theoretical study of sound field reconstruction f.m. fazi p.a. nelson
TRANSCRIPT
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