Report On Chaotic Communication

CHAOTIC COMMUNICATION

KNS Institute of Technology (affiliated to vtu , approved by AICTE) Hegdenagar , Thirumenhalli Bangalore-560064

CERTIFICATE Academic session : Feb 2008 – Feb 2009 This is to certify that the seminar title “CHAOTIC COMMUNICATION ,THEIR APPLICATION AND THEIR ADVANTAGES OVER TRADITIONAL METHOD” as a part of VIII semester Electronics and Communication of Vishveshwaraiah Technological University ,has been successfully done by :

Mr. RUDRAPPA J SHETTI

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Date: Signature of Staff Incharge

Signature of head of the Dept of E&C

A Seminar Report on DEPARTMENT OF E&C

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CHAOTIC COMMUNICATION , THEIR APPLICTION AND ADVANTAGES OVER TRADITIONAL METHOD OF COMMUNICATION Submitted in the partial fulfillment for the award degree of Bachelor of Engineering in Electronics and Communication

Submitted by: RUDRAPPA J SHETTI VISHVESHWARAIAH

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TECHNOLOGICAL UNIVERSITY

KNS Institute of Technology Department of Electronics and Communication 2008-2009

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ACKNOWLEDGEMENT I sincerely acknowledge KNS INSTIUTE OF TECHNOLOGY for providing me with the opportunity to improve my knowledge and presentation skills by giving this seminar.

I thank our Principal , Dr . S . K . Narayana ,for providing us the necessary infrastructure.

I thank our Head of Department of E&C, Prof Nanda Kumar , for his support and motivation .

I thank all the teaching and non teaching staff members of the EC department for providing the necessary help and co-operation.

Last but not the least I would like to thank all my friends for their constant support and valuable suggestion without which this would not have been the success .

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SL NO. 1 2

3

4

5 6

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CONTENTS INTRODUCTION DEFINATION OF CHAOS

CHAOTIC SYSTEM CHAOS CONTROL

CHAOTIC SIGNAL CHAOTIC SHIFT KEYING OFFER SECURE COMMUNICATION

ATTRACTOR

PAGE NO.

2 4 5

6

7 8

12

8

HISTROY OF CHAOTIC COMMUNICATION SYSTEM

14

9

COMPARISION

34

10

ADVANTAGE

11

APPLICATION

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13

CONCLUSION

BIBLIOGRAPHY

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Abstract The discovery of randomness in apparently predictable physical systems have evolved into a new science, the science of chaos. Chaotic systems are unstable and aperiodic, making them naturally harder to identify and to predict. Recently, many researchers have been looking at ways to utilize the characteristics of chaos in communication systems and have actually achieved quite remarkable results. This field of communication is called Chaotic Communication . Chaotic communication signals are spread spectrum signals, which utilize large bandwidth and have low power spectrum density. In traditional communication systems , the analogue sample functions sent through the channel are weight sums of sinusoid waveforms and are linear. However, in chaotic communication systems, the samples are segments of chaotic waveforms and are nonlinear. This nonlinear, unstable and aperiodic characteristic of chaotic communication has numerous features that make it attractive for communication use. It has wideband characteristic, it is resistant against multipath fading and it offers a cheaper solution to traditional spread spectrum systems. In chaotic communications, the digital information to be transmitted is placed directly onto a wide-band chaotic signal . In this paper the concept of chaotic communication is explained together with its applications and advantages over traditional communication methods. The majority of the research carried out so far proves that chaotic communication system has quite a number of advantages over traditional communication system. DEPARTMENT OF E&C

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1.INTRODUCTION In communication, maintaining an ordered discipline has always been a constraint. In order to communicate effectively and efficiently, accurate information has to be sent or received in the correct manner. With computer processing power increasing in the last few decades, scientists have been able to perform complicated calculations in a relatively short period of time to facilitate this. This in turn has given rise to scientific interest in the irregular phenomena around us such as random changes in the weather, the spread of epidemics and the propagation of impulses along nerves. These irregular phenomena are related to the branch of mathematics known as chaotic dynamical systems, which deals with systems having a kind of order without periodicity or nonlinear systems in general. In linear systems, the variables involved appear only to the power of one. These variables are simple and directly related. In nonlinear systems, the variables involved are of powers other than one or even fractional. Such systems are harder to analyze. The chaotic phenomena having no inherent order would appear to have little to do with modern communication where sequence of zeros and ones are sent or received accurately and reliably. One would ask as to why then bother with chaotic communication when the conventional communication system is managing perfectly? The answer is, in recent experiments, digital messages were successfully sent at gigabit per second (Gbps) speeds over 115 km of commercial optical fibre system using chaotic communication with a Bit Error Rate (BER) of one in ten million. The BER was said to be limited by the equipment rather than the technique itself [1].

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In chaotic communication, the nonlinear characteristic of communication devices are utilized instead of being avoided, this eliminates the complicated measures to maintain linearity. As a result, chaotic communication systems can function over a larger dynamical range, with fewer complex components and operate at higher power levels than traditional communication systems.

2.The definition of chaos There is no universally agreed definition of chaos. However, most people would accept the following working definition: Chaos is aperiodic time-asymptotic behaviour in a deterministic system which exhibits sensitive dependence on initial conditions. This definition contains three main elements: 1. Aperiodic time-asymptotic behaviour--this implies the existence of phase-space

trajectories which do not settle down to fixed points or periodic orbits. For practical reasons, we insist that these trajectories are not too rare. We also require the trajectories to be bounded: i.e., they should not go off to infinity. 2. Deterministic--this implies that the equations of motion of the system possess no

random inputs. In other words, the irregular behaviour of the system arises from non-linear dynamics and not from noisy driving forces. 3. Sensitive dependence on initial conditions--this implies that nearby trajectories in

phase-space separate exponentially fast in time: i.e., the system has a positive Liapunov exponent.

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All systems can be basically divided into three types: ➢

Deterministic systems

These are systems for which for a given set of conditions the result can be predicted and the output does not vary much with change in initial conditions

➢ Stochastic systems These systems, which are not as reliable as deterministic systems. Their output can be predicted only for a certain range of values

➢ Chaotic systems Chaotic systems are the most unpredictable of the three systems. Moreover they are very sensitive to initial conditions and a small change in initial conditions can bring about a great change in its output

4.CHAOS CONTROL

Chaos control refers to the situation where chaotic dynamics is weakened or eliminated by appropriate controls; while anti-control of chaos means that chaos is created, maintained, or enhanced when it is healthy and useful. Both control and anti-control of chaos can be accomplished via some conventional and DEPARTMENT OF E&C

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nonconventional methods such as microscopic parameter perturbation, bifurcation monitoring, entropy reduction, state pinning, phase delay, and various feedback and adaptive controls. It has been shown that the sensitivity of chaotic systems to small perturbations can be used to direct system trajectories to a desired target quickly with very low and ideally minimum control energy Chaos may be used to enhance the artificial intelligence of neural networks, as well as increase coding- decoding efficiency in signal and image communications.

5.CHAOTIC SIGNALS •

Chaotic signals has broadband spectrum , hence the presence of information does not necessarily change the properties of the signal .

•

Power output remains constant regardless of information content.

•

It is resistant against multipath fading and offers cheaper solution to traditional spread spectrum systems.

•

Chaotic signals are aperiodic therefore limited predictability.

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•

Chaotic signals are complex in structure and impossible to predict over long time.

• •

chaotic signals appear noise like Hence chaotic signal can be used for providing security at physical level.

6.Chaotic-shift keying offers secure communication OPTICAL ENCRYPTION

Researchers have been making progress toward practical optical encryption systems that exploit laser dynamics. In two recent papers, researchers in France and Japan have shown how feedback-loop-based systems can be best used, and have introduced a new kind of modulation scheme. The separate results demonstrate both the security and ease of decoding of one class of the emerging chaotic-shift keying (CSK) systems, and the applicability of the other to a wide range of systems. Chaotic-shift keying uses fluctuations in wavelength to encode and hide a communications signal. In an optoelectronic implementation, a laser is DEPARTMENT OF E&C

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configured so that its output fluctuates chaotically—that is, in a deterministic way that nevertheless looks random. To change from one bit value to another (1 to 0 or vice versa) the chaotic mechanism is altered slightly. Because the output is still chaotic, an eavesdropper should not see any change in the transmission. However, the receiver detects that the chaos is sometimes synchronized, sometimes not, allowing the signal to be extracted.

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CHAOTIC COMMUNICATION FIGURE 1. In a chaotic-shift keying scheme, the output of a tunable distributed Bragg reflector laser is fed back via a nonlinear spectral filter and time delay, causing a chaotic shift in wavelength. The chaos is modulated by the input signal, which varies the delay between time intervals T0 (511.5 µs) and T1 (543.2 µs). This modulation is then picked up by a receiver matched to T0, and the error signal (where the signal and receiver are unsynchronized) shows the location of the nonmatching bits(ones)

In collaboration between researchers at Georgia Tech Lorraine (Metz, France) and the Laboratoire d'Optique at the Université de FrancheCompté (Besançon, France), researchers have demonstrated that this system can achieve high-fidelity, high-security recordings. In their system, the researchers use a feedback loop with a nonlinear spectral filter to provide the mechanism for chaotic emission from distributed Bragg reflector lasers (see Fig. 1).1

THREE OPTIONS The French team wanted to quantify the advantages of changing different parameters to shift from bit to bit. Initially they considered three options: the mean wavelength (around which the chaos was fluctuating), the bifurcation parameter related to the dynamics of the photodetector and laser diode, and the time delay implemented in the feedback loop.

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FIGURE 2. By representing ones and zeros as different time delays in the laser feedback circuit (top), the data remained securely hidden in the chaotic signal (center), and yet could be recovered without additional signal processing (bottom). (Photo courtesy of Georgia Tech Lorraine)

The least sophisticated option is known as masking: essentially a small wavelength modulation signal is hidden in a large chaotic one. This makes it both difficult to decode and somewhat insecure, since the modulation could be detected by an eavesdropper. Tweaking the bifurcation parameter was also inadequate for the encryption application: for more than 50% of bits to be correctly recognized, the mismatch between the bifurcation parameter had to be less than 0.03. For better than 98% masking (high security), the mismatch had to be more than 0.035.

Changing the time delay in the feedback loop, however, was successful. The researchers showed that the bit seq-uence was well-hidden and easily received (see Fig. 2), and the signal could not be decrypted by looking at the signal spectrum or through autocorrelation. ALTERNATIVE

METHOD

In another project, a collaboration between engineers at Takushoku DEPARTMENT OF E&C

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and Keio Universities (both of Tokyo, Japan), researchers chose to use an acousto-optic modulator (AOM) to degrade CSK synchronization for the nonmatched states.2 This approach does not depend on the type of chaotically emitting laser used, so it could be used with semiconductor lasers that have very fast oscillations. For this scheme to work at high speeds, however, the transient time for synchronization—currently 10 times the AOM frequency—will have to be shortened.

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7.CHAOTIC ATTRACTOR

The Lorentz attractor: the best-known chaotic attractor Also known as a strange attractor, a type of attractor (i.e., an attracting set of states) in a complex dynamical system's phase space that shows sensitivity to initial conditions. Because of this property, once the system is on the attractor nearby states diverge from each other exponentially fast. Consequently, small amounts of noise are amplified. Once sufficiently amplified the noise determines the system's large-scale behavior and the system is then unpredictable. Chaotic attractors themselves are markedly patterned, often having elegant, fixed geometric structures, despite the fact that the trajectories moving within them appear unpredictable. The chaotic attractor's geometric shape is the order underlying the apparent chaos. It functions in much the same way as someone kneading dough. The local separation of trajectories corresponds to stretching the DEPARTMENT OF E&C

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dough and the global attraction property corresponds to folding the stretched dough back onto itself. One result of the stretch-and-fold aspect of chaotic attractors is that they are fractals; that is, some cross-section of them reveals similar structure on all scales.

TYPES OF ATTRACTOR 1. FIXED POINT ATTRACTOR

An attractor that is represented by a particular point in phase space ,sometime called an equilibrium point .As a point it corresponds to a very limited range of possible behaviors of the system.

2. LIMITED CYCLE ATTRACTOR A LIMIT CYCLE IS A PERIODIC ORBIT OF THE SYSTEM IS ISOLATED .

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8.History of chaotic secure communication

Chaos is a very universal and robust phenomenon in many nonlinear systems. Although the area mathematician Pincar´e had noted that some mechanical systems could behave chaotically [21], chaos did not attract wide attention until Lorenz published his paper in 1963[22]. In engineering community, chaos had been mixed with noise for a long time. In 1980’s, the electrical engineers first time “officially” announced the existence of chaos in electrical systems. Since the noise-like behaviors of chaotic electronic circuits, electrical engineers felt uncomfortable to deal with them. It was physicists first showed in 1990 that chaos could be controlled [2].

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Then the synchronization between two identical chaotic systems was reported in1990 [23]. In 1992, the electrical engineering community realized that chaos could used in secure communication systems [24, 25, 26 ] because chaos is extremely sensitive to initial conditions and parameters. The concept of chaotic hardware key for secure communication systems was then gradually realized by engineers and scientists .Since the great potential of applying chaos to secure communication systems, many groups over the world involved in the researches in this field. So far, chaotic communication systems have been updated to the fourth generation. In this paper, theory and structure of the fourth generation is presented. It is useful to provide the reader who is not involved in chaotic secure

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First generation The first generation was developed in 1993 known as additive chaos masking [25] shown in Fig. 1(a) and chaotic shift keying [26] shown in Fig. 1(b). The additive chaos masking scheme shown in Fig.1(a) consists of two identical chaotic systems in both the transmitter and the receiver. The chaotic mask denoted by c(t) is one of the state variables of the chaotic system2 in the transmitter. The message signal m(t), which is typically 20 dB to 30dB weaker than c(t) is added into the chaotic mask signal and gives the transmitted signal s(t). Since the chaotic signal c(t) is very complex and m(t) is much smaller than c(t), one may hope that the message signal m(t) can not be separated from s(t) without knowing the exact c(t).To give the reader a hands-on experience on chaotic secure communication systems, an example of additive chaotic masking scheme is given as follow. From Fig. 1(a) we can see that a chaotic synchronization block is needed in the receiver. Chaotic synchronization is a generalization of “carrier synchronization” in the normal communication systems but it is very different from the latter. We use Chua’s oscillators to demonstrate the chaotic synchronization. A Chua’s Oscillator is shown in Fig. 2(a)

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This scheme was proved that it could not be used under practical conditions because of the following drawbacks. Since the message signal is typically 20dB to 30dB weaker than the “chaotic mask”, this method is very sensitive to channel noise and parameter mismatch between the chaotic systems in the transmitter and the receiver. Furthermore, this scheme has a very low degree of security[8].

Chaotic shift keying shown in Fig. 1(b) also known as chaotic switching was designed to transmit digital message signal. In this scheme, the message signal, which is a digital signal, is used to switch the transmitted signal between two statistically similar chaotic attractors, which are respectively used to encode bit 0 and bit 1 of the message signal. These two attractors are generated by two chaotic systems with the same structure and different parameters. At the receiver end, the received signal is used to drive achaotic system, which is identical to any of the two chaotic filtering and then thresholding the synchronization error signal e(t), which is depicted in Fig. 1(b).

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This scheme is very robust to noise and parameter mismatch. However, it has a low degree of security[7] if the chaotic attractors are too far away in the bifurcation space. However, since this is the first scheme of chaotic digital communication systems, there still exist many possibilities of improving it.

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Second generation: The second generation was proposed during 1993 to 1995 known as chaotic modulation. This generation used two different ways to modulate message signals into chaotic carriers. The first method called chaotic parameter modulation [27] shown in Fig. 7(a) used message signals to change parameters of the chaotic transmitter. The second method called chaotic non-autonomous modulation [28] shown in Fig. 7(b) used the message signal to change the phase space of the chaotic transmitter.

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In Fig. 7(a) the message signal m(t) is used to modulate some parameters of the chaotic system in the transmitter such that its trajectories keep changing in different chaotic attractors. Since the bifurcation space of a chaotic system is very complex, it is very difficult to figure out the way of the changes of the parameters even through the intruder knows some partial knowledge of the structure of the chaotic system in the transmitter. At the receiver end an adaptive controller is used to adaptively tune the parameters of the chaotic system such that the synchronization error approach zero. By doing this, the output of the adaptive controller can recover the message signal. The simulation results are shown in Fig. 8. In this simulation, three message signals are used to tune three different parameters of the chaotic system in the transmitter. Since the chaotic system keeps changing its attractors, the waveform of the transmitted signal as shown in Fig. 8(a) is much more complex than a normal chaotic signal. In Figs.8(b) to (d) we show the three original message signals and the three recovered message signals. Observe that after a transient process of synchronization, the message signals are recovered with some cross talks and small delays. Instead of changing the parameters of the chaotic transmitter, the chaotic non-autonomous modulation shown in Fig. 7(b) used the message signal to perturb chaotic attractor directly in the phase space. Unlike in chaotic parameter modulation where the transmitter is switched among different trajectories in different chaotic attractors, the transmitter in chaotic non autonomous modulation is switched among different trajectories of the same chaotic attractor. Theoretically, chaotic nonautonomous modulation is an error free scheme. The second generation improved the degree of security to some degree but was still found unsatisfactory.

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Third generation: The third generation shown in Fig. 9 was proposed in 1997[9] for the purpose of improving the degree of security to a much higher level than the first two generations. We call this generation as chaotic cryptosystem. In this generation, the combination of the classical cryptographic technique and chaotic synchronization is used to enhance the degree of security. So far, this generation has the highest security in all the chaotic secure communication systems had been proposed and has not yet been broken. In the chaotic cryptosystem the plain text signal p(t) is encrypted by a encryption rule with a key signal, k(t), which is generated by the chaotic system in the transmitter. The scrambled signal is used further to drive the chaotic system such that the chaotic dynamics is changed chaotic system in the transmitter is transmitted to through public channel which can be accessed by the intruder. Since the intruder can not get access to the chaotic hardware key, it is very difficult to find p(t) out from s(t). At the receiver, the received signal r(t) = s(t)+n(t), where n(t) is the channel noise, is used to synchronize both of the chaotic systems in transmitter and the receiver. After the DEPARTMENT OF E&C

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chaotic synchronization had been achieved, the signal k(t) and y(t) can be recovered at the receiver with some noises as denoted by ˜k(t) and ˜y(t). By feeding ˜k(t) and ˜y(t) into the decryption rule at the receiver, the plain text signal can be recovered with some noises as ˜p(t). The simulation result is shown in Fig. 10. Figure 10(a) shows the transmitted signal s(t), from which one can not observe the embedded plain signal. Figure 10(b) shows the recovered and decrypted result at the receiver. Observe that after the transient process of synchronization, the plain text signal is recovered. To show the high security of this scheme, the unmasking method provided in [18] is used to decode the plain text signal. Figure 10(c) shows the unmasked signal ˜y(t) by the intruder, from which it is impossible to retrieve the plain text signal as shown in Fig. 10(d).

Fourth generation: DEPARTMENT OF E&C

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Since the publication of several chaotic cryptanalysis results in low dimensional chaos-based secure communication systems[7, 8, 16, 18], there were concerns that such communication schemes may not be secure enough. To overcome this objection, one approach is to exploit hyperchaos-based7 secure communication systems, but such systems may introduce more difficulties to synchronization. On the other hand, we can enhance the security of low-dimensional chaos based secure communication schemes by combining conventional cryptographic schemes with a chaotic system[9]. To overcome the low security objections against low-dimensional continuous chaos-based schemes, we may use the following two methods. The first method is to make the transmitted signal more complex. The second method is to reduce the redundancy in

the transmitted signal. In [9] we have

presented a method to combine a conventional cryptographic scheme with lowdimensional chaos to obtain a very complex transmitted signal. The impulsive synchronization presented in this paper offers a very promising approach of reducing the redundancy in transmitted signals.

A simple system in baseband. In this section, we combine the results in [9] and impulsive synchronization to give a new chaotic secure communication scheme. The block diagram of this scheme is shown in Fig. 19. From Fig. 19 we can see that this chaotic secure communication system consists of a transmitter and a receiver. In both the transmitter and the receiver, there exist two identical chaotic systems. Also, two identical conventional cryptographic schemes are embedded in both the transmitter and the receiver. Let us now consider the details of each block in Fig. 19. The transmitted signal consists of a sequence

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of time frames. Every frame has a length of T seconds and consists of two regions.

In Fig. 20 we show the concept of a time frame and its components. The first region of the time frame is a synchronization region consisting of synchronization impulses. The synchronization impulses are used to impulsively synchronize the chaotic systems in both transmitter and receiver. The second region is the scrambled signal region where the scrambled signal is contained. To ensure synchronization, we have T < ¢max. Within every time frame, the synchronization region has a length of Q and the remaining time interval T ¡ Q is the scrambled signal region. The composition block in Fig. 19 is used to combine the synchronization impulses and the scrambled signal into the time frame structure shown in Fig. 20.

The simplest combination method is to substitute the beginning Q seconds of every time frame with synchronization impulses. Since Q is usually very small compared with T, the processing time for packing a message signal is negligible. The decomposition block is used to separate the synchronization region and the scrambled signal region within each frame at the receiver end. Then the separated synchronization impulses are used to make the chaotic system in the receiver to synchronize with that in the transmitter. The stability of this impulsive synchronization is guaranteed by our results in Section 4. In the transmitter and the receiver, we use the same cryptographic scheme block for purposes of bi-directional communication. In a bi directional communication scheme, every cellular phone should function both as a receiver and a transmitter. Here, the key signal is generated by the chaotic system .The

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cryptographic

scheme

is

as

follows

[9]:

Fig: Time frame

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9.Conventional Sinusoidal-based Communication vs. Chaosbased Communication Conventional ➢ History:

over 100 years old;

Chaotic less than 10 years old;

matured technology ➢ Industrial

base ➢ Transmission

bandwidth

Heart of world wide

Emerging technology None existing

inform. Tech. Transmit either

Transmit at wide BW

at information BW or at wide BW (Spread Spectrum)

➢ Relation between Distinct separation

theory & Tech.

Many CC sys. schemes

between info. the./ comm. sys./comm.

are described by chaotic circuits

circuits-hardware implementation ➢ Modulation

Formats:

non-BW expansion Anti-podal signals; BPSK; BFSK

BW expansion Chaotic masking; Parametric modulation Dynamic feedback

➢ Synchronization Theory of Sync. For AFC and PLL are well Conventional

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developed; implementational theory is only loss of fraction of

performance due

a dB; No sync. Needed for

implementation is

BSK ➢ Encryption: ➢ Bit Prob. Error:

more significant No

Exist analytic error Expr.

➢ Complexity:

area; loss from

simple

yes No known analytic error Expr. complex

10.Advantages over Traditional Methods

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1. At high speed it is easier to generate strong , high power chaotic signals than periodic signals. 2.

Chaotic

signals

are

not

sensitive

to

initial

conditions

and

have

noise like time series 3. Chaotic transmission has less risks of interception and are hard to detect by eavesdropper. 4. In chaotic communication, then on linear characteristic of are utilized instead of being avoided,

communication devices

this eliminates the complicated measures to

maintain linearity. 5. Chaotic communication systems can function over a larger fewer complex components and operate at

dynamical range, with

higher power levels than traditional

communication systems . 6. The optimal asynchronous CDMA codes using chaotic spread-spectrum

sequences

can support 15% more users than the standard GOLD codes for the same bit error rate (BER) performance. 7. It has auto cross correlation properties, low multipath interference and selfsynchronization property. 8. Power output remains constant regardless of the information content. 9. It is resistant against multi-path fading and offers cheaper solution to traditional spread spectrum systems.

10. Chaotic signal are aperiodic therefore limited predictability.

11.Application of chaos in digital communication

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Digital information

S

received information

source encoder

source decoder

encryptor channel encoder

decryptor CHAOS CHAOS

channel decoder demodulator

modulator Communication channel

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Applications of chaotic communication ✔ Used in secure communication. ✔ Used in Ultra Wide Band radio. ✔ Used in radar and sonar. ✔ Used in oscillator ✔ Used in modulation technique ✔ Used in spread spectrum ✔ Used for secure communication

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12.CONCLUSION A very brief overview on Chaotic Communication has been described, explaining the system setup of synchronised chaotic communication

and

direct

chaotic

communication

with

comparison to traditional communication system setup. A few of the main chaotic modulating schemes have been described, however, it was not possible to explain some of them in depth due to space limitations. The majority of the research carried out so far proves that chaotic communication system has quite a number of advantages over traditional communication Every technology has its own advantages and disadvantages.We also

had

an

over

view

of

history

of

chaotic

secure

communications. We studied about attractors, chaotic systems and

signal.

communication

Comparison and

their

of

conventional

applications.

over

Therefore,

chaotic chaotic

communication has to be used sensibly , it should lead to human integrity and benefit to the mankind.

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BIBLIOGRAPHY [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 [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 [4] Digital communication

using chaos and Non-linear dynamic

book E.Larson,Liu-ming,Liu lew.s

Website referred DEPARTMENT OF E&C

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www.dbebooks.com www.google.com www.tech-faq.com http//en.wikipedia.org/wiki/

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