Noncoherent Correlation-Based Communication
Systems Choosing Different Chaotic Maps
Shintaro ARAI† and Yoshifumi NISHIO†† Department of Electrical and Electronic Engineering, Tokushima University

2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Telephone: +81-88-656-7470, Fax: +81-88-656-7471
Email: {arai, nishio}@ee.tokushima-u.ac.jp
Abstract— This paper proposes a new noncoherent detection
system improved based on the differential chaos shift keying(DCSK) and the correlation delay shift keying (CDSK). In thisscheme, a transmitter changes chaotic maps for generating achaotic sequence efficiently depending on an initial value. Also,
the proposed method can choose the chaotic map by a very simple
algorithm. In order to investigate the proposed method, we carryout computer simulations and observe the performance.

(a)

xi

Chaotic signal
generator

Si
N

Delay Nblxi-N

Information symbols
bl(b)

Delay N

Ri
N/2

ΣRR

Ri+N

I. I NTRODUCTION
Recently, chaos-based digital communication systems arestudied actively [1]− [5]. Especially, it is attracted to develop
noncoherent detection systems which do not need to recover
the basis signals (unmodulated carries) at the receiver. The
differential chaos shift keying (DCSK) [1] and the optimalreceiver [2] are well known as a typical noncoherent system. In
addition, the correlation delay shift keying (CDSK) [3] similar
to the DCSK scheme is also regarded.
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 has aperformance similar to the optimal receiver, and this algorithm
is simpler than the optimal receiver. For the simulation results,
it was observed that its performance was better than that of theconventional transmitter for the suboptimal receiver. From thisresult, we expected that the performances of the noncoherent
detection systems also depend on chaotic maps used.
In this paper, we propose a new noncoherent detection
system improved based on correlation-based noncoherent detection schemes. Specifically, the performance of the noncoherent communication system is improved by changing achaotic map efficiently depending on an initial value in thetransmitter. After describing the motivation and the systemof the proposed method, we evaluate its performance by thecomputer simulation.

Fig. 1.

i=1i

Decoderi+N

Threshold

DCSK operation. (a) transmitter. (b) receiver.

A. DCSK
Figure 1 shows the block diagram of a DCSK transmitter(a) and receiver (b). In this scheme, the transmitter outputs achaotic sequence xi followed by the same sequence multipliedby 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 isgiven by
(1 ≤ i ≤ N )
 xi
Si =
(1)

bl xi−N (N + 1 ≤ i ≤ 2N )Also, the transmitted signal can be written as S =(S1 S2 · · · S2N ) by vector. Since the noise n =(n1 n2 · · · n2N ) is added to the transmitted signal by
the channel, the received signal can be written as R =(R1 R2 · · · R2N ) = S + n.
On the receiving side, it is evaluated by the correlation of
2 signals, which are obtained by dividing the received signals
into two halves (length N ). Thus, the output of the correlation
can be written asN
C1 =Ri Ri+N(2)
i=1

II. S YSTEM OVERVIEW
In this research, we use the differential chaos shift keying(DCSK) [1] and the correlation delay shift keying (CDSK) [3]as the correlation-based noncoherent detection system. Each
operation is described below.
1-4244-0921-7/07 $25.00 © 2007 IEEE.

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, and
it is regarded as the important issue to design DCSK.

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(a)

xi

Chaotic signal
generator

the skew tent map, the skewed Bernoulli shift map and theirreversal maps as chaotic maps, as shown in Fig. 3. In addition,
the chaotic sequence length N is 8. Also, the delay L of CDSK
is 3.

Si
Delay Lblxi-L

Information symbols
bl(b)

Delay L1

1

Xk+1=Xk
N

ΣRR

i=1

Ri

Fig. 2.

Xk+1

Xk+1

Ri-L
i

Decoderi-L

Xk+1=Xk

0.5

0.5

0

0

-0.5

-0.5

Threshold

CDSK operation. (a) transmitter. (b) receiver.

B. CDSK-1-1

Figure 2 shows the block diagram of a CDSK transmitter(a) and receiver (b). In this scheme, the transmitted signal is
the sum of the chaotic sequence xi and its delayed sequencexi−L multiplied by the information symbol bl (±1), where L
is the delay. Thus, the transmitted signal is given by
Si = xi + bl xi−L .

C2 =Ri Ri−L

0a

0.5

1

Xk

-1
-1

(a) Skew tent map.

-0.5

0a

0.5

1

Xk

(b) Skew tent map(reversal).

Xk+1

Xk+1

1

1

Xk+1=Xk

Xk+1=Xk

0.5

0.5

0

0

-0.5

-0.5

(3)

Since the sequence length is N as with DCSK, we can also
write the transmitted signal with the vector. The CDSK receiver is similar to the DCSK receiver. Therefore, the receivercan detect the information by multiplying the received signal
and its delayed signal, as shown in Eq. (4).
N


-0.5

-1
-1

-0.5

0a

(c) Skewed Bernoulli
shift map.

(4)

i=1

Fig. 3.

As in the case of DCSK, the decoded symbol is decided as“+1” or “−1” depending on C2 being larger or smaller than0.
As compared with DCSK, the CDSK transmitter does notneed the switch to generate the transmitted signal. In otherwords, it means that operation of the transmitter can be
performed continuously. Consequently, we can simplify anddesign the transmitter.
III. P ROPOSED M ETHOD
A. Motivation of Proposed Method
Before describing the proposed method, we explain the
motivation to use different chaotic maps.
As described in the introduction, it was observed thatthe effectiveness of the noncoherent detection system whichchanges the chaotic sequence length depends on the initialvalue. From this result, we expected that the performancesof the noncoherent detection systems also depends on chaotic
maps used. In order to investigate this expectation, we carriedout the following computer simulation.
On the transmitting side, the interval of the initial value[−1, 1] is divided into 128 sections. We choose one section
and selected 104 initial points from this section at random. By
using the chaotic sequences starting from these initial points,
we transmit 104 bits of information. In this study, we used

0.5

1

Xk

-1
-1

-0.5

0a

0.5

1

Xk

(d) Skewed Bernoulli
shift map (reversal).

Chaotic maps (a = 0.05).

Figures 4 and 5 show the simulation results of DCSK
and CDSK when Eb /N0 is 10dB, where the horizontal axes
and the vertical axes are the section and the bit error rate
(BER), respectively. From this figure, we can find that the BER
strongly depends on the selected section. For example, in thecase of DCSK, we can see that the BER of the reversal skew
tent map is better than others for the 25th section, as shownin Fig. 4(b). However, in the 35th section, the skew tent map
is better than others. Similarly, it can be also observed thateach BER of chaotic maps changes for every section in thecase of CDSK, as shown in Fig. 5(b). Furthermore, we canfind that the better chaotic map of each section is different for
DCSK and CDSK. These results mean that the performanceof DCSK and CDSK is changed depending on the initial valueof the chaotic sequence. In other words, it can be said that theperformance of the noncoherent detection system is changed
depending on the chaotic map.
From this result, we have devised new DCSK and CDSK
transmitters choosing the chaotic map depending on the initialvalue of the chaotic sequence.
B. Operation of Proposed Method
Figures 6(a) and (b) show the block diagram of the proposed
DCSK and CDSK transmitters, respectively. These transmitters are constructed by attaching two blocks to the conven-1434

Skew tent map
Skewed Bernoulli shift map

Skew tent map
Skewed Bernoulli shift map

Skew tent map (reversal)
Skewed Bernoulli shift map (reversal)

Skew tent map (reversal)
Skewed Bernoulli shift map (reversal)

0.35

0.18
0.16

0.3

0.14

0.25
0.1

BER

BER

0.12

0.08

0.2
0.15

0.06

0.1
0.04

0.05

0.02
0

0
0

20

40

60

80

100

0

120

20

40

60

Section(a) Section (0 ∼ 127) versus BER (Eb /N0 = 10dB).
Skew tent map
Skewed Bernoulli shift map

80

100

120

Section(a) Section (0 ∼ 127) versus BER (Eb /N0 = 10dB).

Skew tent map (reversal)
Skewed Bernoulli shift map (reversal)

Skew tent map
Skewed Bernoulli shift map

0.18

Skew tent map (reversal)
Skewed Bernoulli shift map (reversal)

0.35

0.16

0.3

0.14
0.25
0.1

BER

BER

0.12

0.08

0.2
0.15

0.06
0.1
0.04
0.05

0.02
0

0
20

22

24

26

28

30

32

34

36

38

40

20

22

24

26

Section

30

32

34

36

38

40

Section(b) Section (20 ∼ 40) versus BER (Eb /N0 = 10dB).
Fig. 4.

28(b) Section (20 ∼ 40) versus BER (Eb /N0 = 10dB).

Basis of proposed method (DCSK).

Fig. 5.

tional transmitter. Selection of the chaotic map and generationof the transmitted signal are operated in the following order.
First, one section is determined by the initial value of thechaotic sequence in the 1st block. Next, a better chaotic map
is chosen for the section in the 2nd block. Here, in order tochoose the better chaotic map, we use the result obtained by
the previous simulation in Figs. 4 and 5. Finally, the chaotic
sequence is generated using the better chaotic map in the
3rd block. Also, the operation after choosing is the sameas the conventional operation, namely, the transmitted signal
can be similarly written with Eqs. (1) and (3). Moreover, thereceiver of the conventional DCSK and CDSK can be also used
for the detection of the information. By using the proposed
method, the chaotic map can be changed efficiently and thetransmission capability can be improved.
IV. S IMULATION R ESULTS AND D ISCUSSION
In this section, we study the performance of the proposed
method by computer simulations. The simulation conditions
are as follows. In the transmitting side, the chaotic map ischosen from 4 types, the skew tent map, the skewed Bernoulli
shift map and their reversal maps. Also, the chaotic sequencelength N is 8 or 16, and the interval of the initial value, i.e.,the section is 128 or 1024. In CDSK, the delay L is 3. In the

Section check

Basis of proposed method (CDSK).

Selection of
Chaotic maps

xi

Chaotic signal
generator

Si
Delay NN

blxi-N

Information bits
bl(a) Proposed DCSK transmitter.
Section check

Selection of
Chaotic maps

xi

Chaotic signal
generator

Si
Delay Lblxi-L

Information symbols
bl(b) Proposed CDSK transmitter.
Fig. 6.

Proposed method.

channel, noise is assumed to be only AWGN. Hence, the noiseat the transmitter and the receiver are not considered. On thereceiving side, the same receiver as the conventional DCSK
and CDSK is used. Based on these conditions, the system
performance is evaluated by plotting the BER against Eb /N0
when 104 bits of information are transmitted.
Figure 7 shows the simulation results of the proposed
method. In order to compare the performance of the proposed

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1.0E+00

DCSK
CDSK

BER

1.0E-02

1.0E-03

1.0E-04

0

5

10

15

20

Eb/N0 [dB](a) N = 8.
1.0E+00

Proposed DCSK (Section:1024)
Proposed DCSK (Section:128)
Conventional DCSK
Proposed CDSK (Section:1024)
Proposed CDSK (Section:128)
Conventional CDSK
BPSK

1.0E-01

1.0E-02

1.0E-03

TABLE IAVERAGE TRANSMITTED ENERGY PER 1 CHIP OF CHAOTIC SEQUENCE(N = 8, L = 3, S ECTION = 128, Eb /N0 = 10[ DB ]).
Conventional method
0.333
0.454

Proposed DCSK
Conventional DCSK
Proposed CDSKConventional CDSK
BPSK

1.0E-01

BER

method, the performance of the conventional DCSK and
CDSK is shown together in Fig. 7(a) and (b), respectively.
In addition, Fig. 7(b) shows the comparison of performancefor different values of the sections. From the result, it canbe observed that the performance of these proposed methods
is better than the both conventional system. In particular, theperformance of the proposed CDSK improves significantly.
However, as for the proposed DCSK method, the performance
improve slightly. This reason is considered that transmitted
energy has influenced. Table I shows the average transmitted
energy per 1 chip of the chaotic sequence length. From thistable, we can find that the energy per 1 chip of the proposed
method is increasing. In the case of the DCSK, since the transmitter outputs the double sequence of the chaotic sequencelength N , the overall transmitted energy has increased. On
the other hand, since the transmitted signal of the CDSK
becomes the length with the sequence N added the Delay
L, the sequence length of the CDSK is shorter than that of
the DCSK. Namely, we consider that it influenced the resultthat the noise energy of the proposed DCSK became large thanthat of the conventional DCSK. However, although transmitted
energy increased, the performance of the proposed method
improved. Therefore, if the balance of transmitting energy and
BER is regarded, we expect that the performance may improvefurther.

1.0E-04

Proposed method
0.392
0.518

0

5

10

15

20

Eb/N0 [dB](b) N = 16.
Fig. 7.

Moreover, we can also confirm that the performance withmany sections is better than that with few sections, as shownin Fig 7(b). As a reason, we assume that it is because theselection accuracy of the chaotic map improved by increasing
the section. Thus, we consider that it is possible to improve
the performance by narrowing the interval of this initial value
in the case where the chaotic sequence length increases.
In general, increasing the chaotic sequence length of DCSK
is not necessary to improve the performance. Also, the performance of CDSK changes significantly depending on the
chaotic sequence length or the delay. However, we expectthat these issues are solved by using the proposed method.
Therefore, it can be said that choosing the chaotic map
depending on the initial value is effective.
V. C ONCLUSION
In this paper, we have proposed the new DCSK and CDSK
transmitter changing chaotic maps depending on the initialvalue. We also evaluated the BER performance of the proposed method by comparing with the conventional DCSK and
CDSK. As a result, it was observed that the performance of the
proposed method is better than that of the conventional system,
namely, we obtained the effectivity of the proposed method.

Simulation results.

In this study, we carried out this simulation using the small
chaotic sequence length. Thus, investigating the performanceof the proposed method which increased the chaotic sequencelength is our future work.
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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.
[3] M. Sushchik, L. S. Tsimring and A. R. Volkovskii, “Performance Analysis
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[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication
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[5] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalized correlation-delayShift-Keying Scheme for Noncoherent Chaos-Based Communication Systems,” IEEE Trans. Circuits and Systems Part I, vol. 53, no. 3, pp. 712721, Mar. 2006.
[6] S. Arai and Y. Nishio, “Variable Sequence Length Transmitter for
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