Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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148
V.
1.
Law et
al.
'"
"0
a
~
Qi
"0
c
:::>
e
OJ
- PET web
-+
He
& 0 2
1-
rollers
- PET web
-+
Figure
1:
parallel-plate APP system.
This paper describes how freely available plasma generated acoustic emission
(microphone position at the output
of
the chamber) and non-sinusoidal arbitrary
current and voltage waveforms are captured and processed for real-time non-
invasive performance analysis. The data is collected using a multivariate
analysis tool kit, (developed in LabVIEW software) where the most critical
parameters
of
the process are extracted and decisions on input parameters are
made based on this data [4]. The process is monitored during plasma treatment
of
materials without affecting the actual operating conditions.
2. Multivariate analysis software
The software
is
placed within a laptop computer. For ease
of
use, reduced
equipment setup cost, and portability, the data collection probes (fast digitizers
and microphones) are connected to the computer through
USB ports and a in-
line microphone sockets. The MV A tools comprise a short-term Fourier
transformation
[5]
to produce acoustic spectrogram waterfall, phase space
analysis and principal component analysis
(PCA)
of
electrical waveform data
[4], and non-parametric cluster analysis packarge built from NI Vision Builder
software
[6]
.
The basic methodology
of
the Short-Term Fourier Transform (STFT)
is
to
extract N samples
of
the acoustic signal by windowing and compute the DFT
(Discrete Fourier Transform)
of
the signal without a significant spectral leakage.
In this case a Gaussian window function is used because we are detecting a
repetitive signal, rather than speech patterns. The window length is
4096. This
program cuts the recorded sound into frames and rebuilds them into a block
of
n
frames.
Principal component analysis is performed on the current and voltage
waveforms and dispalyed in the from
of
a Loading plot. The X-axis is the
voltage and the Y-axis the current. The scores are construct from the rms values
Multivariate Analysis Tools for Non-lnvasive Performance Analysis
149
of
current and voltage wavefoms plus the addition
of
the frequency in unit
of
kHz. This process is represented in equations 1 and 2. The high dimensional
space
of
the PCA Loading plot is used to track the plasma performance.
Current wavefonn
-+
Inns
-+
Inns+
f
==
score
(1)
Voltage waveform
-+
V
rm
s
-+
V
rms+
f
==
score
(2)
To perform NPCA, the Loading plot image is exported into NI Vision builder.
This software contains a suite
of
hieratical particle measurement tools and is set
to
look for bright objects within a local threshold. A background correction
method within the 'detect object step' is used. The local threshold algorithm
works by calculating the intensity (I)
of
the pixel under consideration (I =
1)
with its neighbouring pixels
(In
==
1 or
0,
where 0 has
no
data point). For a data
point that
is
sized
as
a single pixel this would mean that 8 neighbouring pixels
would be examined.
If
the 8 neighbouring pixel have an I
==
0, then the pixel
under consideration will be
classiJied as a cluster. The algorithm allows the local
threshold window
to
be varied in both height and width in steps
of
one pixel.
This process
is
schematically illustrated in figure
2.
,...--,---._-.--....--.
Pixel (particle)
under consideration
1--+-
__
,
,~+--j}
Local threshold
w;
odow
\.'-
__
~
_____
.1
y-
Pixel frame
Figure
2:
Schematic
of
window used in the local threshold algorithm.
3. SE-llOO Results
Acoustic:
In
this section acoustic and PCA representations
of
the current
waveforms are presented and the interpretation discussed. The data was taken
from a helium plasma driven at
-25
kHz and applied power
of
500W. Visually,
t11is
plasma is characterised
by
a uniform although spatially non-homogeneous
plasma, with some visible features called column micro discharge (CMD)
present
[4]
. Figure 3 shows two acoustic spectrums for these plasma modes.
150
V.
1.
Law
et
al.
Figure
3:
plasma acoustic spectrum 500 W He plasma. The top spectrum con·esponds to tbe bottom
band cluster in figure 4. The bottom acoustic spectrum
coaesponds to tbe top cluster in figure 4.
A tone analysis
of
the acoustic spectrums reveals a number featmes. Firstly the
fundamental drive frequencies are slightly different at
(fa
= 24.5 and
25
kHz,
respectively) and hence their second overtones 49 and
50 kHz. Secondly the
combinations (or, intennodulation distortion)
is
present but
is
most marked in
figure 3b
as
compared
to
figure 3a. This shou
ld
be
is
excepted
as
we are
measuring sound energy, rather than electromagnet frequencies. Mathematically
this can be expressed
as
fo
±
fl-3
to generate a comb
of
sums and differences
combination tones around
fo
and its second overtone. Table 1 and 2 reveal the
result
of
this analysis for figure 3a and 3b. In Table
1,
the tabulated results
indicate a partial intermodulation with the difference being present for
f~
and the
sums present
of
the second overtone.
Table
1:
Intermodulation distortion analysis for figure
3a
.
Figure 3a
fl
orh
or.f3
-f.
+fx
24.5 kHz
±fl
-3
kHz -21.5 kHz -27.5 kHz
24.5
kHz±h
-5kHz
-18.5 kHz
No
49.5 kHz
±fl
-3
kHz
weak
-53.5 kHz
49.5 kHz
±f
2
-5
kHz weak
-54.5 kHz
Table
2:
lntermodulation distortion analysis for figure 3b.
Figure 3b
.il
orh
orf3
-j~
+.f~
25
kHz
±fl
-1
kHz
-24
kHz
No
25
kHz±h
-4kHz
-21
kHz
-29
kHz
25
kHz
±f
3
-9.5
kHz -15.5 kHz
-34
.5 kHz
50 kHz
±fl
-1
kHz
-49
kHz No
SOkHz±h
-4kHz
-46
kHz
-50
kHz
50 kHz
±.f3
-9.S kHz -40.5 kHz -59.S kHz
Multivariate Analysis Tools
for
Non-Invasive Performance Analysis
151
Table 2 shows that the strong intermodulation distortion relationship. Indeed the
distortion can
be
found for nearly all combinations
up
to
fo
±il-3
Furthermore the
displayed spectral densities for these frequencies are evident with respect to
figure 3a. The measured sum and difference need to have a physical origin,
generally this is thought
to
be a non-linear mechanism, and in this case the
reactance
of
the plasma is the likely cause.
PCA
results: To find the deterministic link between the acoustic data and the
plasma, PCA
of
the plasma generative current and voltage waveforms has been
performed. Figure 4 shows the exploration
of
this link for the first minute
of
the
500 W helium plasma generation. The process starts at the bottom left and
progresses
to
the top right with data points added at are rate
of
1 point per 0.5
seconds. The red-line highlights the trajectory between the clusters. Here it is
seen that two clusters are formed, the clusters elongate in the direction
of
the
arrow and process time. It
is
observed that the acoustic emission spectral density
profile is synchronised with the individual clusters. The bottom cluster
corresponds to the acoustic emission in figure 3a and the upper cluster is
associated with figure 3b. Under these conditions the two clusters are self-
evident. However when the process is run at high powers the cluster number
becomes complex making the classification and interpreted uncertain.
To check that the clusters do not arise from the measurement software or from
the multiplicity
of
chambers two tests have be performed. The software test used
a dual function generator that produced similar input waveforms: the outcome
being a signal cluster with a random spread
of
data point. The multi-chamber
effect was eliminated
as
the cluster effect was found when only one chamber
was used.
With this knowledge a series
of
peA
plots as a function
of
applied power steps
(400, 700, 1000, 1100, 1300 and 1500 W) were made, the outcome
of
this
experiment is show in figure
5.
The data reveals the cluster number increases
with through the
500 to 1300 W power ranges. Above this power the clusters
begin to merge and visual inspection becomes difficult. At these high powers it
known that the plasma
is
operating in the spatially homogeneous pesudoglow
discharge mode. For further cluster analysis non-parametric cluster analysis was
performed.
152
V
J.
La,\, d
al.
Figure
4:
peA
loading plot
of
helium plasma driven at 500 W as a function
of
time.
iB
0
..
~7,5
los..
.:
•
•
"'s
..
'I,;···.
'
....
8 ... '
II
Voltage score range
Figure
5:
Cluster formation for the helium plasma
as
a function
of
6 power steps.
NPCA
results:
To
match the captured
duster
data NPCA on the data sets have
been performed. With the data images exported into Vision builder software, the
local adaptive threshold window XY kernel was iterated so that 2 clusters at low
power where registered and increase in number
in the mid power range before
falling to one
or
two clusters at high power. The best fit was found when the
kemels equaJ; X = 17, and Y:=
17
pixels. The cluster fit is shown
in
figure 6.
Here it is seen that the algorithm does reproduce to a high degree the visual
perspective
of
the data. One
or
two outlying data points not registered can
be
attributed to transients that
do
not belong to
anyone
of
the man Clusters.
Multivariate Analysis Tools for Non-Invasive Performance Analysis
153
~
..
iii
..
,
, ill
400W
(2)
700
W(2)
..
..
-
+
.
1..
.
....
.
.
,
1000
W
(3)
11
00W
(4)
I
., .
A
i
I
I
I
.'
I
300
W
(2)
1500W
(1)
Figure
6:
Cluster calculation
of
Helium plasma
at:
400, 700, 1000, 1100, 1300 a
nd
1500 W.
4. Conclusion
This abstract has described the application
of
data-driven multivariate analysis
(real-time frequency spectral analysis,
PCA and NPCA) to atmospheric pressure
plasma. Three mathematical tools were applied to the acoustic emission
discharge and generative analogue current and voltage waveforms. The result is
a linear transformation
of
the input data into high dimensional space that is
visible to the operator as a function
of
process time. Using this format a
relationship between the acoustic emission and electrical data is found. In the
lower power mode (uniform although spatially non-homogeneous plasma) two
alternating clusters with a time periods
of
-10
seconds are found. When scaling
and the sampling rate are fixed, NPCA can be performed. The
NPCA was
performed using Euclidian distance, rather than angle measurement [7]. The
reason for this approach is to enable the local threshold algorithm within the
National Instruments LabVIEW Vision builder software to be employed with
ease. A local threshold algorithm was utilised.
It
was found that the best fit to
the applied power to cluster has a symmetrical kernel
of
X =
17
: Y =17. The
outcome
of
instrumentation tests and chamber geometry studies point to a
plasma influence in the formation
of
the observed cluster partitioning.
With regard to the input-output data relationship in real-time, the spatial
orientation
of
the displayed data points will allow a surrogate model
of
the
underlying physical process to be generated; this aspect
of
the work is under
investigation.
154
V.
1.
Law et al.
Acknowledgements
This work is partly supported by Enterprise Ireland grant CDT17/IT/304 and the
Science foundation Ireland under grant
08/SRC11141.
References
[1] B. Twomey, M. Rahman, O. Byrne,
A.
Hynes, L.-A.
O'Hare,
L. O'Neill
and D. Dowling. Effect
of
plasma exposure
on
the chemistry and
morphology
of
aerosol-assisted, plasma-deposited coatings. Plasma
Processes and Polymers. 5, 737-744, 2008.
[2] Evaluation
of
real-time non-invasive performance analysis tools for the
monitoring
of
pilot plant atmospheric pressure plasma system.
1.
Tynan,
V.I.
Law, B. Twomey,
A.
M. Hynes, S. Daniels O. Byrne, D. P Dowling.
Meas, Sci,
Technol20
, 115703,2009.
[3]
V.I
. Law, V. Milosavljevic, N. O'Connor,
1.
F.
Lalor, and S. Daniels
Hand-held
flyback driven coaxial dielectric barrier discharge: development
and characterization. Rev, Sci, Inst. 79(9) 094707-1-10,
2008.
[4]
V.I.
Law,
N.
O'Connor,
B. Twomey, D. P. Dowling and
S.
Daniels.
Visualization
of
Atmospheric pressure plasma electrical parameters. Topics
of
Chaotic Systems: Selected Papers
of
Chaos 2008 International
Conference. World Scientific Publishing
2009.
[5]
H. Wu, M. Siegel and P. Khosla. Vehicle sound signature recognition by
frequency vector principal component analysis. IEEE Trans, Inst, Meas.
48(5)
1005-1009. 1999.
[6] Information on National LabVIEW and Vision builder AI3.5 can be found
at
httpll:www.ni.com
[7] N. F. Thornhill, H.
Melb0
and
1.
Wilko
Multidimensional visualization and
clustering
of
historical process data. Ind. Eng. Chern. Res. 45, 5971-5985,
2006.
Chaos Communication: An Overview
of
Exact,
Optimum and Approximate Results Using
Statistical Theory
Department
of
Statistics
University
of
Warwick
Coventry CV 4 7 AL, UK
Anthony
1.
Lawrance
(e-mail:
a.j.lawrance@Warwick.ac.uk)
155
Abstract:
This paper overviews exact, optimum and approximate decoding and
performance results for antipodal chaos shift-keying (CSK)
in
which a bit
is
transmitted by modulating a chaotic segment and decoded by use
of
the corresponding
unmodulated reference segment. Both
single~
and multiple-user versions with both
known- and transmitted-reference segments are considered, the so-called coherent
and non-coherent cases. There are three main themes to the paper (i) the use
of
statistical likelihood theory for deriving optimum or improved decoders, (ii) the
availability
of
mathematically exact theory for BER performance
of
decoders, (iii)
qualitative statistical insights provided by simple Gaussian approximations to BER.
Keywords: communication systems, chaos shift keying, correlation decoding,
likelihood-based decoding, exact calculation
of
bit error rate, Gaussian
approximations, statistical theory.
1. Introduction
Chaos-based communication involves using segments
of
chaotic waves to
carry messages rather than the traditional sinusoidal waves. There are
perceived advantages
of
chaos-based communication in terms
of
spread
spectrum, security and robustness. Research activity in the area can be traced
back to before
2000, but the set
of
papers in a special issue
of
the IEEE
proceedings, Hasler, Mazzini
et aI., 2002[3], was definitive at the time. Since
then there has been much activity in the journals, a pair
of
linked monographs,
Lau and Tse,
2003[9], Tam, Lau et aI., 2007[16], and three edited collections
of
chaos applications in telecommunications, Kennedy, Rovatti et aI., 2000[5],
Stavroulakis, 2006[14] and Larson, Liu
et
aI., 2006[8]. The aim here is to
give an overview
of
exact, optimum and approximate results from the point
of
view
of
statistical theory. There is much engineering leverage to be obtained
from this perspective. Earlier results were transfered from conventional
communication systems
or
derived with simple Gaussian approximations
which did not fully incorporate the new joint chaotic-statistical interplay and
sometimes tended to under-represent
or
over-complicate theoretical matters.
A particular feature
of
chaos communications systems
is
the interplay
of
dynamical and statistical behaviour, and more particularly, the interplay
of
the
156 A.
1.
Lawrance
x2
=(
x2i
l
Channel
x,
=
(x"l
Figure I. Block diagram
of
multi-user chaos shift-keying communication system.
statistical aspects
of
chaotic spreading and channel noise. The exact results
give useful insights as well as opening computational paths, while the
reworked and additional Gaussian approximations, although inaccurate,
direct attention to key quantities determining performance. Modelling in the
paper
is
restricted to employing discrete rather than continuous time, and thus
is discrete time baseband. The models used are mathematical extractions
of
real systems, without electronic intricacies, but never-the-Iess allow
theoretical investigation
of
performance and design issues.
2. Chaos Shift Keying Communication
Chaos shift-keying (CSK) is perhaps the most developed form
of
chaos-
based communication, probably inspired by earlier spread spectrum systems,
and can be traced back to Dedieu, Kennedy
et aI., 1993[2] and Kocarev and
Parlitz, 1995[6]. An engineering perspective on the area is given in the
monographs by Lau and Tse,
2003[9] and Tam, Lau
et
aI., 2007[16].
To appreciate the structure
of
a CSK communication system, the block
diagram in Figure I illustrates for
L-users.
The
binary messages
hi =
±l,l
= 1,
2,
..
,L
are
to
be
simultaneously transmitted
in
a memoryless
channel by the
L users to L receivers. The focus
is
on a single active user,
in
the presence
of
interference by other-users. To transmit a binary message,
user
I has a chaotic generating map rO defined over
(-c,
c) , such
as
logistic
or Bernoulli-shift, which provides a spreading segment, given by the row
vector
xJ
= (xlI ,
···
,X
IN
)
,where
- ( ) -
(I
- I)
()
(I)
()
- { (
t-I)
()}
-1
2 N
X l t - r X
lt
_
1
- r
XII'
r . - r r
.,
t - " ... , .
(1)
The binary bit
hi
is
multiplied into each value
of
the spreading segment and
is
then transmitted as the message segment,
hi
XI
j ' j = 1,2,
...
, N .
All the L bit-modulated spreading segments are transmitted additively in
the same time slot and arrive at all receivers but with additive individual
Exact, Optimum and Approximate Results Using Statistical Theory
157
channel noise, C
li
, j = 1,2, ... , N for the til user; this is white Gaussian noise
with variance
a;.
In
the simplest model, described
as
coherent, additionally
the individual spreading segment
of
the til user is available without noise at
the
[,"
receiver, either by chaotic synchronization or by identical chaotic
generation, and called the reference segment. This segment is compared with
message segment
in
order to decode b
l
,
or
in
statistical language,
to
estimate
b
l
.
In
the non-coherent or differential case, the spreading segments are
separately and additionally transmitted and are received with channel noise or
synchronization error. The main motivation
of
all the subsequent modeling is
calculation
of
the bit error rate.
For the
[,"
and active-user transmitting a single bit, the message segment
containing the bit information arriving at the
[,"
receiver is
L
'ii
=
blx
li
+ I
bkx
ki
+ c
li
' j = 1,2,
...
, N
b'I=1
(2)
for I = 1,2, ... ,
L.
The summation term represents interference by the message
bits
of
other users. The decoding
of
the bit b
l
in the received
[,"
message
segment
is
done
in
conjunction with knowledge
of
the
t"
spreading reference
segment, but without knowledge
of
the other spreading segments at the
receiver. The message bit
b
l
is treated as a statistical parameter and the
optimum decoder
is
the likelihood-based estimate
of
b
l
;
on the other hand, the
bits and spreading segments
of
the other users are not
of
interest to the
t"
user and are treated
as
unknown random variables.
A statistically-based view
of
bit error rate is that it is the sum
of
the
conditional probabilities
of
decoding a
+1
as
-1
and vice versa weighted by
the ± 1 transmission proportions. For most systems and decoders, by
symmetry the conditional probabilities are equal, and
so
no weighting
is
necessary.
3. Single-User CSK: Likelihood Decoding
Developing the single-user case L = I is a useful way to unfold the
primary approach without the complication
of
interference
by
other users; the
suffix I = 1 is henceforth omitted
in
this case. The received message segment
is, by taking L = I
in
(2),
(3)
With the Gaussian white noise channel, the likelihood
of
the transmitted bit b
and the unknown noise variance
a;
is obviously
l(b,a;lr)=(aE5)
exp
-~I(ri-bxy
.
N { 1 N }
2a,
]=1
(4)
Maximum likelihood estimation implies choosing the value
of
b which has
the greater likelihood, and the appropriate likelihood ratio simplifies
as