Digital Chaotic Synchronized Communication System

Abstract

The experimental study of a secure chaotic synchronized communication system is presented. The synchronization betweentwo digital chaotic oscillators, serving as a transmitter-receiver scheme, is studied. The oscillators exhibit rich chaotic behaviorand are unidirectionally coupled, forming a master-slave topology. Both the input information signal and the transmittedchaotic signal are digital ones.

Full text

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Research Article
Digital Chaotic Synchronized Communication System
S.G. Stavrinides*,1, A.N. Anagnostopoulos1, A.N. Miliou2, A. Valaristos2,
L. Magafas3, K. Kosmatopoulos1 and S. Papaioannou4
(1) Physics Department, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece.
(2) Department of Informatics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece.
(3) Department of Electrical Engineering, Technological and Educational Institute of Kavala, Kavala, Greece.
(4) Department of Civil Engineering, Technological and Educational Institute of Serres, Serres, Greece.
Received 19 March 2009; Revised 16 June 2009; Accepted 12 July 2009
Abstract
The experimental study of a secure chaotic synchronized communication system is presented. The synchronization between
two digital chaotic oscillators, serving as a transmitter-receiver scheme, is studied. The oscillators exhibit rich chaotic be-
havior and are unidirectionally coupled, forming a master-slave topology. Both the input information signal and the trans-
mitted chaotic signal are digital ones.
Keywords: Nonlinear circuits, Chaotic circuits, Digital chaotic oscillator, Synchronization.
Journal of Engineering Science and Technology Review 2 (1) (2009) 82-86
JOURNAL OF
Engineering Science and
Technology Review
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Nonlinear oscillator synchronization is a process that is frequently
encountered in nature, explaining relevant phenomena. A signi-
cant property that nonlinear dynamical systems possess is their
ability to be synchronized. As a consequence, chaotic system syn-
chronization is encountered in a variety of scientic elds, from
astronomy and electronic engineering to social sciences.
Chaotic deterministic signals exhibit several intrinsic fea-
tures, benecial to secure communication systems, both analog
and digital ones. Two key features of deterministic chaos are the
“noise-like” time series and the sensitive dependence on initial
conditions [1, 2]. Both of them grant to chaotic signals low prob-
ability of detection in chaotic transmissions and low probability of
decoding, in case of interception [3].
Due to their possible application for secure internet commu-
nications, a number of promising non-linear circuits, demonstrat-
ing chaotic behavior, have been presented in the last decade [4-8].
There are two main issues in studying the control of chaotic elec-
tronic circuits suitable for secure communications [9]. The rst one
is the way a non-linear circuit begins to operate in chaotic mode
(route to chaos) [2, 10, 11] and the second one is the achievement
of synchronization between transmitter-receiver [12, 13].
Since the discovery by Pecora and Carroll that chaotic sys-
tems can be synchronized [12], the topic of synchronization of
coupled chaotic circuits and systems has been studied intensely
[14] and some interesting applications such as broadband com-
munication systems or cryptographic systems have come out of
this research [15-18].
In this paper the system synchronization properties of a cha-
otic communication system, suitable for secure communication,
are examined.
2. Scheme and Circuit description
A very interesting electronic circuit exhibiting chaotic behavior
and with potential applications in secure communications, was
proposed and numerically examined in [19], while its transmitter
has been experimentally studied in [20]. The circuit, of the system
under question, is presented in Fig 1.
Both the transmitter and receiver, of this chaotic communi-
cation system, are second order non-linear non-autonomous elec-
tronic circuits, with their mode of operation depending on the ex-
ternally applied driving frequency. It has already been found that
the transmitter circuit exhibits the period doubling [21, 22] and the
intermittency [23, 24] routes to chaos as well as internal crisis [25],
in different ranges of driving signal frequency M(t).
The main advantage of this circuit is that it is capable of syn-
chronized chaotic communication, suitable for transmission of
digital signal. It should be noted that the transmitted chaotic sig-
nal is not analog but a discrete one. Moreover, there is no need of
transmitting any special synchronization signal. Synchronization is
achieved by the transmitted chaotic information (discrete) signal,
itself.
The transmitter and the receiver are identical circuits [19, 20].
* E-mail address: stavros@physics.auth.gr
ISSN: 1791-2377 © 2009 Kavala Institute of Technology. All rights reserved.
1. Introduction

83
S.G. Stavrinides, A.N. Anagnostopoulos, A.N. Miliou, A. Valaristos, L. Magafas, K. Kosmatopoulos and S. Papaioannou
/ Journal of Engineering Science and Technology Review 2(1) (2009) 82-86
Both the circuits include an integrator-based second-order RC reso-
nance loop, a comparator H (the circuit’s non-linear element), an
exclusive OR gate, with an input M(t), for the external source and
a buffer to avoid overloading of the ΧOR gate. The external excita-
tion M(t), that is necessary for non-autonomous oscillators, can be
either a sequence of square pulses of period T = 2π/ω or a more
complex signal, if one wants to encode an arbitrary message, for
example. This external excitation serves as the system’s informa-
tion signal.
The main target is the exact reconstruction, at receiver’s out-
put, of information signal M(t) applied at the transmitter circuit.
This is achieved by synchronously reconstructing U2 at the trans-
mitter’s-receiver’s analog output.
The principle of operation is demonstrated below. Here the
chaotic pulses U*(t) F(y1,t) drive the resonance loops (analog
part) of both the transmitter and the receiver. The transmitter sys-
tem is governed by the following set of equations:
(1a)
(1b)
(1c)
while the receiver is governed by:
(2a)
(2b)
It should be noted the same driving term αF(y1,t), in equa-
tions (1a) for the transmitter and (2a) for the receiver, which rep-
resents the system’s coupling factor. The following substitutions
have been used in the previous systems of equations, since the
parameters are written in a dimensionless form:
(3a)
(3b)
(3c)
The symbol stands for the XOR operation, while H
S
stands for the shifted Heaviside function H
S
(y) = H(-y+1). M(t) is
the normalized square pulse input signal of period T=2π/ω1.
The circuit demonstrates only damped oscillations, as long
as no excitation is applied to the XOR gate. The amplitude of the
oscillating variables U1 and U2 converges exponentially
to a stable steady state, for all reasonable initial conditions
o r while for a non zero external
drive M(t) the circuit becomes periodically forced, exhibiting
chaos.
Introducing in the set of equations (1), the error variables
Δx=x2-x1 and Δy=y2-y1, we obtain the equations governing the er-
ror dynamics:
(4)
The solution of (4) shows the exponential decrease of the
errors for all possible initial errors Δxo and Δyo. Thus, the synchro-
nization is globally asymptotically stable. This requirement leads
to the conclusion that for Δx→ 0 and Δy→ 0, the corresponding
state variables, are robustly synchronized (x1→x2 and y1→y2).
Consequently, the non-linear functions behave in a synchronous
way H(y2) → H(y1) as well.
This result suggests an extremely simple technique of recov-
ering the signal M(t) at the receiver end. The received signal is
applied to the XOR unit of the receiver. Due to the sum mod2
property, the signal F(y1,t) can be recovered from the chaotic one
without any errors, according to:
(5)
3. Experimental Results
Synchronization between transmitter and receiver was ex-
perimentally veried. Both sub-circuits remained synchronized
under different conditions, regarding the circuit parameters, as
well as, the driving frequency fM, which was provided by a digital
signal generator (HM8130). It should be noted that the driving
frequency represents the external digital information that is fed to
the communication system.
In this section a typical modulation-demodulation procedure
through chaotic synchronization is presented. All signals were
monitored by a digital storage oscilloscope (HP54603B), further
Figure 1. Schematic diagram of the transmitter-receiver system
)

84
connected to a PC for recording and analysis purposes, so that the
proper characterization of the circuit behavior could be achieved.
Appropriate software, built in NI’s LabView environment, was
used in order to control all digital instruments used and process
the signals acquired [26].
The system’s parameters were set to be equal in both the
transmitter and the receiver circuits. In order to operate in a cha-
otic mode, the parameter values, for Uo=350mV and U*=4Vp-p,
were set at α = 6.35 and b = 0.02. For this set of parameter values,
the system exhibits chaotic behavior in various ranges of external
excitation fM (chaotic windows) and undergoes various routes to
chaos [21-25].
In order to study the system’s synchronization while it oper-
ates in a chaotic mode, the driving frequency was set to fM=6,222
ΚHz. In Fig. 2 the transmitter’s phase portrait (U1 vs. U2) is pre-
sented. The chaotic nature of signals U1 and U2 is evident. Next
to the phase portrait, the transmitter’s chaotic characterization has
been already conrmed in [20].
In Fig. 3 the driving pulse signal M(t) is shown, together with
its power spectrum. The periodic nature of M(t) is obvious. In Fig.
4(a) the signal F(y1,t) at the transmitter’s digital output appears,
while in Fig 4(b) the corresponding power spectrum is shown.
The power spectrum conrms the wideband nature of the output
signal. This wideband signal shows the impossibility of detecting
the original information by using simple ltering processes.
According to [27, 28], one can conclude that the present chaotic
communication scheme can be secure. Finally, in Fig. 5(a) the re-
covered signal is presented. Both the signals itself, as well as, the
corresponding power spectrum - Fig. 5(b) - demonstrate the fact
that the recovery procedure is quite exact.
Figure 2. The phase portrait U1-U2 characterizing the transmitter’s chaotic
operation.
Figure 3. a) Digital input signal M(t) and (b) Its power spectrum (FFT)
(a)
(b)
Figure 4. a) The transmitted chaotic signal F(t) and (b) its power spectrum
(FFT)
(a)
(b)
S.G. Stavrinides, A.N. Anagnostopoulos, A.N. Miliou, A. Valaristos, L. Magafas, K. Kosmatopoulos and S. Papaioannou
/ Journal of Engineering Science and Technology Review 2(1) (2009) 82-86

85
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It should be mentioned, that the system security depends
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(a) The frequency of input signal M(t), since it should be in
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Conclusions
A scheme capable of secure chaotic digital communication is
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S.G. Stavrinides, A.N. Anagnostopoulos, A.N. Miliou, A. Valaristos, L. Magafas, K. Kosmatopoulos and S. Papaioannou
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/ Journal of Engineering Science and Technology Review 2(1) (2009) 82-86