Performance Evaluation of Error-Correcting Scheme
Without Redundancy Code for Noncoherent Chaos
Communications
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-its, 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 considers a novel error-correcting
scheme exploiting chaotic dynamics for noncoherent chaos communication. 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 valueof 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 performanceof the receiver. Further, enhanced efﬁciency also comes fromoperating on successively modulated data; by involving lessredundancy in the error correction system, it can be designedwith high coding rate. In this paper, we analyze the scheme’scapability, by examining computational times and accuracy ratesof error correction. bounds on its capability.

ing 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. Further, enhanced efﬁciency also comes fromoperating on successively modulated data; by involving lessredundancy in the error correction system, it can be designedwith high coding rate. Preliminary computer simulations have
conﬁrmed that the advantage gained in BER performance isabout 2–3 dB compared to conventional noncoherent systems.
However, the performances of their error-correcting scheme
has been only conﬁrmed in primary simulations. In this paper,
we analyze the scheme’s capability, by examining computational times and accuracy rates of error correction.

I. I NTRODUCTION

II. S YSTEM OVERVIEW

Chaos communication systems are an interesting topic in
the ﬁeld of engineering chaos [1]– [7]. 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] [9].
In our previous research, we focused on the chaotic dynamics and proposed the error-correcting method using thechaotic dynamics [8] [10]. In standard noncoherent chaos
communication systems, a binary data bit is modulated in atransmitter by a chaotic sequence of chosen length; the bit
is demodulated at the receiver using the modulated sequenceand 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 which are scarcely used forthe demodulation in standard chaos communications.
In our proposed system, two successive chaotic sequences
are generated from the same chaotic map; the second sequenceis generated with an initial value which is the last value of theﬁrst sequence. In this case, successive chaotic sequences hav-

978-1-4244-5309-2/10/$26.00 ©2010 IEEE

We consider the discrete-time binary CSK communication
system with the error correcting, as shown in Fig. 1. Detail ofeach block is described below.
A. 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 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. Our
encoder is designed based on Chaos Shift Keying (CSK) which
is a digital modulation system using chaos. Figure 2 shows our
encoder 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

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AWGN

Transmitter

Data

Fig. 1.
System.

S

Encoderbased on CSK

Ex. N=2, K=4

ReceiverR

Suboptimal
Detector

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

Error-Correcting
based onChaotic Dynamics

OutputBlock Diagram of Discrete-Time Binary CSK Communication

Suboptimal Detector
Detection of symbolfor each block

Ex. N=2, K=4, Data=(1 0 0 1)
Chaotic sequence “1” =Decoded= d0
datax0, x1, x2, ...

Symbol “1”Chaotic Signal
Generator 1...1 0 0 1

“1”

“0”

“0”

“1”

N

N

N

N

Outputy0, y1, y2, ...

Fig. 3.

Encoder based on CSK for error correction.

Sj = (xα , f(xα ), · · · , f(i)(xα ), · · · , f= d0(N −1)(xα ))(2)

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

(3)

d1

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 3rdsymbol is generated by the end value of 1st symbol and 2nd
symbol, respectively.
B. Channel and noise
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 .
C. Receiver
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.

d3

P = (X, Y)
Dm

v0[When the symbol “0” is sent]

where f (i) and g (i) 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 at
random 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.

d2

Operation of proposed error-correcting method.

P0=(x0, y0)

Sj is different for each symbol.
[When the symbol “1” is sent](1)d3

Analysis of Chaotic
Dynamics

Chaotic sequence “0” =Fig. 2.

d2Error Correcting

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”

d1

u

P1
= (x1, y1)

R’=(rα+i, rα+i+1)
Fig. 4.

Calculation of shortest distance.

However, the receiver never knows the initial value of chaos in
the transmitter. Our proposed error-correcting method consistsof the suboptimal detector and the error correction based on
chaotic dynamics, as shown in Fig. 3.
1) Suboptimal detector: First of all, the receiver performs
the noncoherent detection for each received block and demodulates each symbol. In this study, we apply the suboptimal
detection algorithm of Ref. [9] as the noncoherent detection.
Our suboptimal detector 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 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 the point and the map has to be calculated.
Therefore, the receiver can calculate the shortest distance using
the scalar product of the vector.
Any two points P0 = (X0 , Y0 ) 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 ||√2
2=(X − rα+i ) + (Y − rα+i+1 )

(6)

where

2459

unit vector u
v0

P1 − P0||P1 − P0 ||= R0 − P0 .

=

(7)
(8)

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 )

(a)
d0

d2

d3

d1

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

Become the smallest distance(b)

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

Fig. 5.

Analysis of Chaotic Dynamics using Suboptimal Detection.

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 ) is
included, and ﬁnd their summations D1 and∑ D0 . Finally,∑
we decidethe∑decoded symbol as 1 (or 0) forD1 < D0∑(or
D1 >D0 ). The calculation of the shortest distancecan be extended to Nd -dimensional space for Nd ≥ 4.
2) Error-correcting method without Redundancy code:
After demodulation of each symbol, the receiver performs
the error-correcting method. For error correction without the
redundancy code, the receiver uses the received signal samples
again. For ease of explanation, we use Fig. 5 and explain an
operation of the proposed error-correcting scheme. Here, we
use same assumption in the explanation of the encoder (Fig. 2).
Also, we assume that decoded symbols become (1, 0, 0, 1) andthe 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 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= 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 with
Dbasis , 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 is
occurred, some of DR (0)–DR (K − 1) become smaller ascompared 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 DR (j). Thus, the receiver can determine that the detection error is occurred at the 2nd symbol.
Although we consider the case of 1 bit correction in thisassumption, the case of 2 or more bit can be also performedin the same way. In this case, since the number of thesymbols reversed at once, namely the number of conceivablecombinations increases, a computation time also increases.
III. E VALUATION OF PROPOSED ERROR - CORRECTINGMETHODIn this section, we evaluate the performance of the proposederror-correcting method. We carry out computer simulations
and investigate three evaluations a bit error rate (BER), an
accuracy rate of each bit correction and a computation time.
The simulation conditions are as follows. In the transmitting
side, we assume K = 32, 64. The parameter of the skew tent
map is ﬁxed as a = 0.05. The chaotic sequence length per
1 bit is N = 4. For calculation of the shortest distance, we
use 4-dimensional space, namely Nd = N = 4. Based onthese conditions, we iterate the simulation 10,000 times andcalculate BER, the accuracy rate and the computation time.
Figures 6(a) and (b) show the BERs versus Eb /N0 for
K = 32 and K = 64, respectively. We plot the performanceof the proposed error-correcting method and the performanceof the conventional method, namely, the performance without
the error-correcting method in Figs. 6(a) and (b). From theseﬁgures, we can conﬁrm that the advantage gained in BER
performance of the proposed error-correcting method is about
1–2 dB compared to conventional method. However, theperformance of 2 bit correction by using the proposed errorcorrecting method is only slightly better than that of its 1 bitcorrection.
For investigation in detail, we observe the accuracy rate ofeach bit error correction in Figs. 7(a) and (b). In this ﬁgure, theleft vertical axis is a percentage of the number of each bit error,
the right vertical axis is the accuracy rate of error correction.
In general, it is well-known that the percentage of 1 bit error is
statistically larger than that of 2 bit error as shown in Figs. 7(a)and (b). 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. However, we can observethat the accuracy rate of its 2 bit correction is better than that2460

TABLE IC OMPUTATION TIME FOR ERROR CORRECTION (104 SYMBOLS ).

1.0E+00

Conventional method(w/o coding)

BER

1.0E-01

w/o coding
1bit correction
2bit correction

1bit correction

1.0E-02

Computation time [sec]K = 32
K = 64
1.88
5.15
101.7
434.4
1516.712264.2

2bit correction
1.0E-03

1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](a) K = 32.
1.0E+00

Conventional method(w/o coding)

BER

1.0E-01

1bit correction

1.0E-02

2bit correction

1.0E-03

1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0[dB](b) K = 64.
Fig. 6.

BER performance.

40

100
Percentage of 1bit error

90
80

30

Percentage of 2bit error

Accuracy rate of
2bit correction

70

25

60

20

50

Accuracy rate of
1bit correction

15

40
30

10

20

5
0

Accuracy rate of error correction[%]

Percentage of each bit error[%]

35

10
0

2

4

6

8

10

12

14

16

18

0
20

Eb/N0[dB](a) K = 32.
40

100

Percentage of each bit error[%]

Percentage of 1bit error

80

30

Accuracy rate of
2bit correction

Percentage of 2bit error

25

Accuracy rate of
1bit correction

20

60
50
40

15

30

10

20

5
0

70

10
0

2

4

6

8

10

12

14

16

18

Eb/N0[dB](b) K = 64.
Fig. 7.

020

Accuracy rate of error correction[%]90

35

of its 1 bit correction from these ﬁgures. Namely, it can besaid that the 2 bit error correction by using the proposed errorcorrecting method is performed at high accuracy rate although
the performance of its 2 bit correction is only slightly better
than that of its 1 bit correction.
Finally, we summarize computation times of the proposederror-correcting method in Table I. We carry out three simulations the conventional method (w/o coding), the 1 and 2
bit corrections by using the proposed error-correcting methodwith same conditions for calculating BER, and calculate acomputation time required to decode all the symbols with any
ﬁxed Eb /N0 . These simulations are carried out using a PC
with a Core2Duo 2.4 GHz CPU and 2 GB of RAM. From thistable, it can be conﬁrmed that the computational time required
the 2 bit correction is largest in the results. Thus, we consider
that it is important to simplify the algorithm of the proposederror-correcting method and to reduce extra calculation forimproving the computation times.
IV. C ONCLUSIONSIn this study, we have analyzed the proposed errorcorrection’s capability, by examining the computational time
and the accuracy rate of error correction. As results of computer simulations, it can be said that the 1 bit correctionof the proposed error-correcting method is currently moreaccessible than that of its 2 bit correction, although simplifying
the algorithm of the proposed error-correcting method andreducing its extra calculation for improving the computation
times are our future works.
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Accuracy rate of error correction.

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