogosVerlagBerlinGmbH
λογος
Aachener Beitra¨gezurTechnischenAkustik
Editor:
Prof.Dr.rer.nat.MichaelVorla¨nder
InstituteofTechnicalAcoustics
RWTHAachenUniversity
52056Aachen
www.akustik.rwth-aachen.de
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detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de.
D82(Diss.RWTHAachenUniversity,2014)
(cid:13)c CopyrightLogosVerlagBerlinGmbH2015
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ISBN978-3-8325-4090-6
ISSN1866-3052
Vol.22
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Directivity Patterns for Room Acoustical
Measurements and Simulations
VonderFakultätfürElektrotechnikundInformationstechnikder
Rheinischen-WestfälischenTechnischenHochschuleAachen
zurErlangungdesakademischenGradeseines
DOKTORSDERINGENIEURWISSENSCHAFTEN
genehmigteDissertation
vorgelegtvon
Diplom–Ingenieur
Martin Pollow
ausRosenheim
Berichter:
UniversitätsprofessorDr.rer.nat.MichaelVorländer
UniversitätsprofessorDr.-Ing.PeterVary
TagdermündlichenPrüfung:29.September2014
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfügbar.

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Abstract
Acousticsourcesandreceiverspossessdistinctdirectivitypatternsthatquantify
their directional dependent behavior. In common room acoustical applications
these directivity patterns are often neglected, although they are known to have
considerable influence on the obtained results. This thesis presents methods to
obtainandimplementsourceandreceiverdirectivitypatternsforroomacousti-
calmeasurementsandsimulations.Atypicalcomplexscenarioforroomacoustics
consists of a concert hall that is excited by various natural sound sources (such
as musical instruments) with the sound received by human listeners. Therefore,
a large number of natural sound sources has been measured, analyzed and pro-
cessed in order to assemble a directivity database. The sound reception of the
humanlistenersisdescribedbythehead-relatedtransferfunctions(HRTFs)that
are derived from numerical simulations or measurements of artificial heads and
human individuals.
Both source and receiver directivity patterns can be represented as spherical
wave spectra (SWS) using a decomposition of the angular functions into the
set of orthonormal spherical harmonics. In this domain angular interpolation
and range extrapolation can be implemented conveniently, yielding physically
correctresultsforsamplingschemesthataresufficientlydenseinordertoavoid
spatial aliasing artifacts. Suitable regularization allows to derive the SWS of
spatiallyincompletedata(withmissinginformationatsomedirections)ordata
that suffers from measurement uncertainties.
Themeasurementofroomimpulseresponsesforarbitrarydirectivitypatternsis
performed using specialized spherical loudspeaker arrays to provide broadband
excitation in terms of temporal and spatial frequency components. Assuming
linear and time-invariant systems a sequential measurement approach greatly
enhances the maximum resolution of the synthesized directivity patterns. For
room acoustical simulations that include source and/or receiver directivity pat-
ternstheparticlebasedmethodscanbeextendedbymultiplicationinthespatial
domain. Wave based analytical simulations allow to implement arbitrary direc-
tivity patterns by the computation of the Cartesian derivatives of the room’s
eigenmodes. The results obtained in this thesis can be used to enhance the au-
ralizationofroomsandtoanalyzetheperceptionalimpactofsourceandreceiver
directivity in room acoustical applications.
V

Contents
1 Introduction 1
2 Fundamentals and preliminaries 5
2.1 Definitions of directivity . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fourier transform of signals . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Fourier acoustics in spherical coordinates . . . . . . . . . . . . . . . 9
2.3.1 Wave equation and separation of variables . . . . . . . . . . 9
2.3.2 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Spherical wave spectrum . . . . . . . . . . . . . . . . . . . 13
2.3.4 Boundary value problems . . . . . . . . . . . . . . . . . . . 15
2.3.5 Transformation of acoustic spherical fields . . . . . . . . . . 18
2.3.6 Dirac impulse on unit sphere . . . . . . . . . . . . . . . . . 20
2.3.7 Cross-correlation of spherical functions . . . . . . . . . . . . 21
2.3.8 Frequency dependent order truncation . . . . . . . . . . . . 22
2.4 Discrete Fourier acoustics in spherical coordinates . . . . . . . . . . 23
2.4.1 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 Error measure . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Condition number . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 Matrix inversion using regularization . . . . . . . . . . . . . 27
2.5 Spherical sampling schemes and aliasing analysis. . . . . . . . . . . 29
2.5.1 Equiangular quadrature sampling scheme . . . . . . . . . . 30
2.5.2 Gaussian quadrature sampling scheme . . . . . . . . . . . . 31
2.5.3 Sampling scheme using hyperinterpolation . . . . . . . . . . 32
2.5.4 Equiangular sampling schemes with regular spacing . . . . . 33
2.6 Measurement of acoustic transfer paths . . . . . . . . . . . . . . . 34
2.6.1 Multiple exponential sweep method. . . . . . . . . . . . . . 35
2.6.2 Reciprocity of transfer paths . . . . . . . . . . . . . . . . . 35
3 Directivity patterns of natural sound sources 37
3.1 Recording of musical instrument radiation patterns . . . . . . . . . 38
3.1.1 Types of sound excitation . . . . . . . . . . . . . . . . . . . 38
3.1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.3 Measurement equipment . . . . . . . . . . . . . . . . . . . 40
3.1.4 Measurement environment . . . . . . . . . . . . . . . . . . 42
3.1.5 Obtaining recordings for a directivity database . . . . . . . . 42
VII
Contents
3.1.6 Recording audio for auralization . . . . . . . . . . . . . . . 44
3.2 Processing of recorded data . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Analysis of the used array geometry . . . . . . . . . . . . . 45
3.2.2 Averaged magnitude data . . . . . . . . . . . . . . . . . . . 47
3.2.3 Time-dependent analysis . . . . . . . . . . . . . . . . . . . 47
3.2.4 Harmonic peak extraction . . . . . . . . . . . . . . . . . . . 48
3.3 Analysis of spectral smoothness . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Plotting cross-correlation values over frequencies . . . . . . 52
3.3.2 Woodwind instruments . . . . . . . . . . . . . . . . . . . . 54
3.3.3 Brass instruments . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.4 String Instruments. . . . . . . . . . . . . . . . . . . . . . . 58
3.3.5 Statistical evaluation . . . . . . . . . . . . . . . . . . . . . 58
3.4 Acoustic centering methods . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Centering by minimizing the weighted SWS . . . . . . . . . 61
3.4.2 Centering by minimizing phase transitions . . . . . . . . . . 62
3.5 Summary natural sound sources. . . . . . . . . . . . . . . . . . . . 65
4 Directivity patterns of binaural sound receivers 67
4.1 Obtaining head-related transfer functions. . . . . . . . . . . . . . . 68
4.1.1 Simulation of HRTFs . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Sequential measurements of HRTFs . . . . . . . . . . . . . 71
4.1.3 Fast HRTF measurement methods . . . . . . . . . . . . . . 72
4.2 Low-frequency extension of measured HRTFs . . . . . . . . . . . . 75
4.3 Range extrapolation of HRTFs . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Principle of calculation . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Results for measured HRTFs . . . . . . . . . . . . . . . . . 78
4.3.3 Results for simulated HRTFs . . . . . . . . . . . . . . . . . 80
4.3.4 Conclusions range extrapolation of HRTFs . . . . . . . . . . 81
4.4 Reconstruction of missing data of HRTFs . . . . . . . . . . . . . . 81
4.5 Representation of HRTFs with varying focal point . . . . . . . . . . 83
4.6 Summary binaural sound receivers . . . . . . . . . . . . . . . . . . 84
5 Including directivity patterns in room acoustic measurements 87
5.1 Standardized room acoustic measurements . . . . . . . . . . . . . . 88
5.2 Analytic model for spherical loudspeakers . . . . . . . . . . . . . . 88
5.2.1 Aperture function to model membrane vibration . . . . . . . 89
5.2.2 Total surface velocity and acoustic impedance . . . . . . . . 90
5.2.3 Radiated sound pressure . . . . . . . . . . . . . . . . . . . 91
5.3 Design of compact spherical loudspeaker arrays . . . . . . . . . . . 92
5.3.1 Dodecahedron loudspeakers as multichannel source . . . . . 92
VIII