Chaos and Communications

Transcription

1 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 chaosbased 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

2 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 spreadspectrum carrier for communication applications. Furthermore, chaotic synchronization makes it possible for chaos-based communication using the conventional coherent approach. In the remaining part of this chapter, several fundamental chaotic synchronization schemes, and several chaosbased 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 III 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 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

3 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 homoclinic 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 demultiplication (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 system can exhibit nonperiodic solutions. Later, Levinson [7] referred to these solutions as singular behavior. Melnikov [12] introduced his perturbation method for chaotic systems in This method is mainly applied to driven dynamical systems. 3

4 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 This is a model for the time evolution of 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 of nonintegrable Hamiltonian systems in classical 4

5 Chapter 1 Chaos and Commnnications 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 applications and 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 one because 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?oo such 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) n w -: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

6 1.3 Quantifying Chaotic Behavior loss along the attractor. Dimensionality is used to quantify the geometric structure of a chaotic attractor 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 qualitatively 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 this divergence 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 phase space, 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

7 Chapter 1 Chaos and Communications 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 a D([Jn (x(o»v(o)& II 0(&2) = II( D([Jn (x(k» }(O)&II + 0(&2), (1.3) 7

8 1.3 Quautifying Chaotic Behavior where D<p(x(k)) is the Jacobian, i.e., the m x 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)) be k denoted by {d i : 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) 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] 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 of points along the attractor, given an arbitrary initial point. 8

9 Chapter 1 Chaos and Communications 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..,i n is proportional to the information needed to locate the system on a special trajectory with precision s [37, 38]. Therefore, K N KN is the additional information needed to predict which cube the trajectory will be in at (n + 1 )T, given trajectories up to nt. This means that K N 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,(K n + 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 information 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 rate at which the accuracy of a prediction decays as prediction time increases, i.e., the rate at which predictability will be lost [37]. 9

10 1.3 Quantifying Chaotic Behavior Attractor Dimension Long-term chaotic motion in dissipative dynamical systems is confined to a strange 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 In p[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 dimension is important from an experimental viewpoint because it is straightforward to estimate. The mass, p[b x (&)] ' 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

11 Chapter 1 Chaos and Communications (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 1n(l/ c) 0-1m, (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, d L, is defined as (1.13) where k = max {i : ~ Ai > O}. It has been shown that do?cd" d,~dc [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, = d L 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 of motion are known. The relation between KE and Lyapunov exponents is also available. In one-dimensional maps, KE is just the Lyapunov exponent [70]. In higherdimensional 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

12 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

13 Chapter 1 Chaos and Communications 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 of what 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 cannot distinguish 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 selfgenerated. 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

14 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 nonchaotic system follows one of a few typical scenarios, called routes to chaos. 1.5 Chaos-Based Communications Conventional Spread Spectrum In recent years, there has been explosive growth in personal communications, 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 highfrequency, low-power, low-voltage circuitry. The huge demand for communications results in a large number of users; therefore, today's communication 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 communication 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 same or neighboring frequency bands. The optimum strategy in this situation, where every user appears as interference to every other user, is for each communicator's signal to look like white noise which is as wideband as possible. 14

15 Chapter 1 Chaos and Communications 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 spectrumintensive 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 same frequency 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

16 1.5 Chaos-Based Communications 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 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 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

17 Chapter 1 Chaos and Communications by some nonlinear, dynamical means at the receiver. If we do this in realtime, 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 many of the concepts necessary for analyzing synchronous chaos. A crucial progress was made by Pecora and Carroll [80-84], who have shown theoretically 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 the receiver 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)~ll(xl(t), X2(t), x3(t)) ~icit)~lixl(t), x 2 (t), x 3 (t)) xj(i)~13(x 1 (I), xit), x 3 (t)) x 1 (t) y(l) xf (t)~ll (Y (f). x~(t), x3(t)) x2(t)~12( Y (I), x2(t), x{(i)) xj (t)~13( Y (I), x~(t), xl(tll 11(1) Drive system Response system 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 x;(t) 17

18 1.5 Chaos-Based Communications 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 depend on the synchronization capability of chaotic systems, the robustness of se1fsynchronization 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 synchronization [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- 1 produces 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 synchronization set 18

19 Chapter 1 Chaos and Commnnications M = {(X,y) E A/]Rm : y = H(x)} (1.17) exists and is given by some function H:Ar~Ayc]Rm. 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 '>00 (1.18) It was reported in [91] that for a linear bandpass communication channel with additive white Gaussian noise (A WGN), drive-response synchronization is not robust (signal-to-noise ratio, > 30 db is required) and the continuoustime 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, ], chaoticshift-keying (CSK) [ ], and differential chaos shift keying (DCSK) [113], have been proposed. In the following, we will provide a brief summary of these schemes 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

20 1.6 Communications Using Chaos as Carriers 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 communication 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), x 2 (1), x3(1» i:3(t)=ll~[(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), x3 (1)),iej(t)=i 3 ( Y (f), x2(t), x3(1» Response system 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 in (I) 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

21 Chapter 1 Chaos and Communications 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)~ll(xl(t)+m(t), xit), X3(t» x2(t)~lixl(t)+m(t), xit), x3(t)) x3(t)~lixl(t)+m(t), x 2 (t), x3(t» Drive system m(t) xl(t)~ll (y(t), -'2(t), x](t)) xm~liy(t), xw), xj(t)) xw)~13( y(t), x2(t), xj(t) Response system met) 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 communication security 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., a system 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

22 1.6 Communications Using Chaos as Carriers 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) Chaos Shift Keying In binary CKS [ ] 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. The coherent 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 selected range of the parameter values, the system can demonstrate chaotic behavior [22]. 22

23 Chapter 1 Chaos and Communications Transmitter x(t) "0" t Digital infomlation to be transmitted I I : Receiver I I Channel ---- ~L (al Correlator yet) --, Synchronization circuit Symbol duration Synchronization circuit Correlator Digital infomlation received (b) Threshold detector 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

24 1.6 Communications Using Chaos as Carriers with different bifurcation parameters, demodulation can be performed by estimating the bifurcation parameter of the "reconstructed" chaotic signals 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 that of 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 DCSK is comparable to that of a conventional 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

25 Chapter 1 Chaos and Communications 1.7 Remarks on Chaos-Based Communications Security Issues Recent studies [ ] 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 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

26 1.7 Remarks on Chaos-Based Communications issue involving both system level and basic research. The effects of bandwidth 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