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LETTER

BER Estimation of a Chaos Communication System
including Modulation-Demodulation Circuits
Masahiro WADA† , Student Member, Junji KAWATA†† , Yoshifumi NISHIO† ,and Akio USHIDA† , Members

SUMMARY In this study, a simple chaos communication
system including modulation-demodulation circuits is studied.
The inﬂuence of modulation-demodulation circuits to chaos synchronization is investigated. For the estimation of communication quality, bit error rate (BER) is calculated by computer
simulation when a sequential random pulse information signal istransmitted via this proposed system.
key words: chaos, chaos synchronization, AM, communication,
BER

1.

Introduction

amplitude modulation (AM) circuit. The radio-band
modulated signal is demodulated at the receiver side.
We conﬁrm that those nonlinear systems inﬂuence tothe chaos synchronization by numerical simulation. In
order to estimate communication quality, BER for several conditions (e.g. carrier frequency and bit rate aretuned) are obtained by computer simulation. The obtained results indicate adverse inﬂuence to the communication quality of the chaos communication system.
2.

Since a chaos synchronization phenomenon in a coupledchaotic system is reported by Pecora and Carroll [1], avariety of behavior concerned with chaos synchronization phenomena on various coupled systems has beenstudied. Especially chaos communication systems exploiting the feature attract many researchers’ attentions as one of engineering applications. Although avariety of studies on chaos communication systems has
been reported [2]–[4], we consider some problems are remained. In many chaos communication systems, chaos
synchronization plays an important role [5] and theaccuracy of the synchronization directly inﬂuences oncommunication quality. Therefore, it is important toresearch the eﬀect of noise or nonlinearity of communication channel including transmission terminal. On
the other hand, in order to transmit chaotic signals byradio frequency, it is necessary to modulate baseband
chaotic signal to radio frequency. It is generally known
that real modulation and demodulation circuits have a
bad inﬂuence on the quality of communication. However, inﬂuence of modulation and demodulation circuits
in communication systems using chaos synchronization
has not been discussed until now.
In this study, we investigate a chaos communication system including modulation and demodulation
circuits. Two chaotic systems are connected by themodulation-demodulation circuits. We use a simple
chaotic masking communication system using Chua’s
circuits [6]. Information signals are modulated by an
Manuscript received May 24, 1999.
Manuscript revised September 29, 1999.
†The authors are with the Faculty of Engineering, Tokushima University, Tokushima-shi, 770-8506 Japan.
††The author is with the Faculty of Engineering,
Tokushima-Bunri University, Kagawa-ken, 769-2193 Japan.

Circuit ModelIn this study, we use a chaotic masking method fortransmission of signals, which is the simplest chaos
communication system using chaos synchronization
based on Pecora-Carroll concept, and we adopt an AM
system for the modulation-demodulation. Figure 1
shows a concept of a chaos communication system using a chaotic masking method. Figure 2(a) and (b)show a Chua’s circuit for generating an original chaotic
source, and its characterized piecewise-linear resister,respectively. We know that Chua’s circuit has two typesof chaotic attractors, two of them are called as singlespiral and double-scroll chaotic attractors, respectively.
Hereafter the circuit parameters of Chua’s circuit in numerical simulation are given asFig. 1

Concept of a chaos communication system.

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Fig. 4

Fig. 2(a) Chua’s circuit and (b) characterized Chua’s diode.

Fig. 5

3.

Fig. 3

AM modulation system.

• Setting of Chua’s circuit for single-spiralchaotic attractor
C1 = 5.56[nF], C2 = 50[nF], L = 7.14[mH], G =1/R = 0.7[mS], m0 = 0.241[mS], m1 = 0.807[mS], andBp = 1.0[V].
• Setting of Chua’s circuit for double-scroll
chaotic attractor
C1 = 5.0[nF], C2 = 50[nF], L = 7.14[mH], G =1/R = 0.75[mS], m0 = 0.241[mS], m1 = 0.807[mS],
and Bp = 1.0[V].
We can also calculate the natural frequency of the circuit as around 8.4[kHz] from the value of C2 and L in
Chua’s circuit.
The modulation circuit is shown in Fig. 3. It is atypical amplitude modulation system which consists ofa transistor and some circuit elements. We model thetransistor by the Ebers-Moll model for numerical simulation. In this study, we use a chaotic signal obtained
from the terminal of b in Fig. 2(a) for obtaining the attainment of chaos synchronization easily. The demodulation circuit is constructed by a rectiﬁer, an RC ﬁlter
and several circuits as shown in Fig. 4. Further, twosubsystems based on Chua’s circuit are interconnectedby voltage buﬀers to reproduce the original chaotic signal as shown in Fig. 5.

Demodulation system.

Reproducing circuit based on Chua’s circuit.

Inﬂuence to Chaos Synchronization

At ﬁrst, we consider the case that the only chaotic signal obtained from Chua’s circuit is modulated. Please
notice that the transmitted signal does not include general information signals. Only chaotic signal is modulated by the amplitude modulation circuit in Fig. 3,
then the modulated signal is transmitted to the receiver side. We compare the transmitted chaotic signal with the reproduced chaotic signal. Figures 6 and
7 are computer simulated results in the cases of singlespiral and double-scroll chaotic attractors, respectively.
(a-1) shows an original chaotic attractor obtained from
Chua’s circuit of the transmitter side, (a-2) synchronization state between a demodulated chaotic signal
and a reproduced chaotic signal, (a-3) the reproduced
chaotic attractor at the receiver side, (b) the original
chaotic waveform, (c) the modulated chaotic waveform,(d) the demodulated chaotic waveform, and (e) the reproduced chaotic waveform at the receiver side. We
can see that the recver cannot reproduce the original chaotic signal completely, especially for the caseof double-scroll attractor. Because it was known that
Chua’s circuit can achieve chaos synchronization completely without any noise, the synchronization error isdue to the modulation-demodulation circuits. Chaos
synchronization of Chua’s circuit is described in [7].
4.

Estimation of Communication Quality

4.1 Simulation on Transmission of Information
Hereafter, we consider the case of transmission of modulated chaotic signals including information. We use asequential square random pulse for the information signal. Here we carry out the simulation of data transmission on numerical calculation. The computer simulated
results for both single-spiral and double-scroll chaotic

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Fig. 6 Transmission of a chaotic signal (single-spiral chaoticattractor)
for fc  130[kHz]. Horizontal axis of (b)∼(e) is time√(τ × LC2 [sec]).

Fig. 7 Transmission of a chaotic signal (double-scroll chaoticattractor)
for fc  130[kHz]. Horizontal axis of (b)∼(e) is time√(τ × LC2 [sec]).

attractors are shown in Figs. 8 and 9, respectively. (a)is an original chaotic waveform v(C1), (b) is a digital information signal vs , (c) is the sum of the original chaotic
waveform and the digital information signal v(C1)+vs ,(d) is a demodulated waveform vx , (e) is a reproduced

Fig. 8 Transmission of chaotic signal (single-spiral chaotic attractor) including digital information signal for fc  130[kHz]and bit rate 2.65 × 103 [bps].

Fig. 9 Transmission of chaotic signal (double-scroll chaotic attractor) including digital information signal for fc  130[kHz]and bit rate 2.65 × 103 [bps].

chaotic waveform v(C1 ), (f) vx − v(C1 ), and (g) lowpass ﬁltered signal of vx − v(C1 ). We choose suitable
circuit parameters for generating best performance. BitIEICE TRANS. FUNDAMENTALS, VOL.E83–A, NO.3 MARCH 2000

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chaotic attractor, we can conﬁrm that the BER performance is going better as fc increases. Meanwhile
the double-scroll chaotic case, it has only a little transition for contradistinction to the case of single-spiralchaotic attractor. Figure 11 shows BER while bit rateis changing. The single-spiral case BER tends to be violently reduced as bit rate decreases, the other way, thedouble-scroll case BER tends to be slightly reduced.
We can conclude that high voltage E or lower frequency fc cannot produce a good performance and that
especially the double-scroll chaotic case is strongly inﬂuenced by the modulation-demodulation circuits.
Fig. 10 BER vs. carrier frequency fc [kHz] for E = 0.10[V] andbit rate 2.65 × 103 [bps]. Each BER is simulated by 100000 pointsof data.

5.

Conclusions

In this study, we have investigated a simple chaos communication system including modulation-demodulation
circuits. BER performance of the chaos communication
system has been also estimated. We conﬁrmed that
nonlinearity of the modulation-demodulation circuits
inﬂuenced to chaos synchronization and also communication quality. Although we have treated only AM
chaotic masking communication system, the similar results could be obtained from another chaos communication systems based on Pecora-Carroll chaos synchronization.
Acknowledgment
This work was supported by The Mazda Foundation’s
Research Grant.

Fig. 11 BER vs. bit rate[bps] for fc  130[kHz] and E =0.10[V]. Each BER is simulated by 100000 points of data.

rate is chosen as 2.65×103 [bps], namely the bit widthis about 3.78×10−4 [sec]. In Figure 8, we can almost reproduce original information signals obtained from thereceiver circuits by inputting vs to the several circuits.
In Figure 9, some bit error was conﬁrmed.
4.2 BER Estimation
We calculate BER for estimation of communication
quality including the modulation-demodulation circuits. Simulated results of BER are shown in Figs. 10and 11. Figure 10 shows BER while carrier frequencyfc is changing, amplitude E of the information signal vsis ﬁxed as 0.10[V], and bit rate of digital information
is ﬁxed as 2.65×103 [bps]. In the case of single-spiral

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