Adaptive Wave Field Synthesis for Surround Sound Reproduction from an Array of Loudspeakers

1. Adaptive Wave Field Synthesis for Surround Sound Reproduction from an Array of Loudspeakers ECE 463: Adaptive Filters Project Presentation: March 9, 2006 Louis Terry

2. Presentation Overview • Motivation • Problem Statement • Generalized Problem Statement • Mathematical Background • Generalized Solution • Wave Field Analysis • Wave Field Synthesis • Model-Based rendering • Reduction to solution of original Problem Statement • Practical WFS System • Adaptive Adjustment of System • Least Squares Implementation • Questions?

3. Motivation • Create a realistic surround sound experience from a single “speaker” • Yamaha YSP-1000: • Actually 42 small acoustic drivers Single Beam Calibration and Surround Sound Beams Images courtesy of Yamaha

4. Problem Statement • Given: • Linear array of speakers in an enclosed room • Find: • Optimal delay and amplitude per speaker to emulate 5 channel sound (left, right, center, back left, back right) Images courtesy of Yamaha

5. Virtual Sources: outside/inside listening area Plane Source Generalized Problem Statement • Given: • Nonlinear array of speakers in an enclosed room • Find: • Optimal delay and amplitude per speaker to emulate arbitrary point and/or plane sources Images courtesy of Sonic Emotion

6. Mathematical Background • Huygens Principle: • “[T]he wavefront of a propagating wave of light at any instant conforms to the envelope of spherical wavelets emanating from every point on the wavefront at the prior instant” Image courtesy of Mathpages.com

7. Mathematical Background • Kirchhoff-Helmholtz integral Geometry for Kirchoff-Helmholz integral Image courtesy of MS Thesis, Paul D. Henderson

8. Mathematical Background • Explanation of Kirchoff-Helmholtz integral • Given the pressure and pressure gradient on a closed surface, one can recreate the complete wave field inside that closed surface. • Leads to Wave Field Analysis (WFA) • To synthesize the wave field, one can use a continuum of monopole and dipole sources distributed on the enclosing surface. • Leads to Wave Field Synthesis (WFS)

9. Generalized Solution1 • Wave Field Analysis (WFA) • Use WFA to determine acoustic properties of the room • Design a filter to compensate for the acoustics of the room • In general is not minimum phase and the exact inverse can not be calculated • Wave Field Synthesis (WFS) • Use WFS to design a filter to recreate an arbitrary sound field • Assumption: Listening area mostly enclosed by loudspeakers • Final transfer function from input to auralized wave field: 1: From multiple papers authored by S. Spors, A. Kuntz and R. Rabenstein, University of Erlangen-Nuremberg

10. Wave Field Analysis • Idea: Transform pressure field into plane waves with incident angle and intercept time with respect to a reference point (plane wave decomposition) • Use multi-dimensional spatial Fourier transform to decompose pressure field • Radon transformation may also be used • Inherent issues: • Spatial aliasing • Usually only a 2-D analysis can be done • Out of plane sources impossible to mode • Pressure field obtained from discretized Kirchoff-Helmholtz integral

11. Wave Field Synthesis • Idea: Generate loudspeaker driving signals given either a wave field to reproduce (data-based rendering) or sources to emulate (model-bade rendering) • Data-based rendering: • Must use specialized equipment to capture particle velocity as well as pressure field and then extrapolate driving signals from data • Model-based rendering: • Given source types (plane/point) and spectrum can mathematically solve for pressure field • Loudspeaker driving signals can be derived from this information

12. : Location of loudspeaker : Spectrum of point source : Geometrically dependant constant : Distance between loudspeakers : Wavenumber : Location of source Model-based Rendering • For point source:

13. Model-based Rendering • Spectrum of loudspeakers: • In the time domain: • Superposition applies for rendering fields with multiple sources

14. Reduction to original problem statement • Goal: Use array of loudspeakers to emulate 5 channel surround sound • Traditional 5 speaker configuration treats each speaker as a point source to synthesis a coarse wave field • Solution: • Solve for with equal to the audio of channel

15. WFS System W M x N auralized wave field L L x 1 Primary sources q N x 1 Room compensation filters C M x M listening room transfer matrix R L x M Practical WFS System • Can be represented as a series of matrix operations Room dependent!

16. Adaptive Adjustment of System • Adapt room compensation filter to compensate for room transfer function • Need microphone array(s) to measure pressure field in the listening room • For optimizing on a 2-D plane (consistent with previous analysis), a circular array is ideal • Least Squares algorithm is used to adapt

17. Adaptive Adjustment of System • System Diagram: • Cost function:

18. Adaptive Adjustment of System • Plane wave decomposed microphone signals are used in error calculation • Advantage: Complete spatial information about influence of listening room is contained in decomposed wave fields • Advantage: Calculated compensation filters are valid for the complete area inside loudspeaker array • Multichannel Least Squares algorithm utilized • Minimizes the mean squared error over all directions of the plane wave decomposition for every frequency. • Each plane wave component describes the wave field inside the whole listening area for one direction • Minimizing the error for all directions results in filters compensating the whole listening area.

19. Least Squares Implementation • Minimization function: • Generally results in IIR filters! • Introduce regularization factor • New minimization function: • Extra term adds power constraint which limits length of resulting filters • Choice of regularization constant critical for convergence • Coupled with an appropriate delay resulting filters are also causal

20. : Frequency function for regularization weight Least Squares Implementation • Resulting compensation filter:

21. Questions?

22. Thank you!