2009 International Symposium on Nonlinear Theory and its Applications
NOLTA'09, Sapporo, Japan, October 18-21, 2009

M -ary Modulation Scheme Using Separation of Chaotic Dynamics
for Noncoherent Chaos-Based Communications
Shintaro ARAI† , Yoshifumi NISHIO‡ , Takaya YAMAZATO†† and Shinji OZAWA††ITS Laboratory, Faculty of Engineering, Aichi University of Technology
50-2 Manori, Nishihasama-cho Gamagori, Aichi 443-0047, JAPAN‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506 JAPAN††EcoTopia Science Institute, Nagoya University
C3-1, Furo-cho, Chikusa-ku, Nagoya 464-8063 JAPAN
Email: {arai-its, ozawa}@aut.ac.jp, nishio@ee.tokushima-u.ac.jp, yamazato@nuee.nagoya-u.ac.jp
Abstract—This paper proposes a new M -ary modulation scheme for noncoherent chaos-based communications.
A chaotic sequence is successive based on the chaotic dynamics. Without the successive sequence based on the
chaotic dynamics, general noncoherent systems are very
difﬁcult to demodulate the data. However, we consider
that the systematic separation and the reconstruction of thechaotic dynamics can be applied as additional information.
Namely, M -ary data symbols can be expressed by separating the chaotic dynamics purposely. We carry out computer simulations of the proposed scheme and conﬁrm thata saving of over 3–4 dB is realized in transmitter energy byincreasing M .
1. Introduction
Chaotic sequences obtained from a certain class of difference equations are nonperiodic and sensitive to initialconditions, and it is difﬁcult to predict their future behavior
from past observations. Many researchers have focused onthese characteristics of chaos and have developed various
digital communication systems using chaos [1]– [6]. Especially, demodulation techniques using incoming signals
only or chaotic systems are called noncoherent detection
techniques.
In recent years, chaos has also attracted attention in theﬁeld of intelligent transportation systems (ITS), as a UWB
radar for ranging systems and a trafﬁc ﬂow forecasting [7]–[9]. For instance, Dake, et al. focused on the chaotic dynamics which means the speciﬁc rules for generating thechaotic sequence and proposed the UWB chaotic radar for
removing a noise which is called a clutter in the radar [8].
As results, they conﬁrmed that the performance of their
proposed clutter reduction method is better than that of
the conventional method. Furthermore, in our previous research, we proposed the error-correcting method using thechaotic dynamics [11]. As results, we observed the better performance of the proposed error-correcting method.
Thus, we consider that the utilization of characteristics ofchaos is important to improve the performance of chaosbased communication systems and ITS applications.
In this study, we focus on the chaotic dynamics again andpropose a new M -ary modulation scheme for noncoherent
chaos-based communication systems. The M -ary modu-

lation scheme can transmit b coded information bits at atime by using M = 2b distinct waveforms, i.e., one waveform for each of the 2b expresses b-bit sequences [10]. The
chaotic signal samples are successive based on the chaotic
dynamics. General noncoherent systems have performed
modulation and demodulation using the chaotic sequence
generated according to the chaotic dynamics. Especially,in the case of the optimal receiver and the suboptimal receiver [2] [3] [12], to demodulate the information without
the successive sequence based on the chaotic dynamics isvery difﬁcult. We consider that the M -ary information canbe expressed by separating the chaotic dynamics of the successive sequence purposely. A transmitter separates the sequence based on the chaotic dynamics according to a speciﬁc rule, and a receiver reconstructs the original chaotic
sequence. Namely, the separation and reconstruction of thechaotic dynamics can be applied for noncoherent chaosbased communication systems as additional information.
In this paper, we describe a concept and a operation of the
proposed M -ary modulation scheme. Moreover, we carryout computer simulations and evaluate the performance ofthe proposed scheme.
2. Concept of Proposed M -ary Modulation Scheme
In this section, we explain a concept of the proposed
M -ary modulation scheme. Our proposed M -ary modulation scheme applies the chaotic dynamics and its separation. Figure 1(a) shows a general noncoherent detection
method, such as the optimal receiver and the suboptimalreceiver. We take the optimal receiver for instance. The
optimal receiver calculates the probability density function(PDF) between the chaotic sequence which modulated thedata using chaos and the chaotic maps used on the transmitting side and outputs the data by choosing the larger
probability.
Here, let us consider the sequence which rearranged
the order of the chaotic sequence at random, as shown in
Fig. 1(b). In this ﬁgure, each signal sample is not successive depending on its anteroposterior samples. In this case,since the chaotic dynamics of this sequence was lost, thedetection of the data is impossible even if it is the sequence
without the noise. Namely, for detecting the data using the
optimal receiver, the successive chaotic sequence based on- 312 -

Xk

Detection is possible

X0
X4

X6

X2
X1

Outputdata

Noncoherent
detection

k

X3 X5
X7

X=X0, X1, X2, X3, X4, X5, X6, X7
Chaotic sequence (N=8)

(a)
Xk

X0
X4

X0

X6

X2

Separation of
chaotic dynamics

k

X3 X5
X1

X6

X4

X7

X1

X=X0, X1, X2, X3, X4, X5, X6, X7
Chaotic sequence (N=8)

Noncoherent
detection

X5
X3

Detection is
impossible!

X7 X2

X4 X0 X3 X6 X1 X7 X5 X2
X’=X0’, X1’, X2’, X3’, X4’, X5’, X6’, X7’
Sequence which separated chaotic dynamics(b)

Figure 1: Concept of proposed M -ary modulation scheme.
the chaotic dynamics is essential.
However, we consider that the loss of the chaotic dynamics can be applied as additional information for noncoherent chaos-based communications, i.e., it is M -ary modulation scheme using the separation of the chaotic dynamics
which is proposed in this paper. In advance, we provide M
interleavers which separate the chaotic dynamics, and M
deinterleavers which reconstructs the original chaotic sequence. Since the arbitrary symbol is assigned to each pairof the interleaver and the deinterleaver from M -ary symbols, it is possible to design the M -ary modulation scheme.
These are the concept of the proposed M -ary modulation
scheme. In the next section, we describe the operation of
the proposed M -ary modulation scheme in detail.
3. Operation of Proposed M -ary Modulation Scheme
Figure 2 shows a block diagram of the proposed M -ary
modulation scheme. This system consists of a transmitter, a channel and a receiver. In this scheme, the transmitterand the receiver have M interleavers and M deinterleavers,
respectively, for separating and reconstructing the chaotic
dynamics. In advance, the arbitrary symbol is assigned toeach pair of these from M -ary symbols. The operationsof modulation and demodulation are described in detail below.
3.1. 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
(1)
xk+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. To perform the
proposed M -ary modulation, the chaotic dynamics of thechaotic sequence is separated by the m-th interleaver according to the input data m (m : 0, 1, · · · , M − 1). To
transmit each input data m, N chaotic signal samples are
generated. Therefore, the transmitted signal block is denoted by a vector S, which is described by Eq. (2),S

= x0,m , x1,m , · · · , xk,m , · · · , xN −1,m
= S0 , S1 , · · · , Sk , · · · , SN −1 ,(2)where xk,m means the separated signal samples by the mth interleaver.
The transmitted signal arrives at the receiver through the
AWGN channel. The AWGN channel is the most commonly used basic channel model. Here, the noise signal isdenoted by the noise vector n = (n0 , n1 , · · · , nN −1 ).
Thus, the received signal block is given by R =(R0 , R1 , · · · , Rk , · · · , RN −1 ) = S + n.
3.2. Receiver
The receiver recovers the transmitted signal from the received signal and demodulates the information data. Since
we consider a noncoherent detection system, the initialvalue is not known on the receiving side. However, thereceiver has the chaotic map used for the modulation at thetransmitter in its memory.
We describe the operation of the proposed M -ary demodulation. First, the receiver feeds the received signal
into M deinterleavers and rearranges the order of the sequence. Since the interleaver and the deinterleaver corresponding to each data symbol are only 1 pair between
the transmitter and the receiver, the only one of M interleavers can reconstruct the original chaotic sequence. The
received signal rearranged by m-th deinterleaver is denotedby Rm = (R0,m , R1,m , · · · , Rk,m , · · · , RN −1,m ),Next, the receiver measures the chaotic dynamics using
the rearranged received signal. In this study, we calculate
shortest distances between the rearranged received signal
and the chaotic map for the measurement of the chaotic dynamics. The calculation method is based on our suboptimalreceiver in our previous study [12].

- 313 -

Transmitter

Receiver(M-ary modulation using separation of chaotic dynamics (M=2b) )Input data m (b-bit)(m: 0, 1, ..., m , ..., M-1)

AWGN

Separation of
chaotic dynamics

Chaotic signal
generator

S

Interleaver
0nReconstruct ofchaotic dynamics

R

Interleaver
1

Deinterleaver
0

Calculation of D0
with chaotic map

Deinterleaver
1

Calculation of D1
with chaotic map

Interleaverm

DeinterleavermCalculation of Dmwith chaotic map

Interleaver
M-1

Deinterleaver
M-1

Calculation of DM -1
with chaotic map

SelecttheSmallest
Dm

Output data(b-bit)(Dm : Shortest distance of m-th deinterleaver)Figure 2: Block diagram of proposed M -ary modulation scheme using separation of chaotic dynamics.
P = (X, Y)

P0=(x0, y0)

Dm

v0

u

distances is chosen as Dm . The receiver performs these
operations until the last sample∑(i.e., RN −1,m ) is included,and calculates the summation ∑Dm .
Finally, by comparing each
Dm , the receiver selects
the smallest one. Therefore, the decoded data which is assigned m-th interleaver and deinterleaver can be outputted.

P1= (x1, y1)

R’m = (Rk,m, Rk+1,m)

4. Simulation Results and Discussion

Figure 3: Calculation of 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 case, we consider two successive signal samples
R0m = (Rk,m , Rk+1,m ) as coordinate of chaotic map. To
decide which map is closer to the point R0m the shortest
distance between the point and the map has to be calculated. Therefore, the receiver can calculate the shortest distance using the scalar product of the vector.
Any two points P0 = (x0 , y0 ) and P1 = (x1 , y1 )are chosen from each straight line in the Nd -dimensional
space, as shown in Fig. 3. Using Fig. 3, the receiver cancalculate the point P = (X, Y ) closest to R and the shortest distance of m-th deinterleaver Dm using the following
equations.

Dm

P = (X, Y ) = (u · v0 ) u + P0

(3)

= ||P − R0m ||√2
2=(X − Rk,m ) + (Y − Rk+1,m )

(4)

where

P1 − P0||P1 − P0 ||= R0m − P0 .

unit vector u =
v0

(5)
(6)

In the case of 2-dimensional space, there are two straightlines in the space. Therefore, the minimum value from two

To evaluate the proposed M -ary modulation scheme, we
carry out computer simulations under the following conditions. On the transmitting side, 100,000 bits are transmitted with different initial values. We assume N = 32
and 64, and the parameter of the skew tent map is ﬁxed
as a = 0.05. In addition, we assume M = 4, 8, 16, 32
and 64, i.e., b = 2, 3, 4, 5 and 6. On the receiving side,
we use 4-dimensional space (Nd = 4) for calculation of
Dm . Based on these conditions, we change the number ofM and calculate the average of the bit error rate (BER) for
each M -ary.
Figures 4(a) and (b) show the BER versus Eb /N0 (SNR
per bit) for the proposed method with N = 32 and 64, respectively. Also, Figure 4(c) shows the BER performancewith the same SNR per bit. To observe the effectiveness ofthe proposed method, Figs. 4(a)–(c) also show the performance of the suboptimal receiver as the binary system.
From Fig. 4(a) and (b), we can observe that the proposed
scheme can reduce the SNR per bit by increasing the number of M . To achieve BER = 10−4 , the required SNRs per
bit of N = 32 and 64 are about 18 dB for the binary system, but if M is increased to 64 (b =6 bits), the required
SNRs per bit of N = 32 and 64 are about 15 dB and 14 dB,
respectively. Namely, a saving of over 3–4 dB is realizedin transmitter energy required to achieve BER= 10−4 byincreasing to M = 64.
Next, we compare the BER performances of the proposed scheme and the binary system with same SNR per
bit. As shown in Fig. 4(c), the BER performance of the proposed scheme shows gain over the binary system. About
2dB gain in error performance is obtained by using the propose scheme. We consider that the separation of the chaotic- 314 -

1.0E+00
Binary(Conventinal)
1.0E-01
M=4
M=16

BER

dynamics by the interleaver works to the reconstructionof the original chaotic sequence effectively. Therefore, it
can be said that the proposed method is very effective as
the M -ary modulation scheme. However, when the SNR
decreases from 10 dB, the BER of the binary system isslightly better than that of the proposed scheme. As thereason for this, we consider that the successive sequencebased on the chaotic dynamics decrease by the inﬂuence ofnoise. Also, the selection for deciding the data might become difﬁcult by increasing M . For solving this problem,
we would redesign the interleaver and the deinterleaver. In
this study, the separation of the chaotic dynamics by the interleaver is performed at random. In other words, we haveto optimize a parameter which decides the separation of
the chaotic dynamics. Thus, optimizing a parameter of the
proposed scheme is our future work.

1.0E-02

M=8
M=32

1.0E-03M=64
1.0E-04

5. Conclusions

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](a)1.0E+00

Binary(Conventinal)

1.0E-01
M=4
BER

In this study, we have proposed the M -ary modulation scheme using the separation of the chaotic dynamics.
As results of computer simulations, a saving of over 3–
4 dB is realized in transmitter energy required to achieve
BER= 10−4 by increasing to M = 64. Therefore, wehave conﬁrmed that the separation and the reconstructionof the chaotic dynamics are very effective for the noncoherent chaos-based communications. As a future work,
we will develop the ITS application based on our proposed
method.

M=81.0E-02
M=16
M=32

1.0E-03[1] G. Kolumbán, B. Vizvári, W. Schwarz, and A. Abel, “Differential
chaos shift keying: A robust coding for chaos communication,”
Proc. NDES’96, pp. 87-92, Jun. 1996.
[2] M. Hasler and T. Schimming, “Chaos communication over noisy
channels,” Int. J. Bifurcation and Chaos, vol. 10, no. 4, pp. 719736, Apr. 2000.
[3] F.C.M. Lau and C.K. Tse, “On Optimal Detection of Noncoherent
Chaos-Shift-Keying Signals in a Noisy Environment,” Int. J. Bifurcation and Chaos, vol. 13, no. 6, pp. 1587-1597, Jun. 2003.
[4] K.Y. Cheong, F.C.M. Lau and C.K. Tse, “Permutation-BasedM-ary Chaotic-Sequence Spread-Spectrum Communication Systems,” Circuits, Systems and Signal Processing, vol. 22, no. 6, pp.
567-577, November 2003.
[5] L. E. Larson, J-M. Liu and L. S. Tsimring, Digital Communications
Using Chaos and Nonlinear Dynamics, Springer, 2006.
[6] P. Stavroulakis, Chaos Applications in Telecommunications, Talor& Francis Group, 2006.
[7] V. Venkatasubramanian and H. Leung, “A robust chaos radar forcollision detection and vehicular ranging in intelligent transportation systems,” Proc. of ITSC’09, pp 548-552, Oct. 2004.
[8] N. Dake, Y. Uwate, Y. Nishio and T. Yamazato, “Ultra-Wideband
Chaotic Radar and Clutter Reduction Methods,” RISP J. Signal
Processing, vol.9, no.6, pp.473-478, Dec. 2005.
[9] H. Xie and Z. Liu, “Short-term trafﬁc ﬂow prediction based onembedding phase-space and blind signal separation,” Proc. IEEE
CIS’08, pp. 760-764, Sept. 2008.
[10] J. G. Proakis, Digital Communications Fourth Edition, McGrawHill, 2001.
[11] S. Arai, Y. Nishio and T. Yamazato, “Error-Correcting Method
Based on Chaotic Dynamics for Noncoherent Chaos Communications,” Proc. NOLTA’08, pp. 652-655, Sep. 2008.
[12] S. Arai and Y. Nishio, “Suboptimal Receiver Using Shortest Distance Approximation Method for Chaos Shift Keying,” RISP J. Signal Processing, vol. 13, no. 2, pp. 161-169, Mar. 2009.

M=64
1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](b)
1.0E+00
Conventinal(N=16, b=1)
1.0E-01

Conventinal(N=8, b=1)
Proposed(N=32, M=16(b=4))

BER

References

1.0E-02

1.0E-03
Proposed(N=64, M=16(b=4))
1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](c)Figure 4: BER performances: (a) N = 32, (b) N = 64,(c) Comparison with same SNR per bit.

- 315 -