2009 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 11-12, 2009

Error-Correcting Scheme Without Redundancy
Code Using Chaotic Dynamics for Noncoherent
Chaos Communications(Invited Paper)
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{arai, ozawa}@aut.ac.jp† Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPANnishio@ee.tokushima-u.ac.jp‡ EcoTopia Science Institute, Nagoya University
C3-1, Furo-cho, Chikusa-ku, Nagoya 464-8063 JAPANyamazato@nuee.nagoya-u.ac.jp

Abstract— This paper proposes a novel error-correcting
scheme for noncoherent chaos communications. By using thechaotic dynamics which is one of characteristics of chaos our proposed scheme is possible to operate the error-correcting without
the physical redundant code. As results of computer simulations,
we conﬁrm about 1–2 dB gain in BER performance as comparedwith the conventional noncoherent chaos communications.

I. I NTRODUCTION
Chaos communication systems are an interesting topic in
the ﬁeld of engineering chaos [1]– [4]. Especially, many
researchers focused on the development of noncoherent detection systems which are demodulation techniques using signals
modulated by chaos only or chaotic systems. Differential chaos
shift keying (DCSK) [1] and the optimal receiver [2] arewell-known 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].
In our previous research, we proposed the suboptimal receiver using the shortest distance approximation [5]. Instead ofcalculating the PDF, the proposed suboptimal receiver approximates the PDF by calculating shortest distances between thereceived signal and the chaotic maps and performs detection of
the transmitted symbol. As results of the computer simulations,
we conﬁrmed the validity of the proposed suboptimal receiver
as an approximation method of the optimal receiver.
To improve the performance of the suboptimal receiver, weare concerned with an error-correcting scheme. Especially, wefocus on characteristics of chaos to design an error-correcting
method. Chaotic sequences obtained from a certain classof difference equations are non-periodic, sensitive to initialconditions and it is difﬁcult to predict their future behavior
using past observations. In other words, chaotic sequences
are generated according to speciﬁc rules, i.e., the chaoticdynamics.
In this study, we focus on the chaotic dynamics and propose
a novel error-correcting scheme. In our proposed system, two
successive chaotic sequences are generated from the samechaotic map; the second sequence is generated with an initialvalue which is the last value of the ﬁrst sequence. In thiscase, successive chaotic sequences having the same chaotic
dynamics are created. This feature gives the receiver additional
information to correctly recover the information data andthus improves the bit error performance of the receiver. Inaddition, as compared with general error correcting methods,
our proposed scheme is possible to operate the error-correcting
without the physical redundant code.
We carry out computer simulations and evaluate the performance of the proposed method.
II. S YSTEM OVERVIEW
We consider the discrete-time binary CSK communication
system with the error correcting, as shown in Fig. 1. Detail ofeach block is described below.
In the transmitter, binary data are encoded by chaotic
sequences generated by a chaotic map. In this study, we use
a skew tent map which is one of simple chaotic maps, and it
is described by Eq. (1) 2x + 1 − ai(−1 ≤ xi ≤ a)
1+a(1)
xi+1 =
 −2xi + 1 + a (a < x ≤ 1)i1−awhere a denotes a position of the top of the skew tent map. Our
encoder is designed based on Chaos Shift Keying (CSK) which
is a digital modulation system using chaos. Figure 2 shows ourencoder for our error-correcting scheme. To perform the error
correction at the receiver, K information bit are transmitted as
K signal blocks (0, 1, · · · , j, · · · , K − 1), The encoder selects
a chaotic signal generator according to the symbol. If thesymbol “1” is sent, Eq. (1) is used, and if “0” is sent, thereversed function of Eq. (1) is used. Thus, the signal vector
Sj is different for each symbol.
[When the symbol “1” is sent]- 14 -

Sj = (xα , f (1) (xα ), · · · , f (i) (xα ), · · · , f (N −1) (xα ))

(2)

AWGN

Transmitter

Data

Fig. 1.
System.

SEncoderbased on CSK

Ex. N=2, K=4

ReceiverR

Error-Correcting
based onChaotic Dynamics

SuboptimalDetector

R = (r0, r1, r2, r3, r4, r5, r6, r7)
OutputSuboptimal Detector

Block Diagram of Discrete-Time Binary CSK Communication

Detection of symbolfor each block

Ex. N=2, K=4, Data=(1 0 0 1)

Decoded= d0
data

Chaotic sequence “1” =x0, x1, x2, ...

Symbol “1”...1 0 0 1

Chaotic Signal
Generator 1

“1”

“0”

“0”

NN

N

OutputNFig. 3.

y0, y1, y2, ...

(3)

As one can see, the initial value of the 4th symbol and 3rdsymbol is generated by the end value of 1st symbol and 2nd
symbol, respectively.
The 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 R = (r0 , r1 , · · · , rKN −1 ) = S + AW GN .
The receiver recovers the transmitted signals from the received signals and demodulates the information symbol. Also,
the receiver performs the error correction in this study. Since
we consider a noncoherent receiver, the receiver memorizes
the chaotic map used for the modulation at the transmitter.
However, the receiver never knows the initial value of chaos inthe transmitter. Our proposed error-correcting method consistsof the suboptimal detector and the error correction basedon chaotic dynamics, as shown in Fig. 3. First of all, thereceiver performs the noncoherent detection for each received
block and demodulates each symbol. In this study, we apply
the our suboptimal detection algorithm as the noncoherentdetection [5]. After demodulation of each symbol, the receiver
performs the error-correcting method. Here, we introduce theoperation of our suboptimal detection.

d2Dmv0(i)where f and g are the function of the skew tent map forsymbol “1” and “0”, respectively, i is the iteration of f org, α = N × j, xj or yj denotes the initial value of the jth
symbol = “1” or “0” respectively, N is the chaotic sequencelength per 1 bit. When K bit data is transmitted, the lengthof the data becomes K × N . An initial value is chosen atrandom when beginning to make signal blocks and is different
in each chaotic signal generator. In addition, the j-th sequenceis generated from the initial value which is the end value of theformer sequence with same symbol of j-th bit. As an example,
we assume N = 2, K = 4 and the data are (1, 0, 0, 1) shownin Fig. 2. In this case, the transmitted signal vector S is given
as follows.
S = (S0 , S1 , S2 , S3 )= (x0 , x1 , y0 , y1 , y2 , y3 , x2 , x3 )= (s0 , s1 , s2 , s3 , s4 , s5 , s6 , s7 )
(4)

d1

d3

P = (X, Y)

P0=(x0, y0)

[When the symbol “0” is sent]

(i)

= d0

Operation of proposed error-correcting method.

Encoder based on CSK for error correction.

Sj = (yα , g (1) (yα ), · · · , g (i) (yα ), · · · , g (N −1) (yα ))d3Analysis of ChaoticDynamics

Chaotic sequence “0” =Fig. 2.

d2

Error Correcting“1”

S = ( S0, S1, S2,S3 )= (x0, x1, y0, y1, y2, y3, x2, x3)= (s0, s1, s2, s3, s4, s5, s6, s7)

Chaotic Signal
Generator 0
Symbol “0”d1uP1= (x1, y1)

R’=(rα+i, rα+i+1)
Fig. 4.

Calculation of shortest distance.

Our suboptimal detection calculates the shortest distance
between the received signals and the chaotic map in the Nd dimensional space using Nd successive received signals (Nd :2, 3, · · ·). As an example, we explain the case of Nd = 2.
In this case, we consider two successive signal samples R0 =(rα+i , rα+i+1 ) as coordinate of chaotic map. To decide which
map is closer to the point R0 the shortest distance between thepoint and the map has to be calculated. Therefore, the receivercan calculate the shortest distance using the scalar product ofthe vector.
Any two points P0 = (X0 , X0 ) and P1 = (X1 , Y1 ) arechosen from each straight line in the Nd -dimensional space,as shown in Fig. 4. Using Fig. 4, the detector can calculate
the point P = (X, Y ) closest to R and the shortest distanceD using the following equations.

DP = (X, Y ) = (u · v0 ) u + P0

(5)

= ||P − R0 ||√22=(X − rα+i ) + (Y − rα+i+1 )(6)whereunit vector uv0

P1 − P0||P1 − P0 ||= R0 − P0 .

=(7)
(8)

In the case of 2-dimensional space, there are two straightlines 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
perform these operations until the last∑sample (i.e.,∑ rα+N −1 ) isincluded, and ﬁnd their summations D1 and∑ D0 . Finally,∑we decidethe∑decoded symbol as 1 (or 0) forD1 < D0∑(orD1 >
D0 ). The calculation of the shortest distancecan be extended to Nd -dimensional space for Nd ≥ 4.

- 15 -1.0E+00

Ex. N=2,K=4, Data=(1 0 0 1), Error occurs at d2
error
Decoded data=(d0 ,d1, d2, d3)=(1 0 1 1 )Conventional method(w/o coding)

1.0E-01

(a)
d2

d3

d1

BER

d0

Dbasis = D1(r0, r1, r4, r5, r6, r7) + D0(r2, r3)(b)Reversed0=1
d0’=0

DR(0)=D1(r4, r5, r6, r7) + D0(r0, r1, r2, r3)

d1=0

d1’=1

DR(1) = D1(r0, r1, r2, r3 , r4, r5, r6, r7)

d2=1
1

d2’=0
0

DR(2)=D1(r0, r1, r6, r7) + D0(r2, r3, r4, r5)

d3=1

d3’=0

DR(3)=D1(r0, r1, r4, r5) + D0(r2, r3, r6, r7)

1.0E-02

1bit correction

1.0E-03

1.0E-04

Become the smallest distance02

D1(*): Shortest distance between the map of Symbol “1” and (*)D0(*): Shortest distance between the map of Symbol “0” and (*)

Analysis of Chaotic Dynamics using Suboptimal Detection.

III. E RROR - CORRECTING METHOD WITHOUTR EDUNDANCY CODEIn this section, we describe the error-correcting schemewithout redundancy code. For ease of explanation, we use
Fig. 5 and explain an operation of the proposed errorcorrecting scheme. Here, we use same assumption in theexplanation of the encoder (Fig. 2). Also, we assume that
decoded symbols become (1, 0, 0, 1) and the detection error
has occurred at the 2nd symbol (d2 ), namely the case of 1 bitcorrection.
First of all, the receiver sorts the received signal samples
based on decoded symbols and analyzes the chaotic dynamics
of two sequences which are sorted according to decoded
symbols, as shown in Fig. 5(a). If the receiver can detects symbols and sorts blocks correctly, we can obtain two successive
chaotic sequences based on the chaotic dynamics. However,if the detection error is occurred when the receiver detects
symbols, the sorted sequence mixes two chaotic sequences
which differ in the chaotic dynamics. We focus on these characteristics of chaos for the error-correcting. For analyzing thechaotic dynamics, the receiver apply our suboptimal detection
algorithm, i.e., the calculation of the shortest distance between
the chaotic maps and two sorted received sequences. Thus, wedeﬁne a reference distance Dbasis as follows.
Dbasis

6810

12

14

16

18

20

Eb/N0[dB]Fig. 6.

Fig. 5.

4= D1 (Sequence of decoded symbol “1”)+ D0 (Sequence of decoded symbol “0”) (9)where D1 (.) and D0 (.) mean the shortest distance between
the chaotic map of Symbol “1” and “0”, respectively.
Next, we calculate the distance DR (j) for comparing withDbasis , where the subscript R means initial character of“Reverse”. This equation means the shortest distance between
sorted sequence when the j-th decoded symbol is reversed andthe chaotic map corresponding to their sequences. Namely,
we assume the detection error is occurred at j-th symbol andcalculate DR (j). If the receiver can detects symbols and sorts
blocks correctly, DR (j) becomes lager values as comparedwith Dbasis . On the other hands, if the detection error isoccurred, some of DR (0)–DR (K − 1) become smaller as

BER performance (K = 32).

compared with Dbasis . The reason for changing the values
is also to change combinations of the chaotic dynamics.
Therefore, the receiver selects the smallest distance fromDbasis and DR (j) and corrects an error.
For instance, Fig. 5(b) shows conceivable combinations ofsorted sequences and calculates DR (j) when the detection
error has occurred at the 2nd symbol. In this case, DR (2)becomes the smallest distance as compared with Dbasis andother other DR (j). Thus, the receiver can determine that thedetection error is occurred at the 2nd symbol.
IV. S IMULATION R ESULTIn this section, we evaluate the performance of the proposederror-correcting method. We carry out computer simulations
and calculate a bit error rate (BER). The simulation conditions
are as follows. In the transmitting side, we assume K = 32.
The parameter of the skew tent map is ﬁxed as a = 0.05. The
chaotic sequence length per 1 bit is N = 4. For calculationof the shortest distance, we use 4-dimensional space, namely
Nd = N = 4. Based on these conditions, we iterate thesimulation 10,000 times and calculate BER, the accuracy rate
and the computation time.
Figure 6 show the BER versus Eb /N0 for K = 32. We
plot the performance of the proposed error-correcting method
and the performance of the conventional method, namely,
the performance without the error-correcting method in Fig.6.
From these ﬁgures, we can conﬁrm that the advantage gainedin BER performance of the proposed error-correcting method
is about 1–2 dB compared to conventional method.
V. C ONCLUSIONSIn this study, we have focused on the chaotic dynamics andproposed the error-correcting scheme without the redundancy
code. As results, we have conﬁrmed that the 1–2dB gain inerror performance of the proposed method with N = Nd = 4.
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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. 719-736, Apr. 2000.
[3] M. Hasler and T. Schimming, “Optimal and suboptimal chaos receivers,”
Proc. IEEE, vol. 90, Issue 5, pp. 733-746, May 2002.
[4] P. Stavroulakis,Chaos Applications in Telecommunications, Talor & Francis Groupr, 2006.
[5] 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.

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