THEORETICAL STUDY OFSOUND FIELD

RECONSTRUCTION

F.M. Fazi P.A. Nelson

Sound Field Reconstruction

Different Techniques

Least Square Method (LSM)• Based on minimising the error between

the target and reconstructed sound field

High Order Ambisonics (HOA)• Based on the Fourier-Bessel analysis of

the sound filed

Wave Field Synthesis (WFS)• Based on the Kirchhoff-Helmholtz

integral

LSM: basic principle

Loudspeaker complex gains (vector a) are obtained by directly filtering the microphone signals (vector p)

This process can be represented as p=Ca

Cp a

LSM: basic principle

Vector p represents the microphone signals obtained measuring the original sound field.

p represents the microphone signals obtained by measuring the reconstructed sound field.

The target is to chose the loudspeaker gains that minimise

2ˆe = p - p

p p

,k lH

H

LSM: Propagation Matrix

It is possible to compute or measure the propagation matrix H.

Element Hk,l represents the transfer function between the l-th loudspeaker and the k-th microphone

The mean square error is now

Matrix H

2 2ˆ e = p - p p - Ha

Σ is a non negative diagonal matrix containing the singular values of H

U, V are unitary matrices, which represent orthogonal bases

LSM: solution and SVD

HH = UΣV

The solution to the inverse problem (matrix C) is given by the pseudo-inversion of the propagation matrix

Applying the Singular Value Decomposition, the propagation matrix can be decomposed as

The computation of Matrix C becomes:

H -1 H HC = V(Σ Σ) Σ U

+ +C = H a = H p

Linear algebra and functional analysis

1

ˆ ˆN

i

i iv ve e

ˆ os( )ˆ c ii iv ve e

v

ˆ ˆ ijji ee

1

N

i

i ip pY Y

(( ))i p x dxY x

iY p

iji jY Y

p(x)Yi(x)

êi

HEE v HYY p

x

SVD – Linear algebra

w Mvˆ ˆ H

i i

N

i=1

w wg g GG wˆ ˆ i

N

i=1

Hie e v= EEv v

v

w

M

êi

ĝi

ˆ ˆii i=wg e v

HHw =G E v

ˆie v

ˆ ig w

HG MEΛ H M GΛE=

x2

x1 y1

y2

| |p aHUΣV

SVD – Functional analysis

ˆ iu pˆ ˆii i=pu v a

( | )( ) ( )y

ySdp G Sa y yx x

GSx

Sy

ˆ iv a

1ˆˆ

ii i=av u p

1 1

ˆ ˆ( ) ( (1

( )1

)ˆ ()ˆ )x

N N

xSi i

i ii

ii

uv va dSp

iu x xx xy p

x

y

SVD - Encoding and decoding SVD allows the separation of the encoding and decoding

process The regularisation parameter β allows the design of stable

filters

UH

p a

V

12

1

22

2

0

0

ENCODING DECODING

Cp a

LSM: concentric spheres

Spherical Harmonics

| |p aHUΣV

4( )) (

y

jk

ySp

edSa

x y

xy

yx

G

( , )mnY

r1

r2

Spherical harmonics

LSM: concentric spheres

(2)21( ) ( ) ( )n njk h k j krr

Spherical Harmonics

Hankel and Bessel Functions

| |p aHUΣV

4( )) (

y

jk

ySp

edSa

x y

xy

yx

( , )mnY

r1

r2

LSM: concentric spheres

22(21 )

02

12

21

1

( ) (ˆ(

)ˆ(rˆ(r ) r ) )ˆ(r

( ))

N n

Sn m n n n

nmn

m pr

Y dSjk h k j k

ar

Y

r1

r2

|HU p-1Σ| a V

Important Consequences

It is possible to analytically compute the singular values of matrix H.

They depend on the transducers radial coordinates only.

The conditioning of matrix H strongly depends on the microphones radial coordinate.

The singular functions of matrix H and represent the spherical harmonics.

Singular values and Bessel functions

Singular Vectors and Spherical Harmonics

Normalized Mean Square Error

2

2

ˆ, ,

( , ),

S

S

p p dS

ep dS

r r

rr

Microphone radial position

Zero order Bessel function

Limited number of transducers

The presented results hold for a continuous distribution of loudspeakers and microphones (infinite number of transducers).

Problems related to the use of a limited number of transducers:

• Matrices U and V represent not complete bases• Spatial aliasing (affects all methods)• Regular sampling problem• Matrices U and V are not orthogonal if defined

analytically (but are orthogonal using LSM)

Comparison of reconstruction methods

If the number of transducers is infinite LSM, HOA and KHE are equivalent in the interior domain .

The KHE only allows controlling the sound field in the exterior domain, but requires both monopole and dipole like transducers

If the number of transducers is finite, different methods are affected by different reconstruction errors.

( ) ( | )( ) ( | ) ( )

1 if

1 2 if

0 if

yS

i

y

e

p Gp x G p

V

S

V

y x y

x y yn n

x

x

x

Original sound filed

High Order Ambisonics

Least Squares Method

Kirchhoff Helmholt

z Equation

Conclusions The basics of Least Squares Method have been presented. The meaning of the generalised Fourier transform and

Singular Value Decomposition has been illustrated. It has been shown that HOA and the simple source

formulation could be interpreted as special cases of the LSM

Further research To design a device for the measurement and

analysis of a real sound field. To design a system for analysing the sound filed

generated by real acoustic sources. To design a system for the reconstruction and

synthesis of 3D sound fields using an (almost) arbitrary arrangement of loudspeaker.

Original Sound Field

LSM with regularisatio

n

LSM

eccentric

spheres 1

LSM

eccentric

spheres 2

Thank you