Sound Field Reproduction

Peter Goss

Outline• What is sound field reproduction?

• Free-field theory and simulation results

• Reverberant theory

• Implementation issues

• How to simulate reverberation

• Simulation results

• Future directions

Sound Field reproduction

• A sound field is the variation in pressure in a region

• Reproduction mimics an original sound field (say that of a point source) with a different array of speakers

Malham 1998

Trueman 2000

Long distance: point plane (1kHz below)Real Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5

-0.5

0

0.5

Imag. Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5-1

-0.5

0

0.5

1

Real Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5

Imag. Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

10m

Real Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5-0.04

-0.02

0

0.02

0.04

Imag. Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5-0.04

-0.02

0

0.02

0.04

20m

Real Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5

Imag. Theoretical

-2.5 -1.25 0 1.25 2.5

-2.5

-1.25

0

1.25

2.5-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-0.03

-0.02

-0.01

0

0.01

0.02

0.0340m

Applications of Spatialisation• Entertainment

(surround sound)• Virtual reality• Teleconferencing

Strickland 1997

Nintendo.com 2007

Other Systems: Dolby

• Used in most home theatre setups and cinemas

• Described by 5.1, 6.1 7.2 etc.

• Speakers have to be in designated positions

Shockwavesound 2008

Other Systems: WFS

• Wave field synthesis uses Huygen’s principle

• Requires a very large number of speakers

• Has been implemented in a cinema: IOSONO

Dellers 2007

Ambisonics• Similar to the technique used here

• Early versions only take first order expansion

• Generally assumes sources radiate plane waves

• Can’t have speaker close to listener

Binaural techniques

• Rely on pyscho-acoustics

• Control sound at each ear (usually with headphones)

• Specialization of sound due to two things – time delay + HRTF

• HRTF unique to each person

Why only 2D?• 2D requires much less speakers

• Simpler to setup

• Theory is very close

MATHS WARNING!

Disclaimer: the speaker does not take any legal responsibility for physical pain induced due to the following mathematics

SFR Theory• Any sound field

(steady state) must satisfy Helmholtz eq.

• Sol. in 2D is:• Each unique

sound field has a unique set of coefficients

• If we look at ‘incoming’ waves, then the Ys disappear

022 pkp

n

inxnn

inxnn

xx ekrYbekrJap ,x

n

inxnn

xekrJap ,x

22

c

f

ck

Bessel Functions

• Useful property is high-pass nature

• Can ignore higher terms

SFR Theory• Any sound field

(steady state) must satisfy Helmholtz eq.

• Sol. in 2D is:• Each unique

sound field has a unique set of coefficients

• If we look at ‘incoming’ waves, then the Ys disappear

• We can truncate

022 pkp

n

inxnn

inxnn

xx ekrYbekrJap ,x

N

Nn

inxnn

xekrJap ,x

22

c

f

ck

Setup• Speakers arranged

around a circular reproduction area:

• Speakers do not have to be in a circular arrangement

• We want to minimize reproduction error:

Bethelem & Abhayapale 2005

daP

daPP

region

d

region

d

2

2

,

,,

x

xx

Theory• Pressure at each

point is sum of pressure due to each speaker

• Assuming each speaker is an ideal point source: the transfer function is:

• H0(2) is the Hankel

function of the second kind, n=0

L

lll

L

ll

Hg

pp

1

1

,

,,

x

xx

yxx kH )(l

20, H

ziYzJz nn)(

n 2H

Transfer function theory

• H can be expanded in the same way as the field:

• So:

y

x

iny

)(nn

N

Nn

inxnn

)(l

ekrl

ekrJl

kH

2

20,

H

H yxx

L

lnln

L

l

N

Nn

inxnnl

N

Nn

inxnn

L

lll

lg

ekrJlgekrJ

Hgp

xx

1

1

1

,,

xx

Theory cont.

• The reproduction error is:

2)(dnn

N

Nnn kRw

E

K

kR

nn xdxxJkRw0

2

N

Nn

dnn kRwKE

2)(

kK /2

Matrix form:

)(

)(1

)(

)(

dN

dN

dN

d

β

Agβ

N

N

N

1

Lg

g

g

2

1

g

)()(

)()(2)(

dHd

dHdd

nn

N

Nnn kRw

E

K

Wββ

ββWββ

Finally:

• This has its known global minimum at:

• So, by varying βd, we vary the reproduced sound field… theoretically

• To check, simulation code from David Excell’s Thesis was modified

dHH WβAWAAg

1ˆ

Summary of method:1. Define region (pick maximum

radius - R)2. Choose N by rounding up kR3. Need at least L = 2N+1 speakers4. Position speakers around region5. Calculate transfer function

coefficients for each speaker6. Calculate desired coefficients7. Use magical formula:

dHH WβAWAAg

1ˆ

Simulation Results:

Reproduction error: 2.45% Reproduction error: 6.82%

Reprod. Radius = 0.3m, f = 1000Hz, speakers in circle at 2m, 0.3m -> N=6 -> 13 speakers

Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

Simulation Results:

Reproduction error: 8.35% Reproduction error: 0.58%Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-5

0

5

-5

0

5

-5

0

5

-5

0

5

Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1

Point sources Plane wave sources

Error as a function of radius

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Radius (m)

Erro

r (%

)

0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5

6

7

8

9

Radius (m)

Erro

r (%

)

How reverberation affects the method• The transfer function for each

speaker is now unknown

• Will depend on room geometry, speaker position and speaker type (directivity etc.)

• Need some way of measuring these transfer functions

Reverberation theory

• Transfer functions are expanded as before:

N

Nn

inxnnl

xekrJlH ,x

• If these transfer coefficients, α, are measured then method is same as before

Measuring the transfer functions:• Operate just 1

speaker at a time• The field at each

microphone can be given by their harmonic expansion:

• Microphones placed at the edge of the area

N

Nn

inxnn

xekrJp ,x

Measuring the transfer functions:

• Again representing in matrix form:

• Using the Moore-Penrose pseudo-inverse (†) to solve for α(l)

JJJ iNJN

NiJN

NiJN

iNN

NiN

NiN

iNN

NiN

NiN

ekrJekrJekrJ

ekrJekrJekrJ

ekrJekrJekrJ

11

21

212

11

111

222

111

D

N

N

N

l

1α

L

mics

p

p

p

2

1

p

micsmics ll pDαDαp †

Summary of method:1. Define region (pick maximum radius

- R)2. Choose N by rounding up kR3. Need at least L = 2N+1 speakers4. Position speakers around region5. Measure transfer function

coefficients for each speaker: *6. Calculate desired coefficients7. Use magical formula:

dHH WβAWAAg

1ˆ

5. Measure transfer function coefficients for each speaker:

1. Operate just 1 speaker at a time2. Place microphones at edge of

region3. Calculate D matrix based on

(expected) positions of microphones

4. Calculate best coefficients using another magical formula:

micsl pDα †

Possible Issues in Implementing:• Impracticality in ring of

microphones in centre of room

• Inaccuracy in positions of speakers

• Inaccuracy in positions of microphones

• Microphones having different responses

How to simulate Reverberation• Image-Source

method:• In optics, an image

source can be placed at equal distance from a reflector:

• Same principle, except losing some energy (α) + 4 walls – so many image sources

Image-Source Method:

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25-30

-20

-10

0

10

20

30

40

Image Source method data:num reflec B

num sources total N

1 0.837 4 42 0.700 8 123 0.586 12 244 0.490 16 405 0.410 20 606 0.343 24 847 0.287 28 1128 0.240 32 1449 0.201 36 18010 0.168 40 22011 0.141 44 26412 0.118 48 31213 0.098 52 36414 0.082 56 42015 0.069 60 48016 0.058 64 54417 0.048 68 61218 0.040 72 68419 0.034 76 76020 0.028 80 84021 0.024 84 92422 0.020 88 1012

• Using

• Number of sources quickly increases:

• Distance increase + weight decreases, so may cancel out number of sources

)(nnN rrsourcesimage 12

Effect of reverberation

Simulation setup:

• Used same settings as paper for comparison

Results!!• Code done myself –

with equations from Betlehem & Abhayapala + Allen & Berkely’s paper

• Simulating 5 reflections

• Reproduction good, but why 5 reflections?

• Varying the number of reflections get very odd results:

Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

Reproduction error: 8.35%

Plot of error against number of reflections considered:

Simulation setup:

Changing to central position in room:

Corner of room:

Speaker position uncertainty• Vary position of speakers from known

positions by an amount ~N(0,sigma^2), where sigma is varied

• Shouldn’t have effect at all, because the system does not need speaker positions

• Initial results showed no clear trend• Repeating with same setting showed

large variability in error• So, running with a set sd of 5cm, and

varying number of reflections, repeating each value 50 times

50 trials of each number of reflections: (note: log scale!)

0 5 10 15 20 25 30 3510

0

101

102

103

104

105

Basic Settings

Rep

rodu

ctio

n E

rror

(%

)

Number of reflections

Examining the outliers:• Each outlier examined

deviated only at edge• Strongly suggests that

it’s the higher basis functions that are wrong

• Weights given are very high – so suggesting it is the measuring of transfer functions

• Perhaps increasing N would help, or increasing the microphones

Real Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Theoretical

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

Real Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4 -1

-0.5

0

0.5

1Imag. Reproduced

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

-0.5

0

0.5

error: 58189%

Comparison

0 20 4010

-1

100

101

102

103

104

105

106

Basic Settings

0 20 4010

-1

100

101

102

103

104

105

106

0 20 4010

-1

100

101

102

103

104

105

106

Rep

rodu

ctio

n E

rror

(%

)

Mean comparison:

0 5 10 15 20 25 30 3510

0

101

102

103

104

Basic Basic Functions

One extra Basis Functionextra Basis Function and extra Mics

Rep

rodu

ctio

n E

rror

(%

)

Number of reflections

Median Comparison

0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

Basic Basic Functions

One extra Basis Functionextra Basis Function and extra Mics

Rep

rodu

ctio

n E

rror

(%

)

Number of reflections

More work to be done:• Not clear what this effect is due to –

may just be a program error• One theory is certain positions have

very high N+1, N+2 coefficients, which cancel out the effect of the small bessels

• This needs further testing• One possibility in implementing is

ignoring higher orders due to large time delay – possibility to be looked at

Showing general effect of speaker pos uncertainty• Choosing 10 reflections (with extra

N) and varying uncertainty:

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1610

-1

100

101

102

103

104

Effect of Speaker Variation

Means:

Microphone variability• Varied microphone weight again by

normal dist. – assuming zero bias• Showed nice clear results, below is a

95% confidence interval for the mean:

Plans for future:• Extend to Quasi-2D

• Look at unexpected speaker position results in more detail

• Effect of microphone position uncertainty

• Attempt to implement this system