Digitization of Chaotic Signal for Reliable Communication in

Non-ideal ChannelsRupak Kharel, Sujan Rajbhandari, Zabih Ghassemlooy &

Krishna BusawonOptical Communications Research Group

Northumbria Communication Research LabNorthumbria University

UK

Outline

• Chaos – Introduction• Application to communication• Different techniques for chaotic communication• Problem statement• Digitization of Chaotic signals• Results• Final Comments

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Chaos – Introduction

• Deterministic system– no random or noisy inputs or parameters. The

irregular behaviour arises from the system’s nonlinearity rather than from the noisy driving forces.

• Aperiodic long term behaviour – there should be trajectories which do not settle down

to fixed points, periodic orbits or quasiperiodic orbits as t →∞.

• Sensitive dependence on initial conditions– nearby trajectories separate exponentially fast,

meaning the system has positive Lyapunov exponent.

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Chaos Example – Lorenz Equation

• Are dynamical system exhibiting chaotic property

• Can be represented as three dimensional system given as:

• States are evolving in a complex non repetitive pattern over time.

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Chaos – Application in Communication

• Chaotic signal has a broadband spectrum, – hence the presence of information does not necessarily

change the properties of transmitted signal.• Power output remains constant regardless of the

information content.• Resistant to multi-path fading, offering cheaper

solution to traditional spread spectrum systems.• Chaotic signal are aperiodic therefore limited

predictability.• Hence, chaotic signal can be used for providing

security at physical level.

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Chaotic Synchronization – How(???)

• Chaotic systems are very sensitive:– slightly different initial conditions and initial

parameters lead to totally different trajectories.– A small error between transmitter and receiver is

expected to grow exponentially.

Q1: How can one achieve synchronization?Q2: Can this sensitive chaotic system be used in communication?• Pecora & Carroll1 showed that it is possible to synchronize two chaotic

system if they are coupled with common signals.• Cuomo & Oppenheim2 practically utilized chaotic synchronization for

transmitting message signal. 1) L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., 64, pp. 821-824, 1990 2) K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phy. Rev.

Lett., 71, pp. 65-68, 1993.6

Observer Based Synchronization

• Use of observer concept from Control Theory to obtain synchronization between two similar chaotic systems.

• Estimation of all unknown states using one (or more) states.

Cx

hBAxx

y

tyf )ˆ(ˆˆ.

xCKhBxAx tt ytyf Chaotic Oscillator Chaotic Observer

function. forcing a is andfunction smooth a is ,, where hfRyRn x

stable. is -matrix thesuch that chosen is Gain KCAK

Note: The pair (A,C) should be observable to be able to design an observer. 7

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1

1

1

0

0

,1

010

n

naa

0C

BA

Observer Based Synchronization…

• Now, if the error the error dynamics is given as:

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• Since (A - KC) is stable, the error e asymptotically converges to zero ensuring synchronization.

• Other types of observers are also available, such as Sliding-Mode, Integral, Proportional-Integral observers, etc.

Chaotic Communication – Techniques

• Chaotic Masking Technique

• Chaotic Parameter Modulation Technique

• Message Inclusion Technique

• Chaotic Shift Keying (CSK)

Almost all other methods falls into one or more of these categories.

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Chaotic Masking Technique

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• Message signal m(t) is buried in the broad chaos spectrum by adding it to a chaotic mask y(t)

• At receiver, the estimated chaotic mask is removed from received signal to obtain m(t)

Parameter Modulation Technique

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• Some parameters of chaotic system are varied by adding the message signal.

• At receiver an adaptive controller is used to tune the chaotic system parameters to ensure zero synchronization error.

Message Inclusion Technique

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• Rather than changing the chaotic parameter, the message is included in one of the states of the chaotic oscillator. By doing this, the chaotic attractor at phase space is directly changed.

• A transmitted signal will be different than the state where the message will be included.

Chaotic Shift Keying (CSK)

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• For transmitting digital message signal.• Two statistically similar chaotic attractor are respectively used to encode bit ‘1’

or ‘0’.

• The two attractors are generated by two chaotic systems having the same structure but slightly different parameters.

• At receiver, the received signal is used to drive a chaotic system similar to one of the transmitters.

• The message is recovered by thresholding the synchronization error signal.

Problem Statement

• Whenever newer method in chaotic communication is proposed, the assumptions – ideal channel or – very low level of noise

• Hence, the proposed method may seem promising with regards to security but they may not be feasible if subjected to real noisy and dispersive environment.

• Also, digital communication has advanced techniques for error correction and dispersion compensation schemes .

• Hence, it would be logical if future developments in chaotic communication can be built upon existing digital technologies.

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Proposed Method

Digitization of Chaotic Signal• The message signal mt is masked into x1 state of the chaotic

oscillator to produce output yt such that yt = x1 + mt

• yt is digitized using A/D converter and binary data sequences are generated.

• These binary sequences are digitally modulated (OOK). • A matched filter (matched to the transmission filter) followed by a

slicer to regenerate the binary sequence. 15

Proposed Method (contd…)

• An equalizer and error correction code can readily be applied if channel is dispersive and noisy.

• The recovered chaotic signal is used for chaotic synchronization and extraction of the actual hidden message signal.

• The observer for the system (1) can be defined as

where the gain Kp is chosen such that the matrix (A - C Kp) is stable.

• In this method, it is shown that message recovery is possible with a high degree of accuracy at signal-to-noise ratio (SNR) of 14 dB even when the bit error rate is very high.

• The SNR can further be reduced by including error correction codes and digital signal processing.

• Note: Security issues are not taken into consideration in this study and a simple chaotic masking is used to demonstrate the concept of digitization.

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Remarks

• Converting chaos signal into a digital format offers the followings:– Compatibility with existing infrastructure.– Noise, multipath induced distortion and dispersion, and fading

can readily be dealt with in the digital domain.– Dispersion can simply be compensated by means of equalizers

including like linear equalizer, decision feedback equalizers and the more recently reported wavelet and artificial neural network (ANN) based equalizer.

– Once the minimum BER required for message recovery is set, the error control coding, (e.g. convolutional, turbo and a low parity density codes) can be used to improve the BER performance.

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Application using Lorenz Equation

• PCM with uniform quantization level (n) is used for digitization of signal yt.

• On-off Keying (OOK) with 100% duty cycle is used of digital transmission.

MessageTransmitter

• Additive white Gaussian (AWG) channel is employed with varying SNRs.

• Observer is used for synchronization purpose at the receiver and then estimate the hidden message signal.

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Receiver

Simulation Parameters

• Simulation software: Matlab/Simulink• Input message : m(t) = A sin t• Quantization resolution: n = 6 using PCM• Low pass filter: 8th Butterworth filter fcut-off = 2

rads/sec to suppress quantization error• BER: 10-6 is considered to be optimum• It will be shown that the proposed method will be

able to extract message at BER up to 10-4

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Simulation Results/Discussion (contd…)

• Synchronization is possible even if receiver is driven by quantized chaotic signal. The 450 line indicates perfect synchronization illustrating robustness of the observer synchronization

• Message recovery is faithful at BER 10-6.

Fig. 1. Synchronization between states used for masking at Tx and Rx

Fig. 2. Transmitted and recovered message at BER 10-6.

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Simulation Results/Discussion (contd…)

• At BER of 10-4, the message recovery is still possible.• At higher BER the recovered message is distorted.• BER of < 10-4 is the optimum level for reliable communication for the proposed

method.

Fig. 3. Recovered message at different order of BER .

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Conclusions

• A chaotic communication based on digitization was proposed.

• The proposed method was able to extract message at BER of order < 10-4.

• Error control methods can be applied to improve the BER.

• For reducing dispersions and fading, equalizing filters can be employed.

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Future Works

• Performance evaluation in dispersive channels and compensation using equalizers.

• Reducing the effect of quantization on message recovery using DPCM or advanced source coding.

• Implementing the proposed method to be used except in chaotic masking method for enhancing security.

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THANK YOU.

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