2007 RISP International Workshop on Nonlinear Circuits and Signal Processing (NCSP'07)
Shanghai Jiao Tong University, Shanghai, China, Mar. 3-6, 2007

Research on Differential Chaos Shift Keying
Changing Deviation of Chaotic Sequence
Shintaro ARAI and Yoshifumi NISHIO
Department of Electrical and Electronics Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {arai, nishio}@ee.tokushima-u.ac.jp

Abstract

2. Differential Chaos Shift Keying (DCSK)This paper investigates a performance of Differential Chaos
Shift Keying (DCSK) using the chaotic sequence with biased
values. In other words, a transmitted signal is changed by thedeviation of the output of chaos depending on transforming achaotic map. We carry out computer simulations and compare
its performance to the conventional DCSK system.

Figure 1 shows the block diagram of the DCSK transmitter(a) and receiver (b). This system is well-known as the typicalcorrelation-based noncoherent chaos communication. The
modulation scheme and the detection method are describedbelow.

(a)

xi

Chaotic signalgenerator

SiDelay N

1. Introduction
Recently, a noncoherent receiver for digital communication systems using chaos is studied actively[1]-[7]. Especially, it is attracted to develop noncoherent detection systems
which does not need to recover a basis signals (unmodulatedcarries) at the receiver. The differential chaos shift keying(DCSK) [1] and the optimal receiver [2] are well known as a
typical noncoherent system.
In our previous research, we proposed a transmitter changing a chaotic sequence depending on an initial value for a
suboptimal receiver [6], where the suboptimal receiver hasa performance similar to the optimal receiver, and this algorithm is simpler than the optimal receiver. As a simulation
result, it could be observed that its performance was better
than that of the conventional transmitter for the suboptimalreceiver. From this result which changes the performance
depending on the initial value, we expected that the values
included in the chaotic sequence inﬂuenced the noncoherent
chaos communication performance greatly.
In this study, we investigate a performance of the chaotic
sequence with biased values using DCSK as the noncoherent
chaos communication. In other words, a transmitted signal ischanged by the deviation of the output of chaos depending ontransforming a chaotic map. We carry out computer simulations and compare its performance to the conventional DCSKsystem.

(b)

Ri

Delay Nblxi-NN

Information symbols
blN/2
i=1

Ri+N

RiRi+N

DecoderThreshold

Figure 1: DCSK operation. (a) transmitter. (b) receiver.
In this scheme, the transmitter outputs a chaotic sequencexi followed by the same sequence multiplied by the information symbol bl (±1). In order to transmit 1-bit information,
N chaotic signals are generated, where N is the chaotic sequence length. Therefore, the transmitted signal is given by xi
Si =

(1 ≤ i ≤ N )(1)

bl xi−N(N + 1 ≤ i ≤ 2N )Also, the transmitted signal can be written as S =(S1 S2 · · · S2N ) by vector.
In the channel, a noise is assumed to be additive white
Gaussian noise (AWGN) and is denoted by the noise vectorn = (n1 n2 · · · nN ). Thus, the received signal block is given
by R = (R1 R2 · · · R2 N ) = S + n.

- 349 -On the receiving side, it is evaluated by the correlation of
2 signals which are divided from the received signals into thehalf (N ). Thus, the output of the correlation can be written as

C1 =N
∑

Ri RN +i
(1 + r)xk + q + r



1−q






(1 − r)xk + q




q(2)
xk+1 =

i=1

The decoded symbol is decided as “+1” or “−1” dependingon C1 being larger or smaller than 0.
In order to generate the transmitted signal, the DCSKtransmitter needs to switch correctly by the chaotic sequencelength N . Therefore, the sophisticated switch is required, andit is regarded as the important issue to design DCSK.

3. Chaotic Map With Different Slopes(1 − r)xk − q




q






(1 + r)xk − q − r
1−q(−1 ≤ xk ≤ −q)(−q < xk ≤ 0)
(3)
(0 < xk ≤ q)
(q < xk ≤ 1)

As an example, chaotic sequences are shown Figs. 3(a)and (b) in the case of (q, r) = (0.6, 0.2) and (0.8, 0.8).
From these ﬁgures, It can be observed that the chaotic sequence includes many values near 1.0 (or −1.0) as both q
and r approach 1.0, namely, it can be said that deviation wasmade to the values of the chaotic sequence. In addition,
Xk

1

0.8
0.6

Xk+1
1

0.4
0.2

Xk+1=Xk

0-0.2

0.5

-0.4
-0.6

r-0.8

0-1

020

40

60

80

10080

100-r

k(a) (q, r) = (0.6, 0.2).
-0.5

Xk

10.8

-1-1-q -0.5

00.5 q0.6

10.4

Xk

0.20

Figure 2: Chaotic map with different slopes.

-0.2
-0.4
-0.6

In order to generate the chaotic sequence with biased values, we use the chaotic map shown in Fig. 2. This map was
made from the Bernoulli shift map well known as a typical 1dimensional map based on the reference [7] and is describedby Eq. (3). As one can see, the slopes of the map can bechanged by deciding parameters q (0.5 ≤ q ≤ 1.0) andr (0.0 ≤ r ≤ 1.0).

- 350 --0.8-10204060(b) (q, r) = (0.8, 0.8).
Figure 3: Chaotic sequences for each parameter.

kEbwhen (q, r) = (0.5, 0.0), the corresponding map becomes
Figure 4 plots the BERs versus Eb /N0 for each paramethe Bernoulli shift map. In this study, the parameters q and r ter. In order to compare the performance of the chaotic seof this map is changed and the performance of DCSK system quence with biased values, Fig. 4 shows the performance ofis investigated.
the conventional DCSK (i.e. (q, r) = (0.5, 0.0)) together. Asone can see, we ﬁnd that the BER performance improves asboth q and r approach 1.0. That both parameters approach
4. Simulation Result
1.0 means using the chaotic sequence including many values
In this section, we study the performance of the DCSK near 1.0 (or −1.0), as shown in Fig. 3. From this result, wecommunication system using the chaotic map with the differ- consider that it is because the chaotic sequence became strong
ent slopes by computer simulation. The simulation conditions to interference of the noise by including many values near 1.0(or −1.0).
are as follows.
On the transmitting side, the chaotic sequence length N
are 16 and 32. In addition, the parameters q and r decid0.6
ing the slope of map is used the case of (q, r) = (0.6, 0.2)
(q, r) = (0.8, 0.8)
and (0.8, 0.8). The noise is assumed to be only AWGN in
0.5(q, r) = (0.6, 0.2)
the channel. Hence, the noises at the transmitter and receiver(q, r) = (0.5, 0.0)
are not considered. On the basis of these conditions, the sys0.4(Conventional DCSK)
tem performance is evaluated by plotting BER against Eb /N00.3
when 104 bits of information are transmitted. Also we estimate the chaotic sequence of each parameter by calculating
0.2
the autocorrelation and the cross-correlation property.
0.1
1.0E+00

BER

1.0E-01(q, r) = (0.8, 0.8)

0(q, r) = (0.6, 0.2)
(q, r) = (0.5, 0.0)

-0.1

(Conventional DCSK)

2

4

68

10

14

16(a) N = 16.
1.0E-02

0.6(q, r) = (0.8, 0.8)
(q, r) = (0.6, 0.2)
(q, r) = (0.5, 0.0)

0.5

1.0E-03

1.0E-0405

10Eb/N0 [dB]

1520Eb0.4

1.0E+00

0(q, r) = (0.6, 0.2)
(q, r) = (0.5, 0.0)

-0.1

(Conventional DCSK)

51015202530(b) N = 32.

1.0E-02

Figure 5: Autocorrelation property.

1.0E-03

1.0E-04

0.3

0.1(q, r) = (0.8, 0.8)

1.0E-01(Conventional DCSK)

0.2(a) N = 16.

BER

1205

10
Eb/N0 [dB]15(b) N = 32.
Figure 4: Bit Error Rate (BER).

20

Next, let us observe the property of the chaotic sequence
having the deviation of values. Figures 5 and 6 show the autocorrelation and the cross-correlation property, respectively.
As one can see the autocorrelation property, it can be said thatthe property improve as both q and r approach 1.0. Namely,
we can conﬁrm the reason that the BER has been improved
as shown in Fig. 4. However, the cross-correlation property- 351 -

got worse to the contrary. It is because many similar values
included in the chaotic sequence by having many values near
1.0. In other words, in the case where both parameters q andr are very close to 1.0, its sequence will become unsuitable
for multiplexing. Hence, it is an important future problem
to generate the chaotic sequence suitable for communication
which considered the balance of the parameters q and r.

0.001(q, r) = (0.8, 0.8)

Eb[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,Jun. 1996.
[2] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int . Journal of Bifurcation andChaos, vol. 10, no. 4, pp. 719-736, Apr. 2000.
[3] M. Sushchik, L. S. Tsimring and A. R. Volkovskii, “Performance Analysis of Correlation-Based Communication
Schemes Utilizing Chaos,” IEEE Trans. Circuits and Systems Part I, Vol. 47, pp 1684-1691, Dec. 2000.

(q, r) = (0.6, 0.2)
(q, r) = (0.5, 0.0)

0.0005

References(Conventional DCSK)

0[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication Systems, Springer, 2003.

-0.0005[5] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalizedcorrelation-delay-Shift-Keying Scheme for Noncoherent
Chaos-Based Communication Systems,” Proc ISCAS’04,
vol. 4, pp. 601-604, May 2004.

-0.0012468

10121416[6] S. Arai and Y. Nishio, “Variable Sequence Length Transmitter for Noncoherent Chaos Shift Keying,” RISP Journal of Signal Processing, vol. 10, no. 4, pp. 255-258,
Jul. 2006.

(a) N = 16.
0.001(q, r) = (0.8, 0.8)
(q, r) = (0.6, 0.2)
(q, r) = (0.5, 0.0)

Eb0.0005[7] G. Cimatti, R. Rovatti and G. Setti, “Chaos-Based Generation of Optimal Spreading Sequences for DS-UWB Sensor Networks,” Proc NOLTA’06, pp. 583-586, Sep. 2006.

(Conventional DCSK)

0-0.0005-0.001

51015202530(b) N = 32.
Figure 6: Cross-correlation property.

5. Conclusions
In this paper, we investigated the performance of thechaotic sequence with biased values using DCSK. As a results, we found that the chaotic sequence suitable for communication can be generated by conditioning the deviationof the values. Based on this result, investigating further thechaotic map used by this study is our future work.

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