2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Error-Correcting Method Based on Chaotic Dynamics
for Noncoherent Chaos Communications
Shintaro ARAI† , Yoshifumi NISHIO† and Takaya YAMAZATO‡†Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN‡EcoTopia Science Institute, Nagoya University
Furo-cho, Chikusa-ku,Nagoya, 464-8603, JAPAN
Email: {arai, nishio}@ee.tokushima-u.ac.jp, yamazato@nuee.nagoya-u.ac.jp
Abstract—This paper proposes the error-correcting
method based on the chaotic dynamics for noncoherent
chaos communications. We generate successive chaotic sequences from the identical chaotic map. And for the next
sequence we set the initial value to the end value of theformer sequence. By such way we can create the successive chaotic sequences having the same chaotic dynamics.
This feature gives the receiver additional information tocorrectly recover the received noisy signal. Therefore, by
analyzing the chaotic dynamics at the receiver, it is possible to improve the error performance. In other words, error
performance of the receiver can be improved by the utilization of the suboptimal receiver with the analysis of chaotic
dynamics of the successive received sequence. As resultsof computer simulations, we conﬁrm about 3 dB gain in
BER performance compared to the conventional suboptimal receiver.
1. Introduction
Recently, research on digital communications systems
using chaos becomes a hot topic [1]− [8]. Especially, it isattracted to develop noncoherent detection systems which
do not need to recover basis signals (unmodulated carries) at the receiver. The differential chaos shift keying(DCSK) [1] and the optimal receiver [2] are well known
as a typical noncoherent system. In this study, we focus
attention on the optimal receiver.
The optimal receiver performs an optimal detection byusing the probability density function (PDF) between thereceived signals and the same chaotic map of the transmitting side. Thus the receiver can detect the information symbol by comparing the PDFs. However, the optimal receiver
suffers from a computational complexity due to the large
chaotic sequence length. Also, the large chaotic sequencegives rise to a decrease in the performance of the optimalreceiver. Thus, it is important to develop a receiver withperformance equivalent to the optimal receiver using different algorithms, i.e., a suboptimal receiver.
In our previous research, we proposed the suboptimalreceiver using the shortest distance approximation [9]. Instead of calculating the PDF, the proposed suboptimal receiver approximates the PDFs by calculating the shortest
distance between the received signals and the chaotic map.
As results of the computer simulations, we conﬁrmed thevalidity of the proposed suboptimal receiver as an approximation method of the optimal receiver.
Note that the suboptimal receiver shows a slight performance loss to the optimal receiver. In other words, the de-tection characteristic of the suboptimal receiver is not superior to the optimal receiver. However, we expect to improve
the performance of the suboptimal receiver by focusing features of chaos. Because we investigated and analyzed theinﬂuence on chaos communications systems by the chaotic
dynamics and its behavior previously [10]− [12] and conﬁrmed that the chaotic dynamics affect the performance ofchaos communication systems greatly. Therefore, we consider that it is important to develop communication systems
based on features of chaos.
The purpose of this is to propose an error-correcting
method based on the chaotic dynamics for the suboptimalreceiver. Let us consider two successive chaotic sequencesthat are generated from the identical chaotic map. We generate the second sequence whose initial value is the end
value of the former sequence. In this case, the chaotic dynamics of these two successive chaotic sequences are thesame. This feature gives the receiver additional information to correctly recover the received noisy signal. Therefore, by analyzing the chaotic dynamics at the receiver, it ispossible to improve the error performance. In other words,
error performance of the receiver can be improved by theutilization of the suboptimal receiver with the analysis ofchaotic dynamics of the successive received sequence.
As results of computer simulations, we conﬁrm about 3
dB gain in BER performance compared to the conventional
suboptimal receiver.
2. System Overview
We consider the discrete-time binary CSK communication system, as shown in Fig. 1. Detail of each block is
described below.
2.1. TransmitterIn the transmitter, a chaotic sequence is generated by achaotic map. In this study, we use a skew tent map which
is one of simple chaotic maps, as shown in Fig. 2(a), and it
is described by Eq. (1) 2x + 1 − ai(−1 ≤ xi ≤ a)
1+axi+1 =(1) −2xi + 1 + a (a < x ≤ 1)i
1−awhere a denotes a position of the top of the skew tent
map. The information symbol is modulated by Chaos Shift
Keying (CSK) which is a digital modulation system using- 652 -

Transmitter

becomes g (N ) (yj+1 ). Moreover, Seq “1” and Seq “0” arealso a successive sequence based on the chaotic dynamics.
These reasons are described by Sec. 3.

Symbol "1"

Chaotic signal
generator 1

Receiver
AWGN
Rj

Sj

Symbol "0"

Detector

DecodedsymbolS0 S1 ...

Chaotic signal
generator 0... SK-2 SK-1 Seq Seq"1" "0"Data(K bits N)

Figure 1: Block diagram of discrete-time binary CSK communication system.

Redundancy(2N or 4N)

Figure 3: Packet Format.

Xi+1

Xi+1
1

1

Xi+1=Xi

Xi+1=Xi

0.5

0.5

0

0

-0.5

-0.5

-1
-1

Sj

0a

-0.5

0.5

Xi

-1
-1

1

(a) Symbol “1”

-0.5

0a

0.5

2.2. Channel and NoiseThe channel distorts the signal and corrupts it by noise.
In this study, noise of the channel is assumed to be the additive white Gaussian noise (AWGN). Thus, the received
signals block is given by Rj = (Rj,0 Rj,1 · · · Rj,N −1 ) =Sj + AW GN .
Xi

1(b) Symbol “0”Figure 2: Chaos Shift Keying.
chaos. When the transmitter generates the signals, we use
chaotic sequences generated by different chaotic maps depending on the value of an information symbol. If the information symbol “1” is sent, Eq. (1) is used (Fig. 2(a)), andif “0” is sent, the reversed function of Eq. (1) is used (Fig.
2(b)). To transmit a 1-bit information, N chaotic signals
are generated, where N is the chaotic sequence length.
In this study, to perform the error-correcting by the receiver, information are transmitted by a packet which consists of a data part and a redundancy part, as shown in
Fig. 3, where Sj is the transmitted signal vector, j =(0, 1, · · · , K − 1), K is total bit per packet, Seq “1” andSeq “0” are the chaotic sequence generated by Fig. 2(a)and (b), respectively. Note that the transmitted signal vector Sj is different for each symbol. [When the symbol “1”is sent]

2.3. Receiver
The receiver recovers the transmitted signals from thereceived signals and demodulates the information symbol. Also, the receiver performs the error-correcting in thisstudy. Since we consider a noncoherent receiver, the receiver memorizes the chaotic map used for the modulation
at the transmitter. However, the receiver never knows theinitial value of chaos and the information symbol in thetransmitter. Before explaining the error-correcting, we introduce the operation of our suboptimal receiver to be thebasis for the proposed error-correcting.
[Operation of Our Suboptimal Receiver∼Shortest Distance Approximation Method∼ ]R=(Rj, l, Rj, l+1, Rj, l+2)

1

Symbol "1"
0.5

Rj, l+2 0

Sj = (xj f(1)(xj ) · · · f(i)(xj ) · · · f(N −1)(xj ))(2)

Symbol "0"-0.5[When the symbol “0” is sent]
Sj = (yj g (1) (yj ) · · · g (i) (yj ) · · · g (N −1) (yj ))

-1
-1

(3)

where f (i) and g (i) are the function of the skew tent map
for symbol “1” and “0” (Figs. 2(a) and (b)) respectively, iis the iteration of f or g, xj or yj denotes the initial value ofthe jth symbol = “1” or “0” respectively. When K bits data
are transmitted, the length of the data part becomes K ×N .
The redundancy is the sequence which has the known information symbol, and its length is different depending on
the error-correcting method. In addition, the initial valueof the chaotic sequence is also different in each chaotic
signal generator and is updated each packet, namely, thechaotic sequence generated by each generator in the packet
is a successive sequence based on the chaotic dynamics. As
an example, let us assume the jth and (j +3)th symbols are“1” (i.e. the (j + 1)th and (j + 2)th symbols are “0”). In
this case, xj+3 becomes f (N ) (xj ). In the same way, yj+2

1
0.5

-0.5
Rj, l

0

0
0.5

-0.5

Rj, l+1

1 -1

Figure 4: Operation of Our Suboptimal Receiver (Nd = 3).
Our suboptimal receiver calculates the shortest distance
between the received signals and the chaotic map in theNd -dimensional space using Nd successive received signals (Nd : 2, 3, · · ·). As an example, we explain the caseof Nd = 3. Figure 4 shows the 3-dimensional space of the
skew tent map whose coordinates correspond to the threesuccessive received signals R = (Rj,l , Rj,l+1 , Rj,l+2 ) ,where Rj,l denotes the jth symbol and lth signal of thereceived signals (l = 0 1 · · · N − 3). We can calculate
the shortest distance between R and the chaotic map using the scalar product of the vector. Any two points of
P0 = (x0 , y0 , z0 ) and P1 = (x1 , y1 , z1 ) are chosen fromeach straight line in the space of Fig. 4, as shown in Fig. 5.

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P = (X, Y, Z)

P0=(x0, y0, z0)

D

v0

u

P1
= (x1, y1, z1)

...

...

j-1

j

j+1

j+2

S

S

Seq
"1"

Seq
"0"

N

N

N

N

K-1
K-2
"1" or "0" "1" or "0"

Data

R = (Rj, l, Rj, l+1, Rj, l+2)(a)Figure 5: Calculation of the shortest distance.

if

D

=

||P − R||j-1

min D0 ( S

j

j+2

K-1

Seq
"0"

j

S

K-1
K-2
"1" or "0" "1" or "0"

Using Fig. 5, we can calculate P = (X, Y, Z) and theshortest distance D by the following equations.
P = (X, Y, Z) = (u · v0 ) u + P0

Redundancy) +D0 ( S

"1" or "0")= (j-1, j) = (0, 0)

(4)

j-1

min D0 ( S

K-2"1" or "0"

(5)

j+2
Seq
"0"

j

j+1

K-1
"1" or "0"

Seq
"1"

) +D1 ( S)

= (j-1, j) = (0, 1)

where
Unit vector u
v0

P1 − P0||P1 − P0 ||= R − P0 .
=

j-1

(6)

min D1 ( SK-2"1" or "0"(7)

Note that if the point is outside the cube, we calculate thedistance between the point and the nearest edges of themaps.
For the 3-dimensional case, there are four straight lines
in the space. Therefore, the minimum value in four distances is chosen as the shortest distance D1 for symbol “1”.
In the same way, D of symbol “0” is chosen as D0 . We calculate both ∑of D1 and D0 for all l and ﬁnd their summations∑
D1 and
D0 . Finally,
symbol∑∑ we decide∑ the decoded∑
as 1 (or 0) for D1 < D0 (or D1 > D0 ).
The calculation of the shortest distance can be extended
to Nd -dimensional space for Nd ≥ 4.
In our previous research, we observed that the performance of our method corresponds that of the optimal receiver when N = Nd -dimension [9]. Therefore, we conﬁrmed the validity of the proposed suboptimal receiver as
the approximation method of the optimal receiver.
3. Proposed Error-Correcting Method
First of all, we explain a motivation of the proposed
method. To develop the error-correcting method, we focus
attention on successive chaotic sequences generated fromthe identical chaotic map based on the chaotic dynamics.
Let us consider two successive chaotic sequences that are
generated from the identical chaotic map. Here, we assume that the initial value of the second sequence is the end
value of the former sequence. In this case, the chaotic dynamics of these two successive chaotic sequences are thesame. This feature gives the receiver additional information to correctly recover the received noisy signal. Therefore, we consider that it is possible to improve the error
performance by analyzing the chaotic dynamics at the receiver.
To take advantage of these characteristics, we propose
the error-correcting method based on the chaotic dynamics.
Our proposed error-correcting method calculates the shortest distance by the double or triple of the sequence length
N which used for the modulation in the transmitter, i.e. thedouble or triple dimension (Nd = 2N or 3N ). Figure 6(a)

j+1
Seq
"1"

j

j+2

K-1
"1" or "0"

Seq
"0"

) +D0 ( S)

= (j-1, j) = (1, 0)
j-1

min D1 ( S

K-2

j

j+1

K-1

Seq
"1"

j

SK-1

"1" or "0" "1" or "0") +D0 ( S

"1" or "0")= (j-1, j) = (1, 1)(b)Figure 6: Proposed Error-Correcting Method (Nd = 2N ).
shows the image of the operation of the error-correcting. To
simplify the explanation, we assume the redundancy length
to be 2N . The redundancy is the sequence which has the
known information, “1” and “0”. Here, let us expect thatthe j = (K − 1)th bit is “1” or “0”. Since the (K − 1)th
bit is “1” or “0” and the redundancy has “1” or “0” certainly, the (K − 1)th bit sequence also has the relation witheither of two types of the sequence based on the chaotic
dynamics. Moreover the (K − 2)th sequence also has therelation with either of three types of the sequence. Thus we
can consider the j and j − 1th bit as the four pattern, as
shown in Fig 6(b), where D1 (.) and D0 (.) mean the shortest distance between the chaotic map of Symbol “1” and“0”, respectively. Since the sequence connected two types
becomes the double sequence length, we can calculate theshortest distance using Nd = 2N -dimensional space. We
calculate each distance like the pattern of Fig 6(b). Finally,
we choose the pattern of the shortest distance out of thefour patterns, and the (K − 1)th bit is decided accurately.
By using this determined symbol for detection of the next
bit, these operation can be shifted and it can demodulate
to all information. Therefore, the likelihood of information
can improve by expecting the combination of the sequencebased on the chaotic dynamics, and we can detect information in high accuracy. In this case, the coding rate becomesK
K+2 .
In the same way, we can calculate the shortest distanceusing Nd = 3N -dimensional space to perform the errorcorrecting. In this case, the redundancy length is 4N , thecombination of the sequence becomes the eight patterns
Kand the coding rate becomes K+4.

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4. Simulation Results and Discussion
To evaluate the proposed error-correcting method, we
carry out computer simulations with following simulation
conditions. In the transmitting side, the packet consists ofthe data part with K = 64 bits and the redundancy partwith 2N or 4N . In this case, each coding rate becomes
0.96 or 0.94. Here, the parameter of the skew tent map
is ﬁxed as a = 0.05, and we use the chaotic sequencelength N = 3 and 4. To compare the performance of Nd dimensional space to calculate the shortest distance, we use
Nd = 2N -dimensional space, and Nd = 3N -dimensional
space. Based on these conditions, we iterate the simulation
10,000 times and calculate the average of BERs.
1.0E+00

Error-corecting method (Nd=3N)
Error-corecting method (Nd=2N)
Conventional method(w/o coding)

5. Conclusions
In this study, we have proposed the error-correcting
method based on the chaotic dynamics for the suboptimalreceiver. We generate the chaotic sequence from the samechaotic map and make the successive sequences to havethe same chaotic dynamics. We perform the detection by
analyzing the chaotic dynamics of the received sequences.
As results, we have conﬁrmed that the likelihood of the received symbol increases and obtained the 3dB gain in error
performance.

BER

1.0E-01

1.0E-02

1.0E-03

1.0E-040

5

10

Eb/N0[dB]15

20

Error-corecting method (Nd=3N)
Error-corecting method (Nd=2N)
Conventional method(w/o coding)

1.0E-01

References[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.

(a) N = 3.
1.0E+00

not only by the shortest distance between the received signals and the chaotic map but also analyze the chaotic dynamics of the successive symbols, the likelihood of thereceived symbol increase and the performance improvement is achieved. Moreover, although the cording rate isvery high, the proposed method can well correct the error.
Therefore, we can say that the efﬁciency of the proposederror-coding is excellent.
However, we also conﬁrm that the curve of the proposed
method of N = 3 with Nd = 3N -dimensional space corresponds that with Nd = 3N -dimensional, as shown in
Fig. 7(a). We have not clearly understood the reason why
two curves correspond, yet. We would like to investigate
and analyze the chaotic sequence and its behavior in detailwhen our error-correcting method is used.

[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.

BER[3] M. Sushchik, L. S. Tsimring and A. R. Volkovskii, “Performance
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1.0E-02[4] M. Hasler and T. Schimming, “Optimal and suboptimal chaos receivers,” Proc. IEEE, vol. 90, Issue 5, pp. 733-746, May 2002.
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1.0E-04[6] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalized correlationdelay-Shift-Keying Scheme for Noncoherent Chaos-Based Communication Systems,” IEEE Trans. Circuits and Systems Part I, vol. 53,
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0

5

10

Eb/N0[dB]1520(b) N = 4.

[7] L. E. Larson, J-M. Liu, L. S. Tsimring, Digital Communications Using Chaos and Nonlinear Dynamics, Springer, 2006.

Figure 7: Simulation results.

[8] G. Cimatti, R. Rovatti and G. Setti, “Chaos-Based Spreading in DSUWB Sensor Networks Increases Available Bit Rate,” IEEE Trans.
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Figures 7(a) and (b) show the BERs versus Eb /N0 for
N = 3 and N = 4, respectively. To compare theperformance of the error-correcting method, Figs. 7(a)and (b) also show the performance of our suboptimal receiver, namely, the performance without the coding. Fromthese results, we observe that the both BERs of the errorcorrecting method show gain over the system without thecoding. Signiﬁcant improvements are observed for N = 4
with Nd = 3N -dimensional. In this case, since the receivercalculate the shortest distance using Nd = 12-dimensional
space, the computation complexity also increase. As ourreceiver calculates the likelihood of the received symbols[9] S. Arai and Y. Nishio, “Detection of Information Symbols and Sequence Lengths Using Suboptimal Receiver for Chaos Shift Keying,”
Proc NOLTA’06, pp. 799-802, Sep. 2006.
[10] 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.
[11] S. Arai and Y. Nishio, “Noncoherent Correlation-Based Communication Systems Choosing Different Chaotic Maps,” Proc ISCAS’07,
pp. 1433-1436, May 2007.
[12] S. Arai, Y. Nishio and T. Yamazato, “Analysis on Chaotic Sequencewith Biased Values for Noncoherent Chaos Communication,” Proc
NOLTA’07, pp. 144-147, Sep. 2007.

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