1
Numerical Modelling of the Head-Related Transfer Function
Yuvi Kahana
27/11/2000
2
Can we obtain individualised HRTFs
without a single acoustic measurement ?
First Question
Method
Convert the geometry of an object
(head+pinna) into its acoustic response
Solve the wave equation
3
“The rather complicated shape of the pinna makes a rigorous mathematical treatment very difficult - perhaps impossible”
Weinrich (1984)
Shin-Cunningham andKulkarni (1996)
“Theoretically, it is possible to specify the pressure at the eardrum for a source from any location simply by solving the wave equation…Needless to say, this is analytically and computationally an intractable problem”
PREVIOUS WORK
Also Genuit (1986), Katz (1998) and others using simplified techniques
4
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• HRTFs extraction using the SVD and the BEM
• Sound field animations
• Conclusions
CONTENTS
5
PROJECT “BOTTLE-NECKS”
3D modelsScanner / digitiser
(CAD, Cyberware ps, 3030, Hi-rez)
Artificial head
Human head
valid BEM / IFEM modelsBoundary conditionsDecimation / holes
computing resourcesHardware
(PC, parallel (‘super’ computers)
Software
(Sysnoise,Comet, Ansys, Ideas)
validationMeasurementsAnalytical solutions
6
• Develop a tool for accurate analysis of the physical mechanisms
of the external ear
• Analyse pinna-based spectral cues at high frequencies
• Obtain individualised HRTFs by means of an optical sensor
• Use numerical methods for analysis of simple models where
analytical solutions cannot be used
• Visualise sound fields for virtual acoustic systems
NUMERICAL MODELLING OF HRTFs - OBJECTIVES
7
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• HRTFs extraction using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
8
METHODS FOR ACOUSTIC CALCULATIONS
Analytical methods
•Closed form solutions•Only for simple geometry
Geometrical methods
•Ray/beam tracing•Mirror images
Statistical energy methods (SEA)•Energy exchanges between system components
Numerical methods
•Finite Element Method (FEM)•Volume discretisation into finite elements
•Boundary Element Method (BEM)•Discretisation of bounding surface into boundary elements
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∫ ∇−∇=s
dSgppgp nrrrrrrr )].|()()()|([)( 000000
Σ
Vn
n
S
σn
r r0
BEM - DIRECT BOUNDARY INTEGRAL EQUATION (BIE)
)()()( 0vol22 rr Qpk −=+∇
Inhomogeneous Helmholtz equation
||4)|(
0
||0
0
rrrr
rr
−=
−−
π
jkeg
Free space Green function
(harmonic excitation)
3D 2D - computationally inefficient
10
DBEM (Direct BEM)
•Solves the pressure and particle velocity on the boundary surface•Exterior or interior domains•Discretisation, collocation, shape functions, nonsymmetric matrices•Efficient with small to medium size problems
IBEM (Indirect BEM)
•Solves the differences between the outside and inside values of thepressure and particle velocity on the boundary surface
•Exterior and interior domains•Variational formulation, symmetric matrices•Efficient with large problems
Special formulation: symmetric, axisymmetric, baffled models
Non-uniqueness problem (irregular frequencies) regularisation
BEM - PROPERTIES
11
THE “NON-UNIQUNESS” PROBLEMVALIDATION OF THE SPHERE MODEL
Front Rear
Without overdetermination
With overdetermination
Analytical solution
12
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex
models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
13
CALCULATION OF HRTFs USING THE PRINCIPLE OF RECIPROCITY
•Refined ear
•Source positioned close to entrance to ear-canal
u1=U0ejωt
B
p1
A
u2=U0ejωt
B A
p2
14
82
84
86
88
90
92
94
96
0 2000 4000 6000 8000 10000
Frequency [Hz]
Mag
nitu
de [d
B]
Pressure at the ear due to a source at 1;1;1
Pressure at 2mm outside the ear due to asource at 1;1;1Pressure at 1;1;1 due to a source at 2mmoutside the ear
VALIDATION OF THE PRINCIPLE OF RECIPROCITY
•Errors: <0.2 dB
•HRTFs can be calculated with high accuracy for near-field and far-field
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CALCULATION OF HRTFs USING THE PRINCIPLE OF RECIPROCITY (dB scale)
200 Hz1,000 Hz
2,000 Hz5,000 Hz
-2.3 / +2.7 -7.2 / +7.2 -32.7 / +8.3 -39 / +16.5
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MESH DECIMATION
• Preserve shape
• Normalise distances
between vertices
• Minimise number
of vertices
• Edge split, edge collapse
(a)
(b)
e f1f2
v1
v2
vedge-
collapse
edge-splitv1
v2
v3
v4
e
v4
v1
v3
v2e1e3 e2
e4
vv1
v2
shrinkage
expansion
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POLYGON REDUCTION / NORMALISED MESH MODELS -FULL AND HALF MODELS OF KEMAR
No. of elements
300050001000020000
250050001000015000
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HRTF SIMULATION OF LOW-MEDIUM SIZE SIMPLE MODELS
•CORTEX head - with and without torso. Converted from CAD and decimated.•Sphere - r = 8.75 [cm]
•Ellipsoid - rx=9.6, ry=7.9, rz=11.6 [cm].•‘Ear’ positions optimised for minimal errors, and locally refined for reciprocity.
19
HRTFs OF AN ELLIPSOID
HRTFs at horizontal plane (el = 0°, az = 0°-355°,)
HRTFs at elevation (el =45°, az =0°-355°,)
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MATLAB GUI OF NUMERICALLY MODELLED HRTFs OF AN ELLIPSOID
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COMPARISON OF HRTFs OF SIMPLE MODELS-HORIZONTAL PLANE
Left ear / azimuth - 0 deg.
88.00
90.00
92.00
94.00
96.00
98.00
100.000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]
Mag
nitu
de [d
B]
Sphere
Ellipsoid
Head Cortex
Head+Torso Cortex
Head+Torso - smoothed
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COMPARISON OF HRTFs OF SIMPLE MODELS -AT ELEVATION
Right ear /azimuth 45 deg./elev 45 deg.
91.00
92.00
93.00
94.00
95.00
96.00
97.00
98.000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]
Mag
nitu
de [d
B]
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HRTF SIMULATION AND MEASUREMENT ARRANGEMENTS
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HRTFs OF KEMAR (WITH DB60)MEDIAN PLANE MEASUREMENT AND SIMULATION
0 deg. 40 deg.
90 deg. 130 deg.
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COLOUR MAPS OF SIMULATION AND MEASUREMENT OF THE HRTFs OF KEMAR - MEDIAN PLANE
MEASUREMENTSIMULATION
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COMPARISON OF SIMULATION AND MEASUREMENT OF THE ILD IN THE LATERAL VERTICAL PLANE
SIMULATION MEASUREMENT
27
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
28
THE RESPONSE OF THE EXTERNAL EAR -SIMULATION MODEL AND MEASUREMENT APPARATUS
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-15
-10
-5
0
5
10
15
1000 3000 5000 7000 9000 11000 13000 15000 17000 19000
Frequency [Hz]
Mag
nitu
de [d
B]
DB65 - simulation; theta=0, phi=0;
DB65 - measurement; theta=0, phi=0
SIMULATION AND MEASUREMENT OF THE RESPONSE OF A BAFFLED DB65 PINNA - θ=0
30
-10
-5
0
5
10
15
1000 3000 5000 7000 9000 11000 13000 15000 17000 19000
Frequency [Hz]
Mag
nitu
de [d
B]
DB65 - simulation; theta=90, phi=0;
DB65 - measurement; theta=90, phi=0
SIMULATION AND MEASUREMENT OF THE RESPONSE OF A BAFFLED DB65 PINNA - θ=90
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-25
-20
-15
-10
-5
0
5
10
1000 3000 5000 7000 9000 11000 13000 15000 17000 19000
Frequency [Hz]
Mag
nitu
de [d
B]
DB65 - simulation; theta=180, phi=0;
DB65 - measurement; theta=180, phi=0
SIMULATION AND MEASUREMENT OF THE RESPONSE OF A BAFFLED DB65 PINNA - θ=180
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SIMULATION AND MEASUREMENT OF THE RESPONSE OF A BAFFLED DB60 PINNA
Lateral vertical (frontal) plane
Resolution of 1 degree on a linear scale
High accuracy up to 20 kHz
SIMULATION MEASUREMENT
33
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
34
MODES OF A CYLINDER IN AN INFINITE BAFFLE
[dB] 86.710
⎟⎠⎞
⎜⎝⎛+=
RL
PP
822.04
max +=RL
Rλ
2R
L
•Simple theory
L=10mm, 2R=22mm
35
MODES OF AN INCLINED CYLINDER IN AN INFINITE BAFFLE (cont.)
11 kHz
4.1 kHz
amplitude phase
36
MODE SHAPES OF THE EXTERNAL EAR(AVERGAE OF 10 PINNAE)
EXCITATION AT GRAZING INCIDENCE (AFTER SHAW 1997)
37
FREQUENCY RESPONSE OF THE CORTEX PINNA IN AN INFINITE BAFFLE - GRAZING INCIDENCE ANGLES
Omni-directional
‘Vertical’ transverse modes
‘Horizontal’transverse modes
38
BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA –THE FIRST MODE
4.2 kHz
39
BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA – THE SECOND MODE
7.2 kHz
40
BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA – THE THIRD MODE
9.6 kHz
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BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA – ‘HORIZONTAL’ MODES
11.6 kHz
14.8 kHz
17.8 kHz
42
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
43
SPATIAL PATTERNS IN ACOUSTIC SCATTERING
G = UΣVH
p = UΣVHq
Uhp = ΣVHq
qm
pn
(UHU = UUH = I)
or
p = Gq
SINGULAR VALUE DECOMPOSITION
11 12 11 1
221 22 22
1 2
M
M
MN N N NM
G G GP q
qG G GP
qp G G G
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
44
HRTF OF A RIGID SPHERE BASED ON SPHERICAL HARMONICS
2 1 ( )!( , ) ( 1) (cos )4 ( )!
m m m jmn n
n n mY P en m
φθ φ θπ+ −
= −+
(2)*
0 0 (2)0
( ) ˆ ˆˆ( ) ( ) ( , ) ( , )( )
nm mn
n nn m nn
h krp j c q Y Y
h kaρ θ φ θ φ
∞
′= =−
= ∑ ∑r r
45
GREEN FUNCTION MATRIX RELATING POINTS ON A RIGID SPHERE AND SOURCES IN THE FAR FIELD (LARGE SPHERE)
* *1 1 1 1 1 1
0 0
* *2 2 1 1 2 2
0 0
*1 1
0
ˆ ˆ ˆ ˆ( , ) ( , ) . . . . ( , ) ( , )
ˆ ˆ ˆ ˆˆ( | ) ( , ) ( , ) . . . . ( , ) ( , )
ˆ ˆ( , ) ( , ) . . . .
n nm m m m
n n n n n n L Ln m n n m n
n nm m m m
n n n n n n L Ln m n n m n
nm m
n n K K nn m n
f Y Y f Y Y
f Y Y f Y Y
f Y Y
θ φ θ φ θ φ θ φ
θ φ θ φ θ φ θ φ
θ φ θ φ
∞ ∞
= =− = =−
∞ ∞
= =− = =−
∞
= =−
=
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
∑ ∑
G r r
*
0
ˆ ˆ( , ) ( , )n
m mn n K K n L L
n m n
f Y Yθ φ θ φ∞
= =−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
∑ ∑
* * *0 1 1 0 2 2 0
00 1 1 1 1 1 1
10 2 2 2 2 2 2 * *1 1 2
0
ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , )( , ) ( , ) ( , )( , ) ( , ) ( , ) ˆ ˆ ˆ ˆˆ( | ) ( , ) ( ,
( , ) ( , ) ( , )
L Lm Nn Nm N
n N m mN n n
m NNK K n K K N K K
Y Y YfY Y Y
fY Y YY Y
fY Y Y
θ φ θ φ θ φθ φ θ φ θ φθ φ θ φ θ φ
θ φ θ φ
θ φ θ φ θ φ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦
G r r
LL L ML L
OML L
*2
* * *1 1 2 2
ˆ ˆ) ( , )
ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , )
mn L L
N N NN N N L L
Y
Y Y Y
θ φ
θ φ θ φ θ φ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
LM
L
Hˆ ˆ( | ) ( ) ( )N k l=G r r Y r FY r
46
LINEAR TRANSORMATION WITH UNITARY MATRICES
SVD:
Hˆ ˆ( | ) ( ) ( )N k l=G r r Y r FY r
( ) ( )ˆ ˆ( ) ( )
N k k
N l l
==
U Y r T rV Y r T r
H Hˆ ˆ ˆ( | ) ( ) ( ) ( ) ( )N k k N l l=G r r Y r T r T r Y r∑
Hˆ( | ) =G r r USV
47
THE SINGULAR VALUES OF A 32x32 GREEN FUNCTION MATRIX WITH UNIFORMLY SAMPLED SPHERES
48
REAL PARTS OF SPHERICAL HARMONICS AND THE LEFT SINGULAR VECTORS OF THE GREEN FUNCTION MATRIX
Re{ ( )}kU r
Re{ ( )}kY r
49
CALCULATION OF THE UNITARY TRANSFORMATION MATRICES
Im{ ( )}kT rRe{ ( )}kT r
HRe{ ( ) ( )}k kT r T r
50
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
51
THE SINGULAR VALUES OF THE GREEN FUNCTION MATRIX RELATING A BAFFLED CYLINDER AND THE HEMISPHERE
356 ‘field’ points121 ‘source’ points
52
BAFFLED CYLINDER AND THE HEMISPHERE -COLOUR MAPS OF THE SINGULAR VECTORS
Re {v} σ1 - 4.2 kHz Re {v} σ2 - 4.2 kHz Re {v} σ1 - 10.8 kHz Re {v} σ2 - 10.8 kHz
Re {u} σ1 - 4.2 kHz Re {u} σ2 - 4.2 kHz Re {u} σ1 - 10.8 kHz Re {u} σ2 - 10.8 kHz
53
MODELLED PINNAE
DB60 DB65
DB95DB90
KEMAR
YK
B&K CORTEX
54
B&K DB60
DB65 YK
SINGULAR VALUES OF ACCURATE PINNAE
(2825 × 209) (3906 × 209)
(3389 × 209) (3392 × 209)
55
REAL PARTS OF THE SINGULAR VECTORS OF DB60
4.8 kHz
56
REAL PARTS OF THE SINGULAR VECTORS OF DB60
8.8 kHz
57
REAL PARTS OF THE SINGULAR VECTORS OF DB60
10.3 kHz / σ1
58
REAL PARTS OF THE SINGULAR VECTORS OF DB60
10.3 kHz / σ2
59
REAL PARTS OF THE SINGULAR VECTORS OF DB60
13.8 kHz
60
FREQUENCY RESPONSE DECOMPOSITION OF DB60 WITH TRUNCATED MATRICES
* * * *11 12 1 11 11 21 1 1
* * * *2 12 22 2 2 2 21 22 2 2
* *1 2 1 2
1 2
n Nn N
n N n N
n n n nn Nn n m m
N N n nN NN N
v v v vp u u u up u u u u v v v v
p u u u u v v
p u u u u
σσ
σ
σ
1⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
L LL LL L L L
M M M O M M M O M M O M M ML L L
M M M M M O M OL L
1
2
* *
* * * *1 2
mmn mN
MM M MN MN
qv v
qv v v v
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
M
LMM M M M O M
L L
*
1
N
n n nn nm mn
p u v qσ=
= ∑
θ=00 ,ϕ=900
θ=00 ,ϕ=00
61
• Project description
• Overview of numerical modelling techniques in acoustics
• HRTFs and the principle of reciprocity (simple/complex models)
• Frequency response of baffled pinnae
• Acoustic modes of the external ear
• Spherical harmonics and mode shapes
• Extraction of HRTFs using the SVD and the BEM
• Sound field animations
• Conclusions
Where are we?
62
SCATTERED SOUND FIELD AROUND KEMAR DUE TO A MONOPOLE - FREQUENCY AND TIME DOMAINS
63
STEREO-DIPOLE VIRTUAL ACOUSTIC IMAGING SYSTEM
Σ
C(z)Hm,A(z)
u(z)Recorded signals
d(z)Desired signals
e(z)Errorsignals
v(z)Source input
signals
w(z)Reproduced
signals
z-m A(z)
Target matrixModelling delay
Matrix of optimalfilters
Matrix of planttransfer functions
•Frequency domain - DC (1 Hz) to 6400 Hz, steps of 200 Hz
•Time domain - Digital Hanning pulse, impulse response of each field point
•Cross-talk cancellation - inverse problem
[1 0 ]T
64
STEREO-DIPOLE FREQUENCY AND TIME DOMAIN ANIMATIONS
65
CONCLUSIONS
•Numerical modelling of HRTFs is NOT a trivial task.
•HRTFs can be modelled accurately to between10-15 kHz, and the response of baffled pinnae can be modelled accurately up to 20 kHz.
•The accuracy of the laser scanner appeared to be significant for the analysis at high frequency.
•The normal mode shapes, as found by Shaw, were validated and investigated with numerical techniques rather than measurements.
•A connection between orthogonal basis functions and the SVD has been shown.
•“Mode shapes” can be found for any defined Green function matrix.
•The spatial patterns (of the six investigated pinnae) have similar shapes although with differences in magnitude and a slight shift in resonance frequencies.
66
CONCLUSIONS (cont.)
• It is possible to decompose a reduced order frequency response with only a few terms in the series for baffled pinnae.
• It is possible to visualise the sound field in the frequency and time domains for different arrangements of virtual acoustic imaging systems.