2013 International Workshop on Nonlinear Circuits,
Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Performance Analysis of Error-Correcting Method Using Separation of Chaos
for Noncoherent Chaos Communications
Shintaro Arai† , Yoshifumi Nishio‡ and Takaya Yamazato∗‡Tokushima University
2-1 Minami-Josanjima, Tokushima, JAPAN
Email: nishio@ee.tokushima-u.ac.jp†Kagawa National College of Technology
551 Kohda, Takuma-cho, Mitoyo, Kagawa, JAPAN
Email: arai@cn.kagawa-nct.ac.jp* Nagoya University
Furo-cho, Chikusa-ku, Nagoya, JAPAN
Email: yamazato@nagoya-u.jp

Transmitter

Abstract
This paper analyzes the performance of the error-correcting
method, which was proposed in the previous research, using
the separation and the reconstruction of chaos for noncoherent chaos communications. Especially, we focus on a number of transmitting data and a combination of symbols (i.e.,
data pattern). We consider that the capability of the error correction depends on the number of data and the data pattern
because the proposed method performs the error correction
using successive data symbols. This study performs simulations for the number of data and the data pattern, and evaluates BER performance of the proposed method.
1. Introduction
Chaos communication system is an interesting topic in theﬁeld of engineering chaos [1]– [6]. 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 hasa performance equivalent to or similar to the optimal receiver,using more efﬁcient algorithms [3].
In our previous research, we focused on the chaotic dynamics and proposed the error-correcting method using thechaotic dynamics for noncoherent systems [7]. In our errorcorrecting method, two successive chaotic sequences are generated from the same chaotic map; the second sequence isgenerated with an initial value which is the last value of theﬁrst 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 performanceof the receiver. As results of simulations, we have conﬁrmedthat the advantage gained in the bit error rate (BER) performance of our method is about 1–1.5 dB compared to a conventional method (w/o coding).

Data

Encoder

AWGN

Receiver
Suboptimal
Detector

Error Correction

OutputFigure 1: Block Diagram of CSK Communication Systemwith Error-Correcting method.
Furthermore, we improved this method [7] using a separation and a reconstruction of the chaotic dynamics [8]. The
method [7] uses the 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. As resultsof simulations, we have conﬁrmed that the advantage gainedin BER performance of our improved method is about 2–2.5
dB compared to the conventional one (w/o coding). However, we have only conﬁrmed the performance of our method
in primary simulations. In this paper, we focus on a number of transmitting data and a combination of symbols (i.e.,
data pattern), and analyze the performance of our method. It
is considered that the capability of the error correction depends on the number of data and the data pattern because ourmethod performs the error correction using successive data
symbols. This study performs simulations for the number ofdata and the data pattern, and evaluates BER performance ofour method.
2. Overview of Error-Correcting method [8]This section brieﬂy introduces the error-correcting method
in [8]. We consider the discrete-time binary Chaos Shift Keying (CSK) communication system with the error correcting,as shown in Fig. 1.
2.1 TransmitterIn the transmitter, binary data are encoded by chaotic se-- 225 -

Ex. K=4

and “0”, respectively. 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 the order of the sequence having each signal vector
does not change. Therefore, the encoder shufﬂes the chaotic
sequence in each symbol of S0 according to the interleaver
and outputs the second signal vector S as the transmitting sequence.

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

Symbol “1”Data(K symbols)

Chaotic Signal
Generator 1
S’

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

Interleaver

2.2 Channel

U=4
V=0

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
vector R is given by R = S + AWGN.

π0
π1

U=3
V=1

S

π2

U=2
V=2

π3

U=1
V=3

π4

2.3 Receiver
Switch according toa number of “1” & “0”

U=0
V=4

Figure 2: Operation of Encoder (ex. K = 4).
quences generated by a chaotic map. In this study, we use askew tent map which is one of simple chaotic maps, and it isdescribed as follows.
 2x + 1 − ai(−1 ≤ xi ≤ a),
1+axi+1 =(1) −2xi + 1 + a (a < x ≤ 1).
i
1−aWhere i = 0, 1, · · · , N − 1, a denotes a position of the topof the skew tent map. Our encoder is designed based on
CSK which is a digital modulation scheme using chaos. Figure 2 shows our encoder of the error-correcting method whenK = 4, where K denotes the number of transmitting data. To
perform the error correction at the receiver, K information bit
are transmitted as K signal blocks (0, 1, · · · , j, · · · , K − 1).
First, the encoder selects a chaotic signal generator according to the symbol and generates N chaotic signal samples asa chaotic sequence, where N denotes the chaotic sequencelength. When K bit data is transmitted, the amount of thedata becomes K × N . Thus, the amount of the data per bitof the proposed method becomes the same as the standard
CSK. An initial value is chosen at random when beginningto make signal blocks and is different in each chaotic signal
generator. In addition, the j-th sequence is generated fromthe initial value which is the end value of the former sequencewith same symbol of j-th bit. The encoder connects each sequence according to the order of transmitting data and outputs
the ﬁrst signal vector S0 .
Next, the encoder shufﬂes the chaotic sequence in eachsymbol of S 0 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. Their
shufﬂe patterns are different according to U and V , where
U and V denote the number of the transmitting symbol “1”The receiver demodulates the information symbol from thereceived 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 the transmitter.
Our error-correcting method consists of the suboptimal detector and the error correction based on the chaotic dynamics, as shown in Fig. 3. First of all, 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 [7, 8].
Our suboptimal detector calculates the shortest distance between Nd samples of the sequence in each symbol and Nd dimensional space made from Nd successive chaotic signals
generated by the skew tent map (Nd : 2, 3, · · ·), and outputs
the distance, which achieves the minimum shortest distancewithin the set of l. Where l is the number of the straightlines in the Nd -dimensional space (i.e., l = 2Nd −1 ). When
N = Nd , the receiver outputs one shortest distance. When
N > Nd , the receiver shifts the extraction position for calculating the distance within N signal samples and extracts
Nd samples from it. In this case, the number of the shift is
N − Nd . The receiver calculates the shortest distance every extracting samples and outputs the sum of these shortest distances. When two symbols are transmitted, two kindsof the Nd -dimensional space corresponding to each symbol
are made. The receiver calculates two shortest distances∑ using two
differentN-dimensional
space
and
outputs
D1
d
∑
∑
∑
and
D0 . Where
D1 andD0 denote the sum of theshortest distance of the symbol “1” and “0”, respectively. Finally,receiver decides
the∑decoded symbol as 1 (or 0) for∑ the ∑∑
D1 < D0 (or D1 > D0 ).
After demodulation of each symbol, the receiver performs
the error correction. In this study, we consider the 1 bitcorrection in the proposed method. First, the receiver sorts
the received sequence based on decoded symbols. In addition, the receiver counts the number of each decoded symbol
and chooses a deinterleaver πk−1 corresponding to this number. The deinterleaver πk−1 reconstructs the sequence which is- 226 -

Decoded data=(d0, d1, d2, d3)

π−1
0

U=4
V=0

Suboptimal Detector

U=3
V=1

π−1
2

Reverse of
1st symbol

Detection of symbolfor each block

U=2
V=2

Reverse of
2nd symbol

π−1
3

U=1
V=3

Reverse of
3rd symbol

DbasisAnalysis of
Chaotic Dynamics

π−11

Reverse of
0th symbol
R

Deinterleaver

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

Selection of
the smallest D

Switch according toa number of “1” & “0”Ex. K=4

Output=(d0, d1, d2, d3)

π−1
4

U=0
V=4

R

Deinterleaved sequence

Figure 3: Proposed Error-Correcting Method [8] (ex. K = 4).
shufﬂed by the interleaver πk when the number of each symbol in the receiver corresponds with the transmitter one. Here,
we deﬁne the deinterleaved sequence based on decoded symbol “1” and “0” as 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).

(2)

the proposed error-correcting method in [8]. We consider thatthe capability of the error correction depends on the numberof data and the data pattern because our method performs theerror correction using successive data symbols. In this section, we perform the computer simulation of the error correction with following conditions and observe the inﬂuence ofthe number of data and the data pattern.
The simulation conditions are as follows. In the transmitting side, we assume N = 4. 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). In this study,
we perform two kind of simulations. First is the simulation
for the number of data K. Second is the simulation for thetransmitting data pattern. In each simulation, we iterate thesimulation 10,000 times and record BER for various Eb /N0 .

Where GoD1 (M) (or GoD0 (N)) is the shortest distance between M (or N) and the Nd -dimensional space of the symbol“1” (or “0”).
Next, the receiver assumes the detection error has occurredat the j-th symbol and reverses this symbol. Moreover, thereceiver sorts the received sequence and chooses the deinter- 3.1 BER for Number of Data K
leaver according to the reversed symbol and other symbols.
We show two kind of results for K. First, Fig. 4 shows
Using the deinterleaved sequence, we calculate the distanceBERsversus Eb /N0 for different K when N = Nd = 4. In
DR (j) as follows.
Fig. 4, we plot the performance of the proposed method with
K = 4, 8, 16 and 32 and the conventional one (w/o coding).
DR (j) = GoD1 (M(j) ) + GoD0 (N(j) ).
(3) Second, Fig. 5 shows BERs versus K for different E /N .
b
0
In Fig. 5, we plot the performance of the proposed one and(j)(j)
Where M and N denote deinterleaved sequences when the conventional one when E /N =16dB, 14dB and 12dB.
b0
the j-th decoded symbol is reversed. Here, DR (j) means the From these ﬁgures, we can conﬁrm that BER performance
shortest distance between the deinterleaved sequence and the of the proposed method improves as compared with the conchaotic map corresponding to their sequences. If the receiver ventional one when K ≥ 8. Especially, When K = 16, thecan detects symbols and deinterleaves the sequence correctly advantage gained in BER performance of the proposed one
in the ﬁrst step, DR (j) becomes lager values as compared is about 2–2.5 dB compared to the conventional one. Howwith Dbasis . On the other hands, if the error has occurred, ever, we can also conﬁrm that the performance of the prosome one of DR (0)–DR (K − 1) becomes smaller as com- posed method gradually degrades with increasing K whenpared with Dbasis . Therefore, the receiver selects the smallest K > 16. This is because that the deinterleaved sequence
distance from Dbasis and DR (j) and corrects the error.
length increases depending on large K. We consider thatit becomes difﬁcult to analyze the difference of the chaotic
dynamics between symbols due to increasing of the deinter3. Performance Analysis
leaved sequence length. Next, let us focus on the performance
As described in Introduction, this paper focuses on the of K = 4. As one can see, the performance of the convennumber of transmitting data (i.e., K) and the combination of tional method is better than the proposed one when K = 4.
symbols (i.e., data pattern), and analyzes the performance of This is because that the data pattern becomes small with de-- 227 -

creasing K. We further discuss the performance for the data
pattern in Sec. 3.2.
1.0E+00

3.2 BER for Data Pattern

1.0E-01
Conventional method(w/o coding)

BER

Figure 6 shows BERs versus Eb /N0 for the different data
pattern when N = Nd = K = 4. In the case of K = 4,
there are three data patterns: 1) All symbols “1” or “0”, 2)
U = 3, V = 1 or U = 1, V = 3, 3) U = V = 2. As
one can see, the performance of all symbols “1” or “0” degrades as compared with other performances. This reductionof the performance means the failure of the reconstructionof the chaotic dynamics in the deinterleaver πk−1 . When wesend the same data only, the encoder only uses one of chaotic
maps. In this case, the transmitting sequence does not containthe chaotic dynamics (i.e., the additional information for theerror correction) based on the another map. Therefore, theproposed error correction requires the transmitting sequencewhich contains the chaotic dynamics of both symbols.

1.0E-02

K=32
1.0E-03

K=4
K=16
K=8

1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0 [dB]

Figure 4: BER vs. Eb /N0 for K (N = Nd = 4).
4. Conclusions

1.0E+00

Conventional method(w/o coding)

Proposed method
1.0E-01

BER

This paper has focused on the transmitting data and has
analyzed the performance of the proposed error-correcting
method for the number of data and the data pattern. As analysis results, we have observed that it becomes difﬁcult to analyze the difference of the chaotic dynamics between symbols
due to increasing of the number of the transmitting data. In
addition, we have found that the proposed error correction requires the transmitting sequence which contains the chaotic
dynamics of both symbols.

Eb/N0=12dB

1.0E-02

Eb/N0=14dB
1.0E-03

Eb/N0=16dB

References
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10

16

32

64

128

100

1000

Figure 5: BER vs. K (N = Nd = 4).
1.0E+00
Proposed method (K=4)

Data Pattern
All Symbols "1" or "0"

1.0E-01
U: Number of Symbol "1"
V: Number of Symbol "0"1.0E-02[6] G. Kaddoum and F. Gagnon, “Error correction codes for secure chaosbased communication system,” Proc. QBSC’10, pp. 193-196, May2010.

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1.0E-04[8] S. Arai,Y. Nishio and T. Yamazato, “Improvement of Error-Correcting
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8

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BER

[2] M. Hasler and T. Schimming, “Chaos communication over noisy channels,” Int. J. Bifurcation and Chaos, vol. 10, no. 4, pp. 719-736,
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4

1

Other Data Patterns
U=3, V=1
or
U=1, V=3
or
U=2, V=2
0

2

4

6

8

10

12

14

16

18

20

Eb/N0 [dB]

Figure 6: BER vs. Eb /N0 for Data Pattern (N = Nd = K =4).

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