Chaotic Communications, their Applications and

Chaotic Communications, their Applications and

Advantages over Traditional Methods of

Communication

Anjam Riaz and Maaruf Ali

Oxford Brookes University, Department of Electronics & Computing, Wheatley Campus, Oxfordshire, UK

anjamm@gmail.com; maaruf@ieee.org

Abstract — The discovery of randomness in apparently the weather, the spread of epidemics and the propagation predictable physical systems have evolved into a new of impulses along nerves. These irregular phenomena are science, the science of chaos. Chaotic systems are unstable related to the branch of mathematics known as chaotic and aperiodic, making them naturally harder to identify dynamical systems, which deals with systems having a and to predict. Recently, many researchers have been kind of order without periodicity or nonlinear systems in looking at ways to utilize the characteristics of chaos in general. communication systems and have actually achieved quite remarkable results. This field of communication is called

In linear systems, the variables involved appear only to Chaotic Communication. Chaotic communication signals

the power of one. These variables are simple and directly are spread spectrum signals, which utilize large bandwidth

and have low power spectrum density. In traditional related. In nonlinear systems, the variables involved are

of powers other than one or even fractional. Such systems communication systems, the analogue sample functions sent

are harder to analyze. The chaotic phenomena having no through the channel are weight sums of sinusoid waveforms

inherent order would appear to have little to do with and are linear. However, in chaotic communication systems,

the samples are segments of chaotic waveforms and are modern communication where sequence of zeros and ones nonlinear. This nonlinear, unstable and aperiodic are sent or received accurately and reliably. characteristic of chaotic communication has numerous features that make it attractive for communication use. It One would ask as to why then bother with chaotic has wideband characteristic, it is resistant against multi-communication when the conventional communication

path fading and it offers a cheaper solution to traditional system is managing perfectly? The answer is, in recent spread spectrum systems. In chaotic communications, the experiments, digital messages were successfully sent at digital information to be transmitted is placed directly onto

gigabit per second (Gbps) speeds over 115 km of

a wide-band chaotic signal. This paper provides an

commercial optical fibre system using chaotic

overview of chaotic communication, chaotic modulation

communication with a Bit Error Rate (BER) of one in ten

schemes such as: Chaos Shift Keying (CSK), Differential

million. The BER was said to be limited by the equipment

Chaos Shift Keying (DSK), Additive Chaos Modulation

rather than the technique itself [1]. (ACM) and Multiplicative Chaos Modulation (MCM).

Synchronized Chaotic Systems and Direct Chaotic

Communication are also described in this paper. The advantage is that at such a high speed, it is easier to

generate strong, high-power chaotic signals than periodic signals. Chaotic signals are not sensitive to initial

I. INTRODUCTION conditions and have a noise like time series. As a result,

In this paper the concept of chaotic communication is chaotic transmissions have less risk of interception and are explained together with its applications and advantages hard to detect by eavesdroppers. It has also been shown

that optimal asynchronous CDMA codes using chaotic over traditional communication methods.

spread-spectrum sequences can support 15% more users

than the standard GOLD codes for the same bit error rate

In communication, maintaining an ordered discipline (BER) performance [5]. In chaotic communication, the has always been a constraint. In order to communicate nonlinear characteristic of communication devices are effectively and efficiently, accurate information has to be utilized instead of being avoided, this eliminates the sent or received in the correct manner. With computer complicated measures to maintain linearity. As a result, processing power increasing in the last few decades, chaotic communication systems can function over a larger scientists have been able to perform complicated dynamical range, with fewer complex components and calculations in a relatively short period of time to facilitate operate at higher power levels than traditional this. This in turn has given rise to scientific interest in the communication systems. irregular phenomena around us such as random changes in

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CSNDSP08- 22 -Proceedings

Figure 1. Block Diagram of a Coherent Correlation CSK (Chaos Shift Keying) Receiver. To discuss chaotic communication systems further, this report is structured as follows: Section II covers Chaotic Modulation Techniques; Section III introduces Chaotic Synchronisation Technology and in Section IV Direct Chaotic Communication Technology is examined. II. CHAOTIC MODULATION

Several modulating schemes have been used in chaotic communications. The most common ones are:

• Chaos Shift Keying (CSK);

• Differential Chaos Shift Keying (DCSK); • Additive Chaos Modulation (ACM);

• Multiplicative Chaos Modulation (MCM).

In chaotic communication the digital bits or symbols are mapped onto sample functions of chaotic signals derived from chaotic attractors. To avoid periodicity, the symbols are mapped to the actual unstable aperiodic signal outputs of chaotic circuits and not to parameters of certain known sample functions.

The key difference between a conventional carrier and a chaotic carrier is that the sample function for a given symbol is unstable, aperiodic and is different from one symbol interval to the next. As a result, the transmitted waveform is never the same, even if the same symbol is transmitted again and again.

In Chaos Shift Keying (CSK) each symbol is mapped onto a different chaotic attractor. The information that is to be transmitted is carried by the attractor, which produces the sample function but not by the shape of the sample function. On the receiver end, the decision is made on the basis of the received noisy and distorted sample function as to which attractor was most likely to have produced that waveform.

Each attractor produces a function gj(t) and that the elements of the signal set are given by si(t) = gi(t) for i.

N

(1) si(t)=sijgj(t),j=1,2,…,N

Assuming that the autocorrelation of each gj(t) with itself in each symbol interval T is larger than the cross correlation with any of the other basis functions. In this case a correlation receiver will have to be used to identify which attractor is most likely to have generated the signal [2].

In a conventional correlation receiver based on synchronisation, a local synchronised copy of each basis function gj(t) has to be produced in the receiver using an appropriate synchronisation circuit. To produce each basis functions in a chaotic correlation receiver, chaotic synchronisation is required. Chaotic synchronisation is covered in more detail in Section III.

The concept behind the synchronisation process is as follows: The basis function gj(t) produced by the synchronous counterparts of the circuit in the transmitter are used to recover the basis functions in a coherent correlation receiver [3]. The process is shown in Fig. 1.

In Fig. 1, the signal received ri(t) tries to synchronize simultaneously with all of the synchronous chaotic circuits in the receiver. For example, assume that the signal si(t) = gi(t) is transmitted after a synchronisation time Ts, which is analogous to the pull-in time in a phase-locked loop, the

ˆ1(t)converges to g1(t). By contrast, gˆ2(t)fails outputg

to synchronize with g1(t). The decision as to which symbol was transmitted is made on the basis of the

ˆ1(t)is goodness of synchronisation. In the ideal caseg

ˆ2(t)during the more strongly correlated with ri(t) thang

interval [Ts , T]. Therefore zi1 > zi2 and the decision circuit decides that the symbol ‘1’ was transmitted [4]. However, in any real situation the signal that is sent through a medium is always affected by noise n(t), even if the synchronisation was perfect. Therefore, the method of correlation must be used for detection.

III. CHAOTIC SYNCHRONISATION

The purpose of synchronisation is to recover the basis functions described in Section II to maximize the probability to accurately identify the transmitted symbol. The most widely used synchronisation method is the

∑

j=1

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Proceedings- 23 -CSNDSP08

Identical Synchronisation where the receiving system

_ _ _ _ _ converges asymptotically to that of the transmitting

SYSTEM A (Tx) SYSTEM B (Rx) _ _ _ _ _ _ _ system [6]. Other synchronisation techniques include

Generalized Synchronisation [7] and Phase ..Synchronisation [8].

x=f(x) x'=f'(x')

Identical Synchronisation

Figure 2. Two Identically Synchronised Systems.

In Identical Synchronisation two systems are said to be IV. DIRECT CHAOTIC COMMUNICATION synchronised identically if:

limx'(t)−x(t)=0 (2) In Direct Chaotic Communication (DCC), a chaotic t→∞

source generates chaotic signals directly in the selected

Equation (2) is satisfied for any combination of initial communication band. A stream of chaotic radio pulses states x(0) and x’(0). For example System A is a are used to carry the information component. At the transmitter and System B is a receiver as shown in Fig. 2. receiver, the information can be retrieved from the chaotic radio pulses without intermediate heterodyning [12]. Fig. System A transmits signal si(t) which is a linear 3 shows an example of a time domain continuous chaotic combination of basis functions gi(t). System B must signal. These pulses are fragments of the chaotic signal recover the scalar basis function g(t) = h(x(t)) which has whose duration tc is said to be larger than the quasi-period been derived from the state of System A. Having the of the chaotic oscillations Tc (tc >> Tc). The frequency states of the two systems synchronized, to retrieve the bandwidth of the pulse ∆fc is determined by the bandwidth basis function then the read out function h(·) is applied. of the original signal generated by the chaotic source ∆F This means if x’(t) can be made to converge to x(t) then and is independent of the pulse duration.

ˆ(t)=h(x'(t))will change to g(t). the estimate gΔF=Fu−Fl=1 (4)

Generalized Synchronisation

In Generalized Synchronisation System A and System B of Fig. 2 are said to exhibit this type of synchronisation if there exists a transformation M such that:

t→∞

T

Fu is the upper boundary of the chaotic oscillation band and Fl is the lower boundary. Fig. 4 shows an example of the power spectrum of a continuous chaotic signal.

The bandwidth of the chaotic radio pulse ∆fc is:

Δfc=ΔF1≠

The difference between the conventional periodic radio pulse and the chaotic radio pulse is that tc of the chaotic

Equation (3) is satisfied where the properties of the radio pulse can be varied from tc~1/∆F to tc →∞ with tc > transformation are independent of the initial states x(0) 1/2DF, where the power spectrum of a received tributary and x’(0). This type of synchronisation occurs in of chaotic pulses will be similar to that of the original unidirectional coupled chaotic systems where the driven chaotic signal [13]. The relationship of the frequency system is asymptotically stable [9]. To achieve bandwidth ∆Fp and its pulse duration tp is: synchronization, the transformation M is inverted, giving 1

(6) Δf=−1pˆ(t)=h(M(x'(t))) which approaches g(t). In a gtp

coherent receiver, however, the transformation M is not

always invertible, therefore recovering the state x’(t) does

not permit the recovery of the required basis function [2].

Phase Synchronisation

In Phase Synchronisation, in order to have System A synchronized with System B, as shown in Fig. 2, their difference in phases have to be bounded by a constant, where the phase φ(t) is some correctly chosen monotonically increasing function of time [10]. A good example is the angle of rotation about the unstable equilibrium point in a two dimensional projection of the spiral Chua attractor [11]. Figure 3. Example of Time-domain Continuous Chaotic Signal [12].

(3) limx'(t)−Mx(t)=0

1

(5) tc

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CSNDSP08- 24 -Proceedings

Figure 4. Power Spectrum of Continuous Chaotic Signal [12].

Figure 6. Time Domain Chaotic Radio Pulse Modulated by COOK [12].

[1] J. Mullins, “Chaotic Communication”,

(http://www.spectrum.ieee.org/jan06/2574 [Accessed 29/11/2007]. [2] G. Kolumban, M.P. Kennedy, and L. O. Chua, “The role of

synchronization in digital communication using chaos – Part II: Chaotic Modulation and Chaotic Synchronization”, IEEE Trans. Circuits Syst. I, (Special Issue on Chaos Synchronization and Control: Theory and Applications), Vol. 45, No. 11, Nov., 1998. [3] G. Kolumban, M.P. Kennedy, and L. O. Chua, “The role of

synchronization in digital communication using chaos – Part I: Fundamentals of digital communication”, IEEE Trans. Circuits Syst. I, (Special Issue on Chaos Synchronization and Control: Theory and Applications), Vol. 44, pp. 927-936, Oct., 1997.

[4] M.P. Kennedy and H. Dedieu, “Experimental demonstration of

binary chaos shift keying using self-synchronizing Chua’s circuits,” Proc. IEEE Int. Specialist Workshop on Nonlinear Dynamics of Electronic Systems, Dresden, Germany, July 23-24, 1993, pp. 67-72.

[5] J.M. Liu, H.F. Chen, and S. Tang, “Optical Communication

Systems Based on Chaos in Semiconductor Lasers”, IEEE Trans. Circuit Syst. I, (Fundamental Theory and Application), Vol. 48,

V. CONCLUSION No. 12, December 2001.

A very brief overview on Chaotic Communication has [6] M.P. Kennedy, “Three steps to chaos part I: Evolution”, IEEE

Trans. Circuits and Syst. I, (Special Issue on Chaos in Nonlinear been described, explaining the system setup of

Electronics Circuits – Part A: Tutorial and Reviews), Vol. 40, pp. synchronised chaotic communication and direct chaotic

0-655, October 1993.

communication with comparison to traditional

[7] H. Fujisaka and T. Yamada, “Stability theory of synchronized

communication system setup. A few of the main chaotic motion in coupled oscillator systems,” Prog. Theor. Phys., Vol. modulating schemes have been described, however, it was 69, pp. 32-47, 1983.

not possible to explain some of them in depth due to space [8] N.F. Rulkov, M.M. Sushchik, L.S. Tsimring, and H.D. Abarbanel, limitations. The majority of the research carried out so far “Generalized synchronization of chaos in directionally coupled

chaotic systems,” Phys. Rev. E, Vol. 51, no. 2, pp. 980-994, Feb., proves that chaotic communication system has quite a

1995. number of advantages over traditional communication

[9] L. Kocarev and U. Parlitz, “Generalized synchronization, system.

predictability, and equivalence of unidirectionally coupled

dynamical systems,” Phys. Rev. Lett., Vol. 76, no. 11 pp. 1816-1819, Mar 1996.

[10] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, “Phase

synchronization of chaotic osciallators,” Phys. Rev. Lett., Vol. 76, no. 11, pp. 1804-1807, Mar 1996.

[11] M.P. Kennedy, “Three steps to chaos part I: Evolution”, IEEE

Trans. Circuits and Syst. I, (Special Issue on Chaos in Nonlinear Electronics Circuits – Part A: Tutorial and Reviews), Vol. 40, pp. 657-674, October 1993.

[12] F.C.M. Lau and C.K. Tse, Chaos-Based Digital Communication Figure 5. Non-coherent Transceiver.

Systems: Operating Principles, Analytical Methods and Performance Evaluation. Berlin, Germany, Springer-Verlag,

2003.

[13] C.C. Chong et al., “Samsung electronics (SAIT) CFP presentation

for IEEE 802.15.4a alternative PHY,” IEEE 802.15-05-0030-02-

004a, Jan 2005.

Fig. 5 [12] is a non-coherent Chaotic On-Off Key (COOK) transceiver architecture and Fig. 6 shows the COOK modulated time domain chaotic radio pulse.

The receiver measures the signal energy of a prescribed position. This energy is then compared with the previously chosen threshold. The received symbol is a ‘1’ if the energy is above the threshold, otherwise the received symbol is a ‘0’. Two energy distributions are calculated in order to estimate the threshold. One for bit ‘1’ and one for bit ‘0’. The mean of the distribution for bit ‘0’ is equal to the noise variance at the receiver input. For bit ‘1’ both the pulse energy and noise are calculated, its mean is equal to the sum of the mean pulse energy and mean noise energy [12]. The pulse energy varies because the signal is chaotic and it has its own distribution.

REFERENCES

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