Numerical Modelling of the Head-Related Transfer Function · Numerical Modelling of the Head-Related Transfer Function Yuvi Kahana ... Convert the geometry of an object ... Analytical

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







• HRTFs extraction using the SVD and the BEM


• 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

9

∫ ∇−∇=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



• HRTFs and the principle of reciprocity (simple/complex

models)




• Extraction of HRTFs using the SVD and the BEM


• 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

15

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

16

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

17

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°,)

20

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]

23

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

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• Conclusions

Where are we?

28

THE RESPONSE OF THE EXTERNAL EAR -SIMULATION MODEL AND MEASUREMENT APPARATUS

29

-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]




31

-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]




32

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

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• Conclusions

Where are we?

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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

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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

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BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA –THE FIRST MODE

4.2 kHz

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BEM SIMULATION OF THE MODE SHAPES OF THE DB65 PINNA – THE SECOND MODE

7.2 kHz

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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

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• 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









• 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


8.8 kHz

57


10.3 kHz / σ1

58


10.3 kHz / σ2

59


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

qq

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









• 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.

Documents

Numerical Modelling of the Head-Related Transfer Function · Numerical Modelling of the Head-Related Transfer Function Yuvi Kahana ... Convert the geometry of an object ... Analytical