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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
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Imag. Theoretical
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Real Theoretical
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20m
Real Theoretical
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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
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Real Reproduced
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Simulation Results:
Reproduction error: 8.35% Reproduction error: 0.58%Real Theoretical
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Real Reproduced
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Real Reproduced
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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
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0.3
0.35
0.4
0.45
Radius (m)
Erro
r (%
)
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1
2
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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:
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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
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Real Reproduced
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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
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Real Reproduced
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error: 58189%
Comparison
0 20 4010
-1
100
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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
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