Adaptive Wave Field Synthesis for Surround Sound Reproduction from an

Array of Loudspeakers

ECE 463: Adaptive Filters

Project Presentation: March 9, 2006

Louis Terry

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?

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

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)

N

iAi i

Images courtesy of Yamaha

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

N

iAi i

Images courtesy of Sonic EmotionVirtual Sources: outside/inside

listening areaPlane Source

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

Mathematical Background

• Kirchhoff-Helmholtz integral

dSePe

PPS

S

jS

S

j

S

SS

rrn

r

rrnrr

rrrr

),(),(

4

1),(

Geometry for Kirchoff-Helmholz integral

Image courtesy of MS Thesis, Paul D. Henderson

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)

Generalized Solution1

1. 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

2. Wave Field Synthesis (WFS)– Use WFS to design a filter to recreate an arbitrary

sound field – Assumption: Listening area mostly enclosed by loudspeakers

3. Final transfer function from input to auralized wave field:

)(zR)(zC

)(zR

)(zW

)()()( zCzRzW

1: From multiple papers authored by S. Spors, A. Kuntz and R. Rabenstein, University of Erlangen-Nuremberg

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

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

Model-based Rendering

• For point source:

xeej

KQPi i

j

mi

j imi

rrrrr

rrrr

32

)(),(

ir : Location of loudspeaker

)(Q : Spectrum of point source

K : Geometrically dependant constant

x : Distance between loudspeakers

: Wavenumber

mr : Location of source

Model-based Rendering

• Spectrum of loudspeakers:

• In the time domain:

• Superposition applies for rendering fields with multiple sources

xej

KQWmi

j

i

mi

32)()(

rr

rr

kkqkhakw ni

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 BRBLCRL WWWWW , , , , iQ

i

Practical WFS System

• Can be represented as a series of matrix operations

WFSSystem

WM x N

listeningroom transfer

matrix

RL x M

Roomcompensation

filters

CM x M

auralizedwave field

LL x 1

Primarysources

qN x 1

Room dependent!

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

)(zC)(zR

)(zC

Adaptive Adjustment of System

• System Diagram:

– Cost function: LH eezzzJ ~~~,~~1 eee

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.

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

LH

zeezzzJ ~~~ , ~~ min 1 eee

C

)()()(,~~~

)()(~~ min

1 zzzee

zzzzzJ

L

HH

z

wCve

vveeC

Least Squares Implementation

• Resulting compensation filter:

mTT zzzzBzBzzzC ARIRR~~~~ 1111

)(zB : Frequency function for regularization weight

Questions?

Thank you!