THEORETICAL STUDY OF SOUND 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 C pa

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 pp

LSM: Propagation Matrix  It is possible to compute or measure the propagation matrix H.  Element H k,l represents the transfer function between the l-th loudspeaker and the k-th microphone  The mean square error is now Matrix H

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  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:

Linear algebra and functional analysis v p(x) Y i (x) êiêi x

SVD – Linear algebra v w M êiêi ĝiĝi x2x2 x1x1 y1y1 y2y2

SVD – Functional analysis SxSx SySy 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 UHUH p a V ENCODING DECODING C pa

LSM: concentric spheres Spherical Harmonics r1r1 r2r2

Spherical harmonics

LSM: concentric spheres Spherical Harmonics Hankel and Bessel Functions r1r1 r2r2

LSM: concentric spheres r1r1 r2r2

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 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.

Original sound filed High Order Ambisonics Least Squares Method Kirchhoff Helmholtz 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 regularisation LSM eccentric spheres 1 LSM eccentric spheres 2

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