2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Improvement of Error-Correcting Method Based on Chaotic Dynamics
for Noncoherent Chaos Communications
Shintaro ARAI† , Yoshifumi NISHIO‡ and Takaya YAMAZATO†††Dept. of Communication Network Engineering, Kagawa National College of Technology
551 Kohda, Takuma-cho, Mitoyo, Kagawa 769-1192, JAPAN‡Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506 JAPAN††Institute of Liberal Arts and Science, Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8063 JAPAN
Email: arai@cn.kagawa-nct.ac.jp, nishio@ee.tokushima-u.ac.jp, yamazato@nagoya-u.ac.jp
Abstract—This paper focuses on characteristics ofthe chaotic dynamics and improves our previous errorcorrecting method using them for noncoherent chaos communications. Our previous method is performed by using
a chaotic sequence generated according to the chaotic dynamics. In this case, it is very difﬁcult to recover data without a successive sequence based on the chaotic dynamics.
We focus on this feature and consider that an improved
method separates and reconstructs the chaotic dynamics
of the sequence according to a speciﬁc rule. Namely,
the separation and reconstruction of the chaotic dynamics
can be applied for our improved method as additional information. As results of simulations, we have conﬁrmedthat the advantage gained in BER performance of our improved method is about 2–2.5 dB compared to a conventional method (without coding).
1. Introduction
Chaos communication system is an interesting topic in
the ﬁeld of engineering chaos [1]– [5]. Especially, many researchers have focused on the development of noncoherent
detections which do not need to use basis signals (unmodulated carriers) for demodulation at a receiver. Differential
chaos shift keying (DCSK) [1] and the optimal receiver [2]are well-known as typical noncoherent systems. Moreover,
it is also important to develop a suboptimal receiver, which
has a performance equivalent to or similar to the optimalreceiver, using more efﬁcient algorithms [3].
In our previous research, we focused on the chaotic dynamics and proposed the error-correcting method using thechaotic dynamics [6]. In our error-correcting method, two
successive chaotic sequences are generated from the samechaotic map; the second sequence is generated with an initial value 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
and thus improves the bit error performance of the receiver.
Further, our method is operated without a redundancy bit
sequence referred to perform an error correction in standard communication systems. As results of computer simulations, we conﬁrmed that the advantage gained in the bit
error rate (BER) performance of our method is about 1–1.5
dB compared to a conventional method (without coding).
Moreover, we found that the chaotic dynamics had a great
inﬂuence on the demodulation of chaos communications.
In this study, we further focus on the characteristics

Transmitter
Data

AWGN

Encoder

Receiver
Suboptimal
Detector

Error Correction

OutputFigure 1: Block Diagram of CSK Communication Systemwith Error-Correcting method.
of the chaotic dynamics and improve our error-correcting
method using them. Our error-correcting method is performed by using a chaotic sequence generated according
to the chaotic dynamics. In this case, it is very difﬁcult torecover data without a successive sequence based on the
chaotic dynamics. We consider that the chaotic dynamics of the sequence is purposely separated using a speciﬁcrule. A transmitter shufﬂes the order of the sequence according to the speciﬁc rule for separating the chaotic dynamics, and a receiver reconstructs the original chaotic sequence using the same rule. Namely, the separation andreconstruction of the chaotic dynamics can be applied forour error-correcting method as additional information. In
this paper, we describe operations of our improved errorcorrecting method. In addition, we carry out computer simulations and evaluate BER performance of the improved
method.
2. System Overview
We consider the discrete-time binary Chaos Shift Keying(CSK) communication system with the error correcting, asshown in Fig. 1. Detail of each block is described below.
2.1. TransmitterIn 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 as follows.
 2x + 1 − ai(−1 ≤ xi ≤ a),
1+axi+1 =(1) −2xi + 1 + a (a < xi ≤ 1).
1−aWhere i = 0, 1, · · · , N − 1, a denotes a position of
the top of the skew tent map. Our encoder is designed
based on CSK which is a digital modulation scheme us-- 801 -

Ex. N=2, K=4, Data=(1, 0, 0, 1), U=2, V=2
Symbol “1”Chaotic Signal
Generator 1

K symbols
1001

S’ = ( S’0, S’1, S’2, S’3 )= ( X0,Y0, Y1,X1 )= (x0, x1, y0, y1, y2, y3, x2, x3)

Symbol “0”S = ( S 0,
S1, S2,S3 )= ( X0,Y0, Y1,X1 )= (x0, x1, y2, y3, y0, y1, x2, x3)= (s0, s1, s2, s3, s4, s5, s6, s7)

Transmitting sequence

π3

π1

π4

U=3
V=1

π3

U=1
V=3

Switch according toa number of “1” & “0”Figure 2: Operation of Encoder (ex. N = 2, K = 4).
ing chaos. Figure 2 shows our encoder for our errorcorrecting method. To perform the error correction at thereceiver, K information bit are transmitted as K signal
blocks (0, 1, · · · , j, · · · , K − 1). The encoder selects achaotic signal generator according to the symbol. Here, wedenote “f (i) (x0 )” as the skew tent map (Eq. (1)). Whenthe symbol “1” and “0” are sent, f (i) (x0 ) and g (i) (x0 ) (=−f (i) (x0 )) are used, respectively. Also, we denote U and
V as a total number of symbol “1” and “0”, respectively.
Thus, the signal vector Xu and Yv are different for eachsymbol.
When the symbol “1” is sent,
Xu = (xj , f (1) (xj ), · · · , f (i) (xj ), · · · , f (N −1) (xj )).
(2)
When the symbol “0” is sent,(yj ), · · · , g (yj ), · · · , g(i)(N −1)(yj )). (3)Where u = 0, 1, · · · , U − 1, v = 0, 1, · · · , V − 1, xj or
yj denotes the initial value of the jth symbol = “1” or “0”respectively, N is the chaotic sequence length per 1 bit.
When K bit data is transmitted, the amount of the data becomes K × N . Thus, the amount of the data per bit of the
proposed method becomes the same as the standard CSK.
An initial value is chosen at random when beginning to
make signal blocks and is different in each chaotic signal generator. In addition, the j-th sequence is generated
from the initial value which is the end value of the former 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), as shown in Fig. 2. In this case, the signal vector
S0 (= S0 0 , S0 1 , · · · , S0 K−1 ) is given as follows.
S0

U =4
V =0
U =3
V =1
U =2
V =2
U =1
V =3
U =0
V =4

X 0 , X 1 , X2 , X3–
X 0 , X1 , X2
Y0
X 0 , X1
Y 0 , Y1
X0
Y 0 , Y1 , Y2–
Y 0 , Y 1 , Y2 , Y3

X1 , X3 , X0 , X2–
X1 , X0 , X2
Y0
X0 , X1
Y1 , Y0
X0
Y0 , Y2 , Y1–
Y2 , Y0 , Y3 , Y1

π2

U=2
V=2

π4

Yv = (yj , g

After

π0

π0

U=0
V=4

(1)

Beforeπ2Interleaver

U=4
V=0

“1” “0” “0” “1”

Number ofeach symbol

π1Chaotic Signal
Generator 0
Chaotic sequence “0” = x0, x1, x2, ...

Table 1: Example Pattern of Interleaver (πk ): K = 4.

“1” “0” “0” “1”

Chaotic sequence “1” = x0, x1, x2, ...

= (S0 0 , S0 1 , S0 2 , S0 3 ) = (X0 , Y0 , Y1 , X1 )= (x0 , x1 , y0 , y1 , y2 , y3 , x2 , x3 ).

As one can see, the initial value of the 3rd symbol (S0 3 )and 2nd symbol (S0 2 ) is generated by the end value of 0th
symbol (S0 0 ) and 1st symbol (S0 1 ), respectively.
In this study, the encoder shufﬂes the chaotic sequencein each symbol using an interleaver πk (k : 0, 1, · · · , K).
Here, we explain conditions and operations of the interleaver. As an advance preparation, we prepare K + 1
interleavers. These shufﬂe patterns are different according to the number of each transmitting symbol. Each interleaver’s pattern is set at random and does not correspond with other one. In addition, the interleaver only
shufﬂes the signal vector of each symbol. Note that theorder of the sequence having each signal vector does notchange. For example, Tab. 1 shows the patterns of the interleaver when K = 4. The encoder counts the numberof each symbol and chooses the interleaver corresponding to this number. In the case of data=(1, 0, 0, 1), π2
is chosen because the number of the symbol “1” and “0”is two respectively. Therefore, the transmitting sequence
S(= S0 , S1 , · · · , SK−1 ) is given as follows.
S= (S0 , S1 , S2 , S3 ) = (X0 , Y1 , Y0 , X1 )= (x0 , x1 , y2 , y3 , y0 , y1 , x2 , x3 )= (s0 , s1 , s2 , s3 , s4 , s5 , s6 , s7 ).

(4)

As one can see, Y0 switches positions with Y1 accordingto π2 .
2.2. Channel
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
sequence is given by R = (r0 , r1 , · · · , rKN −1 ) = S +AWGN.
2.3. Receiver
The receiver demodulates the information symbol fromthe received sequence. Also, the receiver performs the error
correction in this study. Since we consider the noncoherent
chaos communication, the receiver memorizes the chaotic
map used for the modulation at the transmitter. However,
the receiver never knows the initial value of chaos in thetransmitter. In the next section, we describe operations of
our improved error-correcting method in detail.
3. Improved Error-Correcting method
Our error-correcting method consists of the suboptimal
detector and the error correction based on chaotic dynamics, as shown in Fig. 3.

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π−1
0

U=4
V=0

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

Received sequence
Suboptimal Detector
Detection of symbolfor each block

Deinterleaver

Dbasisπ−11Reverse of
0th symbol

U=3
V=1

Reverse of
1st symbol

π−1
2

U=2
V=2

Reverse of
2nd symbol

π−1
3

U=1
V=3

Reverse of
3rd symbol

DR(0)
DR(1)
DR(2)
DR(3)

Selection of
the smallest D

Decoded data=(d0, d1, d2, d3) = (1, 0, 0, 0 )Switch according toa number of “1” & “0”Analysis of
Chaotic Dynamics

Ex. N=2, K=4, Data=(1, 0, 0, 1 ), Error has occurred at d3

Output=(d0, d1, d2, d3)=( 1, 0, 0, 1 )

π−1
4

U=0
V=4

Received sequence

Deinterleaved sequence

Figure 3: Operation of Improved Error-Correcting Method (ex. N = 2, K = 4).
First, the receiver performs the noncoherent detection
for each received block and demodulates each symbol. In
this study, we apply our suboptimal detection algorithm
as the noncoherent detection [6]. Our suboptimal detector calculates the shortest distance between R0 i and Nd dimensional space made from Nd successive chaotic signals generated by the skew tent map (Nd : 2, 3, · · ·), andoutputs the sum of the distance, which achieves the minimum shortest distance within the set of l, from i = 0 to
N − Nd . Here, R0 i is the Nd successive signals beginningwith i with in the received signals deﬁned as
0

Ri

=

(rα+i , rα+i+1 , ..., rα+i+Nd −1 ),(5)where α = j ×N . Also, l is the number of the straight lines
in the Nd -dimensional space (i.e., l = 2Nd −1 ). When two
symbols are transmitted, two kinds of the Nd -dimensional
space corresponding to each symbol are made. Let us consider the case of Symbol “1”. For calculating the shortest
distance, we ﬁnd the closest point Pl between R0 i and thel-th line using the scalar product of the vector. When bothedges of the l-th line are deﬁned as P0 l and P00 l , as shownin Fig. 4, the closest point Pl is calculated by the following
equation.
Plunit vector ulvl(ul · vl ) ul + P0 l ,P00 l − P0 l=,||P00 l − P0 l ||= R0 i − P 0 l .
=

(6)
(7)
(8)

In the same ways, we can ﬁnd the closest point Ql between
R0 i and the l-th line of the space of Symbol “0”. Then, theabove operations are expressed as
∑

D1

=

N∑
−Nd
i=0

∑

D0

=

N∑
−Nd
i=0

min ||Pl − R0 i ||.
lmin ||Ql − R0 i ||.
l

(9)

Pl

Pl’’
D1

vl

Ri’

Figure 4: Calculation of Shortest Distance.
3rd symbol (d3 ), namely the case of 1 bit correction.
First, the receiver sorts the received sequence based on
decoded symbols. In addition, the receiver counts thenumber of each decoded symbol and chooses a deinterleaver πk−1 corresponding to this number. The deinterleaver πk−1 reconstructs the sequence which is shufﬂed
by the interleaver πk when the number of each symbolin the receiver corresponds with the transmitter one. In
this example, π3−1 is chosen because the numbers of “1”and “0” are one and three, respectively. Here, we deﬁne the deinterleaved sequence based on decoded symbol“1” (or “0”) as M = (m0 , m1 , · · · , mC1 −1 ) (or N =(n0 , n1 , · · · , nC0 −1 )), where C1 and C0 are the total of
M and N, respectively. Next, the receiver analyzes the
chaotic dynamics of M and N. For analyzing the chaotic
dynamics, the receiver applies our suboptimal detection algorithm, i.e., the calculation of the shortest distance between the chaotic maps and two deinterleaved sequences.
Thus, we deﬁne a reference distance Dbasis as follows.
Dbasis = GoD1 (M) + GoD0 (N).

(11)

Where GoD1 (M) (or GoD0 (N)) is the shortest distance
between M (or N) and the Nd -dimensional space of Symbol “1” (or “0”).
GoD1 (M) =

(10)

C1∑
−Nd
i=0

Finally, we
the decoded∑∑ decide ∑∑ symbol as 1 (or 0) forD1 <D0 (or D1 >D0 ).
After demodulation of each symbol, the receiver performs the error correction. For ease of explanation, we usesame assumption in the explanation of the encoder (Fig. 2).
Also, we assume that the detection error has occurred at the

ul

Pl’

GoD0 (N) =

C0∑
−Nd
i=0

min ||Pl − M0 i ||.

(12)

min ||Ql − N0 i ||.

(13)

l

l

Next, the receiver assumes the detection error has occurred at the j-th symbol and reverses this symbol, as
shown in Fig. 5. Moreover, the receiver sorts the received
sequence and chooses the deinterleaver according to the re-- 803 -

Dbasis = GoD1(r0, r1) + GoD0(r2, r3, r6, r7, r4, r5)(x0, x1)(y2, y3, x2, x3, y0, y1)

Reversed0 = 1d0’ = 01.0E+00

DR(0) = GoD0(r2, r3, r6, r7, r0, r1, r4, r5)(y2, y3, x2, x3, x0, x1, y0, y1)d1’ = 1 Deinterleave DR(1) = GoD1(r0, r1, r2, r3) + GoD0(r6, r7, r4, r5)&(x0, x1, y2, y3)(x2, x3, y0, y1)

d2 = 0d2’ = 1d3 = 0d3’ = 1BER

d1 = 0Analysis of
Chaotic
dynamics(3) Conventilnal method(w/o coding)

1.0E-01

DR(2) = GoD1(r0, r1, r4, r5) + GoD0(r6, r7, r2, r3)(x0, x1, y0, y1)(x2, x3, y2, y3)

1.0E-02(2) Previous error-correcting
methodDR(3) = GoD1(r0, r1, r6, r7) + GoD0(r4, r5, r2, r3)(x0, x1, x2, x3)(y0, y1, y2, y3)

1.0E-03(1) Improved error-correcting
methodBecome the smallest distance
1.0E-04

Figure 5: Calculation of DR (j) and Selection of the smallest distance.

4. Simulation Results
We evaluate the performance of the improved errorcorrecting method by computer simulations. The simulation conditions are as follows. In the transmitting side, we
assume K = 16. The parameter of the skew tent map is
ﬁxed as a = 0.05. For calculation of the shortest distance,
we use 4-dimensional space (Nd = 4). Moreover, we only
perform the 1 bit error correction. Based on these conditions, we iterate the simulation 10,000 times and calculate
BER performance.
Figures 6(a) and (b) show the BERs versus Eb /N0 for
N = 4 and N = 8, respectively. We plot three performances: (1) Improved error-correcting method, (2) the previous one [6], (3) the conventional method (without coding). From these ﬁgures, we can conﬁrm that the advantage gained in BER performance of the improved errorcorrecting method is about 2–2.5 dB compared to the conventional method. Moreover, it can be observed that theperformance of the improved error-correcting method isbetter than the previous one. Namely, we achieve that thecapability of our error-correcting method increases by separating and reconstructing the chaotic dynamics using theinterleaver and the deinterleaver.

4

6

8

10
12
Eb/N0 [dB]

14

16

18

20

1.0E+00(3) Conventilnal method(w/o coding)

1.0E-01

BER(14)

Where M(j) and N(j) denote deinterleaved sequences
when the j-th decoded symbol is reversed. Here, DR (j)means the shortest distance between the deinterleaved sequence and the chaotic map corresponding to their sequences. If the receiver can detects symbols and deinterleaves the sequence correctly in the ﬁrst step, DR (j)becomes lager values as compared with Dbasis . On theother hands, if the error has occurred, some one of DR (0)–DR (K −1) becomes smaller as compared with Dbasis . For
instance, we consider that the error has occurred at the 3rd
symbol, as shown in Fig. 5. In this case, DR (3) becomes
the smallest distance as compared with Dbasis and otherDR (j). The reason for this is that the receiver can reconstruct the original chaotic sequences of symbol “1” and “0”from the received sequence by choosing the correct deinterleaver. Therefore, the receiver selects the smallest distancefrom Dbasis and DR (j) and corrects an error. Additionally, in this example, the receiver can determine that theerror has occurred at the 3rd symbol.

2(a) N = 4.

versed symbol and other symbols. Using the deinterleaved
sequence, we calculate the distance DR (j) as follows.
DR (j) = GoD1 (M(j) ) + GoD0 (N(j) ).

0

1.0E-02(2) Previous error-correcting
method

1.0E-03(1) Proposed error-correcting
method
1.0E-04

0

2

4

6

8

10
12
Eb/N0 [dB]

14

16

18

20

(b) N = 8.
Figure 6: BER vs. Eb /N0 (K = 16 and Nd = 4).
5. Conclusions
In this study, we have improved our proposed errorcorrecting method by separating the chaotic dynamics. As
results, it has been achieved that the capability of the improved error-correcting method increases as compared withthe previous one. Therefore, we have concluded that the
separation and reconstruction of the chaotic dynamics arevery effective as additional information to recover data.
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