chaos and communication.book.chapter

8/8/2019 Chaos and Communication.book.Chapter

http://slidepdf.com/reader/full/chaos-and-communicationbookchapter 1/26

Chapter 1 Chaos and Communications

Traditionally, signals (encompassing desired signals as well as interfering

signals) have been partitioned into two broadly defined classes, i.e., stochastic

and deterministic. Stochastic signals are compositions of random waveforms

with each component being defined by an underlying probability distribution,

whereas deterministic signals are resulted from deterministic dynamical

systems which can produce a number of different steady state behaviors

including DC, periodic, and chaotic solutions. Deterministic signals may be

described mathematically by differential or difference equations, depending

on whether they evolve in continuous or discrete-time.

DC is a nonoscillatory state. Periodic behavior is the simplest type of

steady state oscillatory motion. Sinusoidal signals, which are universally

used as carriers in analog and digital communication systems, are periodic

solutions of continuous-time deterministic dynamical systems.

Deterministic dynamical systems also admit a class of nonperiodic signals,

which are characterized by a continuous "noiselike" broad power spectrum.

This is called chaos.

Historically, at least three achievements were fundamental to the acceptance

of communication using chaos as a field worthy of attention and exploitation.

The first was the implementation and characterization of several electronic

circuits exhibiting chaotic behavior in early 1980's. This brought chaotic

systems from mathematical abstraction into application in electronic

engineering.

The second historical event in the path leading to exploitation for chaos

based communication was the observation make by Pecora and Carroll in

1990 that two chaotic systems can synchronize under suitable coupling or

driving conditions. This suggested that chaotic signals could be used for



1.1 Historical Account

communication, where their noise like broadband nature could Improve

disturbance rejection as well as security.

A third, and fundamental, step was the awareness of the nonlinear (chaos)community that chaotic systems enjoy a mixed deterministic / stochastic

nature [1 - 4]. This had been known to mathematicians since at least the

early 1970's, and advanced methods from that theory have been recently

incorporated in the tools of chaos-based communication engineering. These

tools were also of paramount importance in developing the quantitative

models needed to design chaotic systems that comply with the usual

engineering specifications.

The aim of this chapter is to give a brief review of the background theory

for chaos-based communications. Based on several dynamical invariants, we

will quantitatively describe the chaotic systems, and summarize the

fundamental properties of chaos that make it useful in serving as a spread

spectrum carrier for communication applications. Furthermore, chaotic

synchronization makes it possible for chaos-based communication using theconventional coherent approach. In the remaining part of this chapter,

several fundamental chaotic synchronization schemes, and several chaos

based communication schemes will be reviewed. Finally, some open issues

for chaos-based communications will be discussed.

1.1 Historical Account

In 1831, Faraday studied shallow water waves I I I a container vibrating

vertically with a given frequency OJ. In the experiment, he observed the

sudden appearance of sub harmonic motion at half the vibrating frequency

(OJ / 2) under certain conditions. This experiment was later repeated by Lord

Rayleigh who discussed this experiment in the classic paper Theory of

Sound, published in 1877. This experiment has been repeatedly studied since

1960's. The reason why researchers have returned to this experiment is that

the sudden appearance of sub harmonic motion often prophesies the prelude

to chaos.

2



Chapter 1 Chaos and Commnnications

Poincare discovered what is today known as homo clinic trajectories in

the state space. In 1892, this was published in his three-volume work on

Celestial Mechanics. Only in 1962 did Smale prove that Poincare's homoc

linic trajectories are chaotic limit sets [5].

In 1927, Van der Pol and Van der Mark studied the behavior of a neon

bulb RC oscillator driven by a sinusoidal voltage source [6]. They discovered

that by increasing the capacitance in the circuit, sudden jumps from the

drive frequency, say w to w/2, then to w/3, etc., occurred in the response.

These frequency jumps were observed, or more accurately heard, with a

telephone receiver. They found that this process of frequency demul

tiplication (as they called it) eventually led to irregular noise. In fact, what

they observed, in today's language, turned out to be caused by bifurcations

and chaos. In 1944, Levinson conjectured that Birkhoffs remarkable curves

might occur in the behavior of some third-order systems. This conjecture

was answered affirmatively in 1949 by Levinson [7].Birkhoff proved his famous Ergodic Theorem in 1931 [8]. He also

discovered what he termed remarkable curves or thick curves, which were

also studied by Charpentier in 1935 [9]. Later, these turned out to be a chaotic

attractor of a discrete-time system. These curves have also been found to be

fractal with dimension between 1 and 2.

In 1936, Chaundy and Phillips [10] studied the convergence of sequences

defined by quadratic recurrence formulae. Essentially, they investigated the

logistic map. They introduced the terminology that a sequence oscillates

irrationally. Today this is known as chaotic oscillation.

Inspired by the discovery made by Van der Pol and Van der Mark, two

mathematicians, Cartwright and Littlewood [11] embarked on a theoretical

study of the system studied earlier by Van der Pol and Van der Mark. In

1945, they published a proof of the result that the driven Van der Pol systemcan exhibit nonperiodic solutions. Later, Levinson [7] referred to these

solutions as singular behavior.

Melnikov [12] introduced his perturbation method for chaotic systems in

1963. This method is mainly applied to driven dynamical systems.

3



1.1 Historical Accouut

In 1963, Lorenz [13], a meteorologist, studied a simplified model for

thermal convection numerically. The model (today called the Lorenz model)

consisted of a completely deterministic system of three nonlinearly coupled

ordinary differential equations. He discovered that this simple deterministic

system exhibited irregular fluctuations in its response without any external

element of randomness being introduced into the system.

Cook and Roberts [14] discovered chaotic behavior exhibited by the

Rikitake two-disc dynamo system in 1970. This is a model for the time

evolutionof

the earth magnetic field.In 1971, Ruelle and Takens [15] introduced the term strange attractor for

dissipative dynamical systems. They also proposed a new mechanism for the

on-set of turbulence in the dynamics offluids.

It was in 1975 that chaos was formally defined for one-dimensional

transformations by Li and Yorke [16]. They went further and presented

sufficient conditions for so-called Li -Yorke chaos to be exhibited by a

certain class of one-dimensional mappings. In 1976, May called attention to

the very complicated behavior which included period-doubling bifurcations

and chaos exhibited by some very simple population models [17]. In

1978, Feigenbaum discovered scaling properties and universal constants

(Feigenhaum's number) in one-dimensional mappings [18]. Thereafter, the

idea of a renormalization group was introduced for studying chaotic systems.

In 1980, Packard et al. [19] introduced the technique of state-space reconstruction using the so-called delay coordinates. This technique was later

placed on a firm mathematical foundation by Takens. In 1983, Chua [20]

discovered a simple electronic circuit for synthesizing the specific third-order

piecewise-linear ordinary differential equations. This circuit became known

as Chua's circuit (see the following chapter). What makes this circuit so

remarkable is that its dynamical equations have been proven to exhibit chaos

in a rigorous sense. Ott, Grebogi and Yorke, in 1990 [21], presented a method

for controlling unstable trajectories embedded in a chaotic attractor.

At the same time, there was another course of events leading to the field

of chaos. This was the study ofnonintegrable Hamiltonian systems in classical

4




mechanics. Research in this field has led to the formulation and proof of the

Kolmogorov-Amold-Moser (KAM) theorem in the early 1960's. Numerical

studies have shown that when the conditions stated by the KAM theorem

fails, then stochastic behavior is exhibited by nonintegrable Hamiltonian

systems.

Remarks: Today, chaos has been discovered in bio-systems, meteorology,

cosmology, economics, population dynamics, chemistry, physics, mechanical

and electrical engineering, and many other areas. The research direction has

been transferring from fmding the evidence of chaos existence into applicationsand deep theoretical study.

1.2 Chaos

There are many possible definitions of chaos for dynamical systems, among

which Devaney's definition (for discrete-time systems) is a very popular onebecause it applies to a large number of important examples.

Theorem 1.1 [22]

Let Q be a set.f: Q---+ Q is said to be chaotic on Q if:

(1) fhas sensitive dependence on initial conditions, i.e., there exists (» 0

such that, for any x EQ and any neighborhood U of x, there exists Y E U and

n?oOsuch that

IF(x)-F(Y)1 >0;

(2) f is topological transitive, i.e., for any pair of open sets V, We Q

there exists k> 0 such that fk (V) nw :j:. 0;

(3) periodic points are dense in Q.

1.3 Quantifying Chaotic Behavior

Lyapunov exponents, entropy and dimensionality are usually used to quantify

(characterize) a chaotic attractor's behavior. Lyapunov exponents indicate

the average rates of convergence and divergence of a chaotic atiractor in the

state space. Kolmogorov Entropy (KE) is used to reveal the rate of information

5




loss along the attractor. Dimensionality is used to quantify the geometric

structure of a chaotic attractor.

1.3.1 Lyapunov Exponents for Continuous-Time Nonlinear Systems

The determination of Lyapunov exponents is important in the analysis of a

possibly chaotic system, since Lyapunov exponents not only show qualita

tively the sensitive dependence on initial conditions but also give a quantitative

measure of the average rate of separation or attraction of nearby trajectories

on the attractor. Here, we only state the definitions of Lyapunov exponents

for continuous-time and discrete-time (see next subsection) nonlinear systems.

More detailed descriptions of various algorithms for calculating Lyapunov

exponents can be found in references [23 - 33].

In the direction of stretching, two nearby trajectories diverge; while in the

directions of squeezing, nearby trajectories converge. Ifwe approximate thisdivergence and convergence by exponential functions, the rates of stretching

and squeezing would be quantified by the exponents. These are called the

Lyapunov exponents. Since the exponents vary over the state space, one has

to take the long time average exponential rates of divergence (or convergence)

of nearby orbits.

The total number of Lyapunov exponents is equal to the degree of freedom

of the system. If the system trajectories have at least one positive Lyapunov

exponent, then those trajectories are either unstable or chaotic. If the

trajectories are bounded and have positive Lyapunov exponents, the system

definitely includes chaotic behavior. The larger the positive exponent is, the

shorter the predictable time scale of system. The estimation of the largest

exponent therefore assumes a special importance.

For a given continuous-time dynamical system in an m-dimensional phasespace, we monitor the long-term evolution of an infinitesimal m-sphere of

initial conditions. The sphere will evolve into an m-ellipsoid due to the locally

6




deforming nature of the flow. The ith one-dimensional Lyapunov exponent

can be defined in terms of the length of the ellipsoidal principal axis Ii (I):

A; = lim!lnI/JI) I,Hoo 1 1;(0)

(1.1)

whenever the limit exists [24,25]. Thus, the Lyapunov exponents are related

to the expanding or the contracting nature of the principal axes in phase

space. A positive Lyapunov exponent describes an exponentially increasing

separation of nearby trajectories in a certain direction. This property, in tum,

leads to the sensitive dependence of the dynamics on the initial conditions,

which is a necessary condition for chaotic behavior. Since the orientation of

the ellipsoid changes continuously as it evolves, the directions associated

with a given exponent vary in a complicated way through the attractor. We

cannot therefore speak of a well defined direction associated with a given

exponent.For systems whose equations of motion are known explicitly, Benettin el

al. [25] have proposed a straightforward technique for computing the complete

Lyapunov spectrum. This method can be described in principle as follows.

Let an m-dimensional compact manifold M be the state space of a

dynamical system. The system on M is a nonlinear differentiable map ([J:

M ---+M, which can be conveniently described by the following difference

equation:

x(n) = ([J(x(n -1» = ([In (x(o». (1.2)

Let v(O) denote an initial perturbation of a generic point x(O), and & be a

sufficiently small constant. Consider the separation of trajectories of the

unperturbed and perturbed points after n iterations:

II ([In (x(O» - ([In (x(O) +&v(O» II = II D([Jn (x(O»v(O)& II 0(&2)= II(a([Jn (x(k» }(O)&II+0(&2),

(1.3)

7



1.3 Quautifying Chaotic Behavior

where D<p(x(k)) is the Jacobian, i.e., the mx m matrix of partial derivatives

of <P evaluated at the point x(k), 0(&2) is the higher order term, and 11'11

represents the Euclidean norm. Let the distinct eigenvalues D<P(x(k)) bedenoted by {d

i

k: k =0"" , n -1; i =1,' " , m}. Then the Lyapunov exponents of

the dynamical system are defined by

(1.4)

Unfortunately, this method cannot be applied directly to experimental data,

since we do not usually know the underlying dynamical equations. For

experimental data, one has to resort to Eq. (1.1).

1.3.2 Lyapunov Exponent for Discrete-Time Systems

Consider any initial condition xo, and let {Xk } ; ~ o be the corresponding orbit

of a p-dimensional, discrete-time map 1Jf. Let mI(k), m2(k),'" , mik) be the

eigenvalues of DlJfk(xo). The ith one-dimensional Lyapunov exponent of IJf

with respect to Xo can be defined as

(1.5)

whenever the limit exists [29].

1.3.3 Kolmogorov Entropy

The Kolmogorov entropy (KE), also known as metric entropy or Kolmogorov

Sinai entropy, is also an important measure by which a chaotic motion in an

arbitrary dimensional phase space can be characterized (quantified) [34 - 36].

The Kolmogorov entropy of an attractor can be considered as a measure of

the rate of information loss along the attractor or as a measure of the degree

of predictability ofpoints along the attractor, given an arbitrary initial point.

8




Consider the trajectory of a dynamical system on a strange attractor and

divide the phase space into L-dimensional hypercubes of volume SL. Let

P;o,ij,.,i, be the probability that a trajectory is in hypercube io at t= 0, i 1 at t= T,

i2 at t= 2T, and so on. Then the quantity

Kn= - I , P;o,ij, ·.,i, In(P;o,ij, .., i) (1.6)io,i1 .. ··,in

is proportional to the information needed to locate the system on a special

trajectory with precision s [37, 38]. Therefore, KN+1 - KN is the additional infor

mation needed to predict which cube the trajectory will be in at (n + 1 T,

given trajectories up to nT. This means that KN+1 - KN measures the loss of

information about the system from time instant n to the next n+ 1. The KE

is then defined as the average rate of information loss in the following way:

1 N -l

K=lim lim lim -I,(Kn+1 -KJ.T -->0 &-->0+ N -->00 NT n=O (1.7)

The order in which the limits in the above expression are taken is immaterial.

The limit s ----> 0 makes KE independent of the particular partition. The main

properties ofKE are as follows:

(1) The entropy K (averaged) determines the rate of change in infor

mation entropy (i.e., Eq. (1.6)) as a result of a purely dynamical process of

mixing of trajectories in phase space.

(2) The entropy is a metric invariant of the system, i.e., its value being

independent of the way that the phase space is divided into cells and coarsened.

(3) Systems with identical values of entropy are in a certain sense

isomorphic to each other [39, 40], i.e., these systems must have identical

statistical laws of motion.

(4) When applied to prediction, KE can be interpreted as the average rateat which the accuracy of a prediction decays as prediction time increases,

i.e., the rate at which predictability will be lost [37].

9




1.3.4 Attractor Dimension

Long-term chaotic motion in dissipative dynamical systems is confined to astrange attractor whose geometric structure is invariant to the evolution of

the dynamics. Typically, a strange attractor is a fractal object and, consequently,

there are many possible notions of the dimension for strange attractor. Here,

we discuss some well-known and widely accepted definitions of attractor

dimension. We also discuss the simple relationships to Lyapunov exponents

and entropy.

Dissipative chaotic systems are typically ergodic. All initial conditions

within the system's basin of attraction lead to a chaotic attractor in the state

space, which can be associated with a time-invariant probability measure.

Intuitively, the dimension of the chaotic attractor should reflect the amount

of information required to specify a location on the attractor with a certain

precision. This intuition is formalized by defining the information dimension,

d" of the chaotic attractor as

d-1· Inp[Bx (&)]

[- 1m ,<-->0 In&

(1.8)

where p[Bx(&)] denotes the mass of the measure p contained in a ball of

radius & centered at the point x in the state space [2, 41]. Information dimen

sion is important from an experimental viewpoint because it is straight

forward to estimate. The mass, p[Bx(&)] ' can be estimated by

(1.9)

where U(-) is the Heaviside function, and M is the number of points in the

phase space. In typical experiments, the state vector x is estimated from a

delay embedding of an observed time-series [42].

The information dimension as defmed above, however, depends on the

particular point x in the state space being considered. Grassberger and

Procaccia's approach eliminates this dependence [43] by defming the quantity

10




(1.10)

and then defining the correlation dimension de as

de = lim In C(l').B--+O Inl'

(1.11 )

In practice, one usually plots In C( £ ) as a function of In £ and then measures

the slope of the curve to obtain an estimate of de [44 - 66]. It is often the

case that d, and de are approximately equal.

The box-counting dimension (capacity), do, of a set is defined as

d-1· 1n.ll/(c)

0- 1m ,

&-->0 1n(l/c)(1.12)

where .11/(.0) is the minimum number of N-dimensional cube of side length

£ needed to cover the set.

There is a meaningful relationship between the Lyapunov dimension and

Lyapunov exponents for chaotic systems [41, 67, 68]. If AI, ... , AN are the

Lyapunov exponents of a chaotic system, then the Lyapunov dimension, dL ,

is defined as

(1.13)

where k = max {i : + ... + Ai > O}. It has been shown that do ?cd" d , ~ d c [43, 69].

Equation (1.14) suggests that only the first k +1 Lyapunov exponents are

important for specifying the dimensionality of the chaotic attractor. Kaplan et

al. [67, 68] conjectured that d, = dL in "almost" all cases. Clearly, if this is correct,

then Eq. (1.13) provides a straightforward way to estimate the attractor

dimension when the dynamical equations ofmotion are known.

The relation between KE and Lyapunov exponents is also available. In

one-dimensional maps, KE is just the Lyapunov exponent [70]. In higher

dimensional systems, we lose information about the system because the cell

in which it was previously located spreads over new cells in phase space at a

11



1.4 Properties of Chaos

rate determined by the Lyapunov exponents. The rate K at which the

information about the system is lost is equal to the (averaged) sum of the

positive Lyapunov exponents [71], as shown by

(1.14)

where A's are the positive Lyapunov exponents of the dynamical system

being considered.

1.4 Properties of Chaos

It is now well-known that a deterministic dynamical system is one whose

state evolves with time according to a deterministic evolution rule. The time

evolution of the state is completely determined by the initial state of the

system, the input, and the rule. For example, the state of a digital filter is

determined by the initial state of the filter, the input, and a difference

equation which describes the evolution of the state from one time instant to

the next.

In contrast to a stochastic dynamical system, which may follow any number

of different trajectories from a given state according to some probabilistic

rule, trajectories of a deterministic dynamical system are unique. From any

given state, there is only one "next" state. Therefore, the same system started

twice from the same initial state with the same input will follow precisely

the same trajectory through the state space. Deterministic dynamical systems

can produce a variety of steady-state behaviors, the most familiar of which

are stationary, periodic, and quasi-periodic solutions. These solutions are

"predictable" in the sense that a small piece of a trajectory enables one to

predict the future behavior along that trajectory. Chaos refers to solutions of

deterministic dynamical systems which, while predictable in the short-term,

exhibit long-term unpredictability.

Since the initial state, input, and rule uniquely determine the behavior of a

deterministic dynamical system, it is not obvious that any "unpredictability"

12




is possible. Long-term unpredictability arises because the dynamics of a

chaotic system persistently amplifies errors in specifying the state. Thus,

two trajectories starting from nearby initial conditions quickly become

uncorrelated. This is because in a chaotic system, the precision with which

the initial conditions must be specified in order to predict the behavior over

some specified time interval grows exponentially with the length of the

prediction interval. As a result, long-term prediction becomes impossible.

This long-term unpredictability manifests itself in the frequency domain as a

continuous power spectrum, and in the time domain as random "noiselike"

signal.

To get a better idea ofwhat chaos is, here is a list of its main characteristics

(properties) :

(1) Chaos results from a deterministic nonlinear process.

(2) The motion looks disorganized and erratic, although sustained. In fact,

it can usually pass all statistical tests for randorrmess (thereby we cannotdistinguish chaotic data from random data easily), and has an invariant

probability distribution. The Fourier spectrum (power spectrum) is "broad"

(noiselike) but with some periodicities sticking up here and there [72, 73].

(3) Details of the chaotic behavior are hypersensitive to changes in initial

conditions (minor changes in the starting values of the variables). Equivalently,

chaotic signals rapidly decorrelate with themselves. The autocorrelation

function of a chaotic signal has a large peak at zero and decays rapidly.

(4) It can result from relatively simple systems. In nonautonomous system,

chaos can take place even when the system has only one state variable. In

autonomous systems, it can happen with as few as three state variables.

(5) For given conditions or control parameters, chaos is entirely self

generated. In other words, changes in other (i.e., external) variables or

parameters are not necessary.(6) It is not the result of data inaccuracies, such as sampling error or

measurement error. Any specific initial conditions (right or wrong), as long

as the control parameter is within an appropriate range, can lead to chaos.

(7) In spite of its disjointed appearance, chaos includes one or more types

13



1.5 Chaos-Based Commuuicatious

of order or structure. The phase space trajectory may have fractal property

(self-similarity).

(8) The ranges of the variables have finite bounds, which restrict the

attractor to a certain finite region in the phase space.

(9) Forecasts of long-term behavior are meaningless. The reasons are

sensitivity to initial conditions and the impossibility of measuring a variable

to absolute accuracy. Short-term predictions, however, can be relatively

accurate.

(10) As a control parameter changes systematically, an initially non

chaotic system follows one of a few typical scenarios, called routes to chaos.

1.5 Chaos-Based Communications

1.5.1 Conventional Spread Spectrum

In recent years, there has been explosive growth in personal communica

tions, the aim of which is to guarantee the availability of voice and/or data

services between mobile communication terminals. In order to provide these

services, radio links are required for a large number of compact terminals in

densely populated areas. As a result, there is a need to provide high

frequency, low-power, low-voltage circuitry. The huge demand for com

munications results in a large number of users; therefore, today's com

munication systems are limited primarily by interference from other users.

In some applications, the efficient use of available bandwidth is extremely

important, but in other applications, where the exploitation of communica

tion channels is relatively low, a wideband communication technique having

limited bandwidth effciency can also be used.

Often, many users must be provided with simultaneous access to the sameor neighboring frequency bands. The optimum strategy in this situation, where

every user appears as interference to every other user, is for each communi

cator's signal to look like white noise which is as wideband as possible.

14




There are two ways in which a communicator's signal can be made to

appear like wideband noise:

(1) spreading each symbol using a pseudo-random sequence to increase

the bandwidth of the transmitted signal;

(2) representing each symbol by a piece of "noiselike" waveform [74].

The conventional solution to this problem is the first approach: to use a

synchronizable pseudo-random sequence to distribute the energy of the

information signal over a much larger bandwidth to transmit the baseband

information. The transmitted signal appears similar to noise and is therefore

diffcult to detect by eavesdroppers. In addition, the signal is difficult to jam

because its power spectral density is low. By using orthogonal spreading

sequences, multiple users may communicate simultaneously on the same

channel, which is termed Direct Sequence Code Division Multiple Access

(DS/CDMA). Therefore, the conventional solution can:

(1) combat the effects of interference due to jamming, other users, and

multipath effects;

(2) hide a signal "in the noise" by transmitting it at low power; and

(3) have some message privacy in the presence of eavesdroppers.

With rapidly increasing requirements for some new communication

services, such as wideband data and video, which are much more spectrum

intensive than voice service, communication networks are already reaching

their available resource limitation. Some intrinsic shortcomings of the

convenient DS/CDMA have been known. For example, the periodic nature

of the spreading sequences, the limited number of available orthogonal

sequences, and the periodic nature of the carrier, are imposed to DS/CDMA

systems in order to achieve and maintain carrier and symbol synchronization.

One further problem is that the orthogonality of the spreading sequences

requires the synchronization of all spreading sequences used in the samefrequency band, i.e., the whole system must be synchronized. Due to

different propagation times for different users, perfect synchronization can

never be achieved in real systems [75]. In addition, DS/CDMA systems

using binary spreading sequences do not provide much protection against

15




two particular interception methods: the carrier regeneration and the code

clock regeneration detectors [76]. This is due to the binary nature of the

spreading sequences used in binary waveforms.

The intrinsic properties of chaotic signals stated previously provide an

alternative approach to making a transmission "noiselike". Specifically, the

transmitted symbols are not represented as weighted sums of periodic basis

functions but as inherently nonperiodic chaotic signals, which will be

described in the following subsections.

1.5.2 Spread Spectrum with Chaos

The properties of chaotic signals resemble in many ways those of the stochastic

ones. Chaotic signals also possess a deterministic nature, which makes it

possible to generate "noiselike" chaotic signals in a theoretically reproducible

manner. Therefore, a pseudo-random sequence generator is a "practical"

case of a chaotic system, the principal difference being that the chaotic

system has an infinite number of (analog) states, while pseudo-random

generator has a finite number (of digital states). A pseudo-random sequence

is produced by visiting each state of the system once in a deterministic

manner. With only a finite number of states to visit, the output sequence is

necessarily periodic. By contrast, an analog chaos generator can visit an

infinite number of states in a deterministic manner and therefore produces

an output sequence, which never repeats itself. With appropriate modulation

and demodulation techniques, the "random" nature and "noiselike" spectral

properties of chaotic electronic circuits can be used to provide simultaneous

spreading and modulation of a transmission.

1.5.3 Chaotic Synchronization

How then would one use a chaotic signal in communication? A first

approach would be to hide a message in a chaotic carrier and then extract it

16




by some nonlinear, dynamical means at the receiver. If we do this in real

time, we immediately lead to the requirement that somehow the receiver

must have a duplicate of the transmitter's chaotic signal or, better yet,

synchronize with the transmitter. In fact, synchronization is a requirement of

many types of communication, not only chaotic possibilities.

Early work on synchronous coupled chaotic systems was done by Yamada

and Fujisaka [77, 78]. In that work, some sense of how the dynamics might

change was brought out by a study of the Lyapunov exponents of

synchronized coupled systems. Later, Afraimovich et al. [79] exposed manyof the concepts necessary for analyzing synchronous chaos. A crucial

progress was made by Pecora and Carroll [80 - 84], who have shown theore

tically and experimentally that two chaotic systems can be synchronized.

This discovery bridges between chaos theories and communications, and

opens up a new research area in communications using chaos.

The driving response synchronization configuration proposed by Pecora

and Carroll is shown in Fig. 1.1, in which the Lorenz system is used in the

transmitter and the receiver, where Xi or X; (i =1,2,3, r standing for the

response system) is the state variable of the Lorenz system [13], Ii is the ith

state equation, and 11 (t) is additive channel noise. The drive-response

synchronization method indicates that if a chaotic system can be decomposed

into subsystems, a drive system Xl and a conditionally stable response

system (X2' X3) in this example [82], then the identical chaotic system at thereceiver can be synchronized when driven with a common signal. The output

signals x; and x; will follow the signals X2 and X3. For more discussions on

chaotic synchronization, see [83].

~ i : l ( t ) ~ l l ( X l ( t ) , X2(t), x3(t)) x1(t)

~ i c i t ) ~ l i x l ( t ) , x2(t), x3(t))

x J ( I ) ~ 1 3 ( X 1(I), xit) , x3(t))

Drive system

y(l)

11(1)

xf ( t ) ~ l l (Y (f). x ~ ( t ) , x3(t))

x 2 ( t ) ~ 1 2 ( Y (I), x2(t), x{(I))

xj ( t ) ~ 1 3 ( Y (I), x ~ ( t ) , xl(tll

Response system

x;(t)

Figure 1.1 Drive-response synchronization schematic diagram, in which Xi or x;

(i = 1,2,3, r stands for the response system) is the state variable of the Lorenz

system [13], Ii is the ith state equation, and 17 (t) is additive channel noise

17




Based on the self-synchronization properties of chaotic systems, some

chaotic communication systems using chaotic carriers have been proposed.

Since the performance of such communication systems will strongly dependon the synchronization capability of chaotic systems, the robustness of se1f-

synchronization in the presence of white noise needs to be explored [85].

Inspired by Pecora and Carroll's work, many other synchronization schemes

have been proposed, including error feedback synchronization [87], inverse

system synchronization [88], adaptive synchronization [89], generalized syn

chronization [90], etc.

The error feedback synchronization is borrowed from the field of

automatic control. An error signal is derived from the difference between the

output of the receiver system and that received from the transmitter. The

error signal is then used to modify the state of the receiver such that it can be

synchronized with the transmitter.

The operating theory of the inverse system synchronization scheme is as

follows. If a system L with state x(t) and input set) produces an output yet),

then its inverse L-1produces an output Yr(t) =s(t) and its state xr(t) has

synchronized with x(t).

Adaptive synchronization scheme makes use of the procedure of adaptive

control and introduces the time dependent changes in a set of available

system parameters. This scheme is realized by perturbing the system

parameters whose increments depend on the deviations of the system variables

from the desired values and also on the deviations of the parameters from

their correct values corresponding to the desired state.

Generalized synchronization of the uni-directionally coupled systems

x = F(x) (x E ffi.", drive) (1.15)

y = H(y, x) (y E ffi.", response) (1.16)

occurs for the attractor Ax c lR of the drive system if an attracting syn

chronization set

18




M = {(X,y) E A/]Rm : y = H(x)} (1.17)

exists and is given by some function H : A r ~ A y c ] R m . Also, M possesses an

open basin 13 :: J M such that:

lim II y(t) - H(x(t)) 11= 0, \t(x(O), y(O)) E 13.1-'>00

(1.18)

It was reported in [91] that for a linear bandpass communication channel with

additive white Gaussian noise (AWGN), drive-response synchronization is

not robust (signal-to-noise ratio, > 30 dB is required) and the continuous

time analog inverse system exhibits extreme noise sensitivity (SNR > 40 dB

is required to maintain synchronization). Further, recent studies of chaotic

synchronization, where significant noise and filtering have been introduced

to the channel, indicate that the performance of chaotic synchronization

schemes is worse, at low SNR, than that of the best synchronization schemes

for sinusoids [92].

1.6 Communications Using Chaos as Carriers

The use of modulating an aperiodic or nonperiodic chaotic waveform, instead

of a periodic sinusoidal signal, for carrying information messages has been

proposed since chaotic synchronization phenomenon was discovered. In

particular, chaotic masking [85,93], dynamical feedback modulation [94],inverse system modulation [95], chaotic modulation [88, 96 - 10 1], chaotic

shift-keying (CSK) [102 - 112], and differential chaos shift keying

(DCSK) [113], have been proposed. In the following, we will provide a brief

summary of these schemes.

1.6.1 Chaotic Masking Modulation

The basic idea of a chaotic masking modulation scheme is based on chaotic

signal masking and recovery. As shown in Fig. 1.2, in which the Lorenz

19




system is also used as the chaotic generator, we add a noiselike chaotic

signal Xl (t) to the information signal met) at the transmitter, and at the

receiver the masking signal is removed [85, 86,93]. The received signaly(t),consisting of masking, information and noise signals, is used to regenerate

the masking signal at the receiver. The masking signal is then subtracted

from the received signal to retrieve the information signal denoted by m t).

The regeneration of the masking signal can be done by synchronizing the

receiver chaotic system with the transmitter chaotic system. This communi-

cation system could be used for analog and digital information data. Cuomo

et al. [85] built a Lorenz system circuit and demonstrated the performance

of chaotic masking modulation system with a segment of speech from a

sentence. The performance of the communication system greatly relies on

the synchronization ability of chaotic systems. The masking scheme works

only when the amplitudes of the information signals are much smaller than

the masking chaotic signals.

.i:[(I)=I[(x[(t), x2(1), "3(1»

xil)=Iz{x[(I), x2(1), x3(1»

· i : 3 ( t ) = l l ~ [ ( I ) , x2(1), x3(1»

Drive system

l1(t)

.i:[ (1)=/[ (JJ (I), x ~ ( t ) , xj(l))

.i:2(t)=iz( Y (f), x;[(t), x31))

,iej(t)=i3( Y (f), x2t), x31»

Response system

in (I)

Figure 1.2 Block diagram of a chaotic masking modulation communication system,

in which Xi or X; (i = 1,2, 3, r stands for the response system) is the state variable

of the Lorenz system [13], Ii is the ith state equation, 77 (t) is additive channel noise,

met) and m t) are the injected message signal and the recovered message signal

1.6.2 Dynamical Feedback Modulation

To avoid the restriction of the small amplitude of the information signal,

another modulation scheme, called dynamical feedback modulation, has

been proposed in [94]. As shown in Fig. 1.3, in which the Lorenz system is

used again as the chaotic generator, the basic idea is to feedback the

information signal into the chaotic transmitter in order to have identical

20




input signals for both the chaotic transmitter and the receiver. Specifically,

the transmitted signal, consisting of the information signal met) and the

chaotic signal Xl (t), is communicated to the receiver which is identical to the

chaotic transmitter. Since the reconstructed signal x; (t) will be identical to

x(t) in the absence of noise '7(t), the information signal met) can be decoded

from the received signal.

~ i : / t ) ~ l l ( x l ( t ) + m ( t ) , x i t), X3(t»

x 2 ( t ) ~ l i x l ( t ) + m ( t ) , xit), x3(t))x 3 ( t ) ~ l i x l ( t ) + m ( t ) , x2(t), x3(t»

m(t)

x l ( t ) ~ l l (y(t), -'2(t), x](t))

x M ~ l i y ( t ) , xW), xj(t))x W ) ~ 1 3 ( y(t), x2(t), xj(t)

met)

Drive system Response system

Figure 1.3 Block diagram of a dynamical feedback modulation communication

system, in which Xi or X; (i = 1, 2, 3, r stands for the response system) is the state

variable of the Lorenz system [13], (is the ith state equation, 1] (t) is additive channel

noise, met) and m t) are the injected message signal and the recovered message signal

This analog communication technique can be applied to binary data

communication by setting met) = C if the binary information data is one, and

met) = - C if the binary data is zero. Since the feedback information will

affect the chaotic property, the information level C should be scaled

carefully to make the transmitter chaotic to maintain the desired communi

cation security.

1.6.3 Inverse System Modulation

In the inverse system approach [95], the transmitter consists of a chaotic

system which is excited by the information signal set). The output yet) of the

transmitter is chaotic. The receiver is simply the inverse system, i.e., asystem which produces ret) =set) as output when excited by yet) and started

from the same initial condition. If the system is properly designed, the

output ret) will approach set), regardless of the initial conditions.

21




1.6.4 Chaotic Modulation

In chaotic modulation [88,96-101], the message signal is injected into a

chaotic system as a bifurcation "parameter,,(j), with the range of the

bifurcation parameter chosen to guarantee motion in a chaotic region (for

more details, see Sec. 7.2). The main advantage of the chaotic modulation

scheme is that it does not require any code synchronization, which is

necessary in traditional spread spectrum communication systems using

coherent demodulation techniques. The crucial design factor is, however, the

retrieval of the bifurcation "parameter" variation from the receiving spread

spectrum signal, which may be distorted by nonideal channel and contaminated

by noise (one of the goals of this book is to investigate signal reconstruction

techniques at the receiving end such that the bifurcation parameter and

hence the injected message can be recovered).

1.6.5 Chaos Shift Keying

In binary CKS [102-112] as shown in Fig. 1.4 (a), an information signal is

encoded by transmitting one chaotic signal Zj (t) for a "1" and another

chaotic signal zo(t) to represent the binary symbol "0". The two chaotic

signals come from two different systems (or the same system with different

parameters); these signals are chosen to have similar statistical properties.

Two demodulation schemes are possible: coherent and non-coherent. Thecoherent receiver contains copies of the systems corresponding to "0" and

"1". Depending on the transmitted signal, one of these copies will

synchronize with the incoming signal and the other will desynchronize at the

receiver. Thus, one may determine which bit is being transmitted. A

coherent demodulator is shown in Fig. 1.4 (b), in which Zt(t) and zo(t) are

the regenerated chaotic signals at the receiver.

(j) Bifurcation parameters determine the dynamical behavior of a dynamical system. For some selectedrange of the parameter values, the system can demonstrate chaotic behavior [22].

22




Transmitter

x(t)

"0" tDigital infomlationto be transmitted

I I

: Receiver I I Channel---- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ______ _

yet) -- ,----------1

Synchronizationcircuit

Synchronizationcircuit

(al

Correlator

Correlator

Digital infomlationreceived

(b)

Symbolduration

Thresholddetector

Figure 1.4 Chaos shift keying digital communication system. Block diagrams of

(a) the system, and (b) a coherent CSK demodulator

One type of non-coherent receivers requires the transmitted chaotic signals

having different bit energies for "1" and "0". By comparing the bit energy

with a decision threshold, one can retrieve the transmitted source information

signal. Moreover, other non-coherent schemes exploit the distinguishable

property of the chaotic attractors for demodulation, such as in Tse et al. [114].

In particular, if the two chaotic signals come from the same system

23




with different bifurcation parameters, demodulation can be performed by

estimating the bifurcation parameter of the "reconstructed" chaotic signals.

1.6.6 Differential Chaos Shift Keying Modulation

When the channel condition is so poor that it is impossible to achieve

chaotic synchronization, a differential chaotic modulation technique for

digital communication, called DCSK, has been introduced [113]. This

modulation scheme is similar to thatof

the differential phase shift keying(DPSK) in the conventional digital communication except that the transmitted

signal is chaotic. That is, in DCSK, every symbol to be transmitted is

represented by two sample functions. For bit"1", the same chaotic signal are

transmitted twice in succession while bit "0" is sent by transmitting the

reference chaotic signal followed by an inverted copy of the same signal. At

the receiver the two received signals are correlated and the decision is made

by a zero-threshold comparator. The DCSK technique offers additional

advantages over the CSK:

(l) The noise performance of a DSCK communication system in terms of

bit error rate (BER) versus EblNo (Eb is the energy per bit and No is the

power spectral density of the noise introduced in the channel) outperforms

the BER of a standard non-coherent CSK system. For sufficiently large bit

duration, the noise performance of DCSKis

comparableto

that of aconventional sinusoid-based modulation scheme. In particular, EblNo = 13.5

dB is required for BER= 10-3 [115].

(2) Because synchronization is not required, a DCSK receiver can be

implemented using very simple circuitry.

(3) DCSK is not as sensitive to channel distortion as coherent methods

since both the reference and the information-bearing signal pass through the

same channel.

The main disadvantage of DCSK results from differential coding: Eb is

doubled and the symbol rate is halved.

24




1.7 Remarks on Chaos-Based Communications

1.7.1 Security Issues

Recent studies [116 - 118] have shown that communication schemes using

chaotic or hyperchaotic sources have limited security. Therefore, most of the

chaos-based communication schemes are based on the viewpoint that

security is an added feature in a communication system, which may be

implemented by adding encryption/decryption hardware at each end of the

system.

1.7.2 Engineering Challenges

The field of "communications with chaos" presents many challenging research

and development problems at the basic, strategic, and applied levels.

The building blocks with which to construct a practical chaos-based spread

spectrum communication system already exist: chaos generators, modulation

schemes, and demodulators. Nevertheless, further research and development

are required in all of these subsystems in order to improve robustness to a

level that can be comparable to existing conventional system.

Synchronization schemes for chaotic spreading signals are not yet

sufficiently robust to be competitive with pseudo-random spreading sequences.

Nevertheless, they do offer theoretical advantages in terms of basic security

level. Furthermore, an analog implementation of chaotic spreading may

permit the use of simple low power, high-frequency circuitry.

Although an improved scheme, called frequency modulation DCSK

(FMDCSK) [115], shows a better performance under multipath environment,

channel characteristics are not fully taken into account yet, which limits its

realizability in practical environments.

Finally, there are still many practical problems that need to be solved, for

example, the extension of multiple access design is a practical challenging

25



1.7 Remarks on Chaos-Based Communications

issue involving both system level and basic research. The effects ofbandwidth

limitation also presents different problems to the practical imple-mentation

of such systems.

In summary, chaos provides a promising approach for communications. It

should be emphasized here that the field of chaos communications is very

young: much fundamental work as well as practical problems need to be

addressed before high-performance robust chaos-based communication

systems can be generally available.

26

chaos and communication.book.chapter

Documents