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

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

ECE 463: Adaptive Filters

Project Presentation: March 9, 2006

Louis Terry

- 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?

- 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

- 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

Virtual Sources: outside/inside listening area

Plane Source

- 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

- 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

- Kirchhoff-Helmholtz integral

Geometry for Kirchoff-Helmholz integral

Image courtesy of MS Thesis, Paul D. Henderson

- 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)

- Given the pressure and pressure gradient on a closed surface, one can recreate the complete wave field inside that closed surface.

- 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

- 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

- Use multi-dimensional spatial Fourier transform to decompose pressure field

- 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

- Data-based rendering:

: Location of loudspeaker

: Spectrum of point source

: Geometrically dependant constant

: Distance between loudspeakers

: Wavenumber

: Location of source

- For point source:

- Spectrum of loudspeakers:
- In the time domain:
- Superposition applies for rendering fields with multiple sources

- 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

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

- Can be represented as a series of matrix operations

Room dependent!

- 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

- System Diagram:
- Cost function:

- 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.

- Minimizes the mean squared error over all directions of the plane wave decomposition for every frequency.

- Minimization function:
- Generally results in IIR filters!
- Introduce regularization factor

- Generally results in IIR filters!
- 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

- Extra term adds power constraint which limits length of resulting filters

: Frequency function for regularization weight

- Resulting compensation filter: