1. Project Presentation: March 9, 2006
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. Single Beam Calibration and Surround Sound Beams
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: Find:
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. 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
Virtual Sources: outside/inside listening area
Plane Source
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. Model-based Rendering
For point source:
: Spectrum of point source
: Geometrically dependant constant
: Distance between loudspeakers
: Location of loudspeaker
: Location of source
: Wavenumber

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. Practical WFS System
Can be represented as a series of matrix operations
WFS
System W
M x N
listening
room transfer
matrix R
L x M
Room
compensation
filters C
M x M
auralized
wave field L
L x 1
Primary
sources q
N x 1
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. Least Squares Implementation
Resulting compensation filter:
: Frequency function for regularization weight

21. Questions?

22. Thank you!