Sound Field Reconstruction Techniques: A Theoretical Study

1. THEORETICAL STUDY OFSOUND FIELD RECONSTRUCTIONF.M. Fazi P.A. Nelson

2. Sound Field Reconstruction

3. 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

4. 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 p a C

5. 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 p p

6. 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

7. 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 Σ is a non negative diagonal matrix containing the singular values of H U, V are unitary matrices, which represent orthogonal bases • The computation of Matrix C becomes:

8. Linear algebra and functional analysis p(x) Yi(x) v êi x

9. SVD – Linear algebra x2 v M y2 êi w x1 y1 ĝi

10. SVD – Functional analysis Sx x Sy y

11. SVD - Encoding and decoding • SVD allows the separation of the encoding and decoding process • The regularisation parameter β allows the design of stable filters p a C p a UH V DECODING ENCODING

12. LSM: concentric spheres r1 r2 Spherical Harmonics

13. Spherical harmonics

14. LSM: concentric spheres r1 r2 Spherical Harmonics Hankel and Bessel Functions

15. LSM: concentric spheres r1 r2

16. 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 Hstrongly depends on the microphones radial coordinate. • Thesingular functions of matrix H and represent the spherical harmonics.

18. Singular values and Bessel functions

19. Singular Vectors and Spherical Harmonics

20. Normalized Mean Square Error Microphone radial position Zero order Bessel function

21. Limited number of transducers • The presented results hold for a continuousdistribution 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)

22. 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.

23. Least Squares Method Original sound filed Kirchhoff Helmholtz Equation High Order Ambisonics

24. 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 themeasurementand analysis of a real sound field. • To design a system for analysing the sound filed generated by realacoustic sources. • To design a system for the reconstruction and synthesisof 3D sound fields using an (almost) arbitrary arrangement of loudspeaker.

25. Original Sound Field LSM with regularisation LSM eccentric spheres 1 LSM eccentric spheres 2

26. Thank you