2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Two-Bit Modulation and Demodulation Schemes for Chaos-Based
Noncohernet Communications Using Chaotic Dynamics
Shintaro ARAI† , Yoshifumi NISHIO† and Takaya YAMAZATO‡†Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone: +81-88-656-7470
Fax: +81-88-656-7471
Email: {arai, nishio}@ee.tokushima-u.ac.jp‡Nagoya University
Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan
Phone: +81-52-789-2743
Fax: +81-52-789-3173
Email: yamazato@nuee.nagoya-u.ac.jp

error-correcting method [10]. Namely, using the chaotic dynamics is very effective for chaos-based communication sysThis paper proposes new modulation and demodulation tems.
schemes using a characteristics of chaos for chaos-based nonIn this study, we propose new modulation and demoducoherent communication systems. Successive chaotic signal lation schemes using the chaotic dynamics. Previously, thesamples are generated according to the speciﬁc rule, i.e., the transmitter modulates the 1-bit information by Chaos Shiftchaotic dynamics. Based on the chaotic dynamics, the pro- Keying (CSK) [4]. The proposed scheme modulates 2-bit inposed scheme modulates 2-bit information data by the chaotic formation by the chaotic sequence in the transmitting side.
sequence and demodulates the data by using the chaotic dy- Since successive chaotic signal samples are generated accordnamics which the incoming signal has. As results of computer ing to the chaotic dynamics, the receiver can demodulate thesimulations, we conﬁrm that BERs of the proposed scheme data by using the chaotic dynamics which the incoming sigshow gain over a conventional noncoherent system.
nal has. We carry out computer simulations and evaluate theperformance of the proposed method.
Abstract

1. Introduction
2. System Overview
Chaos-based communication systems are an interesting
topic in the ﬁeld of engineering chaos [1]– [8]. Especially,
many researchers focused on the development of noncoherent detection systems that do not need to recover basis signals (unmodulated carriers) at the receiver. Differential chaos
shift keying (DCSK) [1] and the optimal receiver [2] are wellknown typical noncoherent systems. Moreover, it is also important to develop a suboptimal receiver with performanceequivalent to or similar to the optimal receiver using more
efﬁcient algorithms [3].
Analyzing chaotic sequence as well as its behavior is essential for improving the the performance of chaos-based
communications. A Chaotic sequence is a series of nonperiodic signals generated from nonlinear dynamical systems,
i.e., a chaotic dynamics. These signals are sensitive to initial conditions and difﬁcult to predict the behavior of the
future from the past observational signal. Also the chaotic
sequence can be generated from a simple model, such as a
one-dimensional chaotic map. These characteristics are advantages of using the chaotic sequence for communication
systems. Thus, we consider that the utilization of characteristics of chaos is important to improve the performanceof chaos-based communication systems. In our previous research, we focused on the chaotic dynamics and proposed theerror-correcting method using the chaotic dynamics. As results, we observed the better performance of the proposed

Figure 1 shows the chaos-based communication systemwith new modulation and demodulation schemes. This system consists of a transmitter, a channel and a receiver. Eachblock is described in detail below.
2.1. TransmitterIn the transmitter, a chaotic sequence is generated by achaotic map. In this study, the transmitter uses a skew tent
map, a simple chaotic map, which is described by Eq. (1),
2xk + 1 − a(−1 ≤ xk ≤ a)
1+a
xk+1 =(1)
−2xk + 1 + a(a < xk ≤ 1) ,1−awhere a denotes the position of the peak of the skew tent map,k denotes the number of iterations. The input data is modulated by means of the chaotic sequence. To transmit each
input data, N chaotic signal samples are generated,
2.2. Channel and Noise
We assume an additive white Gaussian noise (AWGN)
channel with a mean of zero and a variance of N0 = σ 2 .

- 695 -AWGN channel

Transmitter

Noncoherent receiver

Received signal block
R=R0, R1, R2, ... Ri, ... RN-2, RN-1

nChaotic signal
generator

S

Modulator using
chaotic dynamics

R

Demodulator using
chaotic dynamics

Output data

b^0=0
^

b1=0

Input data
(2 bits of information)

Figure 1: Block diagram of chaos-based communication system for transmitting 2-bit information.
Chaotic
Sequence

X0

Input data
bk =+1 or -1(k=0, 1)

b0

X1
b1

X2

...

... bkb0

Xk ...

XN-2

Multiplier

b^0=1
^

b1=0

S0

S1

If k is even number

S2

...

Sk ...

i=0, If k is odd number^

MultiplierCalculation ofshortest distanceCalculation ofshortest distance

D0,0

D0,1

D1,0

D1,1

XN-1

Comparator(Smallest Db^ ,b^ is determined)

b1

SN-2

b^0=1
b1=1

Calculation ofshortest distance0... b0Multiplier

^

b1=1

Calculation ofshortest distance

Output data

Transmitted
signal blockb^0=0Multiplier

SN-1

b^1

1

b^0

Figure 3: Block diagram of new demodulation scheme.

i=1= (b0 x0 , b1 x1 , b0 x2 , · · · bi xk , · · · , b1 xN −1 )Figure 2: Operation of new modulation scheme.

(3)
The AWGN channel is the most commonly used basic channel model.

The transmitted signal arrives at the receiver through
AWGN channel. Here, the noise signal is denoted by thenoise vector n = (n0 n1 · · · nN −1 ). Thus, the received
signal block is given by R = (R0 R1 · · · RN −1 ) = S + n.

2.3. Receiver
The receiver recovers the transmitted signal from the received signal and demodulates the information data. Since weconsider a noncoherent receiver, the receiver has the chaotic
map used for the modulation at the transmitter in its memory.
However, the initial value is not known on the receiving side.
3. New digital modulation and demodulation schemes fortransmitting 2-bit information

3.2. Demodulation scheme
Figure 3 shows a block diagram of new demodulation
scheme. First, the receiver feeds the incoming signal into
four-multipliers and multiplies its signal by the projected information bits (bˆi = ±1 (i = 0 and 1)) which are preparedwith each multiplier. Thus the received signal block is expressed by Eq. 4–5.
(R0 , R1 , R2 , · · · , Rk , · · · , RN −1 )= (bˆ0 R0 , bˆ1 R1 , bˆ0 R2 , · · · bˆi Rk , · · · , bˆ1 RN −1 )(4) bˆ0 Rk (If k is even number)ˆbi R k =(5) ˆ
b1 Rk (If k is odd number) ,R =3.1. Modulation scheme
Figure 2 shows an operation of new modulation scheme.
Previously, to transmit 1-bit information, the transmitter generated N chaotic signal samples and outputted the chaotic sequence modulated by CSK. In the new modulation scheme ofthis study, to transmit 2-bit information (bi = ±1 (i = 0 and1)). the transmitter generates N chaotic signal samples andmultiplies the even and odd samples of the its signal by b0
and b1 , respectively, as follows Eq. 2.
 b0 xk (If k is even number)bi xk =
(2)
 b x (If k is odd number) ,1 kThus, the transmitted signal block S is expressed by Eq. 3.
S =(S0 , S1 , S2 , · · · , Sk , · · · , SN −1 )Based each multiplied signal, the receiver measures the
chaotic dynamics. In this study, we calculate shortest distances between each multiplied signal and the chaotic map
for the measurement of the chaotic dynamics. The calculation method is based on our suboptimal receiver in our previous study [9].
Here, we introduce the calculation of the shortest distance.
The receiver calculates the shortest distance between the received signal and the maps in the Nd -dimensional space using Nd successive received signal samples (Nd : 2, 3, · · ·).
As an example, we explain the case of Nd = 2. In this- 696 -

Rk+1

Note that if the point is outside of the cube, we calculate thedistance between the point and the nearest edges of the maps.
In the case of using Fig 4, there are two straight lines in
Rk=(Rk, Rk+1)
thespace. Therefore, the minimum value from two distances
0.5
is chosen as the shortest distance D. The receiver performs
these operations until the last sample∑ (i.e., RN −1 ) is included,and calculates the summation
D. Since we prepare four0
multipliers for the receiver in this study, each multiplier outputs each different distance Dbˆ0 ,bˆ1 . Namely, total four-typesof Dbˆ0 ,bˆ1 (D0,0 , D0,1 , D1,0 and D1,1 ) are outputted.
-0.5After calculating shortest distances, each Dbˆ0 ,bˆ1 is fed into
a comparator. The comparator selects the smallest Dbˆ0 ,bˆ1
from four-distances. If the incoming signal is not inﬂuenced
-1
-1
-0.5
0a
0.5
1
by the channel, the shortest distance between the chaotic map
Rk
and its signal which multiplied the transmitted information
Figure 4: 2-dimensional space of skew tent map whose coorbits becomes zero. The reason for this is that successive
dinates correspond to two successive received signal samples.
chaotic signal samples are generated according to the chaotic
dynamics. Namely, when the projected information bits correspond to the transmitted information bits, the incoming sigP = (X, Y)
nal becomes the successive chaotic signal according to theP0chaotic and the distance between its signal and the chaotic
P
1
map becomes zero. On the contrary, since the incoming sig=(x0, y0)= (x1, y1)
nal is not successive according to the chaotic dynamics if theDprojected information bits does not correspond to the trans0
mitted information bits, the distance between its signal andthe chaotic maps shows a signiﬁcantly larger value than zero.
Even if the incoming signal is inﬂuenced by the channel, thechaotic dynamics of its signal is kept partially although it deR = (Rk, Rk+1)
pends on the noise level. Therefore, the comparator can outFigure 5: Calculation of shortest distance.
put the projected information bits bˆ0 and bˆ1 as the decoded
bits.
case, we consider two successive received signal samples
The calculation of the shortest distance can be extended to
R = (Rk , Rk+1 ) as coordinate of chaotic map (Fig 4)/ To de- Nd -dimensional space for Nd ≥ 3.
cide which map is closer to the point R in Fig. 4, the shortest
distance between the point and the map has to be calculated.
Therefore, the receiver can calculate the shortest distance us- 4. Simulation Results
ing the scalar product of the vector.
To evaluate the proposed method, we carry out computer
Any two points P0 = (x0 , y0 ) and P1 = (x1 , y1 ) arechosen from each straight line in the space of Fig. 4, as shown simulations under the following conditions. On the transmitin Fig. 5. Using Fig. 5, the receiver can calculate the point ting side, 100,000 bits are transmitted with different initialP = (X, Y ) closest to R and the shortest distance D using values. We assume N = 32 and 64, and the parameter of the
skew tent map is ﬁxed as a = 0.05. On the receiving side,
the following equations.
to compare the performances for different numbers of dimenP = (X, Y ) = (u · v0 ) u + P0
(6) sions used in the calculation of the shortest distance, we use
2-dimensional space and 4-dimensional space, i.e., Nd = 2
and 4. Based on these conditions, we calculate the average of
BERs.
D = ||P − R||√
Figures 6(a) and (b) show the BER versus Eb /N0 for the22=(X − Rk ) + (Y − Rk+1 )proposed method with Nd = 2 and 4, respectively. To ob(7) serve the effectivity of the proposed method, Figs. 6(a) and(b) also show the performance of the suboptimal receiver as
the conventional noncoherent scheme. Note that the BERs
whereof the conventional scheme with half length of the sequence
P1 − P0
unit vector u =(8) of the proposed method are plotted to compare the perfor||P1 − P0 ||mance with the same Eb /N0 . Namely, it becomes the pairv0 = R − P0 .
(9) of the proposed method with N = 32 and the conventional
1uv- 697 -

1.0E+00

5. Conclusions
proposed N=64

1.0E-01

conventional N=32

conventional N=16
BER

proposed N=32

1.0E-02

We have focused on the chaotic sequence according to thechaotic dynamics and have proposed new modulation and demodulation schemes using the chaotic dynamics. As resultsof computer simulations, it has been conﬁrmed the effectivityof the proposed method for chaos-based noncoherent communication systems.

1.0E-03

1.0E-04

References

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](a)[1] G. Kolumbán, B. Vizvári, W. Schwarz, and A. Abel,“Differential chaos shift keying: A robust coding forchaos communication,” Proc. NDES’96, pp. 87-92,
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[2] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int . Journal of Bifurcation and
Chaos, vol. 10, no. 4, pp. 719-736, Apr. 2000.

1.0E+00
conventional N=32

1.0E-01[3] M. Hasler and T. Schimming, “Optimal and suboptimalchaos receivers,” Proc. IEEE, vol. 90, Issue 5, pp. 733746, May 2002.

BER

conventional N=16
proposed N=32

1.0E-02[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication Systems, Springer, 2003.

1.0E-03

1.0E-04

proposed N=64

0

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16

18

20

Eb/N0[dB](b)
Figure 6: BER performances: (a) Nd = 2, (b) Nd = 4[5] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalized correlation-delay-Shift-Keying Scheme for Noncoherent Chaos-Based Communication Systems,” IEEE
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[6] L. E. Larson, J-M. Liu and L. S. Tsimring, Digital
Communications Using Chaos and Nonlinear Dynamics, Springer, 2006.

scheme with N = 16, and the pair of the proposed methodwith N = 64 and the conventional scheme with N = 32.

[7] G. Cimatti, R. Rovatti and G. Setti, “Chaos-BasedSpreading in DS-UWB Sensor Networks Increases
Available Bit Rate,” IEEE Trans. Circuits and Systems
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From these results, we observe that the both BERs of the[8] J. C. Fang and C. K. Tse, Reconstruction of Chaotic
proposed method show gain over the conventional scheme.
Signals with Applications to Chaos-Based CommunicaEspecially, the BERs of Nd = 4 improves signiﬁcantly. It
tions, Tsinghua University Press and World Scientiﬁcwould be said that the receiver calculates the shortest disPublishing Co. Pte. Ltd., 2008.
tance reﬂecting the chaotic dynamics due to the large successive chaotic signal. Moreover, even if we compare the perfor- [9] S. Arai and Y. Nishio, “Detection of Information Symmances of both methods with same Eb /N0 , the performancebols and Sequence Lengths Using Suboptimal Receiverof the proposed method is better. Essentially, although Eb
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decrease if we transmit the signal with 2-bit information, theSep. 2006.
choice of data doubles as compared with the signal with 1-bit.
In other words, both relationships are the trade-off. However, [10] S. Arai, Y. Nishio and T. Yamazato, “Error-Correcting
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Chaos Communications,” Proc NOLTA’08, pp. 652that of the conventional method even if the choice of data in655, Sep. 2008.
creases, we consider that the chaotic dynamics of the incoming signal operates greatly for the demodulation. Therefore,
it can be said that the proposed method is very effective as
new modulation and demodulation schemes.

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