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Berberkic, Sanjin
Measurement of small signal variations using one-dimensional chaotic maps
Original Citation
Berberkic, Sanjin (2014) Measurement of small signal variations using one-dimensional chaotic
maps. Doctoral thesis, University of Huddersfield.
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Measurement of Small Signal
Variations Using One-Dimensional
Chaotic Maps
Sanjin Berberkic
BEng, MSc
A thesis submitted to the University of Huddersfield
in partial fulfilment of the requirements for
the degree of Doctor of Philosophy
July 2014
Abstract
A novel electronic signal Measurement System (MS) based on one-dimensional
chaotic maps (Logistic Map (LM) and Tent Map (TM)) has been developed,
analysed and tested. Firstly, an in-depth theoretical analysis of each map was
performed using MATLAB based computation, and the results demonstrated that the
high sensitivity, to initial conditions, of each map was suitable for small signal
change detection and measurement. A new 3D representation of chaos map output
for varying initial input was also developed allowing the suitability of any onedimensional chaotic map to be determined.
An electronic implementation of the chaotic maps, using low noise and low cost
components was developed along with a feedback and a series based MS. The
implementations were tested and the experimental results demonstrate a matching
within ±1 %, between theory and the electronic implementations, both maps
exhibiting behaviour identical to the theoretical maps, ranging from fixed point
stability, periodicity and chaos.
Each map implementation was tested separately and as part of a complete MS and
the results reveal that the proposed measurement technique can detect and measure
input signals changes as low as 5
over a 10 V input range, which yields a greater
resolution than a MS using an 20 bit Analogue to Digital Converter (ADC) over the
same input range.
The main advantage of the presented MS is that the accuracy of the measurement is
independent of the input range which is not the case with classical approach to
2
measurement based on conditioning circuitry followed by an ADC as the minimum
detectable change is directly proportional to the input range.
3
Acknowledgements
I would like to express my gratitude towards my main supervisor Peter Mather for
helping me in bringing this work to a conclusion, his guidance and support during
my research and particularly his help during the writing of this thesis are invaluable.
I would also like to express my deepest appreciation to Violeta Holmes for helping
me, not only during my Ph.D. but during all my studies at Huddersfield University. I
would also like to thank Roger Bromley for all the interesting discussions about
chaos and many other subjects.
Beside my supervisors, I have also had valuable help from Chris Daykin during
countless laboratory sessions and from Paul Glendinning during my visits to
Manchester University.
During all the time spent working on this thesis I had the chance of meeting and
working with great people that helped me in numerous ways, in particular I would
like to thank Maythem, Tim, Yousif and Hussam for being great colleagues and
friends, Anthony and Karim for being great housemates and for all the fun we have
had in the last 4 years.
Finally, I would like to thank my family for supporting me throughout all my
studies; my sister Nadja for being my best friend, my father Alija for his financial
support and for providing me with countless bottles of “Prangijaš”, my girlfriend
Yasmine for supporting me when I needed her the most and my mother Jelena for
being the best mother. This thesis is dedicated to them.
4
Contents
Abstract ................................................................................................................................ 2
Acknowledgements.............................................................................................................. 4
List of Figures .................................................................................................................... 10
List of Tables ..................................................................................................................... 14
Glossary of Terms and Symbols ........................................................................................ 15
Acronyms ....................................................................................................................... 15
Symbols ......................................................................................................................... 16
1
2
Introduction................................................................................................................ 19
1.1
Background ......................................................................................................... 19
1.2
Chaos and Measurement ..................................................................................... 20
1.3
Aims and Objectives ........................................................................................... 23
1.4
Original Work ..................................................................................................... 24
1.5
Document Structure ............................................................................................ 26
Theory and Literature Review ................................................................................... 27
2.1
Classic Methods of Measurement ....................................................................... 27
2.1.1
Conditioning Circuitry ................................................................................. 29
2.1.2
Data Acquisition .......................................................................................... 33
2.1.3
Measurement Method Summary.................................................................. 37
2.2
Chaos ................................................................................................................... 38
5
2.3
Discrete One-dimensional Chaotic Maps ........................................................... 40
2.3.1
The Logistic Map ......................................................................................... 41
2.3.2
The Tent Map .............................................................................................. 49
2.3.3
The Bit Shift Map ........................................................................................ 52
2.3.4
Summary ...................................................................................................... 54
2.4
Applied Chaos ..................................................................................................... 55
2.4.1
Electronic Implementation of Chaos ........................................................... 55
2.4.2
Chaos in Cryptography ................................................................................ 57
2.4.3
Chaos Based ADCs...................................................................................... 58
2.4.4
Chaos Based Measurement System ............................................................. 62
2.5
Lyapunov Exponent ............................................................................................ 63
2.5.1
2.6
3
Lyapunov Exponent Estimation .................................................................. 65
Conclusion .......................................................................................................... 67
Proposed Signal Measurement Technique................................................................. 69
3.1
Quantifying Input Signal Change ....................................................................... 72
3.1.1
3.2
System with Feedback ........................................................................................ 84
3.2.1
3.3
System with Feedback Experimental Setup ................................................ 84
Series System ...................................................................................................... 86
3.3.1
3.4
Measurement Errors..................................................................................... 75
Series System Experimental Setup .............................................................. 86
Conclusion .......................................................................................................... 87
6
4
Measurement System Implementation ...................................................................... 90
4.1
4.1.1
Logistic Map Implementation...................................................................... 90
4.1.2
Tent Map Implementation ........................................................................... 95
4.2
Measurement System Implementation ................................................................ 99
4.2.1
Feedback System Implementation ............................................................... 99
4.2.2
Series System Implementation .................................................................. 103
4.3
5
One-Dimensional Maps Implementation ............................................................ 90
Conclusion ........................................................................................................ 104
Performance Analysis of Implemented Chaotic Maps ............................................ 106
5.1
Transfer Characteristic ...................................................................................... 107
5.2
Logistic Map Transfer Characteristic ............................................................... 108
5.3
Tent Map Transfer Characteristic ..................................................................... 109
5.4
Bifurcation Diagram ......................................................................................... 110
5.4.1
Logistic Map Bifurcation Diagram ............................................................ 110
5.4.2
Tent Map Bifurcation Diagram ................................................................. 112
5.5
Time Series ....................................................................................................... 113
5.5.1
Logistic Map Time Series .......................................................................... 114
5.5.2
Tent Map Time Series ............................................................................... 115
5.6
Lyapunov Exponent .......................................................................................... 116
5.6.1
Logistic Map Lyapunov Exponent ............................................................ 118
5.6.2
Tent Map Lyapunov Exponent .................................................................. 119
7
5.7
5.7.1
Logistic Map Noise Measurement ............................................................. 123
5.7.2
Improved Logistic Map Implementation ................................................... 126
5.7.3
Tent Map Noise Measurement................................................................... 131
5.8
6
Noise Measurement........................................................................................... 121
Conclusion ........................................................................................................ 132
Measurement System Results .................................................................................. 133
6.1
Divergence Between Two Signals .................................................................... 134
6.1.1
Logistic Map .............................................................................................. 134
6.1.2
Tent Map .................................................................................................... 141
6.2
Divergence Against Input Signal Range ........................................................... 145
6.2.1
Logistic Map .............................................................................................. 145
6.2.2
Tent Map .................................................................................................... 146
6.3
Divergence Against Input Signal Change ......................................................... 147
6.3.1
Logistic Map .............................................................................................. 148
6.3.2
Tent Map .................................................................................................... 149
6.4
Conclusion ........................................................................................................ 150
7
Discussion ................................................................................................................ 152
8
Conclusion and Further Work ................................................................................. 156
8.1
Conclusion ........................................................................................................ 156
8.2
Further Work ..................................................................................................... 158
References........................................................................................................................ 159
8
Appendices ...................................................................................................................... 166
List of Appendices: ...................................................................................................... 166
Appendix A.................................................................................................................. 167
Appendix B .................................................................................................................. 168
Appendix C .................................................................................................................. 171
Appendix D.................................................................................................................. 175
Appendix E .................................................................................................................. 184
Appendix F .................................................................................................................. 186
Appendix G.................................................................................................................. 214
Appendix H.................................................................................................................. 216
9
List of Figures
Figure 1-1 Simplified Block Diagram of a Chaos Based MS ........................................... 23
Figure 2-1 Block Diagram of the Elements Composing a Measurement System ............. 27
Figure 2-2 Equivalent Input Noise Voltage of a TL071 Operational Amplifier
(Texas-Instruments, 2005)............................................................................ 32
Figure 2-3 Analogue Signal Sampled Using 4 bit and 5 Bit ADCs .................................. 34
Figure 2-4 Undetected Input Signal Variations Due to Limited ADC Resolution ............ 36
Figure 2-5 Transfer Characteristic of the Logistic Map .................................................... 42
Figure 2-6 Computed Bifurcation Diagram of the Logistic Map ...................................... 43
Figure 2-7 Bifurcation Diagram of the Logistic Map from =3.4 to =3.5 ..................... 44
Figure 2-8 Bifurcation Diagram of the Logistic Map from =3.5 to =3.6 ..................... 45
Figure 2-9 Bifurcation Diagram of the Logistic Map for =3.8 to =3.9 ........................ 45
Figure 2-10 Behaviour of the Logistic Map for Different Values of
............................. 46
Figure 2-11 Sensitivity to Initial Conditions of the Logistic Map For A
Change. ......................................................................................................... 48
Figure 2-12 Sensitivity to Initial Conditions of the Logistic Map For A
Change. ................................................................................................ 48
Figure 2-13 Sensitivity to Initial Conditions of the Logistic Map For A
Change. ......................................................................................................... 49
Figure 2-14 Tent Map Transfert Characteristic ................................................................. 50
Figure 2-15 Tent Map Bifurcation Diagram ...................................................................... 51
Figure 2-16 Bit Shift Map Transfer Characteristic ............................................................ 53
Figure 2-17 High Sensitivity to the Initial Conditions of the Bit Shift Map ..................... 54
10
Figure 2-18 Working Principle of the ADC Based on the Bit Shift Map.......................... 59
Figure 2-19 Tent Map Based ADC .................................................................................... 60
Figure 2-20 Variation of the Lyapunov Exponent for the Logistic Map ........................... 64
Figure 2-21 Lyapunov Exponent Estimation Using the Prediction Error Method ............ 66
Figure 3-1 Flowchart of the 3D Divergence Graph ........................................................... 70
Figure 3-2 Sensitivity to Initial Conditions of the Logistic Map for =4. ......................... 71
Figure 3-3 Divergence of the TM for the Entier Input Range ........................................... 72
Figure 3-4 LM divergence Graph With Set Threshold for a
of 0.0001 ........................ 73
Figure 3-5 Divergence Between Two Samples ................................................................. 74
Figure 3-6
Estimation Error ........................................................................................... 77
Figure 3-7 Two Inputs with Identical Output Values ........................................................ 78
Figure 3-8 Two Identical Time Series From Two Different Inputs .................................. 78
Figure 3-9 Number of Nulls for Different Values of N ..................................................... 80
Figure 3-10 Width of a Null for an Input Change of
............................................. 80
Figure 3-11 Delta Estimation Error Against Number of Iterations ................................... 81
Figure 3-12 Delta Estimation Error VS Amount of Divergence for Different
.............. 83
Figure 3-13 Delta Estimation Error for Different Values of the Parameter ................... 83
Figure 3-14 Block Diagram of the Presented Iterated System .......................................... 85
Figure 3-15 Series System Block Diagram ....................................................................... 87
Figure 4-1 Block Diagram of the LM Implementation ..................................................... 92
Figure 4-2 Simplified Schematic of the Modified LM. ..................................................... 93
Figure 4-3 Block Diagram of the TM Implementation ..................................................... 96
Figure 4-4 Construction of the TM Transfer Characteristic .............................................. 97
Figure 4-5 Schematic of the Proposed Implementation of the TM ................................... 97
Figure 4-6 Schematic of the Feedback System for Map Iteration ................................... 102
11
Figure 4-7 Control Signals for the Iterated Measurement System .................................. 102
Figure 4-8 Block Diagram of the Series System ............................................................. 104
Figure 5-1 Transfer Characteristic Measurement Setup. ................................................. 107
Figure 5-2 Measured VS Theoretical Logistic Map Parabola. ........................................ 108
Figure 5-3 Measured VS Theoretical Tent Map Characteristic....................................... 109
Figure 5-4 Measured Bifurcation Diagram of the LM .................................................... 111
Figure 5-5 Computed LM Bifurcation Diagram .............................................................. 112
Figure 5-6 Tent Map Experimental Bifurcation Diagram ............................................... 113
Figure 5-7 Logistic Map Experimental Time Series ....................................................... 115
Figure 5-8 Tent Map Experimental Time Series ............................................................. 116
Figure 5-9 Logistic Map Time Series for r = 3.35 ........................................................... 117
Figure 5-10 Prediction Error for a Non-Chaotic Time Series ......................................... 117
Figure 5-11 Estimation of the Lyapunov Exponent From the Prediction Error .............. 118
Figure 5-12 Tent Map Lyapunov Exponent Estimation .................................................. 120
Figure 5-13 Measured Noise Floor of the HP 3562A DSA............................................. 122
Figure 5-14 Measured Noise of the LM .......................................................................... 125
Figure 5-15 One-Quadrant Precision Analogue Multiplier ............................................. 127
Figure 5-16 Noise of the One-Quadrant Multiplier ......................................................... 129
Figure 5-17 Simplified Schematic of the Improved LM ................................................. 129
Figure 5-18 Noise of the Improved LM .......................................................................... 130
Figure 5-19 Noise of Previous Implementation VS Inproved Implementation ............... 130
Figure 5-20 Noise of the TM Implementation ................................................................. 131
Figure 6-1 Time Series and Divergence for a 500 v Change. ........................................ 135
Figure 6-2 Divergence for Two Different Input Changes. .............................................. 136
Figure 6-3 LM Multiple Runs for a Fixed Input.............................................................. 138
12
Figure 6-4 Use of the “Noise Band” to Measure the Sensitivity of the LM .................... 138
Figure 6-5 Multiple Time Series For Different Input Samples ....................................... 140
Figure 6-6 Noise Band at Iteration 8 for Different Input Samples .................................. 141
Figure 6-7 Divergence Between Two Time Series With a 200
Figure 6-8 Experimental Divergence of the LM for a Change of
Change ..................... 142
V: (a) and (b)
for the Entire Input Range. (c) and (d) for a Range Limited to 0.1-0.9 ........ 146
Figure 6-9 Experimental Divergence of the TM for a 200 V Change for the Entire
Input Range. .................................................................................................. 147
Figure 6-10 Logistic Map Divergence VS Change ......................................................... 149
Figure 6-11 Measurement Error VS Change for the Series Implementation of the TM
Based MS ...................................................................................................... 150
13
List of Tables
Table 5-1 Estimated and Theoretical Lyapunov Exponent for the LM ........................... 119
Table 5-2 Estimated and Theoretical Lyapunov Exponent for the TM ........................... 120
Table 5-3 Equivalent Noise Bandwidth Coefficient ........................................................ 123
Table 5-4 Noise for Components Used in the LM Implementation. ............................... 125
Table 6-1 Practical and Computed Divergence ............................................................... 137
Table 6-2 TM Change Measurement with Feedback System .......................................... 143
Table 6-3 TM Change Measurement with Series System ............................................... 144
Table A-1 Signal Change Estimation Error ..................................................................... 167
Table A-2 Identical Time Series For Two Different Input Values .................................. 167
14
Glossary of Terms and Symbols
Acronyms
C
Microcontroller
1D
One Dimensional
1Q
One Quadrant
3D
Three Dimensional
ADC
Analogue to Digital Converter
BSM
Bit Shift Map
BZ
Belousov–Zhabotinsky
CMOS
Complementary Metal Oxide Semiconductor
CSV
Comma-Separated Values
DAQ
Data Acquisition Device
DSA
Dynamic Signal Analyser
FPGA
Field Programmable Gate Array
FSR
Full Scale Range
GPIB
General Purpose Interface Bus
IC
Integrated Circuit
LCD
Liquid Cristal Display
LE
Lyapunov Exponent
LM
Logistic Map
LPF
Low Pass Filter
LSB
Least Significant Bit
15
MS
Measurement System
Op-amp
Operational amplifiers
PLL
Phase Lock Loop
ppm
parts per million
rms
root mean square
S/H
Sample and Hold
SD
Spectral Density
SS
Spread Spectrum
TM
Tent Map
USB
Universal Serial Bus
Symbols
Position of a potentiometer
Difference between two input signals
Divergence between two time series at a given iteration
Lyapunov exponent
Parameter of the Tent Map
Parameter of the Bit Shift Map
Parameter of a chaos map
A
Analogue input signal
B
Noise bandwidth
b
Binary number
D
Digital output signal
D
Number of Nulls in section 3.1
16
E
Thermal noise voltage
f
Function
Derivative
Highest frequency
Lowest frequency
Nyquist frequency
Sampling rate required for oversampling
Noise bandwidth coefficient
k
Boltzmann constant
k
Number of time steps used for the estimation of the Lyapunov
Exponent in section 2.5
M
Resolution of an ADC
N
Number of possible digital values of the output of an ADC
Q
Voltage step size (LSB)
R
Resistor
R
Chaos map input range in section 3.1
r
Parameter of the Logistic Map
Ref
Reference voltage
T
Absolute temperature
V
Voltage
Full scale voltage of the input of an ADC
Input voltage
Voltage noise
Output voltage
17
Highest analogue value and the input of an ADC
Lowest analogue value at the input of an ADC
State of a chaos map at iteration
Starting point of Lyapunov exponent calculation
Closest neighbour of the point
18
1 Introduction
1.1 Background
Measurement of physical quantities is essential in the majority of technical devices
as almost every electronic system requires the magnitude of one or more parameters
to be determined in order to perform specific tasks. The quantity and quality of
signal measurement required for systems have grown drastically with the advance of
technology. For example, modern cars, although performing the same task as the
older models are fitted with an increasing number of sensors to enhance user comfort
and safety; the measured parameters can range from critical; engine temperature,
collision detection and oil level to less critical; ambient temperature and seat
position. For some applications, the improvement in measurement quality (accuracy,
precision) represents the main factor to the overall system performance; this can be
the case in pollutant, earthquake or explosive detection etc., where the sensitivity and
accuracy of a system is critical.
In order to measure any physical variable a Measurement System (MS) is required.
With the increasing need for high accuracy systems, the ability to detect small
amplitude changes of a given parameter has become a key factor. Low level signal
parameter change measurement is challenging due to practical limitations such as
interferences from external sources, sensor limitations, in terms of accuracy, and
other errors inherent to the electronic circuitry (limited resolution, noise generated
by the components).
19
As MS are designed to achieve improved performances in terms of sensitivity and
accuracy, the overall complexity of the system is increased which generally
increases the overall cost. In a classical approach to a MS, the physical quantity
being measured is converted to a proportional electric signal (voltage or current) and
adapted to a given system value range before being digitalised using an Analogue to
Digital Converter (ADC). This general method gives satisfactory results in the
majority of cases but becomes inefficient when small input signal changes have to be
detected over a large input range, as the minimum detectable change is directly
proportional to the resolution of the ADC being used. This means that higher
resolutions ADCs which are generally more expensive are required to determine
small parameter changes thus increasing the overall cost of the system. To palliate
the loss of resolution with increased input signal range, a specific MS has to be used
with resolution independent from the input range. In this thesis, a chaos based MS
has been developed and tested for the purpose of eliminating the need for expensive
ADCs when a small change of input signal has to be detected over a large input
range.
1.2 Chaos and Measurement
Chaos as a means of measurement is generally perceived to be counterintuitive as the
essence of chaos is the inability to predict the future state of a system with absolute
certainty. Starting from the second half of the 20th century the study of chaos has
brought a new insight on a phenomenon that was long believed to be random
uncorrelated variations (noise), namely the high sensitivity to initial conditions. The
phenomenon was observed in various fields ranging from economics: the evolution
20
of cotton prices, biology: the evolution of population of foxes and meteorology: the
evolution of weather conditions (G.L.Baker and J.P.Gollub, 1990).
The high sensitivity to initial conditions was first mentioned by French
mathematician Henri Poincaré in 1890; while studying a three-body problem where
the motion of three bodies has to be determined at any particular point in time from
initial data (individual mass, position and velocity). Poincaré showed that the motion
of the bodies is highly sensitive to initial conditions. The significance of this
discovery remained unnoticed until 1961 when the mathematician and meteorologist
Edward Lorenz observed a similar phenomenon while running weather prediction
algorithms on a computer (Gleick, 1988). After evaluating a weather prediction
model for many hours, Lorenz wanted to confirm the results by rerunning the
simulation, but in order to shorten the time required for the simulation Lorenz
entered a result from a printout table from the middle of the original dataset.
Limiting the result to three decimal places, Lorenz assumed that the rounding error
would be insignificant. The results obtained from the second run where at first
similar to the initial simulation before diverging completely. Lorenz then realised
that the weather prediction model was highly sensitive to initial conditions and that a
small change between two starting conditions produced, after a number of iterations,
completely different results. This discovery changed meteorology as Lorenz
concluded that accurate weather prediction for long periods of time is subsequently
impossible. The impact of his discovery wasn’t limited to meteorology, but also
influenced other scientific fields where similar phenomena where observed and led
to the creation of a new science named: “chaos theory” (Ingraham, 1991).
Sometimes referred to as the “butterfly effect”, this phenomenon is the main feature
of chaotic systems. The name of the phenomenon came from the title of a talk given
21
by Edward Lorenz in 1972: “Does the flap of a butterfly’s wings in Brazil set off a
tornado in Texas? ” (Lorenz, 1972).
As opposed to stochastic (non-deterministic) systems where the behaviour is
randomly affected by external forces (noise), the behaviour of a chaotic system is
deterministic and is part of its inherent dynamics. What may appear as noise is in
fact following simple rules yet exhibiting complex behaviour. Contrary to a
stochastic process where the following data cannot be predicted but only guessed
using statistical analysis, the following data of a chaotic system can be predicted and
the accuracy of the prediction can be infinite if the initial values of the chaotic
system are known with infinite accuracy. To relate that to the weather models used
in meteorology, the weather could be accurately predicted for years if the model and
the initial data used (i.e. temperature, wind speeds, pressure etc.) were infinitely
accurate with infinite number of points (Ingraham, 1991).
In this work the fundamental idea is to reverse the process of the high sensitivity to
initial conditions and use simple mathematical expressions which exhibit chaotic
behaviour to measure small variation of input signals. If a small change of starting
conditions produces a large change at a later stage, the process can be reversed and
the large change at a given time can be used to accurately determine a small change
in the starting conditions.
The simplified block diagram in Figure 1-1, shows how a chaos based MS can be
structured. The physical variable is first converted into a usable electric signal before
being fed to a chaotic map, which takes advantage of the high sensitivity to initial
conditions to detect small variations of the input signal. This enables the MS to
detect small variations of the physical quantity being measured.
22
Measured
Parameter
Sensor
Chaotic Circuit
Control / Processing
and Storage
FIGURE 1-1 SIMPLIFIED BLOCK DIAGRAM OF A CHAOS BASED MS
The simplest mathematical models exhibiting chaotic behaviour are referred to as
chaotic maps. Chaotic maps have been implemented electronically for various
applications such as; practical approach to the study of chaotic phenomena,
cryptography and algorithmic ADCs. Chaotic maps can be multidimensional or OneDimensional (1D); all maps exhibit high sensitivity to initial conditions with the
main difference being the number of parameters defining the map. Single parameter
(1D) chaotic maps are ideal for electronic implementation due to the general
simplicity of the circuitry required to implement the maps resulting in lower overall
cost and reduced inherent system noise.
1.3 Aims and Objectives
The aim of the present work was to analyse, design, implement and assess a chaos
based MS that can detect and measure small electric signal changes independently of
the input range. The objectives fulfilled during this work were as follow:

Identify the suitable chaotic function to be implemented as part of a MS
One-dimensional (1D) chaotic maps were analysed using computation. The possible
use of the chaotic maps for signal measurement was evaluated with MATLAB
23
simulation and a method to quantify the input signal change was proposed for two
chaotic maps: Logistic and Tent Map.

Implement the chaotic functions and compare the characteristics with
theory
The two selected chaotic maps were implemented electronically: although some
electronic implementations of one-dimensional chaotic maps are available in
literature (Campos-Cantón et al., 2009, Eguchi et al., 2000, Suneel, 2006) the
proposed implementations were designed for this specific application in order to
reduce the noise and obtain a viable MS by increasing the sensitivity of the system.
The electronic implementations performance characteristics were compared with
those of the theoretical maps to ascertain the accuracy and correlation between the
theory and the practical implementations.

Build the MS and assess the performance
A new method of signal measurement using chaotic maps was proposed. The
performance of the designed MS was assessed and the maps were compared to find
the solution that produces the most advantageous performance in terms of signal
detection and measurement.
1.4 Original Work
The fundamental basis of the work presented in this thesis is original and has led to a
patent application: British Patent Application n◦ 1309585.4. The main areas of
original contribution are:
24

Analysis of the high sensitivity to initial conditions of two 1D chaotic maps
with the use of MATLAB computation software.

A novel three-dimensional (3D) representation of the divergence between
two closely located starting conditions, for the entire input range of 1D
chaotic maps has been developed. This will enable the high sensitivity of any
1D chaotic map to be analysed. The MATLAB code developed for this
purpose can be used for any further work associated with the “butterfly
effect” of 1D chaotic maps. The use of the developed 3D representation is
not limited to measurement and electronics as the high sensitivity to initial
conditions is present in many fields such as neurology, meteorology and
biology.

New electronic implementations of the two selected 1D chaotic maps,
specifically designed for the measurement of small signal variations, have
been developed, characterised and optimised. The proposed circuit
implementations could also be used for any application requiring chaotic
behaviour.

Two topologies of the measurement system have been proposed with the
associated hardware and software and the differences between the two
methods were assessed.

A means of detecting or quantifying the magnitude of input signal change
was proposed for each map. Two methods were developed; a calibration
method for the Logistic Map and a mathematical method for the Tent Map.
25
1.5 Document Structure
This document is structured in the following manner:

In the second chapter the classical method of measurement is presented and
the main sources of error are identified, this is then followed by an
introduction to chaos, one dimensional chaotic maps and high sensitivity to
initial conditions of chaotic systems. Methods and techniques used to identify
actual chaotic behaviour are introduced, finally the chapter is concluded with
the application of chaos in electronics in general and particularly in ADCs
and measurement systems.

The 3rd chapter explains the proposed technique for the realisation of a novel
measurement system which utilises the high sensitivity to initial condition
associated with chaotic systems.

Following the explanation of the proposed signal measurement technique, the
4th chapter focuses on the implementation of the chaotic one-dimensional
maps and the chaos based measurement system.

Chapter 5 assesses the electronic implementations developed in the previous
chapter by comparing practical results with theoretical behaviour of each
map.

In chapter 6 the measurement capability of the proposed MS is evaluated
with practical measurements in order to assess the overall performance.


Finally the results obtained in chapters 5 and 6 are discussed in chapter 7
In the last chapter the author suggests some possible further work followed
by a conclusion about the overall achievements.
26
2 Theory and Literature Review
The traditional techniques used to measure physical quantities are initially presented,
followed by the main sources of errors associated with Measurement Systems (MS).
An introduction to “chaos” is then presented followed by One-Dimensional (1D)
chaotic maps and the techniques used to evaluate whether the behaviour achieved in
a practical system is truly chaotic. Finally, a review of chaos applied in electronics,
communications and Analogue to Digital Converters (ADC) is presented.
2.1 Classic Methods of Measurement
The typical MS is generally composed of several elements in order to convert a
physical variable into a measured value that can be displayed or utilised by an
electronic system, as shown in Figure 2-1 (Dyer, 2001, Morris, 2001).
Physical
Variable
Transducer
Conditioning
Element
Data Acquisition
and Processing
Transmission,
Display or Control
FIGURE 2-1 BLOCK DIAGRAM OF THE ELEMENTS COMPOSING A MEASUREMENT SYSTEM
27

Transducer: common to most MSs, usually generates an analogue output
proportional to the physical parameter being measured; covered by the
following definition. A transducer is a device which provides a usable output
in response to a specific physical quantity (Norton, 1969). Some examples of
transducers are: Piezoelectric sensors for vibration measurement, Hall Effect
sensors for magnetic field measurement, thermocouple probes for
temperature measurement, ultra sound sensors for distance measurement etc.
Transducers are an essential part of a MS as the physical parameter being
measured has to be converted to an electric signal before undergoing
conditioning (Morris, 1993, Neubert, 1975).

Variable conditioning element: adapts the signal provided by the transducer
to a suitable format, for example the Piezoelectric transducer generates an
electrical charge in response to mechanical movement, the electric charge
cannot be displayed directly, used to control a system or stored in memory.
Thus, the charge has to be conditioned / converted to a proportional voltage
or a current that can be exploited by the following stage of the MS. The
conditioning circuitry is generally specifically designed for a given
transducer (Morris, 2001).

Data acquisition and processing: in most modern MS the data obtained from
the conditioning circuitry is converted into a digital format. This allows the
data processing, display and storage with the use of Microcontrollers ( C) or
computers (Schmid, 1970). In order to convert the signal from the analogue
to the digital domain the quantisation is performed by an Analogue to Digital
Converter (ADC). ADCs are mainly characterised by resolution which can
range from 1-bit to more than 30 bits. The resolution of the ADC indicates
28
the number of discrete values (digital values) it can produce over the range of
analogue values which means that the accuracy of the MS is dictated by the
resolution of the ADC as any variations lower than one digital step of the
ADC will be undetectable by the MS (Morris, 2001, Dyer, 2001).

Transmission, display or control: once the data is processed, the system can
either transmit the data, display for direct visualisation, store, or control a
variable in the case of a control system. The practical limit of sensitivity for
any MS is determined by specific factors such as the sensor sensitivity, the
intrinsic noise of the circuitry and the resolution of the ADC used for the
digitalisation of the signal (Morris et al., 2012).
The performances of the transducers are inherent to the construction of the
transducer and are generally independent of the other sections of the MS such are the
conditioning circuitry and the ADC. The transmission, display or control section
doesn’t influence the overall performance of the MS as no additional information is
created during that process where the information is transmitted in order to be
accessible to the user or used to control a system. The following sections will discuss
the parts of the MS that affect the performance of the MS to detect small variation of
the input signal, this will be followed by a discussion on the conditioning circuitry
and the limits to measurement set by the noise which is inherent in all electronic
circuitry used for signal conditioning, followed by the limitations of the ADC.
2.1.1 Conditioning Circuitry
The conditioning circuitry, as defined in the previous section is used to adapt the
signal originating from a transducer to a format that can be digitalised by an ADC.
The signal is converted to a voltage with a range scaled to the input range of the
29
ADC used within the MS. The conditioning circuitry inevitably adds noise and errors
to the analogue signal which limits the overall MS performance as data can be
corrupted by the noise. For this reason, the conditioning circuitry requires rigorous
design and the use of high performance, low noise components adapted to the
required performance of the MS. In many applications the conditioning circuitry is
composed of amplifiers, with each component generating inherent noise, which in
turn reduces the performance of the MS. The theoretical limit for a MS in terms of
performance is set by the thermal noise of the resistors used in the circuit (KeithleyInstruments, 2004, Lax et al., 2006) as any signal variation lower than the thermal
noise will be below the noise floor and therefore undetectable by the system. It is
important to avoid the use of high value resistors (R) as the root mean square (rms)
thermal noise voltage (E) is determined by the expression (2-1) known as Johnson
noise (Horowitz and Hill, 1989) so that for a given temperature, the thermal noise is
proportional to the value of the resistor.
√
(2-1)
Where k is the Boltzmann’s constant (
temperature in Kelvin (K) and B is the noise bandwidth in
), T is the absolute
.
Although, the theoretical measurement limit of any physical parameter is set by the
thermal noise the main source of noise in most circuitry is generally due to the active
components such as Operational amplifiers (Op-amps). Op-amp noise is determined
by three main components: shot (current) noise, voltage noise, and thermal noise
(Texas-Instruments, 2009, Horowitz and Hill, 1989). The shot and voltage noise are
30
specific to a particular Op-amp model, whilst the thermal noise is set by the resistors
used to set the gain of the Op-amp.

Current noise: current noise is converted to voltage noise when flowing
through a resistor. Ideally, to minimise the effects of current noise the source
resistors should be kept to a low value. An Op-amp with low voltage noise
can have a high current noise which can generate proportionally more noise
if the source impedance is not minimised.

Voltage noise: Input voltage noise is bandwidth dependent and measured in
√
. The voltage noise is always referred to the input which means that
the noise is multiplied by the Op-amp gain. The voltage noise dominates in
most cases where the source impedance in low as the thermal noise and
current noise will be negligible in comparison.

Source resistors: The resistors connected to the Op-amp generally contribute
significantly to the overall noise content. The first component is the Johnson
noise of the resistor as shown in expression (2-1), and secondly the resistor
can generate additional noise through the conversion of the Op-amp current
noise to voltage noise, which can dominate the overall noise value if the
impedance of the resistor is high.
The characteristic of the Op-amp noise (this is the case for voltage noise and current
noise) is composed of two main sections; a flat region where the noise is constant
over frequency, this region is referred to as white noise region and a region where
the noise is inversely proportional to the square root of the frequency, commonly
referred to as 1/f noise. The main characteristic of interest in Figure 2-2 is the
‘corner’ frequency where the 1/f noise and the white noise are equivalent. Thus it
31
can be observed that for frequencies lower than the corner frequency the overall
noise is dominated by 1/f noise and above by the thermal noise. A lower corner
frequency means that the Op-amp has lower overall noise; this is typically the case
for low frequency applications.
The graph in Figure 2-2 represents the equivalent voltage noise at the input of the
TL071 Op-amp (Texas-Instruments, 2005); this allows the noise at the output of any
Op-amp based amplifier to be calculated by multiplying the noise for a given
bandwidth by the gain of the Op-amp.
Equivalent Input Noise Voltage (nV/ (Hz))
45
40
1/f Noise
35
30
Corner
Frequency
25
20
15
1
10
White Noise
2
10
3
10
Frequency (Hz)
4
10
5
10
FIGURE 2-2 EQUIVALENT INPUT NOISE VOLTAGE OF A TL071 OPERATIONAL AMPLIFIER
(TEXAS-INSTRUMENTS, 2005)
All sources of noise present at the output of the conditioning circuitry are referred to
as the ‘noise floor’; which is the overall inherent noise, thus any input signal above
the noise floor can be identified and conversely any input signal below the noise
floor (lower amplitude) will be concealed by the noise, which will make the
32
measurement challenging as the system will be unable to differentiate between the
noise and the signal.
In addition to noise, other errors are created by the conditioning circuitry such as
offset voltages, offset voltage drifts or gain errors. Just as noise, the offsets are
generally specified in the datasheets of different Op-amps and the use of low offset
Op-amps is required for high accuracy MS. To maintain a low gain error, the
common practise is to use resistors with minimal tolerance and/or trimming
potentiometers.
The gain of amplifiers composing the conditioning circuitry should be set to match
the ADC input range whilst avoiding over amplification, which can lead to
additional errors (saturation of the amplifier or a signal out of the ADC input range)
As identified in this section, the conditioning circuitry is a critical part of any MS
and requires careful design in order to minimise the noise and errors introduced by
different components. The next element that composes the classic MS is the data
acquisition which will be discussed in the following section.
2.1.2 Data Acquisition
The data acquisition is an essential element of the MS as the analogue signal has to
be converted to digital data in order to allow processing. The data quantisation is
generally performed using an Analogue to Digital Converter (ADC). The ADC is
used to convert analogue signals to discrete time digital values, thus the overall
accuracy of the MS is highly dependent of the ADC performance (Razavi, 1995). An
ADC produces a digital output D, as a function of the analogue input A as shown by
expression (2-2).
33
(2-2)
Whilst the input can take an infinite number of values within the specified input
range of the ADC, the output of the ADC is limited to a restricted number of digital
values (codes) set by the resolution of the ADC. Thus an ADC must convert each
input value to an output code which is an approximation of the input. The ADC
conversion of an analogue signal is illustrated in Figure 2-3 (Plassche, 1994), where
the input voltage is sampled using a 4 bit and a 5 bit ADC. Due to the higher
resolution, the 5 bit ADC has a lower voltage step (Q) as defined by expression
(2-3), which allows a more accurate representation of the input signal. Figure 2-3
also shows that a greater ADC resolution (number of bits) will yield a more accurate
digital representation of the analogue input signal.
1.1
1
Analogue Input
4 Bit ADC
5 Bit ADC
Amplitude (V)
0.9
5 Bit ADC
Q
(LSB)
0.8
0.7
0.6
4 Bit ADC
Q
(LSB)
0.5
Time (s)
FIGURE 2-3 ANALOGUE SIGNAL SAMPLED USING 4 BIT AND 5 BIT ADCS
The accuracy of an ADC is directly linked to the voltage step , which is the
minimal voltage change required at the input of the ADC for the output to change by
34
1 binary value, known as the Least Significant Bit (LSB). The voltage step can be
calculated using expression (2-3) and illustrated in Figure 2-3.
(2-3)
Where
is the Full Scale Range input voltage, determined by expression (2-4)
and N is the number of digital values that the ADC output can take for a range of
analogue values at the input and is defined by expression (2-5).
(2-4)
Where
is the highest analogue value and
the lowest analogue value at
the input of the ADC.
(2-5)
Where M is number of bits representing the resolution of the ADC.
For an ADC to detect an input signal change, and thus yield a digital output change,
the value of
needs to be of lower amplitude than the input signal change whilst
ensuring that the
remains large enough for the given application. This can be an
issue for high accuracy applications requiring extensive input ranges (Craig, 1995).
Figure 2-4 illustrates this problem; a varying analogue signal is applied at the input
of an ADC. In cases where the variations in the input signal are smaller than the
of
the ADC the output code generated by the ADC remains constant, which means that
the MS is unable to detect the signal variations.
35
FIGURE 2-4 UNDETECTED INPUT SIGNAL VARIATIONS DUE TO LIMITED ADC RESOLUTION
The solution to this problem is to use an ADC with a higher resolution value which
will increase the cost as the price is proportional to the resolution of the ADC which
will also require resistors with lower tolerances and higher performance op-amps for
the conditioning of the signal thus further increasing the price of the MS. The change
of ADC often requires the redesign of the MS (hardware and/or software). Another
possibility is the use of a techniques called oversampling that has been developed to
increase the resolution of ADCs (ATMEL, 2005). This method involves taking
multiple samples of the input signal to obtain an increased resolution via algorithms
implemented in software. The main issue with the oversampling is the high penalty
on the sampling rate, as a result the minimum sampling rate required for the over
sampling technique is significantly higher than the Nyquist frequency as shown in
expression (2-6).
(2-6)
36
Where n is the additional number of bit of resolution.
Additionally the input signal must contain enough noise to toggle the LSB of the
ADC in order for the oversampling technique to work. In cases where the input
signal is containing low level of noise; artificial noise must be added to the input
signal which requires additional circuitry.
The problem of limited ADC resolution within a MS is further illustrated by the
following example:
A MS equipped with a 10 Bit ADC used to detect a 0.1
signal over a voltage range of
which for a
detect a 0.1
of
, yields a
to
change in the input
. Applying expressions (2-5) N = 1024,
of
. In this case the ADC is unable to
signal change as the minimum signal change required to modify the
output of the ADC is
. Thus a 18 bit ADC is required.
2.1.3 Measurement Method Summary
In the classic approach to MS the main sources of errors have been identified; in the
conditioning circuitry, a particular attention should be given to the noise of each
component in order to optimise the performance of the MS. Similarly, the selection
of the ADC is paramount and the resolution of the ADC should be selected in
relation to the amount of signal variation to be measured.
In cases where a MS, capable of detecting small variations over an extended input
range is required, the use of a high resolution ADC is essential, which will
37
drastically increase the overall cost of the MS and in some cases require an
unrealistically high resolution (>32 bit).
After highlighting the sources of errors associated with the conditioning circuitry and
the limitations in terms of performance associated with the use of ADCs, the next
section will introduce the subject of chaos and show where it has been implemented
in electronic systems. Finally, the next section will also demonstrate how chaotic
behaviour can be used in a MS as a means of detecting small signal variations over a
large input range, thus addressing the issue of ADC resolution and input signal range
previously identified.
2.2 Chaos
Chaos behaviour can be observed in any nonlinear system that exhibits irregularity
and unpredictability as well as high sensitivity to initial conditions commonly known
as the “butterfly effect”(Ingraham, 1991). Although, being deterministic (nonrandom) and sometimes following simple equations chaotic systems can display
complex behaviour which, significantly never repeats. Chaos can be observed in
simple mechanical systems such as the motion of a pendulum, it also materialises
everywhere in nature, from the turbulences of a water flow to the evolution of
populations (Gleick, 1988). When studying dynamical systems in 1913 Henri
Poincaré noted that in some cases a really small change in initial condition can
produce a disproportionally large change at the output of the system, which makes
prediction impossible as the initial conditions of a practical system can never be
known with absolute accuracy (G.L.Baker and J.P.Gollub, 1990). Real interest in the
subject of chaos started in 1963 with the publication by the meteorologist and
38
mathematician Edward Norton Lorenz of the paper called “Deterministic
Nonperiodic Flow” in which the author shows a simplified convection model and
concluded that the sensitivity to the initial conditions makes the long term weather
prediction impossible (Lorenz, 1963).
The mathematical models with chaotic behaviour referred to as “chaotic maps” can
be separated using two main criteria; the time domain (continuous or discrete) and
the number of space dimensions (one, two or three dimensional).
Contrary to continuous maps where the evolution of the map is described using
differential equations, discrete time maps are not continuous time functions and the
solutions of the map can be calculated with the use of iteration, which makes the
computation resource requirements significantly lower.
The number of dimensions of a chaotic map defines the number of parameters
present within the map. From that aspect, one-dimensional (1D) maps are the
simplest form of chaotic maps as only one parameter is used in the map which thus
makes the calculation and implementation relatively simple.
Combining the two criteria of classification of chaotic maps, it can be concluded that
the discrete 1D maps are more suited to electronic system implementation due to the
ease of computation of discrete maps and the low complexity of single parameter 1D
maps.
The next section will thus discuss three different 1D maps which have relatively
simple structures and detail the behaviour of each.
39
2.3 Discrete One-dimensional Chaotic Maps
Discrete 1D chaotic maps also called 1D difference equations or 1D iterated maps
are the simplest mathematical expressions that exhibit chaotic behaviour. For this
reason 1D maps are often used in the study and application of chaos. 1D maps are
mathematical expressions that model the evolution through iteration of a single
variable. A typical one-dimensional map is of the form shown in equation (2-7).
(2-7)
Where
is the state of the system at iteration ,
iteration n+1 and
is the state of the system at
is a parameter which can vary from map to map and in some
cases the value of which can lead the map to chaotic behaviour.
The relative simplicity of discrete 1D chaotic maps makes them an ideal means of
applying chaos, thus discrete 1D maps have been implemented electronically to
obtain means of practically studying chaos (Suneel, 2006, Campos-Cantón et al.,
2009). Discrete 1D maps are also used in different fields, such as biology to describe
biological systems; this is particularly the case for the Logistic Map (LM) which has
been used as a discrete-time demographic model for population modelling in
resource limited environments (May, 1976). In medicine, 1D chaotic maps have
been used to model neurons as it has been shown that neurons can exhibit chaotic
behaviour (Harth, Zeller et al., 1995). In chemistry, 1D maps have been used to
analyse chemical reactions such as the Belousov–Zhabotinsky (BZ) reaction (R. H.
Simoyi, 1982) or more recently as an abstract model for evolution (Usychenko,
2011). Discrete 1D maps have also been used in information processing systems
such as artificial neural networks (Nozawa, 1992, Song et al., 2007), communication
40
encrypting where the high sensitivity to initial condition is exploited to encrypt data
(Kocarev and Jakimoski, 2001, Martinez-Nonthe et al., 2012), electronic logic gates
(Murali et al., 2005) and random number generation (Cristina et al., 2012, Kanso and
Smaoui, 2009, Nejati et al., 2012, Luca et al.). All these applications show that
discrete 1D maps are powerful tools used for modelling and information processing.
More details about applied chaos are available in section 2.4.
In the following sections three commonly studied, discrete 1D chaotic maps are
presented; the Logistic Map, the Tent Map and the Bit Shift Map. To analyse the
behaviour of each chaotic map, two approaches were taken; the bifurcation diagram
and the time series. The bifurcation diagram shows the behaviour of the map for
different values of the map parameter and thus illustrating which parameter values
cause a map to exhibit chaotic behaviour. The time series enables an analysis of the
behaviour of the map for a fixed parameter value allowing a visual analysis of the
complex behaviour.
2.3.1 The Logistic Map
The Logistic Map (LM), given by the difference equation (2-8) was initially
introduced as a discrete-time demographic model by the biologist Robert May to
model the population of rabbits and foxes (May, 1976). The LM is analogous to the
logistic equation created by the mathematician Pierre François Verhulst (Miner,
1933) and is a discrete 1D nonlinear map with the transfer characteristic shown in
Figure 2-5.
(2-8)
41
Where
is the present state of the LM,
is the next state and is the parameter
of the LM that can be set to a value from 0 to 4.
1
Xn+1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Xn
FIGURE 2-5 TRANSFER CHARACTERISTIC OF THE LOGISTIC MAP
The simplicity of the LM combined with complex behaviour makes the LM the ideal
example of how relatively simple mathematical models can exhibit chaotic
behaviour.
A commonly used technique to demonstrate that a map displays chaotic behaviour is
the bifurcation diagram which displays the behaviour of the map for variations of the
parameter .

Bifurcation Diagram
The bifurcation diagram generated for the LM, using MATLAB, is given in Figure
2-6, with the map being iterated 10000 times for every value of from 0 to 4. It can
be observed that for
the output remains at zero and for
output is constantly increasing. The first output signal oscillations appear for
42
the
where the output starts to oscillate between two fixed points. At approximately
the first period doubling occurs, as shown in Figure 2-7 (expanded region
of Figure 2-6), the output is now oscillating between 4 points. The period doubling is
an important feature of the LM and is referred to as the “route to chaos”. This
feature is not only inherent to the LM but it can be observed in other maps and
chaotic systems (G.L.Baker and J.P.Gollub, 1990). The next period doublings are
visible in Figure 2-8 which is an expanded region of the full bifurcation diagram, and
occurs at 3.543 and at 3.564 respectively. As
increases the period doublings get
closer until an infinite number of bifurcation is reached at approximately 3.569,
finally leading to chaos.
FIGURE 2-6 COMPUTED BIFURCATION DIAGRAM OF THE LOGISTIC MAP
Although the LM behaves chaotically after a given value of the parameter
is
reached, the chaotic behaviour stops for periodic gaps named windows of periodicity
in which the output exhibits periodic oscillations, as shown in the bifurcation
diagram Figure 2-9. As is increased from 3.82 towards 3.83 the output leaves chaos
and goes back to periodicity, once
reaches approximately 3.841 the period
43
doubling starts again and the map proceeds to exhibit chaotic behaviour again at
approximately 3.842. As
is further increased, the output remains chaotic and
occupies more of the output range until the output takes up the entire output range at
= 4. If the value of
is greater than 4 the output of the LM will diverge to minus
infinity and will no longer exhibit chaotic behaviour. This phenomenon is often
referred to as “exiting condition” or “extinction” in reference to population
modelling (Miner, 1933).
FIGURE 2-7 BIFURCATION DIAGRAM OF THE LOGISTIC MAP FROM
44
=3.4 TO
=3.5
FIGURE 2-8 BIFURCATION DIAGRAM OF THE LOGISTIC MAP FROM
FIGURE 2-9 BIFURCATION DIAGRAM OF THE LOGISTIC MAP FOR

=3.5 TO
=3.8 TO
=3.6
=3.9
Time Domain Analysis
An analogous analysis of the LM behaviour can be accomplished in time domain by
displaying the output of the map for a given number of iterations and by varying the
45
parameter . Figure 2-10 shows four different graphs plotted for different values of .
The Y axis represents the output
the X axis. In Figure 2-10(a)
of the LM against the number of iterations on
is set to 2.2, it can be observed after three iterations
that the output settles to a constant value, this result is consistent with the bifurcation
diagram shown in Figure 2-6. As
is increased to 3.3 the output starts to oscillate
between two values as displayed in Figure 2-10(b), and oscillates between four
points in Figure 2-10(c), as
is set to 3.52 which further validates the behaviour
observed in the bifurcation diagram. Finally, in Figure 2-10(d) is set to 3.8 and the
random-like evolution of the time series shows that the LM behaves chaotically. At r
= 4 the output occupies the entire output range while for values of r greater than 4
the map no longer exhibits chaotic behaviour and reaches an exiting condition.
FIGURE 2-10 BEHAVIOUR OF THE LOGISTIC MAP FOR DIFFERENT VALUES OF
46

High Sensitivity to Initial Conditions
As previously stated, the main characteristic of chaos is the high sensitivity to initial
conditions, which can be analysed using simulations. Firstly, the map is iterated for a
given input and the values of
are stored to obtain a signature representing the
input, then a relatively small change is applied to the input and the map is reiterated
with the results of the two signatures being subtracted to obtain the difference,
enabling the point of divergence to be ascertained. In Figure 2-11 the input
0.6, the parameter
is set to
is set to 3.8 in order for the LM to behave chaotically, and the
map is iterated 50 times (shown as plot a1). The same process is performed once
again but with a small change of 0.0001 added to the input
and represented by plot
a2. The results of the iterations are then subtracted and displayed as (a0). It can be
observed that from iteration 1 to iteration 11 the two signatures are very close in
amplitude. The divergence between the two signals is visible at iteration 8 and the
divergence grows exponentially after this point. In Figure 2-12 the change between
the two input values is reduced to
, with the point of divergence being observed
after 23 iterations, which demonstrates that the number of the iteration at which the
two time series diverge is larger for smaller difference of initial conditions of the
LM. This is further demonstrated in Figure 2-13 where the change in initial
conditions is as small as
and the two signals diverge after a greater number
(approximately 68) of iterations. Although there is a pattern between the iteration
when the divergence occurs and the amount of change it is impossible to extrapolate
a relationship between the amount of change and the iteration number using this
analysis, due to the fact that the analysis is performed for a single input rather than a
full input range. An extensive analysis of the high sensitivity to initial condition of
the TM using a novel computational analysis is presented in Chapter 3.
47
FIGURE 2-11 SENSITIVITY TO INITIAL CONDITIONS OF THE LOGISTIC MAP FOR A
CHANGE.
FIGURE 2-12 SENSITIVITY TO INITIAL CONDITIONS OF THE LOGISTIC MAP FOR A
CHANGE.
48
FIGURE 2-13 SENSITIVITY TO INITIAL CONDITIONS OF THE LOGISTIC MAP FOR A
CHANGE.
The theoretical behaviour of the LM was explained by analysing the time domain
behaviour, the bifurcation diagram and the chaotic proprieties. This was mainly
performed using MATLAB simulations to compute and present the different aspects
of the LM. The next section will focus on another discrete 1D chaotic map, namely
the Tent Map (TM), as it is relatively simple to implement and just as with the LM, it
exhibits chaotic behaviour.
2.3.2 The Tent Map
The TM is a piecewise linear function composed of two straight lines contained
within the interval
yet the TM exhibits complex behaviour including
periodicity and chaos (G.L.Baker and J.P.Gollub, 1990). The TM function is given
by equation (2-9) where the parameter
sets the slope / gradient of the lines and can
be varied anywhere within the interval 1 to 2. The effect of the value of
49
will be
discussed further in this chapter. The TM was named after its tent-like triangle
transfer characteristic shown in Figure 2-14.
{
(2-9)
FIGURE 2-14 TENT MAP TRANSFERT CHARACTERISTIC

Bifurcation Diagram of the Tent Map
The bifurcation diagram of the TM is shown in Figure 2-15, where the behaviour of
the TM is set with the parameter . As the parameter
is increased from 1 to 2 the
TM displays a complex behaviour ranging from fixed point stability to periodicity
and finally chaos. The bifurcation diagram shows successive branches similar to the
period doubling cascade of the LM (from
= 1 to 1.42), rather than being single
lines the outlines of the TM are chaotic attractors bounded within a subset of the
interval [0,1], this interval is sometimes referred to as Orbit (Lynch, 2004). As the
parameter
is increased towards 2 the output swing increases in range until the
50
entire [0,1] interval is taken. Unlike the LM the bifurcation diagram of the TM does
not exhibit any windows of periodicity which makes it even more suitable for
applications where chaotic behaviour needs to be maintained. Ideally, the parameter
should be set to 2 as this is the point where the sensitivity to initial condition of the
TM is highest (Lynch, 2004). A problem arises in practical implementation when
setting
at exactly 2, as a small amount of noise could produce an exiting condition
in which case the TM will no longer behave chaotically as
or the input signal are
outside the limits due to the noise. This situation can be avoided by setting
close to
2 with a margin greater than the noise of the practical implementation. For example
if the parameter is set to 1.99, a margin of 0.01 (which is in most cases greater than
the noise present in practical implementation as discussed later in section 5.7) will
insure that the TM remains highly sensitive to initial conditions whilst preventing
any exiting conditions.
FIGURE 2-15 TENT MAP BIFURCATION DIAGRAM
51

Time Domain Behaviour and High Sensitivity to Initial Conditions
The TM and the LM are topologically conjugate which means that the behaviour of
both maps is similar under iteration (Alligood et al., 1997). The time domain
behaviour of the TM being similar to that of the LM, hence the time domain analysis
is not shown in this thesis.
Similarly, because of the analogous behaviour under iteration between the TM and
the LM the high sensitivity to initial conditions of the TM produces results
comparable to the LM shown in section 2.3.1, with further analysis carried out in
Chapter 3.
2.3.3 The Bit Shift Map
The Bit Shift Map (BSM) sometimes referred to as Bernoulli map or Doubling map
is
a
discrete
pricewise-linear
1D
chaotic
map
expressed
by
equation
(2-10) with a transfer characteristic shown in Figure 2-16. Contrary to the TM and
the LM where the parameter of the map can be varied, the parameter of the BSM is a
constant set to two. The name of the BSM comes from the fact that if the input of the
map is written in binary form, after each iteration, the output is obtained by shifting
the bits to the right and replacing the last bit on the left by a zero. From this feature it
can be observed that the simulation of the BSM using computation is problematic as
the rounding of the computer will eventually cause any input number to produce a
zero after a given number of iterations as the binary expansion will be limited in
length, this is not an issue when it comes to practical implementation as the values
present at the input will always be irrational (infinite binary expansion). Similarly to
the TM and the LM, the BSM needs an accurate setting of the parameter
52
to two in
order to avoid exiting conditions. Due to a fixed value of the parameter , a
bifurcation diagram cannot be obtained.
(2-10)
1
0.8
Xn+1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Xn
FIGURE 2-16 BIT SHIFT MAP TRANSFER CHARACTERISTIC

Time Domain Behaviour and High Sensitivity to Initial Conditions
In time domain, the output of the BSM takes the entire output range in the interval
[0,1] which is the case for the TM for
and for the LM when
To
demonstrate the high sensitivity to the initial conditions of the BSM, four time series
were computed using MATLAB. Firstly, an input signal was iterated through the
BSM followed by a signal with an added difference ( ) of 0.001. The two time series
are shown in Figure 2-17 (plot a1) and the divergence between them is displayed in
plot a2. It can be seen that the time series are divergent approximately at iteration 8.
Two additional time series were computed but this time with
53
of 0.0000001 as
shown in (b1). The divergence between the two time series is shown in (b2) which
occurs after approximately 17 iterations demonstrating that the divergence between
two time series occurs later when
0.5
0
0
10
20
Iterations
30
0
10
20
Iterations
30
40
10
20
Iterations
30
40
(b2)
Xn
(a2)
0.5
0
0.5
0
40
(b1)
1
1
Xn
1
(a1)
Xn
Xn
1
is reduced.
0
10
20
Iterations
30
0.5
0
40
0
FIGURE 2-17 HIGH SENSITIVITY TO THE INITIAL CONDITIONS OF THE BIT SHIFT MAP
2.3.4 Summary
The investigation of three discrete 1D chaotic maps has shown that all maps, despite
having a simple characteristic equation, exhibit chaotic behaviour and indicates that
the time/iteration of divergence between two neighbouring starting points is
proportionally linked to the amount of change between the two points.
In addition to the similarity between the BSM and the TM, the BSM requires an
accurate parameter setting which is not the case for the LM and the TM where the
parameters can be varied within a given region without systematically leading to
exiting conditions. In practical implementations, the noise and component tolerances
54
limit the accuracy at which the parameter can be set. In that aspect the LM and TM
are superior to the BSM for the purposes of this work.
Since the BSM is a piecewise linear map similarly to the TM, the BSM will not be
considered for implementation in the present thesis, as the work will be focused on
the LM and the TM in order to evaluate the different characteristics.
The next section will focus on a means of ascertaining chaotic behaviour in
dynamical systems, this presented technique will be used further in this work to
identify chaotic behaviour in electronic implementations.
2.4 Applied Chaos
Chaos has been applied across a variety of disciplines, which is particularly the case
for one-dimensional maps, due to the relative simplicity which makes them
particularly suitable for electronic implementations.
2.4.1 Electronic Implementation of Chaos
One-dimensional (1D) chaotic maps have been used in electronics as random
number and noise generators (Díaz-Méndez et al., 2009, Katz et al., Tanaka et al.,
2000, Vázquez-Medina et al., 2009, Kanso and Smaoui, 2009, Nejati et al., 2012,
Callegari et al., 2005). The output of chaotic circuits occupies the entire output
interval in a random-like manner with a relatively flat spectrum response which
makes the 1D maps suitable for random number and white noise generators. For
example, Díaz-Méndez et al. (2009) have developed and simulated an analogue
electronic implementation of the Logistic Map (LM); the spectral density and the
statistical distribution exhibited by the circuit were similar to that for white noise
55
which proves that the LM can be used as a noise generator. The implementations
available in literature for this purpose are using Complementary Metal Oxide
Semiconductor (CMOS) technology to simplify the integration in a single Integrated
Circuit (IC), for example, Tanaka et al. (2000) have successfully implemented noise
generators based on both the LM and the TM in a single IC. The results obtained are
similar to the noise generator proposed by Díaz-Méndez et al. (2009) and
demonstrate the noise like characteristic of the spectral distribution of the electronic
implementations. Another noise generator was proposed by Leonardo et al. (2012),
but this time rather than using analogue circuitry to generate analogue noise, the
authors have implemented a TM based digital noise generator using a FieldProgrammable Gate Array (FPGA). A variation of the TM was implemented in order
to generate a 13 bit representation of a random natural number, the resources
required in the FPGA where limited to 3 multipliers and 2 adder modules. The
results obtained demonstrate that the 1D map based noise generator performs as
expected and that the statistical distribution is uniformly spread.
In Murali et al. (2005) a 1D chaos map was used for an experimental realisation of a
logic gate, although the experiment was successful the main purpose was to prove
that 1D map can be used as basic computational elements. In another instance,
chaos has been used to improve the capture range of a Phase Lock Loop (PLL);
where an external modulating input was used to set the unlocked PLL into a chaotic
regime that overlaps the original capture range. The chaos inducing modulation was
then turned off, allowing the original dynamics of the PLL to capture the signal.
(Bradley, 1993).
Many electronic implementations of chaotic 1D maps are available in literature with
the main purpose being the practical approach to the analysis and study of chaos
56
with no specific application given by the authors (Campos-Cantón et al., 2009,
Eguchi et al., 2000, Suneel, 2006, Edang et al., 2011, Hernandez et al.).
2.4.2 Chaos in Cryptography
Communication systems are used to transmit a message (information) from a
transmitter to a receiver; before transmitting, the message should be formatted for
suitable transmission and encrypted if the security of the transmission is of concern.
The broad spectrum of the chaotic maps used to create random number and noise
generators can also be used to design Spread-Spectrum (SS) communication
systems. In SS communications, the power of the signal is spread out over a wide
frequency band to avoid narrowband interference. The security is increased since the
carrier signal is no longer a single spike in the frequency spectrum, making it
electronically difficult to monitor and detect. A proposed chaos based system was
used as a SS code on the transmitter and receiver end in order to spread the
bandwidth of the transmitted signal, effectively increasing the signal robustness to
noise and signal jamming (Setti et al., 2002).
In Kocarev (2001) the similarities between classical cryptography systems and the
high sensitivity to initial condition of chaotic system are shown. For this reason,
chaos has also been used to create cryptographic algorithms, for example a method
using the LM has been proposed in literature to simplify the design of chaos based
cryptographic systems along with a step by step procedure for designing
cryptographic systems using chaotic maps (Kocarev and Jakimoski, 2001). In
another paper an optical chaos based secure communication system has been
designed using a discrete implementation of the LM, incorporating a pulse position
57
modulation scheme together with the LM chaotic map to encrypt the signal (Singh
and Sinha, 2010).
Additional instances of chaos being used for communication or cryptography include
but are not limited to, image encryption (Pareek et al., 2006, Patidar et al., 2009)
and analysis of cellular neural networks behaviour (Zou and Nossek, 1993).
2.4.3 Chaos Based ADCs
Chaos has been applied to measurement and particularly to implementation of
ADCs, where the ADCs created using chaotic maps are named algorithmic ADCs
which rely on the piecewise-linear characteristic of 1D maps to double and fold the
signal on each iteration. Algorithmic ADCs generate either a binary or a Gray-code
representation of the sampled analogue signal, with the binary ADCs being based on
the Bit Shift Map (BSM) while the Gray-code ADC are based on the TM (Kennedy,
1995).

Bit Shift Map ADC
The BSM can be used as an ADC as the output is a binary expansion of the sampled
analogue signal. To do so a comparator is added at the output of the function so that
the output is compared against a reference signal. If the signal is higher than the
reference the output of the comparator is set to 1, else the output is set to 0. After
each of the iterations the LSB, that has been generated, is shifted by 1 to the right
and the remaining is iterated through the map to generate the next bit. This process is
repeated N times until a N bit binary expansion is obtained. The input analogue
voltage is then expressed as shown in expression (2-11).
58
(2-11)
∑
b
1
b1 = 1
F(x)
F(F(x))
1
x
b2 = 1
0.5
b3 = 0
0
Input
0
b
0
0.25
0.5
X
0.75
1
0
0
0
0
1
1
1
1
b2
0
0
1
1
0
0
1
1
b3
0
1
0
1
0
1
0
1
1
FIGURE 2-18 WORKING PRINCIPLE OF THE ADC BASED ON THE BIT SHIFT MAP
Figure 2-18 illustrates the working principle of the algorithmic ADC based on the
BSM Map. Each iteration adds one binary digit that represents the input voltage
(Kennedy, 1995, Kapitaniak et al., 2000). For example if the value of the input
voltage is between 0.75 and 0.875 the first bit
higher than the threshold of 0.5 the second bit
greater than the threshold. Finally the third bit
will be set to 1 as the input x is
will also be set to 1 as
is set to 0 as
is
is lower than
the threshold. ADCs based on the BSM have been implemented in literature (Qingdu
and Qifeng, 2012).
59

Tent Map ADC
Similarly to the BSM the TM can be used to create an ADC which generates a Graycode expansion of the analogue input signal. After each iteration the output of the
TM circuit is compared to a 0.5 threshold. If the output is lower than the threshold
the bit is set at 1, else the bit is set at 0. After each iteration the bits are shifted to the
left. This process is shown graphically in Figure 2-19.
b
1
x
F(x)
F(F(x))
1
0.5
0
0
0
0.25
0.5
X
0.75
1
b1
0
0
0
0
1
1
1
1
b2
0
0
1
1
1
1
0
0
b3
0
1
1
0
0
1
1
0
FIGURE 2-19 TENT MAP BASED ADC
Gray-code algorithmic ADCs implemented using the TM are considered superior in
term of performance to binary-code ADCs (implemented using the BSM). This is
due to the fact that only one bit is changing at any time in a Gray-code ADC for an
input signal change equivalent to 1 LSB. In a binary ADC a signal change around
the threshold voltage (0.5 V) can cause many bits to change in cascade which can
introduce transient errors that are difficult to correct (Arayawat et al., 2008). Another
60
advantage of the ADCs based on the TM is the improved tolerance to offset error
compared to BSM based ADCs (Signell, 2005, Signell et al., 1997); this is due to the
fact that in a BSM ADC the offset errors are always amplified with a positive gain
which produces an accumulation of the offset errors. Alternatively, in the TM based
ADC the amplification is switching between positive and negative gain which
reduces the offset errors allowing a design of an ADC with improved resolution
compared to BSM based ADCs. Gray-code ADCs have been implemented with
different architecture such as voltage-mode where the ADC circuitry is dealing with
voltages (Arayawat et al., 2008) and current-mode where the signals propagating
through the ADC are currents rather than voltages which allows for a simplified
Integrated Circuit (IC) implementation (Pouliquen et al., 1991, Wilamowski et al.,
2006, Chaikla et al., 2006).

Summary
Algorithmic ADCs produce accurate results only if the parameters that define the
maps behaviour are precisely set; in the case of the TM the parameter
should be set
to exactly 2 and the voltage reference for the threshold to 0.5. If any error is
introduced to the parameters the error will grow exponentially and coding errors will
appear (Kapitaniak et al., 2000, Kennedy, 1995). If the parameter
2 (required) a small amount of noise can set the parameter to
is set to exactly
>2, which will cause
the map to diverge to 0 (exiting condition). The algorithmic ADCs generate accurate
results if the components are ideal and noiseless and the parameters set accurately.
This explains the low resolution of the ADCs developed using this technique.
Wilamowski et al. (2006) proposed a TM based current-mode ADC with an IC
implementation with limited resolution of 5 bits. A similar ADC architecture was
used by Lu (2011) to design and simulate a 10 bit ADC. In general, chaos based
61
ADC are limited to a maximum of 12 bits (Keng and Salama, 1994, Hai et al., 2010)
which makes them impossible to use for application where an extended input range
combined with the ability to detect small variation of input signal is required.
2.4.4 Chaos Based Measurement System
A limited number of chaos based MS are available in literature; in fact the author is
aware of only two main methods of measurement based on chaos (Chernukha, 1996,
Hu and Liu, 2010, Wang et al., 1999). Wang et al. (1999) have shown that a chaotic
Duffing oscillator can be used to detect weak signals and that the system was
immune to noise. The parameters of the chaotic Duffing oscillator are set so that the
oscillator operates near a bifurcation point. If any weak signal is added to the system,
the oscillator will leave the chaotic region and go into a region where it will alternate
between chaos and non-chaos. The length of time spent in the non-chaotic state is
then used to approximate the weak signal present within the system. This principle
was used to increase the sensitivity of a metal detector; due to the high sensitivity to
sinusoidal signals and the immunity to noise the chaotic Duffing oscillator allows the
faster detection of small metal particles present in food (Hu and Liu, 2010). The
drawback of the proposed system is that it can only be used for detection of
sinusoidal signals as the amplitude of the detected signal cannot be accurately
quantified due to the oscillator having only two states: chaotic or periodic.
The second method presented by Chernukha (1996) which the author refers to as
“synergic measurement method” two chaotic systems based on the LM and a chaotic
oscillator are presented in order to improve the measurement accuracy of sensors
parameters. The chaotic circuit is connected between the sensors and the ADC to
provide conditioning circuitry with improved performance over the conventional
62
approach. The physical parameter being measured modifies a given parameter of a
sensor, which in turn varies a parameter of the chaotic circuit. Because of the high
sensitivity to the initial condition of the chaotic system this results in a significant
change at the output of the chaotic system, allowing the MS to detect small
variations of the measured parameter. The drawback of the proposed method is the
increased amount of samples required to increase the accuracy as the author states
that approximately 500 samples are required to improve the resolution by 2 bits.
Unfortunately, other publications written by the same author that fully explains the
above method are not accessible in English. Despite interesting theoretical and
simulation results the two methods mentioned above have not been implemented and
remain theoretical.
2.5 Lyapunov Exponent
The Lyapunov Exponent (LE), named after the Russian mathematician Aleksandr
Mikhailovich Lyapunov, is used to estimate the sensitivity to initial conditions (i.e.
the degree of chaotic behaviour) in chaotic systems (G.L.Baker and J.P.Gollub,
1990). The LE of a map indicates the rate at which two initial states diverge after a
given number of iterations and can be calculated using the expression
(2-12).
∑
Where
is the LE,
|
the iteration number and
|
(2-12)
the derivative of the chaotic
map. If the LE value of a map is negative two neighbouring points will converge and
63
the map will not exhibit chaotic behaviour, if the LE is positive two neighbouring
points will diverge exponentially and the map is highly sensitive to initial condition
and thus chaotic. This makes the LE the ideal tool to ascertain the presence of chaos
in a given dynamical system.
To demonstrate the use of the LE as a means of ascertaining the presence of chaos,
Figure 2-20 shows the LE for the LM as function of the parameter . As
is
increased from 0 to 3.569 the LE remains negative which is expected as the
bifurcation diagram shows that the TM is not chaotic. At
values above 3.569, the
LE becomes positive showing that the map is chaotic. Finally, in the range where
the LE is mainly positive with some areas where the LE goes below
0 due to the windows of periodicity as explained in section 2.3.1 and illustrated in
Figure 2-9.
FIGURE 2-20 VARIATION OF THE LYAPUNOV EXPONENT FOR THE LOGISTIC MAP
64
The following section will explain how the LE can be used to ascertain the presence
of chaos for experimental data which will allow the test of electronic
implementations developed further in this work, to be performed.
2.5.1 Lyapunov Exponent Estimation
In order to estimate the LE from an experimentally obtained time series, different
methods should be used as the parameters and the nature of the chaotic system
cannot be known accurately. The method used to estimate the LE from the time
series is a direct method proposed by Rosenstein et al. (1992) based on the modified
Wolf algorithm (Wolf et al., 1985) developed in (Sato et al., 1987, J.Kurths and
H.Herzel, 1987) and uses the average exponential growth of the prediction error
between two neighbouring points. The expression of the prediction error is given by
equation (2-13) (Parlitz, 1998). For more in depth details, the mathematics behind
the algorithm can be found in the original paper proposed by Rosenstein (1992).
According to (Rosenstein et al., 1992, Wolf et al., 1985) to increase accuracy, a
longer time series is required.
∑
Where
‖
‖
is the closest neighbour of the point
‖
‖
(2-13)
, k is the number of time steps
used for the estimation of the LE and n in the iteration number.
After calculating the prediction error from the experimental data it is plotted and the
exponential region is used to approximate the LE by measuring the slope of the
linear part of the graph. If the slope is negative the separation between the two
65
neighbouring point is not exponential and the system is not chaotic, alternatively if
the slope is positive the separation is exponential, which indicates a high sensitivity
to initial conditions i.e. chaos.
An example of the LE estimation from a prediction error is shown in Figure 2-21,
where the slope of the line shows the theoretical LE of the chaotic time series, whilst
the practical LE can be approximated using the linear region of the estimated LE
curve. This method will be used later in the work to ascertain the presence of chaos
in the proposed electronic implementation of the chaotic maps.
FIGURE 2-21 LYAPUNOV EXPONENT ESTIMATION USING THE PREDICTION ERROR METHOD
The theoretical LE for the LM when the parameter is set to 4 is equivalent to the
BSM LE and the TM LE when the parameter
Eckmann and D. Ruelle, 1985).
66
and is equal to 0.693 (J.-P.
2.6 Conclusion
This chapter introduced the structure of MS and reviewed the limitations inherent to
conditioning circuitry and ADCs. An introduction to chaos followed by a review of
two 1D chaotic maps was shown, with the typical behaviour being evaluated using
bifurcation diagrams and time domain analysis. The next section of the chapter
focused on applied chaos and particularly on the use of chaos in electronics; ADCs
and measurement systems. The main source of error that limits the performance of
chaos based ADCs were evaluated to be the inaccuracy of the parameters and
threshold voltages within the implemented system. The chapter was concluded by an
explanation of the high sensitivity to initial condition of chaotic maps and the use of
Lyapunov Exponent (LE) to determine the presence of chaos from the analysis of
experimental data.
From the literature review carried out in this chapter the following has been
concluded:

Classical measurement systems are limited by the sensor performance, the
noise of the conditioning circuitry and the performance of the ADC. As the
range of the input signal increases the sensitivity of the system decreases due
to the quantisation of the ADC (step size).

Chaos has been used in electronics and particularly in algorithmic ADCs; the
performance of which are limited due to parameter and threshold errors.

Two theoretical measurement systems based on chaotic circuits have been
presented in literature without practical implementation.
67
The next chapter will introduce a novel measurement technique based on the high
sensitivity to initial conditions of one-dimensional chaotic maps that can be used to
measure small variations of input signals utilising a low resolution ADC. The
proposed technique is not reliant on the need for highly accurate parameter values
and the sensitivity to signal change is independent of the input range.
68
3 Proposed Signal Measurement
Technique
The proposed Measurement System (MS), based on a one-dimensional (1D) chaotic
map, uses the high sensitivity to initial condition, as presented in the previous
chapter, to measure small input signal changes. The technique differs from a
conventional approach to signal measurement by quantifying the difference between
two signals rather than determining the absolute value of a sample. This enables the
accurate detection and measurement of small input signal changes which is totally
independent of input signal range. In the previous chapter the analysis of a single
input with different signal changes was performed in order to demonstrate that the
moment (iteration) of divergence between the two input time series (signals) was
apparently inversely proportional to the amount of change. In order to ascertain the
feasibility of the proposed MS, the fundamental requirement for the linearity of
divergence needed to be evaluated for a given change throughout the entire input
range. The linearity of divergence is not verifiable with a single divergence graph as
given previously (Figure 2-11), which shows only the divergence between two time
series representing two samples. In order to visualise the divergence over the full
normalised input range (0 to 1) a Matlab simulation has been carried out to generate
the 3D graph that illustrates the divergence between the time series throughout the
entire input range. The developed Matlab code used to generate the 3D graphs is
available in Appendix B. This novel 3D visualisation technique allows the analysis
of the high sensitivity to initial conditions of any chaotic map throughout the entire
input range, which makes it a powerful tool for the study of chaotic maps.
69
The simulation was performed by taking 10000 input sample points evenly spaced
throughout the entire input range. The maps were iterated and each time series
obtained was subtracted from time series, for the same points plus a given signal
change ( ). The divergence for each sample was then displayed to produce a 3D
graph, as shown in Figure 3-2 for the Logistic Map (LM). The flowchart of the
algorithm used to generate the 3D graph is show Figure 3-1.
Start
Input = 0
Iterate map N times
Store 1st signature
Input = Input - Δ +
1/resolution
Input = input + Δ
Iterate map N times
Store 2nd signature
No
Subtract signature 1
from signature 2
and store result
Input ≥ 1 ?
Display all
signatures in a 3D
graph
END
FIGURE 3-1 FLOWCHART OF THE 3D DIVERGENCE GRAPH
70
The parameter , given in equation (2-8) for the LM, was set to 4 (chaotic region)
and a change ( ) of 0.0001 was applied for every input value. The graph shows that
whilst being relatively linear the divergence varies between inputs. Any signal lying
in the range of 0 to 0.1 and 0.9 to 1 will diverge earlier (iteration 3 to 5) than signals
in the range of 0.1 to 0.9 ; this issue can be avoided by only utilising the region 0.10.9 of the normalised input range. Additional graphs showing different view angles
for different values of
are shown in Appendix B.
FIGURE 3-2 SENSITIVITY TO INITIAL CONDITIONS OF THE LOGISTIC MAP FOR =4.
In order to determine the difference in the sensitivity to the initial conditions for the
TM based system, the 3D divergence graphs shown in Figure 3-3 was produced in
the same manner as the LM based graph. The divergence of the TM is linear
throughout the entire input range, for a TM parameter
set to 1.99 (chaotic region)
and a change of 0.0001 applied for 10000 points throughout the whole input range.
71
The two graphs (Figure 3-3 (a) and (b)) represent the same data viewed from
different angles where the divergence is visible from iteration 6 for any input value.
In this aspect the TM is more suitable for the design of the small signal change
detection MS as the full input range can be utilised. Additional 3D graphs are
available in Appendix C presenting the behaviour of the TM to different values of .
FIGURE 3-3 DIVERGENCE OF THE TM FOR THE ENTIER INPUT RANGE
3.1 Quantifying Input Signal Change
The LM and TM implementations are both capable of detecting small variations of
input signals, but in order to define a viable measurement system the amount of
change between the two signals should be quantifiable. To quantify the amount of
change between two input signals when using the LM, a threshold has to be set on
the amount of divergence at a given iteration. Thus the iteration sample at which the
threshold is reached is used to determine the amount of change between the two
samples. The calibration can be performed using the 3D divergence graph previously
developed as shown in Figure 3-4.When
72
is varied the threshold is reached at
different iteration numbers, by using the relationship between
and the number of
iterations when the threshold is reached the system can estimate the change between
two input signals.
FIGURE 3-4 LM DIVERGENCE GRAPH WITH SET THRESHOLD FOR A
OF 0.0001
The disadvantage of this method is that the threshold has to be defined using
simulation and calibration techniques. However, this requirement is removed when
the TM is used since the simple piecewise linear characteristic of the map means that
the difference between two input signals can be calculated for the divergence at a
given iteration using the formula given by expression (3-1).
(3-1)
Where
is the amount of change between the two input signals,
between the two time series at iteration
and
is the parameter of the TM. The
equation has been derived using the following process:
73
is the divergence
The gain of the TM is set by the parameter , the amplification factor, which means
that the total amplification after N iterations is
. Thus, if two input values,
separated by a difference ( ) are separately iterated through the map, the
be amplified by
divergence
. Hence to determine the value of
will aslo
between two signals, the
at a given iteration N should be divided by the overall gain obtained
through the iteration process.
To illustrate the process of calculating ,the parameter
was set to 1.99, a random
input was taken and the time series was displayed in Figure 3-5. The dotted green
and blue lines show the values for the original input and input with an added 10 V
change respectively. The divergence between the two time series is represented by
the red full line.
1
1st Sample
2nd Sample
0.9
Divergence ()
0.8
0.7
Xn
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Iteration (N)
12
14
16
FIGURE 3-5 DIVERGENCE BETWEEN TWO SAMPLES
With an ideal MS, the difference between two signals could be estimated with an
unlimited accuracy, however, in practical applications different errors and noise will
74
limit the performance of the proposed MS. The following section will identify all the
errors associated with the chaos based measurement system and the impact of each
error will be quantified.
3.1.1 Measurement Errors
The errors inherent to the proposed MS have been identified as being associated with
the following aspects:

Errors due to the resolution of the ADC: As shown in the previously section,
the estimation of the difference between two input signals is performed using
the divergence between the signatures representing the signals. The
resolution of the ADC will introduce errors (
step size) as the output of
the chaotic map is sampled after each iteration to produce the signatures
(time series).

Errors due to the TM symmetry: The TM being symmetrical, some errors are
possible when two signals are symmetrically placed on each side of the TM
triangle transfer characteristic.
In addition to the sources of errors shown above, the following aspects of the
measurement process have been identified as possible further sources of errors:

Error due to the number of iteration: The measurement should be taken after
a given number of iteration to avoid the loss of information due to the folding
of the chaotic map. This will allow an accurate estimation of .

Error due to the amount of divergence between two signatures: Similarly to
the number of iterations, the divergence between two signatures used to
75
estimate
is important to reduce measurement error as the folding of the
map can produce significant errors.
In this section, the sources of errors are analysed and the optimal values for the
number of iterations and the amount of divergence are calculated in order to obtain
an accurate measurement by minimising measurement errors.

Errors due to ADC resolution
With unlimited number of decimals the calculated , using expression (3-1), will
match the real
but in real applications, the data used for the estimation of
will be
limited in terms of accuracy by the resolution of the ADC used within the MS. The
error between the simulated
and the
calculated, taking into account the
resolution of the ADC is shown in Figure 3-6. The sample was limited to 3 decimal
places to take into account the restricted resolution of the ADC (10 bit), used to
sample the signals in the practical implementation. The error as a percentage of the
estimated value of
, is displayed with the dotted line while the error in volts is
shown using a full line. For the first 6 iterations the divergence (
than one LSB of the ADC the calculation yields
unable to calculate
being smaller
which meant that the system is
(change undetectable by the ADC). At iteration 7,
is greater
than 1 LSB and therefore large enough to be measured by the ADC; the
is
calculated with an error of 19.1%. The error decreased significantly as the number of
iterations increased, this is due to the
being considerably greater than the LSB of
the ADC. By iteration 13 the error has reduced to less than 1% and reaches the
lowest value at iteration 15 with an error of -0.02% representing an error of only
2
for a
of 10
. After iteration 15 the folding of the TM causes the
measurement of the divergence to be inaccurate due to the error increase. The data
76
used to create Figure 3-5 and Figure 3-6 is available in Table A-1 in Appendix A.
These results demonstrate that a
of 10
could be detected to within
using
the proposed technique utilising a low resolution ADC.
x 10
2
100
1
0
0
Error(V)
Error(%)
50
-50
0
5
FIGURE 3-6

-5
10
Iterations
15
-1
20
ESTIMATION ERROR
Errors due to the Tent Map symmetry
Given the symmetrical nature of the TM, two different input signals that are
equidistant from of the peak of the TM triangle will produce identical time series.
For example, the input value 0.49 and the input value 0.51 will produce the same
time series and only the first values (the initial input value) are different, given by
equation (4-9) and illustrated in Figure 3-7. The two time series for
are shown
in Figure 3-8, the data used to produce the graph is available in Appendix A, Table
A-2. Although the two inputs are separated by 0.02, the two time series are identical
from iteration 1 meaning that the divergence will not occur (in a noiseless system).
77
1
0.995
Xn+1
0.99
0.985
X: 0.49
Y: 0.98
X: 0.51
Y: 0.98
0.98
0.975
0.48
0.49
0.5
Xn
0.51
0.52
FIGURE 3-7 TWO INPUTS WITH IDENTICAL OUTPUT VALUES
1
0.9
0.55 X: 0
0.8
Y: 0.51
0.5
0.7
Xn
0.6
X: 0
Y: 0.49
0.45
0
0.5
0.4
0.1
0.3
0.05
0.2
0.1
0
X: 2
Y: 0.04
0
0
1
2
3
4
5
Iteration (N)
2
6
7
8
9
FIGURE 3-8 TWO IDENTICAL TIME SERIES FROM TWO DIFFERENT INPUTS
This feature of the TM behavior can cause errors in the estimation of , however
Matlab was used to compute the divergence
78
at iteration N for a fixed change
throughout the entire input range. This allowed the effect on the estimation of
analysed. In Figure 3-9 the amount of divergence
6 for a
of 50
to be
is shown for iteration 1, 2, 3 and
. A point of significant reduction in divergence , which will be
referred to as “Null” (D) is visible in iteration 1 then the number of nulls increases
with the number of iterations following expression (3-2).
(3-2)
The width of the null is equal to the amount of change
Figure 3-10, where
was set to
at the input as shown in
. The fact that the width of a null is equal to
the change between two input signals, the probability of being at a null can be
calculated using expression (3-3).
(3-3)
Where
is the input range,
the input change and
the number of nulls for a given
iteration .
From expression (3-3), in order to minimise the probability of being at a null,
should be decreased by reducing the number of iteration whilst the input range
should be maximized. For a
of 100
the probability P(Null) is 0.06% when the
measurement is taken at iteration 6. As shown in Figure 3-8, the null occurs when
the time series reach a value close to one which on the next iteration (iteration 2 in
Figure 3-8) generates values close to zero. This pattern can be used by the MS to
79
detect the presence of nulls in the measurement and thus compensate and identify the
signal change more accurately given further investigation.
-4
Divergence (  )
Divergence (  )
2
0
0.25
-4
0.5
Input
0.75
Divergence (  )
0.25
0.5
Input
0
0.25
-3
4
2
0
2
0
1
4
0
4
N=3
x 10
N=2
x 10
4
0
Divergence (  )
-4
N=1
x 10
0.75
0.75
1
0.75
1
N=6
x 10
2
0
1
0.5
Input
0
0.25
0.5
Input
FIGURE 3-9 NUMBER OF NULLS FOR DIFFERENT VALUES OF N
-4
Divergence (  )
2.5
x 10
2
1.5
1
0.5
0
0.49985 0.49990 0.49995
Input
0.5
0.50005
FIGURE 3-10 WIDTH OF A NULL FOR AN INPUT CHANGE OF
80

Errors associated with the number of iterations
The number of iterations performed by the system is directly linked with the
accuracy of the measurement: Figure 3-11 shows error magnitude in the calculation
of
against the iteration number for different values of . As expected the lower the
difference ( ) between two inputs the higher the iteration (N) required to determine
with an acceptable degree of accuracy. For a change of 10
the error decreases
to an acceptable 1% at iteration 13. It takes 10, 9 and 8 iterations to reach the same
level of accuracy for a change of 50
Furthermore for a change of 0.5
, 100
and 200
respectively.
can be estimated to within 0.5% from iteration
1 as the change ( ) can be detected with the ADC without any iteration.
can be
estimated with an error of less than 0.02% even when the time series used for the
calculation are sampled with an ADC that limits the values to 3 decimals. The
accuracy of the
estimation is independent of the amount of change between the
input signals.
100
Delta Estimation Error (%)
80
60
40
20
0
10u
-20
50u
100u
-40
200u
-60
500u
2
4
6
8
10
Iterations (N)
12
14
16
FIGURE 3-11 DELTA ESTIMATION ERROR AGAINST NUMBER OF ITERATIONS
81
As shown in expression (3-1), to estimate
the MS requires three parameters; the
value of the TM parameter , the iteration number N and the value of the divergence
between the two samples at any given N. In the case of an autonomous MS, the
system should be able to determine the measurement, which is closest to the actual
value of , yielding the highest accuracy possible. For this reason, an analysis to
determine when the MS should estimate
has been carried out showing the errors
associated with each parameter.

Errors due to the amount of divergence between two signatures
The following analysis graph (a) in Figure 3-12 shows the error in estimating
against the amount of
between the two time series for different values of . If
below 0.03 the error is greater than
decreases to less than
1% and as
is
increases towards 0.3 the error
0.1%. Finally, above 0.3 the folding of the TM occurs and
increases again. The graph (b) Figure 3-12 shows an enlarged section of the “low
error area” demonstrating that the lowest error is achieved when
is in the interval
[0.2-0.3]. This value can be used to configure the MS to estimate
as soon as
enters the low error interval.
In Figure 3-13 the effect of the parameter
parameter
on the estimation of
was initially set to 1.99 and varied with a
is shown. The
change to simulate
any error that can arise in the real electronic implementations, between the desired
set value of the parameter
and the real value. Sample difference ( ) was set to 50
and the effect on the estimation error was simulated: it appears that in the low
error interval the error on the estimation of
corresponds to an estimated
of 50
remains within
.
82
, which
(a)
100
(b)
3
80
Delta Estimation Error (%)
Delta Estimation Error (%)
2
60
40
20
0
-20
Folding
-40
Folding
Low Error Area
1
0
10u
50u
100u
200u
500u
-1
-2
-60
-80
0
0.2
-3
0.4
0.6
Divergence  (V)
0.8
0
0.1
0.2
Divergence  (V)
0.3
100
Delta Estimation Error (%)
Delta Estimation Error (%)
FIGURE 3-12 DELTA ESTIMATION ERROR VS AMOUNT OF DIVERGENCE FOR DIFFERENT
Low error area 0.2<  <0.3
50
0
2
1
X: 12
Y: 0.4378
0
-1
-2
11
12
Iterations (N)
13
 = 1.989
5
10
Iterations (N)
 = 1.990
 = 1.991
15
FIGURE 3-13 DELTA ESTIMATION ERROR FOR DIFFERENT VALUES OF THE PARAMETER
A trade-off has to be achieved between the accuracy of the measurement and the
probability of reaching a “null”, which will make the estimation inaccurate. If the
estimation is performed once
remain lower than
reaches 50 mV the error on the estimation of
will
1% so that for a range of 10 V the probability of reaching a null
will be 0.5%. Although the probability of obtaining a “null” is not zero, the MS can
identify when a null occurs as the output value of the TM is close to zero.
83
In this section, the mechanism of quantifying the change of input signal was
discussed and a new means of measuring small variations of input signal using onedimensional (1D) chaotic maps was proposed followed by the analyses of sources of
measurement errors associated with the proposed system. Finally, the error analysis
enables the optimal number of iterations and amount of divergence to be determined
and used by the MS in order to obtain an accurate estimation of the input signal
change. The next section will show the different topologies used for the proposed
MS.
3.2 System with Feedback
In order for the MS to enable successive iterations to be performed, through an
electronic implementation, feedback is generally utilised.
3.2.1 System with Feedback Experimental Setup
The individual maps were placed in a feedback system to allow iterations and data
storage, as shown in Figure 3-14. The sampling frequency was set to 100Hz. The
system operates as follows:

The system samples the 1st input and iterates the chaotic map (N iterations);
the result of every iteration is converted to a digital word and stored
developing a signature or data set related to the input sample.

A 2nd input is sampled and iterated, converted and stored (2nd signature).

The signature (data set) for the first sample is then subtracted from the
signature (data set) for the 2nd sample.
84

The resulting difference signature obtained is then used to determine the
amplitude of the change between the two successive samples, as the
number of iterations, before the two signatures diverge, is proportional to
the relative difference between the samples.

The sample difference is measured and not the absolute value.

The data storage and the subtraction are all performed by a microcontroller
( C).
The system implementation and technical details are explained in section 4.2.1.
Chaotic Map
Input
Switch
Feedback
Low resolution
ADC
Signature
Storage
Control Circuitry
Microcontroller
FIGURE 3-14 BLOCK DIAGRAM OF THE PRESENTED ITERATED SYSTEM
The configuration given in Figure 3-14 has been used to assess the performance of
the chaotic maps and as the main implementation of the MS. The series
implementation has been used to demonstrate that higher levels of sensitivity can be
achieved by eliminating the errors generated due to the feedback process.
85
3.3 Series System
In the series system feedback circuitry is not required as the signal is propagated
through consecutive chaotic maps implementation, which enables faster response
times due to the removal of the need for clocking and eliminates errors generated by
the feedback circuitry (sample and hold errors). The main drawback of the series
system compared to the iterated system is that the circuitry required to perform a
given number of iteration is larger and increases with the number of maps placed in
series. Also the number of calculations (series propagation) is set by hardware which
prevents any software modifications of the number of iterations.
3.3.1 Series System Experimental Setup
The series system operates in a similar way as the iterated system, the main
difference is that the central circuitry is required to route the output of each map to
the ADC via multiplexing. In Figure 3-15 a block diagram of a series system with
three identical chaotic maps is illustrated. The block diagram has been limited to
only 3 maps for clarity. The series system works as follows:

The 1st measurement is taken by sampling the output of each map starting
with map A and continues through to C. This is performed sequentially by
multiplexing the analogue input being sampled in software.

The data obtained from sampling the output of each map is used to form a
signature (data set) for the 1st signal sample.

The same procedure is performed for the next input signal sample.
86

The signature from the first sample is then subtracted from the 2nd sample
signature.

The modulus of the result obtained is then used to determine the difference
between the two samples, as the number of iterations, before the two
signatures diverge, or the amount of divergence after N iterations is
proportional to the relative difference between the samples.

The sample difference is measured and not the absolute value.

The data storage and the subtraction are all performed by a Microcontroller
( C).
Chaotic Map A
Chaotic Map B
Chaotic Map C
Input
Multiplexer or Analogue Inputs
Low resolution ADC
Signature Storage
Control Circuitry
Microcontroller
FIGURE 3-15 SERIES SYSTEM BLOCK DIAGRAM
3.4 Conclusion
This chapter analysed how the high sensitivity to initial conditions of onedimensional chaotic maps can be used for the measurement of small signal changes.
The working principle of the proposed technique was defined and analysed with the
following significant points identified:
87

A novel way of visualising the divergence of one-dimensional chaotic maps
was proposed using a three dimensional (3D) graph, developed specifically
for this application.

From the 3D representations of the divergence for the LM and the TM it was
determined that the TM yielded a relatively linear response throughout the
entire input range. Whereas the input range of the LM has to be limited to
avoid early divergence due to non-linearity near the extremities of the input
range as illustrated in Figure 3-2 and Figure 3-3.

A method of quantifying the change between two input samples has been
proposed in expression (3-1), which is related to the number of iterations and
the size of signature divergence.

The sources of errors associated with the proposed system have been
identified and the impact of each error has been assessed and methods of
minimising the errors have been proposed.

The analysis has also shown that “nulls” (point of lower divergence) can
appear on the response of the TM around the threshold voltage.

The number of nulls presents over the entire input range and the probability
of reaching a null during a measurement have been expressed by equations
(3-2) and (3-3) respectively. The probability of reaching a null during a
measurement has been identified as iteration dependent.

A method of identifying when nulls occur has been presented to avoid flawed
measurements.
Finally two possible implementations of a chaos based MS where proposed which
are based on the feedback and series system.
88
The following chapter will discuss the practical implementation of the proposed
maps and the overall MS.
89
4 Measurement System
Implementation
This chapter focuses on the electronic implementation of the Measurement System
(MS) proposed in Chapter 3. Firstly, an electronic implementation of the Logistic
Map (LM) and the Tent Map (TM) are proposed followed by the full implementation
of the two MS topologies; feedback and series.
4.1 One-Dimensional Maps Implementation
This section will discuss the electronic implementation of the LM and the TM,
which are two of the simplest mathematical expressions exhibiting chaotic behaviour
making them ideal for electronic implementation. As the circuitry required to
implement the maps is relatively simple, a low noise implementation is achievable,
which is paramount for MS.
4.1.1 Logistic Map Implementation
For the design and implementation of the LM; the initial step was to investigate an
electronic circuit based on the implementation available in literature (Suneel, 2006).
The electronic circuitry was then assessed to ascertain that the system exhibits the
characteristics and behaviour identified in the theoretical LM model. Subsequently,
the behaviour was established by collecting systems measurements to obtain the LM
parabola, the bifurcation diagram and the time series.
90
The prototype of the LM was implemented using readily available low cost
electronic components as shown in Figure 4-2. The LM equation implemented is the
modified version to enable an input range of 0 V to 10 V instead of the normalised 0
V to 1 V input range of the classic LM given in expression (2-8). The modified LM
shown in equation (4-1) is identical to the original LM the only difference being the
scaling factor.
(4-1)
Where
can vary from 0 to 0.4 which is equivalent to a variation of from 0 to 4 in
the classic LM shown in equation (2-8).
Scaling up the LM does not change the behaviour but solely the input and output
range which allows the Measurement System (MS) to detect variation of input signal
over a wider input range. In order to obtain the expression in equation (4-1), using
electronic circuitry, three main circuit blocks were used as shown in Figure 4-1; one
subtractor and two multipliers. The first multiplier (multiplier (A)) multiples the
input by the parameter
V reference to obtain
to obtain
. The subtractor subtracts the input from a 10
. Finally, the second multiplier (B) multiples the
results from the previous blocks to generate the modified LM expression (4-1).
91
FIGURE 4-1 BLOCK DIAGRAM OF THE LM IMPLEMENTATION
Figure 4-2 shows the circuit diagram of the proposed implementation; which uses
three OP27 operational amplifiers (Analog-Devices, 2003) and one AD633JN
analogue multiplier (Analog-Devices, 2012) along with several passive components.
The passive components used for power supply decoupling and noise filtering are
omitted from the schematic to keep the schematic clear. The 10 V reference voltage
was produced using a low noise AD587 voltage reference IC (Analog-Devices,
2007); not shown in the simplified schematic. The full schematic is shown in
Appendix G and some notes regarding the selection of components are available in
Appendix H.
92
FIGURE 4-2 SIMPLIFIED SCHEMATIC OF THE MODIFIED LM.
In Figure 4-2, U1 is an op-amp configured as a unity gain buffer to insure that R7
and RV1 are not loading the input
. The input
is scaled by the constant
set by the potential divider configuration R7 and RV1 following the expression
(4-2);
can be set anywhere between 0 and 0.4 by varying the position of RV1.
(4-2)
Where α is a variable representing the position of the potentiometer ranging from 0
(minimum) to 1 (maximum). U3 is an operational amplifier configured as a
difference
amplifier
with
a
transfer
function
given
by the
expression
(4-3).
(
93
)
(4-3)
The resistors R1 to R4 are set to identical values so that the signal at the inverting
input of U3 is subtracted from the signal at the non-inverting input. Hence, in this
case the expression (4-3) can be simplified to the expression (4-4), with the reference
voltage (Ref) set to 10 V.
(4-4)
The outputs from U1 and U3 are multiplied using an AD633JN analogue multiplier
IC (U2). With a transfer function according to the manufacturer datasheet (AnalogDevices, 2012) given in expression (4-5).
(4-5)
In the LM implementation, X2, Y2 are connecter to ground since the inputs of the
multiplier are single ended (not differential). Z is also connected to ground as no
offset is required. This simplifies the expression (4-5) to the expression (4-6).
(4-6)
The input X1 is connected to
and the input Y1 to
, which yields the
expression (4-7).
(4-7)
Since the internal circuitry of the AD633JN multiplier (U2) divides the output by 10,
is amplified by U4, configured as a non-inverting amplifier with a gain set to
94
10, in order to obtain the modified LM. The transfer characteristic of the overall
circuit is expressed by (4-8).
(
)(
)
(4-8)
Resistors R6 and R5 are set to obtain a gain of 10 which gives the final transfer
characteristic of the whole circuit to match the modified LM as expressed in (4-1).
In this section an electronic implementation of the modified LM was presented;
using readily available components the transfer characteristic obtained is identical to
the modified LM expression in (4-1). The full schematic of the LM implementation
is shown in Appendix G. Contrary to other implementations available in literature
such as the circuit presented by Suneel (2006) the proposed implementation uses a
single multiplier which simplifies the circuit and reduces the inherent noise as
discussed further in section 5.7.
4.1.2 Tent Map Implementation
The TM circuit was also constructed using readily available low noise components;
the block diagram of the proposed TM implementation is shown in Figure 4-3 and is
constructed using four main circuit blocks; level shifter, half wave rectifier, amplifier
and adder. The input to the circuit is propagating through two paths shown as (1) and
(2). Path (1) is composed of an amplifier to create the line shown as (1). Path (2) is
composed of the level shifter and half wave rectifier to create the transfer function
shown as (2) in Figure 4-4. This section of the circuit implements the 0.5 V threshold
voltage (set by the reference voltage) required to generate the overall transfer
characteristic of the TM; the two transfer characteristics obtained from (1) and (2)
95
are inverted and added to obtain the overall transfer characteristic of the TM. The
schematic of the proposed implementation is shown in Figure 4-5. Some components
are omitted to simplify the schematic. The full schematic is available in Appendix G
along with note regarding the components selection in Appendix H.
Tent Map
(2)
Half Wave
Rectifier
Level Shift
(1) + (2)
Input
∑
(1)
Gain
Gain
FIGURE 4-3 BLOCK DIAGRAM OF THE TM IMPLEMENTATION
96
Output
2
1.5
(1)+(2)
Output (V)
1
0.5
0
(2)
-0.5
-1
(1)
-1.5
-2
0
0.2
0.4
0.6
0.8
1
Input (V)
FIGURE 4-4 CONSTRUCTION OF THE TM TRANSFER CHARACTERISTIC
(2)
Vref
U1
Vin
(1) + (2)
U2
Vu1
Vu2
U4
(1)
U3
Vout
Vu3
FIGURE 4-5 SCHEMATIC OF THE PROPOSED IMPLEMENTATION OF THE TM
The upper branch of the circuit (shown as (2) in Figure 4-3 and Figure 4-5) is
composed of two blocks constructed around the level shifter (U1) and the half wave
rectifier (U2) in Figure 4-5. The input signal is applied to a summing amplifier (U1)
97
to shift by an amount equal to Vref. The resistor values R1, R2 and R4 are identical
so that the transfer function can be expressed as (4-9).
(4-9)
The output of the half wave rectifier constructed around U2 has two conditions;
when the input is greater than 0 V the output is 0 V and when the input is below 0 V
the output is inverted and amplified by a gain set using RV2, R5 and R3. Because of
the shifting introduced by the previous stage (U1) the threshold is moved from 0 V
to -Vref giving an overall response expressed by equation (4-10).
{
|
|
(4-10)
For the lower path (1) the transfer function given by equation (4-11) is achieved
using an inverting amplifier with the gain set by RV1, R6 and R7.
(4-11)
Paths (1) and (2) are summed using the summing amplifier constructed around U4 as
expressed in (4-12).
(4-12)
Finally, substituting expressions (4-10) and (4-12) for
gives expression (4-13).
98
and
respectively
{
|
(|
)
(4-13)
Where R3 and R5 are set to obtain a gain close to -2, while the resistors R6 and R7
are set to obtain a gain of approximately 4. Inverting and adding the two gains (from
paths (1) and (2)) gives a gain of -2 on the second half of the TM characteristic.
RV1 and RV2 are used to vary the gains of the two paths (1) and (2) respectively in
order to set the parameter of the TM. The reference voltage was produced using a
low noise ADR130 voltage reference IC (Analog-Devices, 2013); not shown in the
simplified schematic. Vref is set to -0.5 V which gives the original expression of the
TM shown in expression (4-14).
{
(4-14)
After implementing the LM and the TM the next section will discuss the
implementation of the two different topologies of Measurement System (MS).
4.2 Measurement System Implementation
In this section the feedback and series measurement system (MS) are implemented.
4.2.1 Feedback System Implementation
The working principle of the feedback system was explained in section 3.2.1 and
was implemented electronically as per the block diagram presented in Figure 3-14,
99
the schematic implementation is shown in Figure 4-6. The system design was
divided into three sections; a switch between the input and the feedback, a feedback
path and the control and storage circuitry. The control circuitry, ADC and memory,
are all embedded within a PIC32MX460L C: on a Mikroelektronika LV32MX v6
development board, and programmed using Microchip MPLAB environment along
with a C32 compiler. A graphic Liquid Crystal Display (LCD) with touch screen
provides a graphical user interface by displaying the state of the system. Appendix H
discusses the requirements of the C. For the digital outputs and analogue inputs the
following configuration was used:

Two digital outputs are used to control the switching between the input and
the feedback.

Further digital outputs are used to control the sample and hold circuitry of the
feedback and to discharge (reset) the sample and hold capacitors before the
next sample is taken.

An analogue input was used to sample the chaotic map output and convert to
digital data using the internal 10 bit ADC.
The sampled data was stored in the internal memory and transferred to an external
USB memory in a Comma-Separated Values (CSV) format to allow the data to be
analysed and displayed using MATLAB.
The switching and the sampling process were implemented as follows:

The Switching circuit block (A) between the input and the feedback (B) was
implemented using the DG211 analogue switch IC (Maxim, 2006). The
DG211 being a normally closed switch, a logic "0" closes the switch whilst a
logic "1" opens the switch.
100

The feedback block (B) was implemented with two LF198 Sample and Hold
(S/H) circuits (U6, U7) along with two low leakage 0.1 F capacitors to
reduce errors in the feedback path. The switches SW1 and SW2 are used to
connect and disconnect the input or the feedback to the input of the chaotic
map using pins RC2 and RC3 of the C.

The S/Hs are controlled via the C pins (RC4 and RC5). The S/H capacitors
where selected to provide low leakage and low dielectric absorption error
(Texas-Instruments, 2000). The dielectric absorption can produce significant
errors by offsetting the following sample (Analog-Devices, 2008).
In
addition to the use of polypropylene capacitors, to eliminate any errors due to
remaining charge in the S/H capacitors two N-channel MOSFETS (Q1 and
Q2) were connected in parallel with C1 and C2 to provide full discharge of
the capacitors. A comparator (U1) is used to drive Q1 and Q2, as C output
pins are unable to drive the MOSFETs directly due to insufficient output
voltage. After each series of iterations, before the next input in fed to the
chaotic map the C triggers the comparator and switches on the MOSFETs,
C1 and C2 are short-circuited which removes any remaining charge.
101
(A)
(B)
FIGURE 4-6 SCHEMATIC OF THE FEEDBACK SYSTEM FOR MAP ITERATION
FIGURE 4-7 CONTROL SIGNALS FOR THE ITERATED MEASUREMENT SYSTEM
The signals of a typical run of the iterated MS are shown in the oscilloscope screen
capture in Figure 4-7. The signal number is shown on the left hand side of the graph
while the scale for each signal (in volts per division) is displayed at the top. Signals
102
(2) and (4) represent the control signals for the S/H (RC4 and RC5) respectively.
Signal (3) is used to control the switching between the input and the feedback (RC2);
RC3 is not represented as it is the opposite of RC2. Finally, signal (1) represents the
output of the comparator U1 (RC6). The iterative process follows the steps described
below:

Firstly the input is connected to the chaotic map and the resultant output
signal is stored in the capacitors by sending a pulse to the S/Hs with RC4
then RC5.

The input is then disconnected before the feedback is connected to the
chaotic map.

After a delay specified for the signal to propagate through the map circuit the
output signal is stored in the S/H capacitors. The pulses for the S/Hs are
repeated N times which generate the iterations.

After each iteration the output signal is sampled, digitised and stored in the
internal memory of the C. Once the set number of iterations N is reached
the input is reconnected and the feedback disconnected.

The MOSFETS are activated to remove any remaining charge from the
capacitors.

The MS is then ready for the next measurement.
4.2.2 Series System Implementation
In the series implementation no switching of feedback is required; the input of the
system and the outputs of each map are connected to a separate analogue input of the
ADC as shown in Figure 4-8. The development board and the C used to perform
the ADC function and the measurement are the same as for the feedback system as
103
shown in section 4.2.1. The schematic of the given map is copied and connected in
series as shown in the block diagram of the series implementation in Figure 4-8: the
actual schematic of the series system is not shown due to its simplicity. The C used
has 16 analogue inputs which mean that the MS is able to perform up to 16
measurements of chaos map outputs, without an additional multiplexer. The
propagation of the signal is not controlled by the C as the signal propagates freely
through each map. The signature is constructed by sampling the output of each map.
Input
Chaotic Map 1
Chaotic Map 2
Analogue
input 2
Analogue
input 1
Chaotic Map N
Analogue
input N
PIC32MX460L
FIGURE 4-8 BLOCK DIAGRAM OF THE SERIES SYSTEM
4.3 Conclusion
The first section of this chapter focused on the practical implementation of two onedimensional chaotic maps, the Logistic and the Tent map respectively. The design
process was undertaken to develop an electronic system based on the mathematical
expression of each map; to do so, a block diagram was produced followed by the
electronic implementation of the maps. The transfer function of each functional
block was used to analyse the overall transfer characteristic of the proposed
implementation and to ascertain the matching accuracy between the presented circuit
and the associated mathematical expression.
104
High performance, yet readily available components were used to obtain a circuit
that yields increased overall MS performance, by reducing the overall noise whilst
avoiding expensive and hard to source components.
The second section explains the implemented feedback and series MS. The feedback
system requires switching circuitry and S/Hs for the iterations whereas the series
system requires multiple identical implementation of the map circuitry which creates
a relatively cumbersome MS if a large number of stages are required.
The
following
chapter
will
assess
the
implementations.
105
performance
of
the
developed
5 Performance Analysis of
Implemented Chaotic Maps
This chapter focuses on the performance analysis of the LM and the TM. To assess
the performance, the electronic implementations will be compared to the theoretical
maps using the following measurements:

Transfer characteristic: this will assess the accuracy of the practical
implementation compared to the theory (transfer characteristic).

Bifurcation diagram: the bifurcation diagram will allow the behaviour of
the proposed implementations to be visualised for variations of the map
parameter. The bifurcation diagram obtained practically for each map should
match the theoretical bifurcation diagrams shown in section 2.3 and exhibit
all the characteristic of the theoretical maps ranging from single point
stability, periodicity and chaos.

Time series: the time series will be measured and analysed to demonstrate
the behaviour over time of the proposed hardware implementations.

Lyapunov Exponent: the data of the time series will be used to estimate the
Lyapunov Exponent (LE) of the maps. The estimation of the LE from the
experimental time series will allow the chaotic nature of the proposed
implementations to be ascertained.

Noise measurement: the noise measurement will be used to quantify the
amount of noise inherent to each implementation. This is critical to any MS
106
as the magnitude of the noise components will determine the performance
limit of the MS in terms of signal change that can be accurately determined.
5.1 Transfer Characteristic
To obtain the transfer characteristic of each map, the input of the electronic chaos
map implementation under test was connected to a signal generator which was set to
produce a sinusoid voltage signal. A DC offset was applied to enable the signal to
swing between 0 and the highest voltage of the chaos map input range, which
corresponds to a 10 V pk-pk input signal with a 5 V offset for the LM and a 1 Vpkpk input signal with a 0.5 V offset for the TM. A Data Acquisition Device (DAQ)
from National Instruments NI 6254 (National-Instruments, 2007) with 16 bit
resolution was used to sample the input and output of the circuit. The setup for the
measurement is illustrated in Figure 5-1.
16 Bit DAQ
Channel 1
16 bit DAQ
Channel 2
Map Under Test
SineWave
with DC offset
FIGURE 5-1 TRANSFER CHARACTERISTIC MEASUREMENT SETUP.
The DAQ software was used to retrieve the data from the two channels as a CommaSeparated Values (CSV) file which was then plotted using Matlab and compared to
107
the ideal transfer characteristic of each map to measure the error between the
practical and the computed ideal characteristic.
5.2 Logistic Map Transfer Characteristic
The experimental data was plotted to obtain a graph of the input signal versus the
output, which represents the transfer characteristic of the electronic implementation.
The theoretical parabola representing the LM for r = 4 (normalised) was plotted on
the same graph to enable a comparison between the two plots, shown in Figure
5-2(a). With the two plots being closely matched, no significant differences were
observed, hence for this reason the error between the practical results and the
theoretical parabola is measured and displayed separately in Figure 5-2(b).
FIGURE 5-2 MEASURED VS THEORETICAL LOGISTIC MAP PARABOLA.
From Figure 5-2(b) it can be noted that the maximum error in the LM practical
results is 1.5% and that the majority of the errors are contained within a ±1% band,
108
which demonstrates the accurate fit between the measured characteristic and the
theory. The fact that the transfer characteristics are closely matched proves that the
electronic implementation of the LM designed in section 4.1.1 is performing as
defined by the LM equation as in expression (2-8).
5.3 Tent Map Transfer Characteristic
The parameter
(from equation (2-9)) was set as near to the value of 2 as could be
achieved and the measured transfer characteristic is shown in Figure 5-3 (a) along
with the computed ideal transfer characteristic of the TM. The error between the two
graphs representing the mismatch between the ideal characteristic and the measured
characteristic was obtained by subtracting the ideal characteristic from the measured
characteristic. The error given in Figure 5-3 (b) contained within
across the
whole input range which once again demonstrates the close fit of the practical
implementation to that specified by the theory.
FIGURE 5-3 MEASURED VS THEORETICAL TENT MAP CHARACTERISTIC
109
5.4 Bifurcation Diagram
The Bifurcation diagram, as discussed in section 2.3, illustrates the behaviour of a
chaos map for different values of the associated scaling parameter. For every value
of the scaling parameter the map under test was placed into the feedback system as
presented in section 3.2 and iterated 100 times. The first 10 iterations where
discarded in order to display the long term behaviour of the map and remove any
initial transient values. The map was iterated using a C board and the output was
measured using the internal 10 bit ADC. Finally the measurement of each iteration
was stored as a CSV file and displayed using MATLAB. The bifurcation diagram
obtained experimentally was compared with the computed bifurcation diagram of
each map to quantify the matching between the practical electronic implementation
and the theory.
5.4.1 Logistic Map Bifurcation Diagram
The bifurcation diagram of the electronic implementation of the LM shown in Figure
5-4 is indicating all the characteristics of the theoretical map ranging from fixed
point stability, periodic oscillations and chaos. The parameter
(in equation (2-8))
was varied from 2 to 4 (normalised) by increments of 0.05. Figure 5-5, shows the
theoretical bifurcation diagram of the LM computed using MATLAB. The practical
data is closely relates to the data generated by computation, with the main difference
being the unclear bifurcation from fixed point to 2 point oscillation at about
= 2.9.
As described in (Erguler and Stumpf, 2008) the effect of noise on the behaviour of
the LM is visible on the bifurcation diagram in a way that the bifurcation points
become "blurred". This effect is clearly visible on the transition between single point
110
stability and periodic oscillations. An additional effect of noise on the LM behaviour
is the possibility that the system theoretically diverges to ±∞ which can be triggered
when the parameter
is set to exactly 4. However, in a practical system, noise can
increase the input signal to exceed the input range, which can lead to an exiting
condition. In this work the parameter was set close to 4 but not exactly 4 in order to
use the chaotic behaviour of the LM whilst avoiding exiting conditions.
FIGURE 5-4 MEASURED BIFURCATION DIAGRAM OF THE LM
111
FIGURE 5-5 COMPUTED LM BIFURCATION DIAGRAM
5.4.2 Tent Map Bifurcation Diagram
The proposed implementation of the TM was designed to operate with a parameter
set to 2 or close to 2. For this reason it is impossible to vary
throughout the whole
input range. The values for the resistors and potentiometers were selected to enable
the parameter
to be varied from 1.945 to 2, and resulted in the practical limitation
which restricted the measurement range of the parameter ; the partial bifurcation
diagram is shown in Figure 5-6(b). The TM was iterated 100 times, the first 10
samples were removed to suppress any transient response and keep the long term
behaviour. As expected from the theoretical bifurcation diagram shown in Figure
5-6(a) when the parameter
is set to 2 the range of the output signal occupies the
entire output. The amplitude of the output signal decreases as the parameter
reduced from 2 to 1.995 and further reduces as the parameter
112
is
is decreased towards
1.94. The experimental results are consistent with the theoretical bifurcation
diagram.
FIGURE 5-6 TENT MAP EXPERIMENTAL BIFURCATION DIAGRAM
5.5 Time Series
The time series of chaotic maps displays the evolution of the output signal over a
given number of iterations N. The value of the parameter ( for the LM and for the
TM) sets the behaviour of the maps. The output could be stable, periodic or chaotic
113
as explained in section 2.3.1. To generate the time series, each map was placed in the
feedback system presented in section 3.2. The input of the map under test was
connected to the KROHN-HITE 511 DC voltage reference/calibrator which provides
a stable and accurate DC voltage (see Appendix E). The output of the KROHNHITE 511 was set to an arbitrary value within the input range of the map under test
and the map iterated multiple times. The output after each iteration was sampled
using a 10 bits ADC available in the C, and the data stored as a CSV file and
displayed using MATLAB. The parameter of the map under test was set to a value
that allows chaotic behaviour and the obtained time series used to ascertain the
presence of chaos, first visually and then through the use of Lyapunov Exponent
(LE).
5.5.1 Logistic Map Time Series
The parameter
is set to 3.99 (0.399 for the modified map) so that the behaviour of
the map would exhibit chaos; Figure 5-7 shows the experimental time series
obtained from the electronic implementation of the LM with the output showing non
periodic behaviour and is contained between 1 and 0. This result is consistent with
the bifurcation diagram of the system shown in Figure 5-5.
114
1
Output (V)
0.8
0.6
0.4
0.2
0
0
20
40
60
Iteration (N)
80
100
FIGURE 5-7 LOGISTIC MAP EXPERIMENTAL TIME SERIES
5.5.2 Tent Map Time Series
The time series of the TM output, in Figure 5-8, shows the evolution of the map
under iteration with the parameter
set to 2, in order to obtain a chaotic behaviour;
Figure 5-8 shows that the experimental time series occupies the entire output range
going from 0 to 1 V which is expected for a parameter set to 2 as shown by the
bifurcation diagram of the TM in Figure 2-15. The experimental time series obtained
was used in the following section to ascertain the presence of chaos by estimating
the LE value.
115
1
Output (V)
0.8
0.6
0.4
0.2
0
20
40
60
Iteration (N)
80
100
FIGURE 5-8 TENT MAP EXPERIMENTAL TIME SERIES
5.6 Lyapunov Exponent
A simple visual inspection of the time series is not sufficient to ascertain the chaotic
behaviour of the electronic implementation; hence the LE method is used, as
explained in section 2.5.1.
If the LE of a time series is positive then the system is chaotic (G.L.Baker and
J.P.Gollub, 1990). To estimate the LE it is assumed that the equation emulated by
the electronic implementation is unknown, thus the estimation will take into account
any errors between the electronic implementation and the theoretical map. To
demonstrate that the method used for the LE is valid, a non-chaotic time series was
initially analysed. The map used was the LM and the parameter
was set to 3.35 to
be in the region of non-chaotic behaviour. The measured time series is shown in
Figure 5-9 and the estimated prediction error in Figure 5-10. It can be observed (in
Figure 5-9) that after a short transient the output of the LM oscillates between two
points, and also in Figure 5-10 it is not possible to estimate the value LE, since there
116
is no exponential separation between the two neighbouring points. Additionally, the
LE is not consistently positive, which confirms that the system is not sensitive to
initial conditions, and thus not chaotic.
FIGURE 5-9 LOGISTIC MAP TIME SERIES FOR R = 3.35
FIGURE 5-10 PREDICTION ERROR FOR A NON-CHAOTIC TIME SERIES
117
5.6.1 Logistic Map Lyapunov Exponent
The number of samples taken for the LE estimation was set to N = 500 as
recommended in literature (Rosenstein et al., 1992, Wolf et al., 1985), and analysed
using MATLAB to generate the prediction error plot shown in Figure 5-11. After a
transition from k = 0 to k = 2 (k is the number of time steps used for the estimation
of the LE) the two neighbouring points start to diverge exponentially before
plateauing and decreasing from k = 7, which separation is due to the folding of the
map. The slope m is positive and the result obtained indicates that the LE of the
electronic implementation is 0.79, for a parameter
of 3.99, as shown in Figure
5-11.
FIGURE 5-11 ESTIMATION OF THE LYAPUNOV EXPONENT FROM THE PREDICTION ERROR
118
TABLE 5-1 ESTIMATED AND THEORETICAL LYAPUNOV EXPONENT FOR THE LM
Number Of
Simples
Estimated
Theoretical
% Error
Lyapunov exponent Lyapunov Exponent
N = 500
= 0.79
= 0.693
13.9
The error between theoretical and estimated LE is 13.9%; this degree of accuracy is
acceptable considering that the time series was obtained experimentally with limited
data and measurement noise. Similar results were obtained in literature (Rosenstein
et al., 1992). The estimated LE is positive which demonstrates that the electronic
implementation of the LM exhibits high sensitivity to initial conditions and therefore
that the system is chaotic.
5.6.2 Tent Map Lyapunov Exponent
To ascertain if the TM implementation behaves chaotically the time series from
Figure 5-8 was used to estimate the LE. The graph of the LE estimation is shown in
Figure 5-12, where the slope of the prediction error equals 0.80 from k = 2 to k = 10
(linear region). After k = 10 the folding of the map occurs causing the prediction
error to fluctuate, this has no effect on the LE estimation as the measurement is taken
on the linear region.
119
10
Prediction Error (p)
8
6
4
m = 0.80
2
0
0
5
10
Time Steps (k)
15
20
FIGURE 5-12 TENT MAP LYAPUNOV EXPONENT ESTIMATION
Table 5-2 shows the results error of the positive LE estimation, yielding an error of
15.4% which is also consistent with the results obtained in literature (Rosenstein et
al., 1992) and defines the system as chaotic.
TABLE 5-2 ESTIMATED AND THEORETICAL LYAPUNOV EXPONENT FOR THE TM
Number Of
Estimated
Theoretical
% Error
Samples
N = 500
Lyapunov exponent Lyapunov Exponent
= 0.80
= 0.693
120
15.4
5.7 Noise Measurement
To assess the performance of the chaos map implementation the noise of the system
was quantified, as the inherent noise will determine the sensitivity of the system.
The noise of the electronic implementations was measured using the following
procedure;

The input of the map under test was grounded and the noise at the output was
measured using a HP 3562A Dynamic Signal Analyser (DSA) (Agilent,
1985).

In order to minimise external noise sources the circuit under test was placed
in an earthed metallic box to provide adequate shielding.

To prevent any DC offset being generated by the map under test, a 20 F
decoupling capacitor was connected between the circuit and the signal
analyser.

The measurement data was extracted using a General Purpose Interface Bus
(GPIB) to USB interface cable.

The noise floor of the DSA was measured by connecting a 50 Ω BNC
terminator to the input of the DSA, and measured over 3 decades ranging
from 10

to 10
(Figure 5-13).
In order to obtain an accurate reading a separate measurement was taken for
each decade and the results were combined into a single graph. The
averaging process was set to 50 samples to obtain an accurate measurement
whilst keeping the measurement time within a reasonable limit.
121
The noise floor of the 3562A DSA given in Figure 5-13, appears to have a step-like
characteristic which is due to the use of different internal circuitry for each decade.
In the low frequency region (10 – 100 Hz) some noise spikes are visible; the highest
being due to the mains pickup at 50 Hz. The noise floor of the DSA is contained
√
within 100
and thus for a noise measurement to be accurate the noise being
measured should be at least 1
Equivalent Voltage Noise (V/(Hz))
10
√
(an order of magnitude above the noise floor).
-6
50 Hz
mains
interference
10
10
-7
-8
10
1
10
2
10
3
10
4
Frequency (Hz)
FIGURE 5-13 MEASURED NOISE FLOOR OF THE HP 3562A DSA
When the noise signal being measured was lower than
√
, a low noise
preamplifier with a gain of 1000 was used to avoid any measurement errors due to
the contribution of the DSA noise.
To calculate the overall noise
in terms of Voltage Root Mean Square (
from the noise spectrum the formula in expression (5-1), can be applied.
122
)
√∫
Where
and
(5-1)
are the high and low frequencies of interest respectively and
SD is the noise spectral density.
In the case when the value is relatively constant for all the frequencies, the
expression (5-1) can be simplified as shown in expression (5-2).
√
Where
is the bandwidth and
(5-2)
the noise bandwidth coefficient which is set
according to the order of filter used to limit the bandwidth ( ) as shown in Table 5-3.
TABLE 5-3 EQUIVALENT NOISE BANDWIDTH COEFFICIENT
Filter Order
Equivalent noise bandwidth
coefficient (K)
1st Order
1.57
2nd Order
1.11
3rd Order
1.05
Ideal Filter (Brick Wall)
1
5.7.1 Logistic Map Noise Measurement
The equivalent voltage noise measurement for the LM is shown in Figure 5-14 with
the noise centred between
the mains (50 Hz) visible.
√
and
123
√
and several harmonics from
Applying expression (5-2) to the data within Figure 5-14, for a bandwidth of 1 kHz
limited by a 2nd order filter gives:
351
which converts to
, applying a factor of 6.6; (the standard deviation on the noise Gaussian
distribution). The high sensitivity to initial conditions of the LM makes the
detection, with certainty, of any signal changes lower in amplitude than the inherent
noise of the system, difficult as the noise causes the time series to diverge before the
input signal. In order to identify the major component noise contribution and thus
reduce the noise of the overall system, the noise of each component has been
evaluated given in Table 5-4. The information was determined from the datasheets or
calculated using the relevant expressions (Analog-Devices, 2003, 2012, 2013). The
resistors used are of low value (maximum 10
) to maintain low Johnson voltage
noise as given by expression (2-1) (Horowitz and Hill, 1989).
From Table 5-4 it is clear that the AD633JN is the main source of noise in the LM
implementation; as explained in the section 4.1.1 the AD633JN divides the product
of the two inputs by a factor of 10 before the output. This, therefore forces the use of
an amplifier with a gain of 10 resulting in the noise being amplified by a value of 10.
Hence, the AD633JN contributes 185
which is of the same order of
magnitude as the LM overall noise measured using the DSA (
124
).
-3
Equivalent Voltage Noise (V/(Hz))
10
50 Hz mains
interference
Value used
for the
calculation
-4
10
-5
10
-6
10
1
2
10
3
10
10
4
10
Frequency (Hz)
FIGURE 5-14 MEASURED NOISE OF THE LM
TABLE 5-4 NOISE FOR COMPONENTS USED IN THE LM IMPLEMENTATION.
Component
Equivalent noise voltage
Name
√
OP27
AD633JN
10
Resistor
AD587
-
0.1
0.7
√
28
185
√
0.45
3
-
4
The noise measurement for the LM electronic circuit demonstrates that the overall
noise of the implementation is predominantly due to the AD633JN analogue
125
multiplier, thus, in order to reduce the overall noise, the next section discusses an
improved version of the TM implementation using discrete analogue multipliers.
5.7.2 Improved Logistic Map Implementation
To improve the sensitivity of the LM, alternatives to the AD633JN were investigated
in order to reduce the overall noise. Alternative analogue multiplier ICs are available
but none presents a significant improvement in terms of noise over the AD633JN.
However the fact that signals being multiplied in the LM are positive, allows the use
of a simpler multiplier, namely the One-Quadrant (1Q) multiplier. Compared to the
four-quadrant configuration, used by the AD633JN, where the input and output
signals can be negative or positive the 1Q multiplier can only multiply positive
signals. The main advantage of the 1Q multiplier is the simplicity of the circuitry.
The solution adopted to replace the AD633JN is a semi-discrete 1Q multiplier
implemented using two matched transistor pairs and four precision operational
amplifiers; the simplest form of analogue multiplier shown in Figure 5-15. This
configuration uses a Gilbert cell to compute the product of two numbers (AnalogDevices, 2009).
126
R1
Input
(X)
-
+
+
Vout
LM194
LM194
Input
(Y)
-
-
+
+
Input
(Z)
FIGURE 5-15 ONE-QUADRANT PRECISION ANALOGUE MULTIPLIER
The transfer characteristic for the multiplier in Figure 5-15 is given by expression
(5-3) (National Semiconductors, 1994).
(5-3)
In the LM implementation, the input (Z) is connected to a 1V reference, therefore the
transfer characteristic becomes the product of (X) and (Y). The noise of the
multiplier was measured using the DSA and is shown in Figure 5-16, but as the noise
floor was below that of the DSA, a low noise amplifier with a gain of 1000 was used
to amplify the noise from the multiplier to reduce measurement error. A Low Pass
Filter (LPF) with a cut-off frequency of
was added to the multiplier to reduce
127
the overall noise. Besides harmonics due to the mains interference, the noise
spectrum is constant from
to
which enables expression (5-2) to be used
to calculate the rms noise voltage. The noise for a bandwidth of
order LPF is approximately 35
and a 2nd
which is in the region of a fifth of the
AD633JN noise. Another advantage of this multiplier is that the product is not
divided by a factor of 10, thus eliminating the need for amplification, which further
reduces the noise. Hence the overall noise floor has been reduced, compared to the
AD633JN, by a factor of 50. Figure 5-17, shows the improved LM circuit
implementation.
To demonstrate the gain in terms of noise performance the noise of the improved
LM was measured as given in Figure 5-18. The noise floor is now equivalent to the
noise of the one-quadrant multiplier, which means that the multiplier is still the main
source of noise in the circuit. By applying expression (5-2) the noise is determined as
7
or 46.3
. Figure 5-19 shows the equivalent voltage noise of the two
implementations (AD633JN multiplier against the 1Q multiplier).
128
-5
10
Equivalent Voltage Noise (V/(Hz))
50 Hz Mains interference
Mains harmonics
-6
10
-7
10
-8
Low Pass Filter Roll-off
10
-9
10
1
10
2
3
10
10
Frequency (Hz)
FIGURE 5-16 NOISE OF THE ONE-QUADRANT MULTIPLIER
FIGURE 5-17 SIMPLIFIED SCHEMATIC OF THE IMPROVED LM
129
4
10
-5
10
Equivalent Voltage Noise (V/(Hz))
50Hz mains
interference
Mains harmonics
-6
10
-7
10
-8
10
Low Pass Filter Roll-off
-9
10
1
10
2
3
10
10
4
10
Frequency (Hz)
FIGURE 5-18 NOISE OF THE IMPROVED LM
-3
10
Equivalent Voltage Noise (V/(Hz))
-4
10
AD633JN based LM
implementation
-5
10
-6
10
-7
10
-8
10
1Q analogue multiplier LM
Implementation
-9
10
1
10
2
3
10
10
4
10
Frequency (Hz)
FIGURE 5-19 NOISE OF PREVIOUS IMPLEMENTATION VS INPROVED IMPLEMENTATION
130
5.7.3 Tent Map Noise Measurement
The equivalent voltage noise spectrum of the TM is shown in Figure 5-20; by
applying the expression (5-2) the noise at the output of the TM is 1.4
9.3
or
. This peak to peak noise figure is significantly lower than the noise of
46.3
for the improved LM implementation by a factor of 5. This
improvement is due to a circuitry used to implement the TM that does not require an
analogue multiplier and the noise roll-off is due to low-pass filters implemented
within the TM to reduce the overall noise. The lower noise will enable the TM to
detect input signal change significantly lower than those of the LM.
-6
10
Equivalent Voltage Noise( V/ (Hz))
Mains 50 Hz interference
Mains 3rd and
4th Harmonics
-7
10
-8
10
Average Noise
Spectral Density
-9
10
1
10
2
3
10
10
Frequence (Hz)
FIGURE 5-20 NOISE OF THE TM IMPLEMENTATION
131
4
10
5.8 Conclusion
The performance of the proposed implementations of the LM and the TM has been
analysed and it has been shown that the practical performance closely matches the
theoretical maps in terms of transfer characteristic, bifurcation diagrams, time series
and LE.
The transfer characteristics for the LM and the TM are matched to the theory, within
1.5% and 1% respectively while the bifurcation diagrams display the same behaviour
as the theoretical maps. The LE for each map was used to ascertain the presence of
chaos and the results demonstrate a positive LE for each implementation, which
proves the presence of chaos in the electronic circuits.
Finally, the noise measurements for, the original implementation of the LM, are
relatively high at 2.3
. Hence, to reduce the overall noise, the main source
of noise in the circuit was identified (AD633JN analogue multiplier) and an
improved implementation of the LM was presented using a 1Q analogue multiplier.
The overall noise was reduced by a factor of approximately 50 to 46.2
The noise measurement for the TM yields a noise value of 9.3
.
which is
significantly lower than the noise of the LM due to the simplicity of the TM
circuitry, where there is no requirement for an analogue multiplier. Therefore, the
low noise of the TM circuit, compared to the LM implementation, enables the TM to
measure input signal changes significantly lower than those of the LM.
The following chapter will discuss the measurement capability and the performance
of the proposed Measurement System (MS).
132
6 Measurement System Results
In this chapter the Logistic Map (LM) and the Tent Map (TM) are integrated into the
proposed MS to assess the overall performance in terms of divergence measurement,
which is the basic criterion for small signal measurement and detection. The
divergence measurements are as follows:

Divergence between two signals: This measurement will verify the working
principle of the MS as it will enable the ability of the different
implementations of the MS to detect and/or measure small variations of the
input signal to be evaluated.

Divergence through the entire input range: This measurement will gauge
the ability of the MS to maintain a consistent measurement for the same
amount of input change throughout the entire input range. An ideal MS should
have a perfect linearity throughout the whole input range.

Divergence for different magnitudes of input change: This measurement
will show the accuracy of the MS for different input signal amplitude
changes.
For each input signal the MS system was iterated and the result of each iteration was
sampled and stored using the C. The same operation was then performed for each
input signal with an additional input change. These system measurements were
individually performed for each LM and TM implementation in a feedback based
MS and then for a series implementation.
133
This chapter is divided in two main sections, firstly the LM is placed in the MS and
the performance is measured. This is then followed by the performance of the MS
based on the TM.
6.1 Divergence Between Two Signals
In order to measure the divergence between the input signals, and ensure
consistency, repeatable and accurate signals are required, the input voltage was set
using a DC voltage reference/calibrator (KROHN-HITE 511) with the following
characteristics:




Absolute accuracy of 10 ppm (0.001%)
Voltage range from 100 V to 10 V
Resolution of 1 ppm of the range (0.0001%)
2
noise
The full specifications can be found in Appendix E.
The resolution of the 511 DC reference enables voltage increments of 100 V, which
is more than one order of magnitude lower than the smallest signal change being
measured by the MS.
6.1.1 Logistic Map
To determine how the LM diverges for a known input change the LM was placed in
the feedback MS as explained in section 3.2, the voltages were normalised, (i.e.
scaled down by a factor of 10), to enable a direct comparison between the simulation
and the practical results.
134
To test the LM implementation an arbitrary input value of 0.3430 V was set and after
running (iterating) the system and recording the time series the input voltage was
then increased by 500
and the system was re-run and the data stored. In Figure
6-1(a) the two time series are displayed with the divergence displayed in (b). The
divergence between the two time series is obtained by subtracting the second time
series from the first. Two signals are considered to be divergent after a separation of
more than 5
(for a 0-1
normalised range). The divergence threshold was set so
that the ADC (with a Least Significant Bit (LSB) of 1 mV) used can detect a
divergence without any ambiguity due to noise and ADC errors. It can be observed
that the two time series start to diverge from iteration 4, and the size of the
divergence increases until iteration 8, at iteration 9 the two time series are
uncorrelated and the information is irrecoverable.
Time Series
Output (V)
1
(a)
0.5
0
0
5
10
15
20
Divergence (V)
Divergence
1
(b)
0.5
0
0
5
10
Iteration (N)
15
20
FIGURE 6-1 TIME SERIES AND DIVERGENCE FOR A 500 V CHANGE.
135
Figure 6-2 displays the divergence for a
divergence for
= 200
= 500
(green) along with the
(red). As expected the time series diverge at an earlier
iteration when the signal change ( ) is increased. Table 6-1 shows the iteration at
which the divergence occurs and also shows that practical results and theory yield
the same results. To ascertain the minimum detectable change the LM was tested
with the following procedure; the DC reference was set to a fixed value and the LM
was run with the same sample multiple times. Ideally the time series obtained should
match exactly but as discussed in the previous section (5.7) the noise will limit the
LM sensitivity and the time series will diverge for a constant input signal, due to
system noise. The amount of divergence between the time series enables the
sensitivity of the system namely the minimum input change that can be detected, to
be established.
Divergence
1000
900
800
Divergence (mV)
700
600
500
400
300
200
100
0
0
5
10
Iteration (N)
15
20
FIGURE 6-2 DIVERGENCE FOR TWO DIFFERENT INPUT CHANGES.
136
TABLE 6-1 PRACTICAL AND COMPUTED DIVERGENCE
Computed Divergence
Practical divergence
(Iteration Number)
(Iteration Number)
200 µV
5
5
500 µV
4
4
Input Change ( )
Firstly, the LM was run 5 times with a fixed input and the respective time series
obtained are shown in Figure 6-3. It can be observed that during the first 14 iteration
there is no visible divergence between the time series but at iteration 15 the
divergence becomes detectable i.e. above 5 mV. The minimum detectable change
can be quantified by running the system multiple times for a fixed input, the two
time series that are the most divergent can be taken in order to establish a “noise
band”. Then a known change is applied to the input signal and if the resultant time
series obtained is outside the “noise band” the input change can be detected. The
change is reduced until the time series enters the “noise band” which means that the
system is unable to discriminate between the input change and the noise. The
amplitude of input change before the time series enters the “noise band” is defined as
the sensitivity of the system as is illustrated in Figure 6-4.
137
1000
900
800
Output (mV)
700
600
500
400
300
200
100
0
0
5
10
Iteration (N)
15
20
FIGURE 6-3 LM MULTIPLE RUNS FOR A FIXED INPUT
FIGURE 6-4 USE OF THE NOISE BAND TO MEASURE THE SENSITIVITY OF THE LM
138
To determine the sensitivity of the LM based MS using the “noise band”, the LM
was iterated 8 times for 4 different input signals and the process was repeated 10
times for each input signal. A reference signal set at 0.2 V and three signals
separated by a
0.2001
of 50
and 0.2002
, 100
and 200
yielding voltages of 0.20005 ,
respectively. The time series obtained are presented in Figure
6-5. To visualise the noise bands for each sample, Figure 6-6 shows an amplified
section of the noise band for each sample at iteration 8. The clearance between the
reference signal noise band and the noise band of the signal with
24
= 200
is
, which is a greater amplitude than the noise bands contained within a 15
region. This indicates that the system can detect a
of 200
without any
ambiguity. The clearance between the reference sample noise band and the noise
band for
= 100
is 8
which is lower than the noise band of 15
. In this
case the system is unable to differentiate a divergence due to inherent noise from a
divergence that occurred from a
of 100
. However, due to the clearance of 8 mV
being in the region of the 15 mV noise band, the system can detect a
of 100
by
the use of averaging. Moreover, a number of samples can be taken successively and
averaged for each iteration, to reduce the noise brand and enables the detection of
= 100
. Finally, Figure 6-6 shows that for a
with the reference input voltage and the 100
a 50
of 50
the noise band overlaps
which means that the detection of
change would require extensive use of averaging technique. However, the
main drawback of averaging is that the time required for the system to perform one
measurement is proportional to the number of samples taken. This means that a
measurement taken at 1000 samples per second with 10 averaging samples is
equivalent to a measurement at 100 samples per second. This is not an issue for
139
applications where the input signal is close to DC but it can be a problem for signals
with frequencies of 1
and above.
1000
900
800
Output (mV)
700
600
500
400
300
200
100
0
1
2
3
4
5
Iteration (N)
6
7
8
FIGURE 6-5 MULTIPLE TIME SERIES FOR DIFFERENT INPUT SAMPLES
140
FIGURE 6-6 NOISE BAND AT ITERATION 8 FOR DIFFERENT INPUT SAMPLES
6.1.2 Tent Map
This section focuses on the divergence of two time series for two input samples for
the TM and enables the proposed technique, developed in 3.1, to be tested
experimentally. Samples are taken randomly across the input range for different
input changes in order to test the theory, and particularly to test the performance of
the feedback MS based on the TM by using equation (3-1). An example of two time
series taken with a 200
difference is shown in Figure 6-7(a) while the divergence
between the two time series is displayed in (b). The exponential growth of the
divergence is clearly visible from iteration 6 to iteration 12; however after iteration
12 the folding of the map occurs and the information is consequently irrecoverable.
141
Time Series
Output (mV)
1000
500
0
(a)
2
4
6
8
10
12
14
Divergence (mV)
Divergence
1000
500
0
(b)
2
4
6
8
10
Iteration (N)
12
14
FIGURE 6-7 DIVERGENCE BETWEEN TWO TIME SERIES WITH A 200
CHANGE
From the analysis carried-out in section 3.1, in order to minimize errors the equation
(3-1) should be applied when divergence ≥ 50
, thus for the measurement in
Figure 6-7 the divergence at iteration eight is equal to 54
calculated input change ( ) of 210.9
200
. An error of 10.9
which yield a
for a real change ( ) of
represents a 5.5% mismatch between the measured value and the real . To
ascertain the overall performance of the TM, further results for different input
samples were evaluated and shown in Table 6-2. For each value of
measurements were taken. It can be noted that for
of 5
and 1
two separate
the error
remains at zero since the noise of the system is significantly lower than the change
being measured thus divergence occurs before the noise can be significantly
amplified. With a
of 500
the error occurs as the
the noise of the system. The error reaches 120% for a
being measured is closer to
of 20
, which means that a
change of that magnitude cannot be detected by the system. Thus according to the
142
noise measurement performed in the previous section the noise of the TM should not
exceed 9.3
but the measurement of
shows much greater error than expected,
which is due to the errors introduced by the feedback loop of the system. This
feedback error is not measurable for the LM since the noise produced by the LM is
of greater amplitude than the errors introduced by the sample and holds (S/H) used
in the feedback loop.
TABLE 6-2 TM CHANGE MEASUREMENT WITH FEEDBACK SYSTEM
Real Change
(
Iteration of Divergence
(N)
4
Measured Change
(
5
Error Between Real
and Measured
0%
5
4
5
0%
6
1
0%
6
1
0%
8
510
2%
8
516
3.2%
8
211
5.5%
8
213
6.5%
10
70
40%
10
66
32%
11
44
120%
11
37
85%
1
500
200
50
20
Since the errors introduced by the feedback system are significant in amplitude
compared to the intrinsic noise of the TM, the next measurements performed on the
TM MS will be completed using the series implementation of the MS, Table 6-3
143
shows the measurement results obtained with the series system: the error has
decreased to 19 % for a
of 20 µV compared to 120% error on the system with
feedback loop. The system can detect a input signal change of 50 µV with a high
degree of confidence as the error is limited to 8 %.
TABLE 6-3 TM CHANGE MEASUREMENT WITH SERIES SYSTEM
Real Change
(
Iteration of Divergence
(N)
4
Measured Change
(
5
Error Between Real
and Measured
0%
5
4
5
0%
6
1
0%
6
1
0%
8
504
0.8%
8
503
0.6%
8
202
1%
8
203
1%
10
54
8%
10
53
6%
11
24
19%
11
24
19%
12
14.3
43%
12
13.9
39%
1
500
200
50
20
10
144
6.2 Divergence Against Input Signal Range
To evaluate the divergence of the practical implementation across the input signal
range, the following procedure was carried out: the input signal was varied from 0 to
1 (normalised) with steps of 0.05 V and with a
of 500
. The two time series
obtained for each step and for the step plus change were subtracted to obtain the
divergence, which allows the absolute value of the divergence for the entire input
range to be mapped.
6.2.1 Logistic Map
As shown in Figure 3-2; the divergence of the LM for a given input change is not
uniform throughout the entire input range; the time series diverge earlier at the
extremities of the input range. This feature of the LM map can be avoided by
reducing the input range to 0.1-0.9 of the normalised input range. Figure 6-8(a) and
(b) show the experimental data of the LM divergence where the whole input range is
represented for an input change of
. The iteration at which divergence occurs
is constant (iteration seven) from 0.1 to 0.9 of the input range, and for input regions
0 to 0.1 and 0.9 to 1, divergence occurs at iteration five. Figure 6-8(a) and (b)
represent the same data with two different angles in order to allow a better
visualisation of the non-linearity at the extremities of the input range. From Figure
6-8(b) it can be seen that the LM present a high degree of linearity of divergence
from 0.1 to 0.9 of the normalized input range. To further demonstrate this Figure
6-8(c) and (b) display the same experimental data but with a reduced input range (0.1
to 0.9).
145
FIGURE 6-8 EXPERIMENTAL DIVERGENCE OF THE LM FOR A CHANGE OF
V: (A) AND (B)
FOR THE ENTIRE INPUT RANGE. (C) AND (D) FOR A RANGE LIMITED TO 0.1-0.9
6.2.2 Tent Map
In section 6.1.2, the measurements in Table 6-2 have shown that the errors
introduced by the feedback in the iterated MS limits the performance of the TM,
therefore the measurement of divergence for the entire input range has been carriedout only on a series configuration. This allows for improved performances by
eliminating noise added by the S/H circuitry.
As shown in Figure 6-9, the divergence for a given input change is consistent
throughout the entire input range which is ideal for a MS. This feature of the TM
previously shown with the aid of computation in Figure 3-3, is a significant
advantage over the behaviour of the LM. The divergence for a 250
146
change is
visible from iteration six and reaches more than 60
at iteration eight. The
measurement for the estimation of the input change should be carried out at iteration
seven or eight as this will be the most appropriate trade-off between estimation
accuracy and probability of reaching a null reduction as explained in section 3.1.
FIGURE 6-9 EXPERIMENTAL DIVERGENCE OF THE TM FOR A 200 V CHANGE FOR THE ENTIRE
INPUT RANGE.
6.3 Divergence Against Input Signal Change
The final measurement carried out on each MS was the divergence against the
amount of change between two input signals. The test procedure is different for each
map as the method used to determine

varies.
LM based MS measurement procedure: the input was set to an arbitrary value
within the linear range of the map (0.1 to 0.9). The time series obtained for the
given input signal was stored then the signal was increased by a given amount
( ) and resultant time series obtained and stored. The divergence between the
two samples was calculated by subtracting the two time series and the
iteration at which the absolute value of the divergence exceeded a set
threshold was recorded and plotted on a graph. This process was repeated for
147
different input changes until a graph of the divergence against the amount of
change was produced.

TM based MS measurement procedure: different points where taken
throughout the entire input range, with a
the error between the measured and real
ranging from 10
to 1
and
value was calculated for each input.
This will allow the sensitivity of the system to be determined.
6.3.1 Logistic Map
Figure 6-10 shows the iteration at which the divergence occurs for a given ; the full
line represents the computed divergence whilst the dotted line shows the practical
results. As the divergence is exponential the x axes is set to a logarithmic scale
which means the “divergence versus change” data forms a straight line. Ideally the
divergence of the practical implementation should follow that same line as the
computed, however the practical results follow the computed divergence line until a
change of approximately 100
is applied. At this point the noise inherent to the
systems makes the time series diverge before the signal change ( ). This result is
consistent with the “noise band” measurement discussed in section 6.1.1, where it
has been shown that the LM is limited to approximately 100
due to the inherent noise of the electronic implementation.
148
of detectable change
11
Measured
Computed
10
Iterations (N)
9
8
7
Measurement limit
6
5
4
3
-5
10
-4
10
Change  (V)
10
-3
FIGURE 6-10 LOGISTIC MAP DIVERGENCE VS CHANGE
6.3.2 Tent Map
The proposed implementation of the TM has shown excellent divergence linearity
throughout the entire input range hence the following measurements will determine
the performance of the series based TM MS across different values of input change
( ). Different points have been taken with
error between the measured and real
ranging from 10
to 1
and the
was calculated for each measurement and
displayed as in Figure 6-11. The measurement error remains negligible for a
1
as the noise of the system is insignificant compared to the
For
values lower than 1
the error starts to rise as the
of
being measured.
being measured is of a
similar magnitude to the inherent noise of the TM implementation. The error remains
below 19% for a
of 20
before rising to over 40% as
which represents a noise floor of approximately 4
149
.
is reduced to 10
60
Measurement Error (%)
50
40
30
20
10
0
-4
10
Change  (Volts)
10
-3
FIGURE 6-11 MEASUREMENT ERROR VS CHANGE FOR THE SERIES IMPLEMENTATION OF THE
TM BASED MS
6.4 Conclusion
In this chapter the LM based MS was tested, followed by the TM MS and the results
obtained were as follows:

LM based system can detect an input signal change of 200
ambiguity and a 100
without
signal change with the use of averaging. The input
range has to be reduced to 0.1- 0.9 to obtain a linear divergence throughout
the input range.

The TM base system demonstrates a greater sensitivity to input signal change
in the series configuration as the noise introduced in the feedback limits the
performance of the MS. The results demonstrate that the MS based on the TM
150
can measure input signal changes of 100
with an error of 3% and 20
with an error of 19% with the use of the series implementation which
represents a sensitivity of 4

With a sensitivity of 4
.
independently of the input range, the voltage step
size of the proposed chaos based MS is superior to an 20 bit ADC over a 10 V
input range.
The next chapter will discuss the results obtained from the MS and the performance
of the proposed implementations.
151
7 Discussion
The theoretical analysis of the one-dimensional chaotic maps demonstrates that the
high sensitivity to initial conditions, which characterises chaotic behaviour, can be
used to detect or measure small variation between signal samples. Two different
approaches in order to quantify the change of input signal for two one-dimensional
chaotic maps have been proposed. For the LM, the magnitude of change can be
quantified by setting thresholds on the amount of divergence; this technique is more
suitable for a detection system that would require only a few thresholds rather than a
measurement system as the thresholds are set experimentally for each
implementation which can be time consuming. For the TM; the equation (3-1),
proposed by the author can be used to measure the difference between two signals
which does not require any calibration making the MS based on the TM superior to
the one utilizing the LM.
The results obtained from the newly developed electronic implementations of the
two one-dimensional chaotic maps, demonstrate a ± 1% matching compared to the
theoretical maps. The practical results show an identical behaviour in all aspects;
transfer characteristics, bifurcation diagrams, and time series. The chaotic
performance of each map was demonstrated by estimating the Lyapunov Exponent
applying the Wolf method available in literature (Wolf et al., 1985).
The experimental results obtained for the proposed MS demonstrate that the LM and
the TM can be used as part of a MS to detect or measure small changes of input
signals. The LM implementation is capable of detecting signals in the order of
hundreds of
whilst the TM can measure signals changes higher than 4
152
. The
sensitivity of the MS is independent of the input range which means that an increase
in the input range does not affect the sensitivity of the measurement. This feature of
the proposed MS makes it superior to the existing alternatives which are based on
linear amplifier coupled to an ADC. In the classical approach to signal measurement,
the sensitivity of the MS is directly proportional to the input range, for instance if the
input range is doubled, the sensitivity of the system is reduced by the same factor. In
general, to palliate the loss of sensitivity, the ADC is replaced by an ADC with
higher resolution which increased the cost of the MS or requires the replacement of
existing equipment. Alternatively, the suggested MS can be used with an input range
as high as permitted by the power supply and components ratings without suffering
any degradation of its performance.
The two topologies for MS investigated in the work were the feedback and series
based system. The feedback system or iterated system uses feedback to iterate the
initial input signal multiple times through the same electronic implementation of the
map. This allows a flexible control over the number of iterations. In contrast the
series implementation is configured so that the signal propagates through multiple
circuit implementation of the same map. From the results for the prototype
implementations using low cost electronics, the series implementation provides
higher performance than the feedback system as the noise produced by the feedback
limits the overall performance of the MS. The feedback error is not an issue when
the LM is used as the inherent noise of the implementation is higher than the errors
introduced by the feedback. To take advantage of the lower noise implementation of
the TM, a MS based on the TM should be constructed using the series topology, the
drawback for improved performance is the increased amount of components and the
reduces flexibility when it comes to setting the number of iterations. The
153
performance of the MS could be further improved by different methods such as
replacing some of the electronics with improved ICs or by integrating multiple maps
on a single IC.
A novel signal measurement system (MS) based on one-dimensional chaotic map
has been proposed for the measurement and detection of small signal changes. This
thesis considers for the first time the electronic hardware implementation of a
complete MS based on chaotic maps and develops the theory associated with such a
system. The developed MS has led to a patent application (n◦ 1309585.4). The
proposed MS is radically different from the classic approach to signal magnitude
measurement as the measurement performed results in a measure of a signal change
rather than an absolute value. Such a system could be used in many applications
including:

Relative value measurement: applications where the absolute value is not of
importance but where a small difference between two signal samples needs to
be measured. The main advantage over a classical approach is that the
detectable change of signal (resolution) is independent of the input range thus
allowing for greater dynamic range/ resolution ratio. For example, a strain
gauge can be used to detect small strain variations over a wide initial signal
range thus effectively increasing the resolution compared to a classic MS
using the same ADC. Time series representing user defined strain thresholds
can be set and the input signal monitored in order to detect when the strain
drifts above or under the fixed limit. This example is not limited to strain
measurement and can be applied to any physical quantity such as temperature
or pressure. The proposed MS could also be used in safety applications where
154
thresholds are set on critical parameters in order to trigger an alarm or stop
machinery when the working conditions are outside the set limits.

Increased data acquisition capability: the proposed MS can be used to
improve the detection capability of existing data acquisition systems. For
example, an acquisition board equipped with an existing low resolution (8-bit
or 12-bit) ADC can be expanded, with the addition the proposed MS, to
obtain signal change detection levels in the order of 20-bits by effectively
increasing the detectable step size of the system. The added advantage is that
the relative resolution of the overall acquisition system will be increased as
well as the approximate absolute measurement values still being available.
The results obtains are very promising and given further development could lead to a
MS that offers significant performance and/or cost advantage over other
measurement techniques for many applications.
155
8 Conclusion and Further Work
8.1 Conclusion
The main conclusions of this thesis are:
a) One-dimensional chaotic maps as a mean of measurement have been
successfully investigated using MATLAB.
b) A novel 3D graphical representation of the high sensitivity to initial condition
of one-dimensional chaotic map has been presented allowing a graphical
visualisation of the phenomena. The developed visual representation in not
limited to the Logistic and Tent maps and can be used to analyse the high
sensitivity to initial conditions of any one-dimensional chaotic map.
c) Two one-dimensional chaotic maps (Logistic map and Tent map) have been
analysed and a novel low noise electronic implementation of the maps has
been developed and presented.
d) A mean of quantifying the change of input signal was proposed for each map
and the advantages and drawback of using each map have been assessed.
e) The proposed implementations have been tested using the transfer
characteristic, bifurcation diagram, time series and Lyapunov Exponent
showing a matching between theory and experimental results. The results
156
demonstrate a matching within
between theory and the practical
implementations.
f) Two measurement systems topologies have been proposed and designed; the
series and the feedback topology. The two topologies where assessed and the
following conclusions were drawn: the feedback topology develops higher
inherent noise levels than the series topology. The series topology requires
more hardware and offers less flexibility than the feedback topology.
g) The developed electronic circuit representing the chaotic maps have been
integrated into a measurement system.
h) The sources of errors associated with the proposed measurement system have
been identified and assessed. Means of mitigating some of the identified
errors have been proposed.
i) The proposed measurement systems have been assessed practically and the
results shown that the small variation of input signals can be measured. The
results demonstrate that the MS based on series implementation are superior
in terms of detectable input signal change compared to a feedback system.
The most favourable performance have been obtained using the TM based
series topology: the measurement system was able to detect signal changes as
low as 4
using a 10bit ADC with a step size (LSB) of 10mV for a 10V
input range.
157
8.2 Further Work
The author of this thesis would like to propose the following further work:

Investigate possible improvement to the electronics implementations of the
TM in order to reduce noise which would increase the sensitivity of the
proposed MS. Moreover, the possibility of implementing the MS using
switched capacitor technique could be investigated; this could possibly lead to
single IC integration thus reducing cost and potentially increase the sensitivity
of the MS by lowering the noise for the feedback system.

Further investigation regarding the behaviour of one dimensional chaotic map.
This could potentially lead to alternative methods of using the high sensitivity
to initial conditions as a mean of measuring small signals.

Investigating the “nulls” that theoretically appear in the behaviour of the TM,
and quantify the effects in the implemented system.

Investigate the possibility of improving the performance of chaos based ADC
using the proposed MS.
158
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165
Appendices
List of Appendices:
Appendix A: Tables of data used to generate graphs in Chapter 3.
Appendix B: MATLAB code developed to generate the 3D divergence graph for the
LM along with divergence graphs for different input signal change.
Appendix C: MATLAB code developed to generate the 3D divergence graph for the
TM along with divergence graphs for different input signal change.
Appendix D: Main C program developed for the PIC32MX460L using MPLAB.
Appendix E: Datasheet of the KROHN-HITE 511 DC voltage reference/calibrator.
Appendix F: Patent files submitted as part of the patent application n◦ 1309585.4.
Appendix G: Full schematics of the LM and TM electronic implementation.
166
Appendix A
The data generate using MATLAB and used to create the graphs for the estimation
error in section 3.1.1(Figure 3-5 , Figure 3-6 and Figure 3-7) and the time series for
two different input signals are shown:
TABLE A-1 SIGNAL CHANGE ESTIMATION ERROR
Iteration
Number (N)
1st Sample
2nd Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.37690
0.7500
0.49740
0.98990
0.02009
0.03998
0.07957
0.15835
0.31513
0.62710
0.74205
0.51331
0.96849
0.06268
0.12475
0.24825
0.49403
0.98312
0.37691
0.75005
0.49739
0.98982
0.02025
0.04030
0.08019
0.15959
0.31758
0.63200
0.73231
0.53269
0.92992
0.13944
0.27748
0.55220
0.89111
0.21668
Divergence
Calculated
using data
rounded to
the 3rd
decimal place
0.00001
0.00002
0.00004
0.00008
0.00016
0.00031
0.00062
0.00124
0.00246
0.00489
0.00974
0.01938
0.03857
0.07675
0.15274
0.30395
0.39708
0.76644
Error (%)
between
the real
and the
calculated
0
0
0
0
0
0
0
8.091E-06
8.132E-06
1.022E-05
1.027E-05
9.803E-06
1.011E-05
1.003E-05
1.002E-05
1E-05
6.564E-06
6.364E-06
100
100
100
100
100
100
100
19.1
18.7
-2.16
-2.68
1.97
-1.12
-0.32
-0.17
-0.02
34.4
36.4
TABLE A-2 IDENTICAL TIME SERIES FOR TWO DIFFERENT INPUT VALUES
Iteration
(N)
0
1
2
3
4
5
6
7
8
9
0.49
0.51
0.98
0.98
0.04
0.04
0.08
0.08
0.16
0.16
0.32
0.32
0.64
0.64
0.72
0.72
0.56
0.56
0.88
0.88
167
Appendix B
MATLAB code developed to generate the 3D divergence graph (Chapter 3) of the
LM is presented. The code was written using MATLAB 7.11 (R2010b). Some
divergence graphs generated using the code are also shown.
Matlab code:
% Clear command window and workspace
clc;
clear;
%********************* setting simulation parameters
******************
% set the parameter of the LM
r=4;
change = 0.0002; % set the change
resolution = 100000; % set the number of points
addnoise = 0; % simulation with noise If 1
averaging = 1;
% set to 1 for no averaging
%
loopvar = 1;
%
n = 1;
averageloop = 0;
%
finalresult = zeros(51,100000); % initialise variables
finalresultsinglerun = zeros(51,100000); %
signatures = zeros(1,resolution); %
signaturesT = zeros(1,resolution); %
if addnoise
noiseresult = zeros(1,resolution*40+1000);
while (n < resolution*40+1000)
% generate noise if
required
x = (mean(randn(100,1))*0.000012);
x = x';
noiseresult(n) = x;
n = n + 1;
end
n=1;
end
%**************************** Main Loop
*******************************
while(loopvar<=(resolution)-1) % set the main loop
a1=[0:50];
n=1;
x= (loopvar/resolution);
for averageloop = 1:averaging
n=1;
168
x = (loopvar/resolution);
while(n<=51)
if addnoise == 1;
%check if noise is to be added
noise = noiseresult(loopvar*20-20+(n*averageloop)); %take
%noise from
%the noise
%matrix
else
noise = 0;
end
x = x+ noise;
a1(averageloop,n) = r*(x)*(1-x); % iterate the equation and
add
% noise if required
x = a1(averageloop,n);
n = n + 1;
end
end
if averaging > 1
% apply averaging if required
a1 = mean (a1);
end
a2 = [0:50];
n=1;
x= ((loopvar/resolution)+change);
for averageloop = 1:averaging
n=1;
x= ((loopvar/resolution)+change);
while(n<=51)
if addnoise == 1;
%check if noise is to be
added
noise
noise = noiseresult(loopvar*20+(n*averageloop));
%from the
%noise
%matrix
else
noise = 0;
end
x = x+ noise;
a2(averageloop,n) = delta*(x)*(1-x);
x = a2(averageloop,n);
n = n + 1;
end
end
if averaging > 1
a2 = mean (a2);
end
%**************** Add the iterations of the divergence
****************
end
n=1;
suma = a1-a2;
suma = suma';
finalresult(:,loopvar) = suma;
finalresultsinglerun (:,loopvar) = a1;
loopvar = loopvar + 1;
end
n=1;
resolution = resolution - 1;
169
%take
surf(finalresult(1:20,1:resolution10),'DisplayName','finalresult(1:20,1:resolution)');figure(gcf)
% generate 3d graph of the divergence
Divergence Graphs for different values of
and input ranges:
LM divergence full normalised range with
= 0.0001
LM divergence 0.1 to 0.9 normalised
range with = 0.0001
LM divergence full normalised range with
= 0.00001
LM divergence 0.1 to 0.9 normalised
range with = 0.00001
170
Appendix C
MATLAB code developed to generate the 3D divergence graph of the TM (Chapter
3) is presented followed by 3D graphs representing the divergence of the TM.
Matlab code:
%clears the command window and the
clc;
clear; %workspace b4 every run
%********************* setting simulation parameters
******************
delta=2;
change = 0.0001;
resolution = 10000;
addnoise =0;
loopvar = 1;
n = 1;
testvariable = 1;
%sets the parameter of the tent map
%sets the change
%number of simulation points
%0 = no noise/ 1 = with noise
%some variables used for the loops
finalresult = zeros(40,200000); %declaration of tables
signatures = zeros(1,resolution);
StartingSignatures = zeros (40,resolution);
StartingSignatures2 = zeros (40,resolution);
%*********** Generate noise and store it in a table (only if
addnoise = 1)****************
if addnoise
noiseresult = zeros(1,resolution*80+1000);
while (n < resolution*80+1000)
x = (mean(randn(100,1))*0.000012);
x = x';
noiseresult(n) = x;
n = n + 1;
end
n=1;
end
%************ Main Loop ********************************************
while(loopvar<=resolution-1)
a1=[0:39];
n=1;
x= (loopvar/resolution);
%sets the input value
while(n<=40)
171
from
if addnoise == 1;
%check if noise should be added
noise = noiseresult(loopvar*20-20+(n));
%take noise
%the noise matrix
else
noise = 0;
end
x = x ;
if x < 0.5
x1 = delta*x+ noise;
x2 = 0;
end
if x >= 0.5
x2 = delta*(1-x)+ noise;
x1 = 0;
end
x = x1+x2;
a1(n) = x;
n = n + 1;
end
a2 = [0:39];
n=1;
x= ((loopvar/resolution)+ change);
while(n<=40)
if addnoise == 1;
%check if noise is to be added
%take noise from
noise = noiseresult(loopvar*20+(n));
%the noise matrix
else
noise = 0;
end
if x < 0.5
x1 = delta*x+ noise;
x2 = 0;
end
if x >= 0.5
x2 = delta*(1-x)+ noise;
x1 = 0;
end
x = x1+x2;
a2(n) = x;
n = n + 1;
end
if loopvar == 3769;
% extract one signature for analysis
signature(:,4) = a1; % loopvar sets the signature to be
analysed
signature(:,5) = a2;
signature(:,6) = a1 - a2;
172
signature(:,7) = abs(signature(:,6));
end
n=1;
StartingSignatures (:,loopvar) = a1;
StartingSignatures2 (:,loopvar) = a2;
suma = a1-a2;
suma = suma';
finalresult(:,loopvar) = suma;
divergence
loopvar = loopvar + 1;
end
%storage of the
%********Add the iterations of the divergence********************
while (n<resolution)
signatures(1,n) = sum (abs(finalresult(1:1,n)));
n=n+1;
end
n=1;
while (n<resolution)
signaturesIT(1,n) = (abs(finalresult(1,n)));
n=n+1;
end
n=1
%***************** create the 3D divergence Graph****************
resolution = resolution - 1;
surf(finalresult(1:20,1:resolution10),'DisplayName','finalresult(1:25,1:resolution)');figure(gcf)
173
Divergence Graphs for different values of :
TM divergence full normalised range with
= 0.001
TM divergence full normalised range
with = 0.0001
TM divergence full normalised range with
= 0.00001
TM divergence full normalised range
with = 0.000001
174
Appendix D
The main C program developed for the PIC32MX460L using MPLAB is given. The
program uses the I/Os to control the chaotic MS, the internal ADC to sample the data
and the USB mass storage capability to store the data on an external USB memory.
A colour LCD display with an integrated touch screen was used, to create a graphical
user interface. The block diagram of the
system is shown below.
USB MASS STORAGE
LCD & TOUCH
SCREEN
I/Os
Switches and
Sample &
holds
ADC
1D Chaotic
Map
PIC32MX460L
Main Program:
//List of includs
#include "MainDemo.h"
#include "MainDemoStrings.h"
#include "USB/usb.h"
#include "USB/usb_host_msd.h"
#include "USB/usb_host_msd_scsi.h"
#include "MDD File System/FSIO.h"
/////////////////////// Configuration bits/////////////////////////////
#pragma config FPLLODIV = DIV_1, FPLLMUL = MUL_20, FPLLIDIV = DIV_2, FWDTEN =
OFF, FCKSM = CSECME, FPBDIV = DIV_8
#pragma config UPLLEN
= ON
// USB PLL Enabled
#pragma config UPLLIDIV = DIV_2
// USB PLL Input Divider
#pragma config OSCIOFNC = ON, POSCMOD = XT, FSOSCEN = ON, FNOSC = PRIPLL
#pragma config CP = OFF, BWP = OFF, PWP = OFF
175
////////////////////////Variable declaration//////////////////////////
SHORT startscreen=0;
SHORT chaosscreen=0;
// alternative yellow style
GOL_SCHEME*
yellowScheme;
scheme
int channel8; // conversion result as read from result buffer
unsigned int channel15;
unsigned int offset; // buffer offset to point to the base of the idle buffer
int x;
int tempo;
int iteration;
int y;
int z;
int testvar;
char set,log=0,RUN=0,column=0;
char temp[10];
int signatur[500];
int NumberOfIterations=50;
int pbClk; // Variable for UART
FSFILE * myFile;
BYTE myData[512];
size_t numBytes;
volatile BOOL deviceAttached;
extern const FONT_FLASH GOLMediumFont; // medium font
extern const FONT_FLASH GOLSmallFont; // small font
void StartScreen();
void ChaosScreen();
// draws intro screen
// draws chaos screen
/////////////////////////////////////////////////////////////////////////////
//
DEMO STATES
/////////////////////////////////////////////////////////////////////////////
typedef enum {
intro,
chaos
} SCREEN_STATES;
/////////////////////////////////////////////////////////////////////////////
//
GLOBAL VARIABLES FOR DEMO
/////////////////////////////////////////////////////////////////////////////
SCREEN_STATES screenState =
intro; // current state of main demo state
machine
SETTIME_STATES settimeState =
defo; // current state of main demo state
machine
//////////////////////////////////////////////////////////////////////////////
//
IMAGES USED
//////////////////////////////////////////////////////////////////////////////
// internal flash image
extern const BITMAP_FLASH batimage;
extern const BITMAP_FLASH bat2;
extern const BITMAP_FLASH sun;
//////////////////////////////////////////////////////////////////////////////
/
//
MAIN PROGRAM
//////////////////////////////////////////////////////////////////////////////
/
int main(void)
{
176
TRISD = 0x0000; ////PORT D set as output for the control of the Chaotic
map***************************
PORTD = 0;
#if defined(__PIC32MX__)
{
int value;
value = SYSTEMConfigWaitStatesAndPB( GetSystemClock() );
// Enable the cache for the best performance
CheKseg0CacheOn();
INTEnableSystemMultiVectoredInt();
value = OSCCON;
while (!(value & 0x00000020))
{
value = OSCCON;
// Wait for PLL lock to stabilize
}
//PORTD=0xffff;
}
#endif
deviceAttached = FALSE;
//Initialize the stack
USBInitialize(0);
// GOL message structure to interact with GOL
GOL_MSG msg;
INTEnableSystemMultiVectoredInt(); // USE best performance
SYSTEMConfigPerformance(GetSystemClock());// get clock and set best
performance
AD1PCFG = 0xccff; // ADC pins configuration
GOLInit();
// initialise graphics library &
// initialise the touch panel
TouchInit();
AD1PCFG = 0xccf8; // reconfigure ADC pins
yellowScheme = GOLCreateScheme(); // create yellow style scheme
yellowScheme->Color0 = BRIGHTYELLOW;
yellowScheme->Color1 = YELLOW;
yellowScheme->EmbossDkColor = RGB565CONVERT(0xFF, 0x94, 0x4C);
yellowScheme->EmbossLtColor = RGB565CONVERT(0xFD, 0xFF, 0xB2);
yellowScheme->TextColor0 = RGB565CONVERT(0xAF, 0x34, 0xF3);
yellowScheme->TextColor1 = RED;
PORTB=0x4000;
//and set it at one to power the USB port (make sure that the
corresponding switch is one (SW12))
StartScreen(); // display the main screen
log=0;
while(1)
{
if (RUN==0)
{
//USB stack process function
USBTasks();
//if thumbdrive is plugged in
if(USBHostMSDSCSIMediaDetect())
177
{
deviceAttached = TRUE;
//now a device is attached
//See if the device is attached and in the right format
if(FSInit())
{
//Opening a file in mode "w" will create the file if it
doesn't
// exist. If the file does exist it will delete the old file
// and create a new one that is blank.
myFile = FSfopen("test.csv","w");
//Write some data to the new file.
FSfwrite("Signature",1,4,myFile);
FSfwrite(",",1,1,myFile);
FSfwrite("Signature",1,2,myFile);
FSfwrite("\r\n",1,4,myFile);
FSfclose(myFile);
RUN=2;
}
}
} // end if (log=0)
if (RUN ==1)
{
//////////////////////////Start of iterations///////////////////
//FeedBack disconnected
PORTDbits.RD8 = 1;
for(tempo=0;tempo<100;tempo++);
//input connected
PORTDbits.RD9 = 0;
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD10 = 1; //Master S&H is sampling
for(tempo=0;tempo<200000;tempo++);
PORTDbits.RD10 = 0; //Master S&H is holding
for(tempo=0;tempo<1000;tempo++);
PORTDbits.RD11 = 1; //Slave S&H is sampling
for(tempo=0;tempo<200000;tempo++);
PORTDbits.RD11 = 0; //Slave S&H is holding
for(tempo=0;tempo<80000;tempo++);
channel8 = ADCGetPot();
signatur[0] = channel8;
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD9 = 1;
//input disconnected
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD8 = 0;
//Feedback connected
for(tempo=0;tempo<100000;tempo++);
////////////////////SETS the Number of Iterations for the Chaotic Map
178
for (iteration=0;iteration<NumberOfIterations;iteration++)
{
PORTDbits.RD10 = 1; //Master S&H is sampling
for(tempo=0;tempo<200000;tempo++);
channel8 = ADCGetPot();
signatur[iteration+1] = channel8;
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD10 = 0; //Master S&H is holding
for(tempo=0;tempo<1000;tempo++);
PORTDbits.RD11 = 1; //Slave S&H is sampling
for(tempo=0;tempo<200000;tempo++);
PORTDbits.RD11 = 0; //Slave S&H is holding
for(tempo=0;tempo<50000;tempo++);
for(tempo=0;tempo<50000;tempo++);
}
//Feedback disconnected
PORTDbits.RD8 = 0;
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD9 = 0;
//input connected
for(tempo=0;tempo<100;tempo++);
PORTDbits.RD15 = 1;
//Discharge the Sample and Hold Capacitors
for(tempo=0;tempo<500000;tempo++);
PORTDbits.RD15 = 0;
//USB stack process function
USBTasks();
if(USBHostMSDSCSIMediaDetect())
{
deviceAttached = TRUE;
//if thumbdrive is plugged in
//now a device is attached
//See if the device is attached and in the
if(FSInit())
{
myFile = FSfopen("data.csv","a");
//right format
FSfwrite("\r\n",1,4,myFile);
for (iteration=0;iteration<NumberOfIterations+1;iteration++)
{
//Write the data on the USB drive
signatur[iteration] = signatur[iteration]*3.22265625;
FSfprintf (myFile, "%#2u",signatur[iteration]);
FSfwrite("\r\n",1,4,myFile);
}
FSfclose(myFile);
RUN=2;
}
}
}
if(GOLDraw()) {
// Draw GOL objects
// Drawing is done here, process messages
TouchGetMsg(&msg);
// Get message from touch screen
179
GOLMsg(&msg);
}
}
}
// END while(1) LooP
// END MAIN
/****************************GOL Msg CallBack
subroutine*************************/
WORD GOLMsgCallback(WORD objMsg, OBJ_HEADER* pObj, GOL_MSG* pMsg)
WORD objectID,ident;
objectID = GetObjID(pObj);
ident = GetObjID(5); //get the static text ID
switch(objectID)
{
case 15 :
screen
if(objMsg == BTN_MSG_PRESSED)
{
screenState = intro;
}
return 1;
// If button pressed go back to main
}
return 1;
case 10 :
if(objMsg == BTN_MSG_PRESSED)
{
screenState = chaos;
//chaos BUTTON IS PRESSED
}
return 1;
case 56 :
//Run BUTTON IS PRESSED
if(objMsg == BTN_MSG_PRESSED)
{
RUN = 1;
}
return 1;
case 60 :
if(objMsg == BTN_MSG_PRESSED)
{
NumberOfIterations++;
//iteration + BUTTON IS PRESSED
}
return 1;
case 61 :
if(objMsg == BTN_MSG_PRESSED)
{
NumberOfIterations--;
// iteration - BUTTON IS PRESSED
}
return 1;
break;
}
180
return 1;
// Process message by default
}
/*********************End GOL Msg CallBack
subrouine***************************/
WORD GOLDrawCallback()/*****GOL Draw CallBack
subroutine***********************/
{
switch(screenState){
case intro:
if (startscreen==0) {StartScreen();} // create window and buttons
x = GOLFindObject(5);
t
DmSetValue(x, voltage);
SetState(x,DM_DRAW);
voltage = (channel8*0.0322265625);
y = GOLFindObject(7);
DmSetValue(y, voltage2);
SetState(y,DM_DRAW);
voltage2 = (sec);
return 1; // draw objects created
case chaos:
if (chaosscreen==0) {ChaosScreen();
}
y = GOLFindObject(62);
DmSetValue(y, NumberOfIterations);
SetState(y,DM_DRAW);
return 1;
}
return 1;
}
/************************End GOL Draw CallBack
subroutine*********************/
void StartScreen()/*************Main Screen
Subroutine***************************/
{ GOLFree();
WndCreate(1, // ID
0,0,GetMaxX(),GetMaxY(), // whole screen dimension
WND_DRAW, // set state to draw all
NULL,
// icon
"Chaos MS tool",
// text
NULL);
// use default GOL scheme
BtnCreate(10,
5, 160, 210, 195,
0,
BTN_DRAW,
NULL,
"Choatic Map",
// object’s ID
// object’s dimension
// radius of the rounded edge
// draw the object after creation
// no bitmap used
// use this text
181
yellowScheme);
// use yellow style scheme
// ID
DmCreate(5,
// dimension
30,60,200,90,
SLD_DRAW, // has frame and centre aligned
// to display the value of voltage
voltage,2,1,
NULL);
// use given scheme
// ID
DmCreate7,
30,90,200,120,
// dimension
SLD_DRAW, // has frame and centre aligned
voltage,2,0,
// to display 078.9
// use given scheme
NULL);
startscreen=1;
chaosscreen=0;
}
/***************************END Main screen
subroutine***************************/
/***************************USB application
handler******************************/
BOOL USB_ApplicationEventHandler( BYTE address, USB_EVENT event, void *data,
DWORD size )
{
switch( event )
{
case EVENT_VBUS_REQUEST_POWER:
// The data pointer points to a byte that represents the amount of power
// requested in mA, divided by two. If the device wants too much power,
// we reject it.
return TRUE;
case EVENT_VBUS_RELEASE_POWER:
// Turn off Vbus power.
// The PIC24F with the Explorer 16 cannot turn off Vbus through software.
//This means that the device was removed
deviceAttached = FALSE;
return TRUE;
break;
case EVENT_HUB_ATTACH:
return TRUE;
break;
case EVENT_UNSUPPORTED_DEVICE:
return TRUE;
break;
case EVENT_CANNOT_ENUMERATE:
***** USB Error - cannot enumerate device
return TRUE;
break;
case EVENT_CLIENT_INIT_ERROR:
182
USB Error - client driver initialization error
return TRUE;
break;
case EVENT_OUT_OF_MEMORY:
USB Error - out of heap memory return TRUE;
break;
case EVENT_UNSPECIFIED_ERROR:
USB Error - unspecified
return TRUE;
break;
// This should never be generated.
default:
break;
}
return FALSE;
}
}
//END OF PROGRAM
183
Appendix E
The datasheet of the KROHN-HITE 511 DC voltage reference/calibrator is
presented.
184
185
Appendix F
This Appendix contains the patent files submitted as part of the patent application.
British Patent Application n◦ 1309585.4
Patent application:
Improved Method of Measuring Signal Change and System
Including the same
The present invention relates to a high precision measurement
system for measuring changes in one or more signals.
Although the following invention relates to measuring signal
changes in relatively low value sensors, the person skilled in the
art will appreciate that the present system can be applied to any
device that generates an output signal.
In conventional measurement systems, low amplitude parameter
change measurement is a challenge due to noise and inherent
measurement system errors. The traditional approach to the
problem is for the measurement system to include a sensor to
convert the physical parameter into a signal, typically a voltage
signal, followed by conditioning circuitry to adapt to the
appropriate input range of a high resolution analogue to digital
converter (ADC).
The practical limitation of accuracy for any measurement is
determined by specific facto rs such as sensor sensitivity,
intrinsic noise and ADC performance in terms of bit resolution
and range.
186
Therefore, small changes in input signal can only be detected
and/or measured using relatively exp ensive and sensitive
equipment.
It is therefore an aim of the present invention to provide an
improved method of measurement that address es the
abovementioned problems.
It is a further aim of the invention to provide a sensor
apparatus or system that address es the abovementioned
problems.
It is a yet further aim of the invention to provide a method of
signal change measurement that address es the abovementioned
problems.
In a first aspect of the invention there is provided a method of
measuring signal change including the steps of:
- performing at least one calculation or iteration step based on
one or more chaos functions on a first input signal, or sample
of a signal, to produce a first iteration value;
- performing a second iteration step by repeating the at least
one calculation or iteration step based on one or more chaos
functions on the first iteration value to produce a second
iteration value;
- performing a third and/or further iteration steps whereby the
previous or earlier iteration value undergoes the at least one
calculation or iteration step based on one or more chaos
functions to generate the third and/or further iteration values;
- storing iteration values generated from the first input signal
or sample of said first input signal ; and
- performing at least one calculation or iteration step based on
one or more chaos functions on at least a second input signal,
187
or sample of a second signal, to produce a first iteration value
from the second input signal;
- performing a second iteration step for the second input signal
by repeating the at least one calculation or iteration step based
on one or more chaos functions on the first iteration value
from the second input signal to produce a second iteration
value;
- performing a third and/or further iteration steps for the
second input signal whereby the previous or earlier iteration
value undergoes the at least one calculation or iteration step
based on one or more chaos functions to generate the third
and/or further iteration values;
- storing iteration result values generated from the second input
signal, or part of said second signal ;
- subtracting one set of iteration values generated from either
the first or second input signal from the corresponding
iteration values generated from the other input signal wherein
the number of iterations before the difference between the
iteration result values increases is proportional to the relative
difference between the first input signal and the second input
signal and/or samples thereof.
Typically the stored iteration values generated from the first
input signals, or portions thereof, form a first sample signature
and/or the stored iteration values generated from the second
input signals, or portions thereof, form a second s ample
signature.
Further typically, implementing the abovementioned method
using electronic circuitry and/or an integrated circuit allows
high detection resolution that is independent of the input
range. As it is the difference between the inputs that is
measured and not the absolute value of each input, relatively
188
low cost sensors can be modified to enable small signal
changes, for example in the region of 20 V, to be accurately
measured.
Preferably the iteration values and/or sample signatures are
converted to digital data or a digital word. Typically the
iteration values are converted to any one or any combination of
digital words, binary codes, reflected binary codes ( Gray codes)
and then stored. Further typically an analogue to digital
converter (ADC) is used to convert the v alues.
In one embodiment the stored digital words for each ite ration
form a signature or data set related to that particular input
signal or sample of said signal.
Typically it is the number of iterations before the two
signatures diverge that is proportional to the relative difference
between the input signals or samples of said signals.
Typically, it is possible to obtaining a higher resolution than
that which can be achieved using a standard, comparably
priced, ADC based system. The ADC, in this system, is not
directly sampling the input signal or data, but rather the data at
the output of the chaotic function. This allows a detection of
changes smaller than if the ADC was connected in a
conventional way to the input, using a linear amplification and
ADC combination.
In one embodiment the one or more chaos functions includes a
one dimensional (1D) discrete chaotic map. Preferably the
Chaos function is a Tent Map as shown below.
Tent Map Equation.
189
{
equation (1)
Where Xn represents the normalised input (typically 0 to 1), r is
the fixed multiplying factor in the range (typically 1.8 to 2)
In one embodiment the chaos function includes the Logistic
Map function.
Logistic Map Equation
Xn+1 = rXn(1 – Xn)
equation (2)
Where Xn and Xn+1 are the current and next input values
respectively. r is the scaling factor set to make the function
chaotic and avoid windows of periodicity , typically between
3.97 and 4.
The fundamental advantage of this signal measurement system,
over typical ADCs, is that the size of the signal change that can
be measured is independent of the range, thus increasing input
signal range increases the overall resolution .
This high precision signal change detection and measurement
system utilises the fundamental characteristic of high sensitivity
to initial conditions, exhibited by th e chaotic function. This
normalised behaviour of the function means that an equivalent
high detection resolution can be achieved that is independent
of input range.
190
Typically the calculations or iteration steps are repeated ten
times. Further typically the calculations are repeated for ten
iterations to form the input signal/sample signature.
In a second aspect of the invention there is provided an
apparatus to measure the difference between a first and at least
a second signal and/or samples of a signal, said apparatus
including circuitry to;
- perform at least one of calculation based on one or more
chaos functions on a first input signal sample , to produce a
first iteration value whereby said apparatus performs an
iteration calculation by repeating the at least one calculation
based on one or more chaos functions on the first iteration
value to produce a second iteration result value, perform a
third and/or further iteration steps whereby the previous or
earlier iteration result value undergoes the at le ast one
calculation based on one or more chaos functions to generate
the third and/or further iteration result values ,
- perform at least one of calculation based on one or more
chaos functions on at least a second input signal sample, to
produce a first iteration value from the second input signal , and
to perform an iteration step for the second input signal by
repeating the at least one calculation based on one or more
chaos functions on the first iteration value from the second
input signal to produce a second iteration result value, perform
a third and/or further iteration steps for the second input
signal whereby the previous or earlier iteration result value
undergoes the at least one calculation based on one or more
chaos functions to generate the sec ond and/or further iteration
result values; and further includes;
memory means to store the iteration result values generated
from the first and second input signals; and
191
- either subtract the iteration result values from the first input
signal sample from the corresponding iteration values generated
from the second input signal sample, or vice versa wherein the
number of iterations before the difference between the
iteration values increases is proportional to the relative
difference between the input samp les.
In one embodiment the iteration steps are performed by
circuits in series rather than repeating the c alculation on the
same circuit.
In one embodiment the data storage and/or subtraction is
performed by a microcontroller. The person skilled in the art
will appreciate that the data storage and/or subtraction steps
can be performed with any device/component capable of data
storage and simple mathematical operations. This includes any
one or any combination of Field Programmable Gate Arrays
(FPGAs), microcontrollers, microprocessors with additional
memory, and specialised Integrated Circuits (IC).
In one embodiment the system is substantially integrated into a
single IC and/or connected to a data acquisition board to store
the signatures and/or perform the calculations using at least
one computer means.
In one embodiment the apparatus is coupled to and/or
integrated with a sensor output. An example of such a sensor
is a strain gauge. Typically the gauge output is connected to
the apparatus signal input.
In an alternative embodiment the apparatus is coupled to
and/or integrated with a data acquisition board.
In a third aspect of the invention there is provided a
measurement system comprising taking a first input sample and
performs calculations based on a chaos function, said
192
calculations are repeated for a number of iterations with the
result of each iteration being converted to a digital reference or
word and stored, the stored digital words, for each iteration,
form a signature or data set related to the input sample, and a
second input sample is taken and iterated, and the data is
stored giving a second signature wherein the signature from the
first sample is subtracted to the signature from the second
sample the result of which is then used to determine the
difference between the two samples as the number of iterations
before the two signatures diverge is proportio nal to the relative
difference.
Specific embodiment of the invention are now described with
reference to the following figures, wherein:
Figure 1 shows a system illustrating the ite rations of a tent map
function;
Figure 2 shows a schematic of a chaos function based
measuring system with feedback in accordance wi th one aspect
of the invention;
Figure 3 shows one embodiment of the implementation of a
tent map in accordance with the invention;
Figure 4 shows one embodiment of the implementation of a
logistic map in accordance with the invention;
Figure 5 shows a graph of determining sample difference in
accordance with the invention;
Figure 6 shows the divergence of the tent map for a 50 µV
signal change across a full normalised input range;
Figure 7 shows the theoretical and practical divergence of the
system versus the input change;
Figure 8 shows a series implementation of the Tent Map
function in accordance with one embodiment of the invention;
193
Figure 9 shows a hybrid series/feedback implementation of the
Tent Map function in accordance with o ne embodiment of the
invention;
Figure 10 shows a schematic representing a strain gauge;
Figure 11 shows a circuit diagram for a Tent Map electronic
circuit implementation.
Figure 12 shows a graph demonstrating sensitivity to initial
condition change of 1x10 - 4 for the Logistic Map function;
Figure 13 shows a graph illustrating signature deviations for
input signals and a 1x10 - 4 change for the Logistic Map;
Figure 14 shows a circuit diagram for a Logistic Map elec tronic
circuit implementation.
Figure 15 shows a graph of measurement system input change
detection compared to theoretical analysis.
Chaos theory is based on functions/systems that have widely
diverging outputs for small differences in initial conditions,
often termed the ‘butterfly effect’. This means that the long term response of the system cannot be predicted with any
degree of certainty due to very small parameter changes.
However, the divergence between two chaotic responses can be
used to accurately define the difference between that two input
signals. The Tent Map (TM) function exhibits such behaviour
and has been investigated and implemente d, electronically, in
this work.
There have been previous implementations of the TM and other
chaotic functions using electronic circuits, however, some of
these implementations are purely educational with no real
194
application. Figure 1 shows a system ill ustration of a TM
function as used in associated literature.
The TM function, along with the Bit Shift/Doubling Map
(BSM), has been used to measure signals in the form of an
Analogue to Digital Convertor (ADC). These methods rely on
the piecewise-linear characteristic of the one dimensional (1D)
maps to double and fold the signal on each iteration. After each
iteration the digital output is shifted to the left until a binary
word that represent the input signal is obtained (binary word
for the BSM and Gray-code for the TM). However, inherent
practical system errors grow exponentially limiting the
resolution. A chaos function scaling factor above the ideal, due
to noise, would cause the output to diverge to either supply rail
- known as the ‘exiting condition ’ or ‘extinction’.
The fundamental aspect of the method, presented here, of
signal measurement is that it relies on the difference between
two input signals rather than the absolute value of a given
signal, which makes it robust to any inaccuracy within th e TM
parameters.
This approach adopted herein is not to design an ADC but to
detect small signal changes using an implementation that takes
into consideration noise and errors encountered within
practical systems. Rather than relying on producing a logic
output of a 0 or 1 after each iteration (or stage) the output is
sampled using a low resolution ADC. The digital word for each
iteration is stored in the system memory so that after N
iterations a unique N Bytes signature is obtained for a given
input. This signature can then be compared with any other
signature to determine the difference between them using the
divergence. Errors introduced by the feedback loop can be
eliminated by implementing the system in a cascaded/series
configuration.
195
The present system takes the 1st input sample and performs
calculations based on the chaos function, as shown in figure 2.
These calculations are repeated , via the feedback loop, for a
number of iterations (typically 10) with the result of each
iteration being converted to a digital word and stored. The
stored digital words, for each iteration, form a signature (data
set) related to the input sample. A second input sample is
taken and iterated, and the data is stored giving a second
signature.
The signature from the first sample is then
subtracted from the second sample signature.
The result
obtained is then used to determine the difference between the
two samples as the number of iterations before the two
signatures diverge is proportional to the relative difference.
The sample difference is measured and not the absolute value.
The data storage and the subtraction are all performed by a
microcontroller.
The discrete, 1D Chaotic map, implemented electronically and
tested, was the Tent Map, containing the basic system
functional blocks, as shown in figure 3
The main advantage, of this system, is the possibility of
obtaining a higher resolution than that which can be achieved
using a standard, comparably priced, AD C based system. The
ADC, in this system, is not directly sampling the input data but
rather the data at the output of the Tent map. This allows a
detection of changes smaller than if the ADC was connected in
a classic way, using a linear amp lification and ADC
combination.
In figure 4 5, a0 represents the Tent ma p behaviour over 20
iterations, for a normalised initial input condition of x = 0.6 , a1
is the behaviour for an initial input condition of x = 0.60005,
a2 is the difference between a0 and a1; showing the divergence.
The number of iterations before a0 diverges from zero enables
the initial sample difference to be determined.
196
Figure 5 6, shows that the number of iterations required, to
measure a specified input signal change, is constant across the
full input signal range. Hence, the measureme nt system is input
signal amplitude independent.
The Tent map is not the only, one dimensional (1D), discrete
chaotic map that can be use in the given system. Successful
implementation of the 1D, discrete Logistic Map (LM)
containing the basic system funct ional blocks, as shown in
figure 6, for the same application, has also been achieved.
However, the LM and other related functions generally require
multiplication circuitry, in the practical implementation. The
multiplication circuitry generates relatively large quantities of
noise, compared to other system blocks used, and thus
introduces high noise levels into the overall system.
Consequently, the induced noise distorts the output and
reduces the sensitivity of the system to small input signal
changes. This means the system output signature divergence
occurs earlier (fewer iterations) than predicted by theory, for a
small input signal change.
The simplicity of the TM (no multiplication circuitry required)
enables basic electronic circuitry to be used, whic h only
introduces relatively low levels of noise into the system. Hence
the achieved performance of the practical implementation is
close to the ideal, mathematically simulated, response of the
TM.
The current discrete component implementation of the syst em
can detect changes in input samples of approximately 50 V,
hence for a 10 V range (typical input voltage range for the
system), the resolution is higher than that which can be
achieved using a conventional 16 -bit resolution ADC (10/216 =
152.6 µV per step). This system requires only a lower
resolution, low cost 8-bit ADC to convert the TM response to a
digital signal for storages and processing.
197
Although the design is implemented using a microcontroller
with a built-in ADC and discrete components, the full system
could be designed on a single IC. This would further reduce
induced noise from circuitry within the system, meaning that
smaller input signal changes could be accurately determined.
Figure 7 shows that the theoretical minimum signal change that
can be measure, by the system, is limited by the noise of the
practical implementation and not the TM function.
By removing the need for feedback the detectable input change
as low as 20μV have been achieved. This was made possible by
eliminating the errors introduced by the feedback, namely the
sample and hold circuit. The circuit without feedback is shown
in Figure 8. Multiple TM circuits have been placed in series so
that the signal, instead of being iterated can propagate through
the circuits. The signature is obtained by sampling between
each TM circuit..
Hybrid series/feedback implementation: Combining the
feedback and series systems into one system, shown in figure 9,
using 2 to 8 series stages and feedback will enable flexibility
and sensitivity to be optimised for different applications. The
feedback system enables flexibility in iterations without the
need to modify the circuitry, whilst the series stages, enables
high levels of sensitivity to be achieved whi lst operating at
higher speeds.
Applications of the invention include:
Small Change Physical Parameter Measurement
A typical area of application for the signal change measuring
system is that where a small variation in the output signal from
a physical parameter sensor is required as opposed to absolute
198
values. For example, a strain gauge can be used to detect small
strain variations over a wide initial signal range. The method
normally used for this application is shown in Figure 10.
The change in strain, to be measured, creates a change of
resistance at the terminals of the strain gauge, which is
relatively small (typically in the order of tens of mΩ) compared
to the typical nominal value of the strain gauge of 350 Ω, for
example. The gauge is placed in a Wheatstone bridge resistor (R
Ω equal to the nominal strain gauge resistance) configuration in
order to convert the strain gauge resistance variation, which is
proportional to the strain applied, into a voltage. The voltage is
then amplified by an instrumentation amplifier (low noise and
high precision) before being adapted/conditioned to the input
range of the ADC. This conditioning circuit is generally
application and ADC dependent and requires relatively high
cost precision amplifiers. However, the system is able to
determine absolute value measurements to high precision levels
but requires expensive high resolution ADCs.
If a small change in the strain, equivalent to an extension of
0.00001% over a 10% extension/compression range, occurs
then the minimum resistance variation of 0.0007 Ω over a range
of ±35 Ω needs to be detected. The maximum voltage at the
output of the Wheatstone bridge in the, given example, was
calculated to be ± 0.119V. Using a 16 -bit ADC the minimum
voltage change that can be detected is determined as being
equal to 76.3 µV for a 5 V reference (5/2 1 6 ). Given the
conditioning circuitry, the minimum ADC step sizes equates to
3.66 µV bridge output voltage (minimum required to maintain
full ADC range), which is equivalent to a 1mΩ change, which in
turn is equivalent to a 1.43 micro strain change. However, since
the measurement accuracy of the new system is 20 µV a strain
change of less than 1 micro strain can be detected. Hence,
because of a need for detecting small changes over a wide
199
absolute range, the 16-bit resolution of the ADC is not high
enough. In fact for the conventional system a more costly 18 bit ADC to attain the same level of accuracy (1 micro strain)
would be required.
Increased Data Acquisition Capability.
Initial investigations indicate that the new sig nal change
measurement system can be used to improve the detection
capability of existing data acquisition systems. For example, an
acquisition board equipped with an existing 8 -bit or 12-bit
ADC can be expanded, with the addition the new system, to
obtain resolution levels in the order of 18 -bits. The added
advantage is that the resolution of the overall acquisition
system will be increased as well as the approximate absolute
measurement values still being available.
The skilled person will appreciate t hat the electronic circuit
implementation of the Tent Map chaotic function, using low
cost components, as shown by the simplified circuit in figure
13 enables signal changes in the region of 20 V to be
accurately measured. The potential range of applicati ons for
this system is very large. The integration of the measurement
systems onto a single IC will improve the level of accuracy and
lower costs, thus increasing the scope of pote ntial applications.
In a final example a high precision signal measurement system
utilising the Logistics Map chaos function is presented.
Essentially a novel, high precision signal change measurement
system based on the Logistic Map (LM) has been developed,
analysed and tested. The measurement technique utilises the
high sensitivity to initial conditions characteristic of the one
dimensional Chaotic function. Investigations into the behaviour
of the LM function, using Matlab, demonstrated that the
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deviation between successive output iterations for two or more
input signal samples is proportional to the size of the
difference between them. An electronic prototype of the LM
based measurement system has been developed, using low cost
electronic devices, and the results demonstrate a strong
relationship to the simulations, thus input signal changes can
be accurately detected and quantified. Analysis of the
measurement system has shown that input signal changes of
100 µV can be determined, equivalent to 16 -bit ADC
resolution, over a 10V input range. The fundamental
characteristic difference compared to typical ADC devices is
that the size of the signal change that can be measured is
independent of the input range, thus increasing the input signal
range increases the resolution. This system is highly suited to
applications where the detection of low amplitude signal sample
change is of higher importance than the absolute value.
Introduction: To accurately observe, test and control any physical
variable, a high resolution ‘measurement system (MS)’ is
required. In most engineering systems, low amplitude parameter
change measurement is challenging due to practical noise
limitations and inherent measurement system errors . In the
classical approach to the problem, the MS consists of a sensor
to convert the physical parameter into typically a vo ltage signal,
followed by conditioning circuitry to adapt to the appropriate
input range of a high resolution ‘analogue to digital converter
(ADC)’. The practical limitation of accuracy for any MS is
determined by specific factors such as sensor sensitivit y,
intrinsic noise and ADC performance in terms of bit resolution
and range. This high precision signal change detection and
measurement system utilises the fundamental characteristic of
high sensitivity to initial conditions, exhibited by the chaotic
function. This normalised behaviour of the function means that
an equivalent high detection resolution can be achieved that is
independent of input range.
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Chaos: Chaotic behaviour can be observed in many non -linear
systems that exhibit irregularity and unpred ictability and show
high sensitivity to initial conditions, commonly known as the
‘butterfly effect’. Although, deterministic and commonly
following simple algorithms, chaotic systems display complex
behaviour, which in contrary to a linear system, the res ultant
divergence between two close starting parameter values is
exponential. This property is thus used to detect small changes
in the initial input conditions taken from a sensor. The simplest
way of investigating this phenomenon is to use a discrete ‘on edimensional (1D)’ chaotic map – ‘logistics map (LM)’.
Logistics map: The LM, is analogous to the logistics equation
created by the mathematician Pierre François Verhulst and
given by the difference equation (2).
The behaviour of one dimensional chaot ic functions has been
widely studied and a number of implementations and
applications have been proposed, over a wide range of different
disciplines, such as optics, communications and electronic
engineering. In a simple electronic implementation of the LM
has successfully been used to design a secure communication
system.
The 1D tent map and dyadic functions have been used to
measure signals in the form of an ADC, where the piecewise linear characteristics are doubled and folded. The binary word
is obtained from each successive iteration (logic 1 above the
threshold and 0 below). These ADC signal measurement
methods would only produce high resolution digital outputs, if
the respective scaling parameter and the threshold are ideal
values. However, inherent practical system errors grow
exponentially limiting the resolution . Furthermore, a scaling
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factor above the ideal, due to noise, would cause the output to
diverge to either supply rail - known as the ‘exiting condition’
or ‘extinction’.
This presented implementation is the first instance of a chaos
function being used for signal change detection and is
insensitive to syste m noise and parameter accuracy.
Signal change measurement technique: Matlab simulations were used
to determine the validity of the measur ement system based on
the 1D LM chaos function. Firstly, the input value X n is applied
to the LM function and the resulting output fed back (iterated)
a number of times, with the corresponding signature (output
value per iteration) stored. A small change i s applied to the
input and the analysis re-run, where the resultant signature of
the second analysis is subtracted from the first to obtain the
difference signature. The iteration point where divergence
occurs between the respective signatures was found to be
proportional to the amplitude of the change in su ccessive input
signal samples.
In figure 12, the input X n was set to 0.6 of the normalised input
range and the map iterated 20 times (a1). The process was
repeated for X n = 0.6001, and iteration signature (a2)
subtracted from (a1) giving the difference (a0). It can be
observed that from iteration 1 to 6 there is no significant
deviation in the two signatures a1 & a2; however divergence
increases for subsequent iterations. By reducing the amount of
deviation between successive input signal samples, it was noted
that the iteration at which divergence occurs is proportional to
the size of the sample difference. Simulations demonstrated
that sample changes of magnitudes 1x10 - 1 2 or less, exhibit
signature divergence proportional to the input sample change.
In order for this signal change, measurement technique to be
valid, two successive samples taken at different times and at
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different amplitudes, but with the same difference should
diverge at the same iteration with the same magnitude range.
To determine the consistency and repeatability of the
technique, a three dimensional simulation graph was developed,
as shown in figure 13, where the signature divergence over the
full normalised input range, given a 1x10 - 4 change, is shown. It
can be observed that the iteration where the divergence is
above a specified threshold is constant in the normalised input
signal range of 0.1 to 0.9. The non -linear regions  0.1 and 
0.9, exceed the threshold at an earlier iterat ion, hence these
input ranges were avoided in the implemented signal
measurement system.
System implementation: The LM function, given in equation 1, was
implemented utilising readily available low cost electronic
components, as shown by the simplified c ircuit in figure 14,
using equation (3); the scaled LM equation for a 10 V input
range.
X n + 1 = rX n (10-X n )
equation (3)
This LM circuit is incorporated into a microcontroller based
feedback system, which utilises a low 8-bit ADC to enable
storage of the signatures but not used to detect the input signal
change. Extensive analysis of the practical system demonstrates
a strong correlation with the Matlab simulations, with the
minimum change that can accurately be de termined in the
practical system, after 6 iteration, being 1x10 - 4 , as shown in
figure 15. The computed and measured results in figure 15,
show the number of iterations where the accumulated deviation
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between successive signatures, due to input signal diffe rence,
exceeds a specified threshold.
The signal change of 100 µV that can currently be detected, is
limited by the practical system induce noise, which is
dominated by the multiplier circuitry.
The theoretical Matlab analysis, shown in figure 15,
demonstrates that by reducing induced signal noise through
improved circuit implementation, signal change detection an
orders of magnitude lower is possible. Further improvements in
the multiplier design and increased signal range, should enable
signal changes in the region of < 1x10 - 5 – 20-bit resolution, to
be achievable.
Conclusions:
A signal change measurement system based on the 1D LM
chaos function has been successfully developed using low cost
electronic devices. The system can accurately and consistently
measure signal changes of 100 µV in the region of 1 V to 9 V
of a 10 V input range, equating to a 16 -bit ADC resolution. The
fundamental advantage of this signal measurement system, over
typical ADCs, is that the size of the signal change that can be
measured is independent of the range, thus increasing input
signal range increases the overall resolution.
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Figures for the patent application:
Output
Tent Map
Feedback for
Iteration
Control
Figure 1
Chaos Function
Input
Low resolution
ADC
Switch
Feedback
Control Circuitry
Figure 2
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Signature
Storage
Tent Map
Half Wave
Rectifier
Level Shift
Input
∑
Gain
Figure 3
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Output
Figure 4
Figure 5
208
Logistic Map
Amplifier/
Attenuator
Output
Input
Multiplier
Subtractor
Reference
Figure 6
Figure 7
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Input
Signal
Tent Map
Tent Map
Tent Map
Tent Map
Multiplexer
Low Resolution ADC
Signature Storage
Control Circuitry
Figure 8
Logistic Map
Logistic Map
Logistic Map
Low resolution
ADC
Input
Switch
Feedback
Control Circuitry
Figure 9
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Signature
Storage
R
Strain
Gauge
R
Instrumentation
Amplifier
Signal Conditioning
Circuitry (differential
output)
High Resolution ADC
Eg. 16-Bit ±5V CMOS
ΣΔ ADC
R
Digital
Output
High Precision Bandgap
Voltage Reference
Figure 10
Figure 11
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Figure 12
Figure 13
212
Figure 14
Figure 15
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Appendix G
The full schematics of the LM and the TM are shown, starting with the LM:
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Schematic of the TM:
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Appendix H
The component selection process is shown.
Operational amplifier:
The OP27 was selected for the low input noise characteristic compared to FET input
op-amps, the low voltage drift and high availability; the OP27 is easily sourced from
any component supplier and is available in most labs. The table below shows the
comparison of key parameter against another commonly found op-amp; the TL071.
Parameter
OP27
TL071
Input offset Voltage
10 µV
3 mV
3 nV/√Hz
18 nV/√Hz
0.2 μV/°C
18 μV/°C
Input Noise at
1kHz
Temperature
Voltage Drift
Analogue Multiplier:
Analogue multipliers are seldom used which limits their availability compared to
other components such as op-amps. Besides being easy to find the AD633JN has
similar performance compared to other analogue multipliers.
Voltage references:
The voltage sources used for the electronic implementations of the chaotic maps are
the ADR130 and the AD587. The two main criteria of selection were; low noise and
high precision. The ADR130 and AD587 have a typical output noise of only 3 μV
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and 4 μV respectively which is low compared to other voltage reference ICs.
Additionally, both voltage references possess an optional noise reduction mechanism
allowing for increased noise performance by the use of an external capacitor. Low
noise performance combined with high accuracy (±0.35% for the AD130 and
±0.05% for the AD587) makes the two voltage sources ideal for chaos map
implementations.
Sample and Hold:
The sample and hold (S/H) LF198 ICs were selected for their performance and
availability. Some of the characteristics are;
-
Low gain error (0.0002%)
-
Low output noise
To increase the performance of the feedback system, polypropylene capacitors were
used to reduce the error during sampling along with reset MOSFETS to reduce
dialectic absorption errors.
Microcontroller:
The microcontroller used is the PIC32MX460L. The choice was mainly guided by
practicability as a development board with the PIC32MX460L was already available
in the laboratory. More information about the development board can be found in
Appendix D. Any microcontroller with the following characteristics could be used
instead of the PIC32MX460L to implement the proposed Measurement System
(MS):
-
At least 4 digital outputs for the feedback and switching control
-
An internal low resolution ADC
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-
Internal RAM to store signal signatures
Additionally the microcontroller could have a user interface and external data
storage if required.
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