IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.7 JULY 1999

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PAPER

Performance Comparison of Communication Systems
Using Chaos Synchronization
Junji KAWATA† , Yoshifumi NISHIO†† , Members, Hervé DEDIEU††† , Nonmember,and Akio USHIDA†† , Member

SUMMARY
In this paper, the performance of some communication systems using chaos synchronization is evaluatedand compared. A new channel model taking the attenuation,
impedance mismatch and noise into account, is proposed for the
performance evaluation. The evaluation of bit error rate is done
for both ideal and non-ideal conditions using the channel model.
It is conﬁrmed that some chaos-based communication systems
have a good performance compared with conventional analog
communication schemes.
key words: chaos-based communication system, chaos synchronization, transmission line, channel model, bit error rate

1.

Introduction

Recently chaos and its possible application to communications have received a great deal of attention. At the
beginning vigorous eﬀorts for communication schemes
utilizing chaos synchronization [1]–[5] have been paid,
while in the most recently studies on digital communications employing the property of wide-band and uncorrelated chaotic signals [6]–[9] have attracted attention. For realization of chaos-based communication systems, in either case, it is of signiﬁcance to ﬁnd or develop communication systems with a practically usable
performance. In this sense performance evaluation ofchaos-based communication systems and those comparison are very important. However performance comparison of chaos-based communication systems has beenreported only in [6], [9]. Further it has been pointed
out that chaos synchronization can not be used forcommunications because of the eﬀect of a noisy andband-limited channel [6], [7]. However this has not been
conﬁrmed for all communication systems using chaos
synchronization.
Therefore in this study we evaluate the performance of some communication systems using chaos synchronization and compare them in order to conﬁrm
Manuscript received September 7, 1998.
Manuscript revised December 1, 1998.
†
The author is with the Department of Information Science and Systems Engineering, Tokushima Bunri University, Kagawa-ken, 769–2193 Japan.
††
The authors are with the Department of Electrical andElectronic Engineering, Tokushima University, Tokushimashi, 770–8506 Japan.
†††
The author is with the Department of Electrical Engineering, Swiss Federal Institute of Technology, Lausanne,
CH-1015 Switzerland.

whether the assertion in [6], [7] is true or not and which
system has best performance. As representative examples of communication systems using chaos synchronization, we select the chaotic masking [1], the chaotic
coding [2] and the chaotic modulation [4] systems frommany systems proposed so far.
On the other hand, the eﬀect of noise on chaosbased communication systems has been investigatedrecently. However, there are other eﬀects such as attenuation, impedance mismatch, band limitation, nonlinearity of transmission property and so on, in realcommunication channels. Because we can expect thatchaos-based communication systems are considerablyaﬀected by such eﬀects than conventional communication systems, thus we propose a channel model using
a transmission line in order to consider attenuation,
impedance mismatch and noise, in this paper. In thismodel, the transmission line is matched with its matching impedance, where an additive white Gaussian noiseis added to the transmission signal similarly to [7], [9].
The signal aﬀected by attenuation is then compensatedby an ampliﬁer proposed in our previous study [10].
In [9] the author has assumed that attenuation of channel can be compensated by an automatic gain control
circuit and has not considered it. (In [7] channel conditions in simulations have not been explained.) It should
be noted that in our channel model propagation signals are attenuated really and then compensated actually and that the performance of the systems is evaluated under the conditions. In this paper the eﬀect
of impedance mismatch is left out of consideration forbrevity.
For the performance evaluation the bit error rate
(BER) is simulated for not only lossy and noisy communication channel but ideal one, because the BER ofchaos-based communication systems is not always zeroeven for noise-free case. The BER evaluation of threesystems is done and it is found that the performanceof chaotic coding and modulation systems have a good
performance.
2.

Chaotic Communication Systems

For the performance evaluation of chaotic communication systems, the chaotic masking [1], chaotic coding [2]and chaotic modulation [4] systems are taken up in this

KAWATA et al: PERFORMANCE COMPARISON OF COMMUNICATION SYSTEMS USING CHAOS SYNCHRONIZATION

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Fig. 1

Fig. 2 Chaotic coding system.
controlled dependent source.

Chaotic masking system.

The diamond shaped current source is a current-

paper. In this section, we summarize brieﬂy the communication systems. The transmitter and receiver inall systems are based on the well-known Chua’s circuit.
2.1 Chaotic Masking System
Chaotic masking system [1] is shown in Fig. 1. The circuit equations of the transmitter are described as follows:
vC2 − vC1
dvC1=
− g(vC1 )C1dt
R
vC1 − vC2
dvC2
=
+ iL
C2
dt
R
diL
= −vC2Ldt(1)where a function g(·) is a piecewise linear function andis deﬁned by
1g(v) = m1 v + (m0 − m1 )[|v + Bp | − |v − Bp |].(2)2An information signal s(t) is simply added to thechaotic signal vC1 generated by the transmitter andthen transmitted. Based on the drive-response concept,
the receiver is conﬁgured and it is then governed by thefollowing equations:v̂C1 − vC2
dvC2
=
+ iL
dt
R
di

L L = −vC2
dt

C2

C1



v  − vC1
dvC1

= C2
− g(vC1
).
dt
R

(3)

where v̂C1 is the received signal through the channel.
If the transmitter and receiver synchronize, the information signal s (t) can be recovered ass (t) = v̂C1 − vC1.

(4)

2.2 Chaotic Coding System
The chaotic communication system proposed in [2] may
be regarded as a kind of the chaotic modulation system,
so we call the system chaotic coding system to discriminate from the chaotic modulation systems [3], [4].
An information signal s(t) is coded by the chaotic
signal vC1 using a coding function. The system using
the coding function c = vC1 + s(t) is shown in Fig. 2.
In this system the amplitude of s(t) must be small inorder to ensure that the transmitter remains chaotic.
The circuit equations of the transmitter are givenbyvC2 − vC1
dvC1=
− g(vC1 + s(t))
dtRvC1 − vC2
dvC2
C2
=
+ iL
dt
R
diL
= −vC2 .
L
dt
C1

(5)

The modulated signal vR (t) is transmitted to the receiver. The dynamics of the receiver is

IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.7 JULY 1999

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Fig. 3v  − vC1
dvC1

= C2
− g(vR)dt
Rv  − vC2
dv 
+ iL
C2 C2 = C1
dt
R
di

L L = −vC2.
dt

Chaotic modulation system.

C1

(6)

The recovered information signal can be obtained as
the potentials (t) = vR
− vC1 .

(7)

Fig. 4

2.3 Chaotic Modulation System
Two types of chaotic modulation systems have beenreported in [3], [4]. We take the system [4] in this paper
since the circuit conﬁguration is simple. The chaotic
modulation system [4] is shown in Fig. 3.
The circuit equations for the transmitter are givenbys(t) − vC1
vC2 − vC1
dvC1=
− g(vC1 ) +
C1
dt
R
RS
vC1 − vC2
dvC2
=
+ iL
C2
dt
R
diL
= −vC2Ldt(8)where a current signal ii (t) which corresponds to thelast term of the ﬁrst equation in (8) is injected into
the transmitter and modulates the chaotic signal vC1 .
The modulated signal vC1 is then transmitted to thereceiver which is described byv  − vC1
dvC1
v

= C2
− g(vC1) + C1
dt
R
RS



vC1
dvC2
− vC2

=
+ iL
C2
dt
R
di

L L = −vC2
.
dt

C1

(9)

The recovery of information can be achieved by detecting the current id (t) ﬂowing into the receiver; that is,s (t) = RS id (t)v  − vC1
dv 
= RS C1 C1 − C2dt
R

Channel model using transmission line.

+g(vC1
)−



vC1
.
RS

(10)


/dt in (10) can
Please note that the derivative C1 dvC1
be evaluated by diﬀerence approximation in numericalexperiments.

3.

Channel Model

The transmitter and receiver in all systems are connected with a communication channel. In the systems
we can expect that if the channel is ideal (i.e. lossless,perfectly matched, noiseless and so on) then the transmitters and receivers will synchronize. But real communication channels are lossy, noisy, and often imperfectly matched, the performance of the systems mustbe evaluated under such conditions. For the purpose
proper channel models are required. In [7], [9] AWGN
(additive white Gaussian noise) channel models havebeen used.
We will propose and use a model as shown in
Fig. 4. This model contains a transmission line and its
matching impedance. The transmission line is further
modeled by convenient models [11], [12] shown in Fig. 5
which are widely used for the transient simulation oftransmission lines. An additive white Gaussian noiseis added to the transmission signal similarly to [7], [9].
An ampliﬁer is used to compensate the eﬀect of attenuation and this compensation technique has been proposed in [10]. Note again that in this model transmission signals aﬀected by attenuation are compensatedreally and this is diﬀerent from the relevant studies reported so far.

KAWATA et al: PERFORMANCE COMPARISON OF COMMUNICATION SYSTEMS USING CHAOS SYNCHRONIZATION

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(a)

(b)
Fig. 5 (a) Characteristic model of lossless transmission line and (b) modeling of lossy
transmission line.

If the transmission line is lossless, it is replacedby the characteristic model as shown in Fig. 5(a) [11].
e1 (t) and e2 (t) are the waveform generators to simulate the reﬂection. Let Z0 and τ be the characteristic
impedance and the time delay respectively, then the
waveform generators of the lossless line are calculated
as follows:e1 (t) = 2v2 (t) − e2 (t − τ )e2 (t) = 2v1 (t) − e1 (t − τ )(11)Remark that e1 (t) and e2 (t) must be stored over theperiod τ .
On the other hand, if the transmission line is lossy,
it can be modeled as shown in Fig. 5(b) [12]. The model
consists of some segments called cell. Each cell is composed of four lumped resistors and a lossless line which
is further replaced with its characteristic model. The
number of cells, which depends on the modeling error,
must be adjusted according to the magnitude of loss
and frequency of transmission signal. This model ismore accurate than other models even if a transmission line is not distortionless.
4.

Performance Comparison

In this section we investigate the bit error rate (BER)
for the performance evaluation of the systems. The
performance is evaluated for not only lossy and noisy
communication channel but also ideal one, because the
BER of chaos-based communication systems is not always zero even for noise-free case [7].
We must simulate the systems with the use of numerical integration methods, because they utilize analog chaotic signal generated by the Chua’s circuit whichis governed by nonlinear diﬀerential equations. The
forth order Runge-Kutta method is used in the perfor-

mance estimation. A non-return-to-zero binary waveform is generated from a random binary sequence, when
the amplitude and bit rate are given. After the transmitter and receiver synchronize, the binary waveform
is injected to the systems at time Ts . Decision of recovered date sequence is done by detecting the sign of therecovered signal s (t). The decision process is repeatedat the moment t satisfyingt − τ − Ts = nTb − h,n = 1, 2, · · · , N

and then a recovered bit is compared with a corresponding transmitted bit. Where τ is the time delay of transmission line, N is the number of transmitted bits, 1/Tb
is bit rate and h is step-size in the simulations. In our
experiments, N = 106 and h = 10−4 . The reason why
detecting at just before of bit switching is as follows. In
the chaotic masking system desynchronization of chaos
often occurs due to change of information binary signal. Therefore, bit error probability is expected to be
the smallest at the decision timing. In the decision process the output of a low-pass ﬁlter (LPF) may be usedinstead of the recovered signal to reduce unwanted effects described later.
First of all, in order to evaluate the performance
fairly, we ﬁx the parameters in all systems as follows:
R = 1700[Ω], C1 = 10[nF], C2 = 100[nF]
L = 18[mH], m0 = −0.75[mS],
m1 = −0.41[mS], Bp = 1[V].

(12)

Now we begin to estimate the ideal BER performance of the communication systems for ideal channel (i.e. the channel is lossless, perfectly matched andnoiseless). The channel parameters are Z0 = Zm =50[Ω], τ =0.5[msec] and K = 1, where K is the gainof the ampliﬁer in our channel model. The ideal performance is shown in Fig. 6 The chaotic masking system can cause undesirable transients at bit switching of

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(a)

(b)

tude is only shown because the system became unstable
for larger one. In contrast to the chaotic coding system,
only larger amplitude is used for the chaotic modula/dt intion system. In the system the quantity C1 dvC1
(10) must be estimated to detect the current id in thesimulation. We can evaluate it only by applying diﬀerence approximation, but this caused “spike-like” error
/dt| becomes larger.
in the recovered signal when |dvC1
Smaller amplitude is aﬀected fairly by this error, thus
relatively larger amplitude is used.
In the chaotic masking system the BER becomes
better if bit rate is lower. On the other hand, in the
chaotic coding and modulation systems, the BERs areless than 10−6 and remain unchanged for the change of
amplitude and bit rate. They have a good performancefor ideal conditions.
Next noise performance for the carrier-to-noise ratio (CNR) is evaluated regarding the chaotic signal
generated in the transmitters with no input signal as
the carrier signal. In general the signal-to-noise ratio(SNR) is used for performance evaluation, but the average power of information signal in the three systems is
diﬀerent because of the above mentioned reason. Consequently the performance comparison using SNR is notfair, thus we think that it is better to use CNR than
SNR. From our simulations the average power of thecarrier in all systems is about 3.586 for the parametersof the transmitters (12). Using the value the noise level(namely the variance of Gaussian noise) according to a
CNR is calculated. In BER evaluations the additivenoise is summed up to transmission signals at intervalsof step-size.
In the estimation of noise performance, a 1[km]lossy transmission line with the parameters
Rd = 20[Ω/km], Ld = 0.25[mH/km]
Cd = 100[nF/km], Gd = 0[S/km]
is used, where we decided the number of cells N as 25.
The value of matching impedance is calculated by thefollowing equation: R + jω L 

d
c d 
Zm = |Z0 (jωc )| =  Gd + jωc Cd (c)Fig. 6 Ideal performance in the chaotic communication systems. (a) chaotic masking system, (b) chaotic coding system, (c)chaotic modulation system. For only chaotic masking system a
LPF with the time constant 0.2[msec] is used. The dark shaded
mark denotes a BER less than 10−6 .

information digital signal, which often result in desynchronization (bit error). So a LPF is introduced in order to reduce the transients and its output is used for
the BER evaluation. Note that it is not shown in Fig. 1.
In the chaotic coding system the BER for smaller ampli-

where ωc is the natural angular frequency of the Chua’s
circuit. In this paper the eﬀect of impedance mismatchis not taken account for simplicity. The gain K of theampliﬁer is set to 1.06 (see [10] for details).
Also it can be expected that the recovered signals
are aﬀected by noise and so on, thus a LPF is introduced in order to reduce such eﬀects. The LPF withthe time constant 0.2[msec] is used for the following
BER estimation, which is not shown in Figs. 1–3.
Figure 7 shows the BER obtained. Further the
performance for small parameter mismatch on linear
resistor R is also investigated and illustrated, becauseKAWATA et al: PERFORMANCE COMPARISON OF COMMUNICATION SYSTEMS USING CHAOS SYNCHRONIZATION

1327

(a)

becomes better as noise decreases. In particular the
BERs of the chaotic coding and modulation systems
are less than 10−6 for a region of CNR, similarly toideal case. Moreover this is also applied to the casethat there is a small parameter mismatch between thetransmitter and receiver. It has been mentioned in [13]that the SNR required to get BER 10−6 in analog communication is grater than 60[dB]. Therefore we can conclude that the chaotic coding and modulation systems
have good performance compared with analog communication schemes. Also chaos synchronization canbe used for communications if transmission conditions
are not so poor. In the point of implementation the
chaotic modulation system has advantages, because thechaotic coding system has the current-controlled dependent source and it is diﬃcult to realize ideal one.
5.

(b)Conclusions

In this paper we have evaluated the performance ofsome communication systems using chaos synchronization and compared them. Further we have proposed achannel model taking loss, noise and impedance mismatch into account, and shown how to evaluate BER
for chaos-based communication systems in detail. The
chaotic coding and modulation system have a good performance even if their parameters have a small mismatch. Although these systems are utilizing chaos synchronization, they will oﬀer a practically usable performance. The chaotic modulation system has advantages
in the sense of facility for realization.
We will evaluate the performance of other chaosbased communication systems, e.g., the chaotic inverse
system and consider to design a chaos-based communication system reliable under poor conditions.
Acknowledgement
This work was partly supported by The Mazda Foundation’s Research Grant.
References(c)Fig. 7 Noise performance in the chaotic communication systems.
(a) chaotic masking system (amplitude=0.1[V], bit
rate=200[bps]), (b) chaotic coding system (amplitude=0.01[V],bit rate=2000[bps]), (c) chaotic modulation system (amplitude=0.5[V], bit rate=2000[bps]). For all systems a LPF withthe time constant 0.2[msec] is used. The dark shaded mark denotes a BER less than 10−6 .

it is diﬃcult to match perfectly the component values of the transmitter and the receiver in real circuits.
For larger noise level (smaller CNR) the chaotic masking system has a best performance, while the performance of the chaotic coding and modulation systems[1] L. Kocarev, K.S. Halle, K. Eckert, and L.O. Chua, “Experimental demonstration of secure communication via chaotic
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Junji Kawata
was born in Kagawa,Japan, on 1967. He received the B.E. and
M.E. degrees form Tokushima University,
Tokushima, Japan, in 1990, 1992, respectively. He is currently working towards
the Ph.D. degree. His research interests
is in circuit simulation and chaotic communication. Mr. Kawata is a member of
the IEEE.

Yoshifumi Nishio
received the B.E.,
M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama
Japan, in 1988, 1990, and 1993, respectively. In 1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima Japan, where he is currently an Associate Professor. His research interests
are in chaos and synchronization phenomena in nonlinear circuits. Dr. Nishio is amember of the IEEE.

Hervé Dedieu
received his Ph.D.
degree from the Institute National Polytechnique de Toulouse, France, in 1988.
From 1984 to 1988 he worked with the
Signal Processing Laboratory of the Ecole
Nationale Superieure d’Electronique,d’Electrotechnique et d’Hydraulique deToulouse where his research was concentrated on adaptive ﬁltering. During thesame period he worked also as a contractual engineer for the French Spatial
Agency (CNES). In 1989 he joined the Circuits and Systems
group of the Swiss Federal Institute of Technology in Lausanne
where his research interests include identiﬁcation of nonlinear
systems, applications of nonlinear dynamics in signal processing
and speech recognition. He co-authored two books “Electrical
Filters” with C. Dehollain, M. Hasler and J. Neirynck and “Nonlinear Eﬀects in Digital Filters” with R. Boite and M. Hasler (inFrench, Ed. Presses Polytechniques Romandes). He is currently
lecturer at the Swiss Federal Institute of Technology of Lausanne.

Akio Ushidareceived the B.E.
and M.E. degrees in electrical engineering from Tokushima University in 1961
and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He wasan associate professor from 1973 to 1980
at Tokushima University. Since 1980 hehas been a Professor in the Department
of Electrical Engineering at the university.
From 1974 to 1975 he spent one year as
a visiting scholar at the Department of Electrical Engineeringand Computer Sciences at the University of California, Berkeley. His current research interests include numerical methods
and computer-aided analysis of nonlinear system. Dr. Ushida isa member of the IEEE.