NOLTA, IEICE
Paper

Error-correcting scheme based on
chaotic dynamics and its
performance for noncoherent chaos
communications
Shintaro Arai 1 a) , Yoshifumi Nishio 2 b) ,and Takaya Yamazato 3 c)
1

ITS Lab., Faculty of Engineering, Aichi University of Technology
50-2 Manori, Nishihasama-cho Gamagori, Aichi 443-0047, Japan

2

Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan

3

EcoTopia Science Institute, Nagoya University
C3-1 Furo-cho, Chikusa-ku, Nagoya 464-8063, Japana)arai-its@aut.ac.jp
nishio@ee.tokushima-u.ac.jpc)yamazato@nuee.nagoya-u.ac.jpb)Abstract: This paper proposes a novel error-correcting scheme using chaotic dynamics fornoncoherent chaos communications. In our proposed system, two successive chaotic sequences
are generated from the same chaotic map; the second sequence is generated with an initial value
which is the last value of the first sequence. In this case, 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 thereceiver. As results of the computer simulation, we confirm that the advantage gained in
BER performance of the proposed error-correcting method is about 1–1.5 dB compared toconventional method. In addition, we achieve that the proposed error-correcting scheme isperformed without the new additional redundancy code by using the chaotic dynamics.
Key Words: chaos, noncoherent chaos communications, error-correcting scheme, chaotic dynamics

1. Introduction
Chaos has received a great deal of attention in the past few years from a variety of researchers,
including mathematicians, physicists and engineers [1, 2]. The research direction has been transferringfrom finding the evidence of chaos existence into applications and deep theoretical study in recent
years. Chaotic sequences obtained from a certain class of difference equations are nonperiodic and196[Received: January 11, 2010. Revised: May 8, 2010. Published: October 1, 2010.][DOI: 10.1588/nolta.1.196]

Nonlinear Theory and Its Applications, IEICE, vol. 1, no. 1, pp. 196N206©IEICE 2010

sensitive to initial conditions, and it is difficult to predict their future behavior from past observations.
It was also confirmed that even simple one-dimensional maps can generate chaotic sequences. Since
it is becoming comparatively easy to implement chaotic system, many researchers in the field of
nonlinear circuits and systems have largely been interested for the development of chaos applications.
Chaos communication systems are one of interesting topics in the field of engineering chaos [3–12].
Especially, many researchers have focused on the development of noncoherent detections which do notneed to use basis signals (unmodulated carriers) for demodulation at a receiver. In standard communication systems, which are categorized as coherent detections, basis signals need to be reproduced for
demodulation at the receiver. Thus, it is said that the standard communication systems are difficult
to operate the noncoherent detection. On the other hands, the noncoherent detection using chaos candemodulate the information without basis signals since chaos and chaotic sequences have unique features itself. Therefore, we consider that the noncoherent detection is unique detection method using
chaos. Differential chaos shift keying (DCSK) [3] and the optimal receiver [4] are well-known typicalnoncoherent systems. In this study, we assume the noncoherent detection as chaos communication
systems and focus on the optimal receiver.
The optimal receiver calculates the probability density function (PDF) between the received signal
and the chaotic maps used on the transmitting side and detects the information symbol by choosing
the larger probability. However, the optimal receiver suffers from high computational complexity.
Also, using long chaotic sequence decreases the performance of the optimal receiver. Thus, it isimportant to develop a receiver with performance equivalent to or similar to the optimal receiverusing more efficient algorithms, i.e., a suboptimal receiver.
In our previous research, we proposed the suboptimal receiver using the shortest distance approximation [13]. Instead of calculating the PDF, the proposed suboptimal receiver approximates the PDF
by calculating shortest distances between the received signal and the chaotic maps and performs detection of the transmitted symbol. As results of the computer simulations, we confirmed the validityof 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 otherwords, the detection characteristic of the suboptimal receiver is not superior to the optimal receiver.
However, we expect to improve the performance of the suboptimal receiver by using characteristics ofchaos. Because we investigated and analyzed the influence on chaos communications systems by the
chaotic dynamics and its behavior previously [14, 15] and confirmed that the chaotic dynamics affectthe performance of chaos communication systems greatly. Therefore, we consider that it is important
to develop and advance communication systems based on characteristics of chaos.
For advancing the noncoherent chaos communication system in this paper, we focus on the chaotic
dynamics, which is one of characteristics of chaos, and propose a novel error-correcting scheme basedon the chaotic dynamics. In standard noncoherent chaos communication systems, a binary data bit
is modulated in a transmitter by a chaotic sequence of chosen length; the bit is demodulated at thereceiver using the modulated sequence and the transmitted unmodulated corresponding sequence.
The demodulation of each bit is performed only with the chaotic sequence associated with that bit.
We have paid attention to the successive chaotic sequences between symbols which are scarcely used
for the demodulation in standard chaos communications. Namely, we utilize the chaotic dynamics
among multiple symbols for the error correction.
In our proposed system, two successive chaotic sequences are generated from the same chaotic map;the second sequence is generated with an initial value which is the last value of the first sequence. In
this case, 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, the proposed error-correcting scheme is operated
without a redundancy bit sequence referred to perform an error correction in standard communication
systems. Note that the chaotic sequence already has a kind of redundancy since chaos is well-known as
the wideband signal and is expressed as multiple bits. Thus, the proposed method does not require newadditional redundancy bit sequence since the error correction applies the chaotic dynamics having thetransmitted signal blocks. We carry out computer simulations and evaluate the bit error rate (BER)197

x1
Be not successive basedon chaotic dynamics

x3
Chaotic map “1”x0x2
Chaotic sequence “1”y0Chaotic signal samples
are alternately mixed forevery 2 samples

x1

y3
y0
x3

y3
y2

x0

y1
Chaotic map “0”Be successive basedon chaotic dynamics

y1

y2

x2

Mixed chaotic sequence

Chaotic sequence “0”Fig. 1.

Concept of proposed error-correcting method.

performance of the proposed method.
In Sect. 2, a concept of our proposed error-correcting scheme is introduced. In Sect. 3, a systemmodel and an operation of our proposed error-correcting scheme are described. In Sect. 4, we discuss
computer simulations that were performed to evaluate its performance. Finally, we conclude thispaper in Sect. 5.

2. Concept of proposed error-correcting scheme
In this section, we explain a concept of the proposed error-correcting scheme using Fig. 1. Our
proposed error-correcting scheme applies the chaotic dynamics. A left side of Fig. 1 shows 2 different
chaotic maps and sequences. In this figure, xi and yi (i = 0, 1, · · ·) denote a sample of the chaotic
sequence “1” and “0” generated by the chaotic map “1” and “0” respectively.
Here, let us consider the case that the signal samples of these sequences are alternately mixed
for every 2 samples, as shown in a right side of Fig. 1. In this mixed chaotic sequence, the chaotic
dynamics of some successive samples is lost. For instance, since y0 is not generated by x1 , x1 andy0 are not successive based on the identical chaotic dynamics. Namely, the mixed chaotic sequence
becomes a train of sequence which is not successive based on the chaotic dynamics.
However, we consider that the loss of the chaotic dynamics can be applied for communication
systems. Now, let us assume that we do not know the rule of how to mix 2 different chaotic sequences.
Although we do not know this rule, we know that the chaotic signal samples before mixing aresuccessive according to the chaotic dynamics. For example, x1 influences generation x2 . Namely,
even if the rule of how to mix the sequences is never known, we reconstruct the original chaotic
sequences by analyzing the chaotic dynamics of samples of the mixed chaotic sequence.
We use this feature as an error correction for noncoherent chaos communications. This feature
gives the receiver additional information to correctly recover the information data and thus improves
the bit error performance of the receiver. In the next section, we describe a system model and an
operation of the proposed error-correcting scheme in detail.

3. System model with proposed error-correcting scheme
We consider the discrete-time binary CSK communication system with the error correcting, as shownin Fig. 2. Detail of each block is described below.

3.1 TransmitterIn the transmitter, binary data are encoded by chaotic sequences generated by a chaotic map. In thisstudy, we use a skew tent map which is one of simple chaotic maps, and it is described by Eq. (1)
198

AWGN
Transmitter

Data

ReceiverS

Encoderbased on CSK

Fig. 2.
tem.

R

Suboptimal
Detector

Error Correction
Based onChaotic Dynamics

OutputBlock Diagram of Discrete-Time Binary CSK Communication Sys-

xi+1 =

2xi + 1 − a




1+a

(−1 ≤ xi ≤ a)

(1)


−2x
+
1
+
a
i


(a < xi ≤ 1)1−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 3 shows our encoder forour error-correcting scheme. To perform the error correction at the receiver, K information bit aretransmitted as K signal blocks (0, 1, · · · , j, · · · , K − 1). The encoder selects a chaotic signal generator
according to the symbol. Here, we use “f ” to refer to the function of the skew tent map of Eq. (1).
If the symbol “1” is sent, f is used, and if “0” is sent, g (= −f ) is used. Thus, the signal vector Sj isdifferent for each symbol.
When the symbol “1” is sent
Sj = (xα , f (1) (xα ), · · · , f (i) (xα ), · · · , f (N −1) (xα )).

(2)When the symbol “0” is sent
Sj = (yα , g (1) (yα ), · · · , g (i) (yα ), · · · , g (N −1) (yα )).

(3)where i is the iteration of f or g, α = N ×j, xj or yj denotes the initial value of the j-th symbol = “1”or “0” respectively, N is the chaotic sequence length per 1 bit. When K bit data is transmitted, theamount 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 eachchaotic signal generator. In addition, the j-th sequence is generated from the initial value which isthe 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) shown in Fig. 3. In this case, the transmitted signal vector S isgiven as follows.
S = (S0 , S1 , S2 , S3 )= (x0 , x1 , y0 , y1 , y2 , y3 , x2 , x3 )= (s0 , s1 , s2 , s3 , s4 , s5 , s6 , s7 ).

(4)

As one can see, the initial value of the 4th symbol and 3rd symbol is generated by the end value of
1st symbol and 2nd symbol, respectively.
In this scheme, the transmitted signal blocks do not follow a redundancy blocks referred to perform
an error correction in standard communication systems. Note that the chaotic sequence already hasa kind of redundancy since chaos is well-known as the wideband signal and is expressed as multiple
bits. We focus on the redundancy of chaos as the important characteristic of chaos, i.e. the chaotic
dynamics. Thus, the proposed method does not require new additional redundancy bit sequences
since the error correction applies the chaotic dynamics having the transmitted signal blocks.

3.2 Channel and noise
The channel distorts the signal and corrupts it by noise. In this study, noise of the channel isassumed to be the additive white Gaussian noise (AWGN). Thus, the received signal block is given
by R = (r0 , r1 , · · · , rKN −1 ) = S + AWGN.
199

Ex. N=2, K=4, Data=(1 0 0 1)
Chaotic sequence “1” =x0, x1, x2, ...

Symbol “1”K symbols

Chaotic Signal
Generator 1

1001

“1”

“0”

“0”

“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”

N

N

N

N

Chaotic sequence “0” =y0, y1, y2, ...

Fig. 3.

Encoder based on CSK for error correction.

Ex. N=2, K=4
R
=
r0
r1
r2
r3
r4
r5
r6
r7

Decodeddata

Outputd0d0Suboptimal Detectord1
Detection of symbolfor each blockd2

Proposed Error
Correction Based on
Chaotic Dynamics

d3

Fig. 4.

d1
d2
d3

Operation of proposed error-correcting method.

3.3 Noncoherent receiver with proposed error correction
The receiver recovers the transmitted signal blocks from the received signal blocks and demodulates
the information symbol. Also, the receiver performs the error correction in this study. Since weconsider the 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 in the transmitter. Our
proposed error-correcting method consists of the suboptimal detector and the error correction based on
chaotic dynamics, as shown in Fig. 4. First of all, the receiver performs the noncoherent detection foreach received block and demodulates each symbol. In this study, we apply our suboptimal detection
algorithm as the noncoherent detection [13]. After demodulation of each symbol, the receiver performs
the error-correcting scheme. Before explaining the operation of the proposed error correction, we
describe the operation of our suboptimal detector to be the basis for the proposed scheme.

Suboptimal detector
Our suboptimal detector calculates the shortest distance between R0 i and Nd -dimensional space madefrom Nd successive chaotic signals generated by the skew tent map, (Nd : 2, 3, · · ·), and outputs thesum of the distance, which achieves the minimum shortest distance within the set of l, between i = 0
and N − Nd . Here, R0 i is the Nd successive signals beginning with i with in the received signals, andl is the number of the straight lines in the Nd -dimensional space, which are defined as
R0 i= (rα+i , rα+i+1 , ..., rα+i+Nd −1 ).
Nd −1

l = 2

.

(5)
(6)

When two symbols are transmitted, two kinds of the Nd -dimensional space corresponding to eachsymbol are made. Let us consider the case of Symbol “1”. For calculating the shortest distance, we
200

ri+1
1

Pl

ul

Pl’

0.5

Pl’’
Ri’=(rα+i, rα+i+1)

0

D1

vl
-0.5

Ri’
Fig. 5.

-1-1

Calculation of shortest distance.

Fig. 6.

-0.5

0a

0.5

ri

1

Nd =2-dimensional space.

find the closest point Pl between R0 i and the lth line using the scalar product of the vector. Whenboth edges of the lth line are defined as P0 l and P00 l shown in Fig. 5, the closest point Pl is calculatedby the following equation.
³´(l) (l)(l)Pl =p0 , p1 , · · · , pNd −1 = (ul · vl ) ul + P0 l .
(7)
where
unit vector ulvl

P00 l − P0 l.
||P00 l − P0 l ||= R0 i − P0 l .
=

(8)
(9)

In the same ways, we can find the closest point Ql between R0 i and the lth line of the space of Symbol“0”.
Then, the above operations are expressed as
X

D1

=

NX
−Nd

min ||Pl − R0 i ||.

(10)

min ||Ql − R0 i ||.

(11)

l

i=0

X

D0

=

NX
−Nd

l

i=0

P
P
P
P
Finally, we decide the decoded symbol as 1 (or 0) forD1 < D0 (orD1 > D0 ).
As an example, we explain the case of N = 3, Nd = 2. Figure 6 shows the Nd =2-dimensional
space of the skew tent map whose coordinates correspond to the two successive received signals
PP
R0 i = (rα+i , rα+i+1 ). In this case, we can calculate
D1 andD0 , as follows.
X
D1 = min ||Pl − R0 0 || + min ||Pl − R0 1 ||
l
l
)
)
(r
(r
³
´2 ³
´2
³
´2 ³
´2
(l)
(l)
(l)
(l)
p0 − r0 + p1 − r1
+ min
p0 − r1 + p1 − r2
. (12)
= min
l

X

D0

l

= min ||Ql − R0 0 || + min ||Ql − R0 1 ||
l
l
)
(r
³
´
³
´
= min
l

(l)
q0 − r02+(l)q1 − r1

2

)
(r
³
´2 ³
´2
(l)
(l)
+ min
q0 − r1 + q1 − r2
. (13)
l

3.4 Operation of proposed error-correcting method
After demodulation of each symbol, the receiver performs the error-correcting method. For error
correction without the new additional redundancy code, the receiver uses the received signal samples
again. For ease of explanation, we use Fig. 7 and explain an operation of the proposed error-correcting201

Ex 1. N=2,K=4, Data=(1 0 0 1), Error occurs at d2
error
Decoded data=(d0 ,d1, d2, d3)=(1 0 1 1 )

d0

d2

d3

Reversed0=1
d0’=0

DR(0)=GoD1(r4, r5, r6, r7) + GoD0(r0, r1, r2, r3)

d1=0

d1’=1

DR(1) = GoD1(r0, r1, r2, r3 , r4, r5, r6, r7)

d2=1
1

d2’=0
0

DR(2)=GoD1(r0, r1, r6, r7) + GoD0(r2, r3, r4, r5)

d3=1

d3’=0

DR(3)=GoD1(r0, r1, r4, r5) + GoD0(r2, r3, r6, r7)

Become the smallest distanced1

Dbasis = GoD1(r0, r1, r4, r5, r6, r7) + GoD0(r2, r3)= GoD1(m0, m1, m2, m3, m4, m5) + GoD0(n0, n1)= GoD1(M) + GoD0(N)

(a)

(b)

Fig. 7. Analysis of Chaotic Dynamics using Suboptimal Detection for 1 bitcorrelation: (a) Calculation of Dbasis , (b) Calculation of DR (j).
Ex 1. N=2,K=4, Data=(1 0 0 1), Error occurs at d2

Transmitted Signal Blocks S =(x0, x1, y0, y1, y2, y3, x2, x3)
Decoded=(1, 0, 1, 1)
symbols

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

d2=1

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

Be not successive based on chaotic dynamics
Influence the calculation of GoD1(M)Fig. 8.

d2’=0(y0, y1, y2, y3)

Be successive based on chaotic dynamics
GoD1(M(2)) and GoD0(N(2)) become the smallest distances

Reason why DR (2) becomes the smallest distance.

scheme. Here, we use same assumption in the explanation of the encoder (Fig. 3). Also, we assume
that the detection error has occurred at the 2nd symbol (d2 ), namely the case of 1 bit correction.
First of all, the receiver sorts the received signal samples based on decoded symbols, as shown in
Fig. 7(a). Here, we define the sequence sorted samples 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 andN, respectively. Next, the receiver analyzes the chaotic dynamics of M and N. 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 occurs when the receiver detects symbols, thesorted sequence mixes two chaotic sequences which differ in the chaotic dynamics. We focus on thesecharacteristics of chaos for the error-correcting. For analyzing the chaotic dynamics, the receiverapplies our suboptimal detection algorithm, i.e., the calculation of the shortest distance between the
chaotic maps and two sorted received sequences. Thus, we define a reference distance Dbasis as follows.
Dbasis = GoD1 (M) + GoD0 (N).

(14)

where GoD1 (M) (or GoD0 (N)), which is short for “Group of D1 (or D0 )”, is the shortest distance
between M (or N) and the Nd -dimensional space of Symbol “1” (or “0”) defined as
GoD1 (M) =

C1X
−Nd
i=0

GoD0 (N) =

C0X
−Nd
i=0

min ||Pl − M0 i ||.

(15)

min ||Ql − N0 i ||.

(16)

l

l

Next, we calculate the distance DR (j) for comparing with Dbasis , where the subscript R means
initial character of “Reverse”. This function means the shortest distance between sorted sequencewhen the j-th decoded symbol is reversed and the chaotic map corresponding to their sequences.
Namely, we assume the detection error occurs at the j-th symbol and calculate DR (j) as follows.
202

Ex 2. N=2,K=4, Data=(1 0 0 1), Error occurs at d0 and d2
error
Decoded data=(d0 ,d1, d2, d3)=(0
0 0 1 1)

d2

d3

d0

d1

Reverse

d0=0

d0’=1

d1=0

d1’=1

d0=1

d0’=0

d2=1

d2’=0

d2=1

d2’=0

d3=1

d3’=0

DR(0, 1)=GoD1(r0, r1, r2, r3, r4, r5, r6, r7)
Become the smallest distanceDR(0, 2)=GoD1(r0, r1, r6, r7) + GoD0(r2, r3, r4, r5)

Dbasis = GoD1(r4, r5, r6, r7) + GoD0(r0, r1, r2, r3)= GoD1(m0, m1, m2, m3) + GoD0(n0, n1, n2, n3)= GoD1(M) + GoD0(N)DR(2, 3)=GoD0(r0, r1, r2, r3, r4, r5, r6, r7)

(a)

(b)

Fig. 9. Analysis of Chaotic Dynamics using Suboptimal Detection for 2 bitcorrection: (a) Calculation of Dbasis , (b) Calculation of DR (j, k).

DR (j) = GoD1 (M(j) ) + GoD0 (N(j) ).

(17)

where M(j) and N(j) denote sorted sequences when the j-th decoded symbol is reversed. If thereceiver can detects symbols and sorts blocks correctly in the first step, DR (j) becomes lager values
as compared with Dbasis . On the other hands, if the detection error occurs, some one of DR (0)–DR (K − 1) become smaller as compared with Dbasis . For instance, Fig. 7(b) shows conceivablecombinations of sorted sequences and calculates DR (j) when the detection error occurs at the 2nd
symbol. In this case, DR (2) becomes the smallest distance as compared with Dbasis and other DR (j).
Thus, in this example, the receiver can determine that the detection error occurs at the 2nd symbol.
The reason for this is that the receiver reconstructs the original chaotic sequences of symbol “1”and “0” from the received signal blocks. Figure 8 shows the transmitted signal blocks of this example
and sequences sorted according to the rule of calculating Dbasis and DR (2). As one can see, thesequence of decoded symbol “1” for calculating Dbasis is not successive based on the identical chaotic
dynamics because y2 and x2 are not generated by x1 and y3 , respectively. In other words, since thereceived signal samples generated by the different chaotic map are successive, the influence between
these symbols appears in the calculation result of the shortest distance. Thus, these samples influence
the calculation of GoD1 (M) of Dbasis . On the other hands, 2 sequences for calculating DR (2) become
the successive sequences based on each chaotic dynamics because the receiver reverses the false symbol
and reconstructs the original sequence according to the reversed symbol. Hence, GoD1 (M(2) ) and
GoD0 (N(2) ) of DR (2) become the smallest distances as compared with those of Dbasis and otherDR (j). Therefore, the receiver selects the smallest distance from Dbasis and DR (j) and corrects anerror.
Although we consider the case of the 1 bit correction in this assumption, the case of 2 or more bit
can be also performed in the same way. As a difference between the 1 bit correction and the 2 or more
bit correction, the number of the symbols reversed at once increases according to the number of bit
for the correction. Figure 9 shows an example of an analysis for 2 bit correction. In the calculationof DR (∗ ) for 2 bit correction, we assume the detection errors occur at the j-th and the k-th symbol
simultaneously. Thus, we calculate DR (j, k) as follows.
DR (j, k) = GoD1 (M(j,k) ) + GoD0 (N(j,k) ).

(18)

4. Evaluation of proposed error-correcting method
In this section, we evaluate the performance of the proposed error-correcting method by computer
simulations. 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. For calculation of the shortest distance, we
use 4-dimensional space, namely Nd = 4. Moreover, we perform the proposed error-correcting scheme
203

1.0E+00
Conventional method(w/o coding)

BER

1.0E-01

1.0E-02
1 bit correction
1.0E-03
1 bit and 2 bit correction
1.0E-04

0

2

4

6

8
10 12
Eb/N0[dB]

14

16

18

20

(a) N = 4.

1.0E+00
Conventional method(w/o coding)

BER

1.0E-01

1.0E-02
1 bit correction
1.0E-03
1 bit and 2 bit correction
1.0E-04

0

2

4

6

8
10 12
Eb/N0[dB]

14

16

18

20

(b) N = 8.
Fig. 10.

BER vs. Eb /N0 (K = 32 and Nd = 4).

until 2 bit error correction. Based on these conditions, we iterate the simulation 10,000 times andcalculate BER performance.
Figures 10(a) and (b) show the BERs versus Eb /N0 for N = 4 and N = 8, respectively. We plot
the performance of the proposed error-correcting scheme and the performance of the conventional
method, namely, the performance without the error-correcting scheme in Figs. 10(a) and (b). Fromthese figures, we can confirm that the advantage gained in BER performance of the proposed errorcorrecting method is about 1–1.5 dB compared to conventional method. Namely, we achieve that theerror-correcting scheme is performed without new additional redundancy blocks by using the chaotic
dynamics. However, the performance of 1 bit and 2 bit correction is only slightly better than that of
1 bit correction. In general, it is well-known that the percentage of 1 bit error is statistically larger
than that of 2 bit error. Thus, the effect of 1 bit correction by using the proposed error-correcting
method on BER performance is also larger than that of its 2 bit correction. This problem might besolved by increasing the number of bit for the correction, On the other hand, since the number ofconceivable combinations also increases, more computation time is needed. Therefore, it can be said
that the 1 bit correction is currently more accessible than that of 1 bit and 2 bit correction.
204

1.0E+00
Eb/N0=12 dB
Conventional method(w/o coding)

1.0E-01
BER

1 bit correction

1.0E-02
Eb/N0=14 dB

1.0E-03
Eb/N0=16 dB

1 bit and 2 bitcorrection

1.0E-04

1

4

8 10

16

32

64 100128

1000

Chaotic sequence length N
Fig. 11.

BER vs. Chaotic sequence length N (K = 32 and Nd = 4).

Figure 11 plots the BERs versus the chaotic sequence length N for different Eb /N0 values. In theregion of small N , we can confirm the effect of the proposed scheme. Especially, when Eb /N0 = 16,
the BER of N = 8 improves from 1 × 10−3 to 1 × 10−4 using the proposed error-correcting scheme.
However, in the region of large N , the effect of the proposed scheme decreases. The reason for thisis that central samples of each sequence influence the error correction when N becomes large. In the
proposed scheme, we assume the detection error occurs at the arbitrary symbol and sort the received
sequences. In addition, we calculate DR (j) and DR (j, k) using the sorted sequences. Note that thejoint of the sorted sequence most influences the calculation of DR (j) and DR (j, k). In other words,
since central samples of each sequence are successive based on the chaotic dynamics, the influence ofthe chaotic dynamics having central samples is stronger than that of the joint of the sorted sequencein the case of large N , Thus we consider that the effect of the proposed scheme decreases according
to increase in N . Therefore, to design the error-correcting scheme with central samples of chaotic
sequences is our future problem.

5. Conclusions
We have proposed the novel error-correcting scheme for noncoherent chaos communications. We haveconsidered that the chaotic dynamics can be applied as additional information to correctly recover
the information data. Further, the proposed method does not require new additional redundancy bit
sequence since the error correction applies the chaotic dynamics having the transmitted signal blocks.
As results of the computer simulation, we have confirmed that the advantage gained in BER performance of the proposed error-correcting method is about 1–1.5 dB compared to conventional method.
Namely, we have achieved that the error-correcting scheme is performed without new additional redundancy bit sequence by using the chaotic dynamics. However, we have also found that the effectof the proposed scheme decreases according to increase in N . The reason for this is considered that
central samples of each sequence influence the error correction when N becomes large. Therefore, todesign the error-correcting scheme with central samples of chaotic sequences is our future problem.
Moreover, we consider that analyzing the scheme’s capability, such as the computational cost, is also
important future work.

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