No Redundant Error-Correcting Scheme Using
Chaotic Dynamics for Noncoherent Chaos
Communications
Shintaro ARAI† , Yoshifumi NISHIO† and Takaya YAMAZATO‡† Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN{arai, nishio}@ee.tokushima-u.ac.jp‡ EcoTopia Science Institute, Nagoya University
Furo-cho, Chikusa-ku, Nagoya, 464-8603, JAPAN,
yamazato@nuee.nagoya-u.ac.jp
Abstract— This paper proposes the error-correcting schemewithout redundancy sequences based on the chaotic dynamics
for noncoherent chaos communications. We generate successive
chaotic sequences from the identical chaotic map. And for the
next sequence we set the initial value to the end value of theformer sequence. By such way we can create the successive
chaotic sequences having the same chaotic dynamics. This feature
gives the receiver additional information to correctly recover
the received noisy signal. Therefore, by analyzing the chaotic
dynamics at the receiver, it is possible to improve the error
performance without redundancy signal. As results of computer
simulations, we confirm about 3 dB gain in BER performance ascompared with the conventional suboptimal receiver when using
the short chaotic sequence length per 1 bit.

I. I NTRODUCTION
Research on digital communications systems using chaos
becomes a hot topic [1]− [7]. Especially, it is attracted todevelop noncoherent detection systems which do not need torecover basis signals (unmodulated carries) at the receiver. The
differential chaos shift keying (DCSK) [1] and the optimalreceiver [2] are well known as a typical noncoherent system.
In this study, we 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 optimalreceiver suffers from high computational complexity. Also,using long chaotic sequence decreases the performance of the
optimal receiver. Thus, it is important to develop a receiverwith performance equivalent to or similar to the optimalreceiver using more efficient algorithms, i.e., a suboptimalreceiver.
In our previous research, we proposed the suboptimal receiver using the shortest distance approximation [8]. Instead of
calculating 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 optimal and suboptimalreceiver, we are concerned with an error-correcting scheme fornoncoherent chaos communications. Especially, we focus on
characteristics of chaos to design an error-correcting method.
Chaotic sequences obtained from a certain class of difference

978-1-4244-3828-0/09/$25.00 ©2009 IEEEequations are non-periodic, sensitive to initial conditions andit is difficult to predict their future behavior using past observations. In other words, chaotic sequences are generated
according to specific rules, i.e., the chaotic dynamics.
In this study, we focus on the chaotic dynamics and propose
a no redundant error-correcting scheme. Let us consider two
successive chaotic sequences that are generated from theidentical chaotic map. We assume that the second sequence isgenerated from the initial value which is the end value of theformer sequence. In this case, the chaotic dynamics of these
two successive chaotic sequences are the same. This feature
gives the receiver additional information to correctly recover
the received noisy signal. Namely, the error performance of thereceiver can be improved by analyzing the chaotic dynamics
of the successive received sequences. In addition, as comparedwith general error correcting methods, our proposed scheme
is possible to operate the error-correcting without the physicalredundant signal.
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.
A. TransmitterIn the transmitter, a chaotic sequence is generated by achaotic map. In this study, we use a skew tent map which
is one of simple chaotic maps, as shown in Fig. 2(a), 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.
The information symbol is modulated by Chaos Shift Keying(CSK) which is a digital modulation system using chaos.
When the transmitter generates the signals, we use chaotic
sequences generated by different chaotic maps depending on
the value of an information symbol. If the information symbol“1” is sent, Eq. (1) is used (Fig. 2(a)), and if “0” is sent, the2633

Transmitter
Symbol "1"
Receiver

Chaotic Signal
Generator 1

AWGN
Sj

Rj

Suboptimal
Detector

Symbol "0"

Decoded
SymbolAnalysis of Chaotic Dynamics
for Error-Correcting

Chaotic Signal
Generator 0

Fig. 1.
System.

Block Diagram of Discrete-Time Binary CSK Communication
Xi+1
1

Xi+1
1
Xi+1=Xi

Xi+1=Xi

0.5

0.5

0

0

-0.5

-0.5

C. Receiver (Suboptimal Detection)
The receiver recovers the transmitted signals from the received signals and demodulates the information symbol. Also,
the receiver performs the error-correcting 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 and theinformation symbol in the transmitter. Before explaining theerror-correcting, we introduce the operation of our suboptimalreceiver to be the basis for the proposed error-correcting.

R’=(Rα+i, Rα+i+1, Rα+i+2)

1

Symbol "1"
-1
-1

-0.5

0a

0.5

Xi

-1
-1

1

(a) Symbol “1”Fig. 2.

-0.5

0a

0.5

Xi

0.5

1

(b) Symbol “0”Rα+i+2 0

Chaos Shift Keying.
Symbol "0"-0.5

reversed function of Eq. (1) is used (Fig. 2(b)). To transmit a
1-bit information, N chaotic signals are generated, where N
is the chaotic sequence length.
In this study, to perform the error-correcting at the receiver,
K information bits are transmitted as K signal blocks, as
shown in Fig. 3, where Sj is the transmitted signal vector,
j = (0, 1, · · · , K − 1), Note that the transmitted signal vector
Sj is different 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 f (i) and g (i) are the function of the skew tent map forsymbol “1” and “0” (Figs. 2(a) and (b)) respectively, i is theiteration of f or g, α = N × j, xj or yj denotes the initialvalue of the jth symbol = “1” or “0” respectively. When K
bits data is transmitted, the length of the data becomes K ×N .
An initial value is chosen at random when beginning to makesignal blocks and is different in each chaotic signal generator.
In addition, the j-th sequence is generated from the initialvalue which is the end value of the former sequence withsame symbol of j-th bit. As an example, let us assume thejth and (j + 3)th symbols are “1” (i.e. the (j + 1)th and(j + 2)th symbols are “0”). In this case, xj+3 = f (N ) (xj ),and yj+2 = g (N ) (yj+1 ).

S0 S1 ...

Sj

1
0.5

-0.5

0

0

Rα+i

Fig. 4.

-0.5

0.5

Rα+i+1

1 -1

Operation of Our Suboptimal Receiver (Nd = 3).

Our suboptimal receiver 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 = 3.
Figure 4 shows the 3-dimensional space of the skew tent map
whose coordinates correspond to the three successive received
signals R = Rα+i , Rα+i+1 , Rα+i+2 . We can calculate theshortest distance between R and the chaotic map using thescalar product of the vector.
P = (X, Y, Z)
P0=(x0, y0, z0)
DuP1= (x1, y1, z1)v0Fig. 5.

R’ = (Rα+i, Rα+i+1, Rα+i+2)
Calculation of Shortest Distance.

Any two points of P0 = (x0 , y0 , z0 ) and P1 = (x1 , y1 , z1 )are chosen from each straight line in the space of Fig. 4, asshown in Fig. 5. Using Fig. 5, we can calculate P = (X, Y, Z)
and the shortest distance D by the following equations.
P = (X, Y, Z) = (u · v0 ) u + P0... SK-2 SK-1

D

Data
(K bits N)
Fig. 3.

-1-1=||P − R ||

(4)
(5)

where

Signal Blocks.

Unit vector u

B. Channel and NoiseThe 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 signals block is given by
Rj = (Rα , Rα+1 , · · · Rα+i · · · , Rα+N −1 ) = Sj + AW GN .

v0

P1 − P0||P1 − P0 ||= R − P0 .
=

(6)
(7)

Note that if the point is outside the cube, we calculate thedistance between the point and the nearest edges of the maps.
For the 3-dimensional case, there are four straight lines inthe space. Therefore, the minimum value in four distances

2634

R0

is chosen as the shortest distance D1 for symbol “1”. In thesame way, D of symbol “0” is chosen as D0 . We perform these
operations until the last sample (i.e.,
Rα+N −1 ) is included, andfind their summations
D1 andD0
. Finally, 
we decidethedecodedsymbol
as
1
(or
0)
for
D
<
D0 (or1D0 ). The calculation of the shortest distance canD1 >be extended to Nd -dimensional space for Nd ≥ 4.
In our previous research, since we observed that the performance of our method corresponds that of the optimal receiver
when N = Nd -dimension [8], it can be said that the validity ofthe proposed suboptimal receiver as the approximation methodof the optimal receiver.

R1

R6

Fig. 7.

R4

R7Sort of Received Signal Blocks according to Decoded Symbols.
DBase = D1(Seq“1” (R0, R2, R5) ) + D0(Seq“0”(R1 , R3, R4, R6, R7))

ReverseL
1: Symbol “1”0: Symbol “0”Chaotic Sequence

Transmitted
Signal Blocks =R3

Seq“0”(R1, R3, R4, R6, R7) = (RN,… R2N-1, R3N, … R4N-1, R4N, …, R5N-1,
R6N, …, R7N-1, R7N, …R8N-1)

Ex. K=8, Data = (1, 0, 1, 1, 0, 1, 0, 0)-1

R5

Seq“1” (R0, R2, R5) = (R0, … , RN-1, R2N, …, R3N-1, R5N, …R6N-1)

III. P ROPOSED E RROR -C ORRECTING S CHEME WITHOUTR EDUNDANCY S EQUENCES

Xi

R2

S 0 S 1 S 2 S3 S 4 S 5 S 6 S 7

R0 =1

0

D1,0(0) = D1(Seq“1” (R2, R5) ) + D0(Seq“0”(R0, R1 , R3, R4, R6, R7))

R1 =0

1

D1,0(1) = D1(Seq“1” (R0, R1, R2, R5) ) + D0(Seq“0”(R3, R4, R6, R7))

R2 =1

0

D1,0(2) = D1(Seq“1” (R0, R5) ) + D0(Seq“0”(R1, R2, R3, R4, R6, R7))

R3 =0

1

D1,0(3) = D1(Seq“1” (R0, R2, R3, R5) ) + D0(Seq“0”(R1, R4, R6, R7))

R4 =0

1

D1,0(4) = D1(Seq“1” (R0, R2, R4, R5) ) + D0(Seq“0”(R1, R3, R6, R7))

R5 =1

0

D1,0(5) = D1(Seq“1” (R0, R2) ) + D0(Seq“0”(R1, R3, R4, R5, R6, R7))

R6 =0

1

D1,0(6) = D1(Seq“1” (R0, R2, R5 , R6) ) + D0(Seq“0”(R1, R3, R4, R7))

R7 =0

1

D1,0(7) = D1(Seq“1” (R0, R2, R5 , R7) ) + D0(Seq“0”(R1, R3, R4, R6))

Fig. 8.

Analysis of Chaotic Dynamics using Suboptimal Detection.

AWGN

Received
Signal Blocks =R 0 R1 R 2 R3 R 4 R 5 R 6 R 7
Symbol Detector

Decoded( 1
Symbols =

0

1

0 0
Error

1

0

0)

Seq“1” (R0, R2, R5) Seq“0”(R1, R3, R4, R6, R7)

Analysis of Chaotic Dynamics
for Error-Correcting

Seq“1” (R0, R2, R3, R5) Seq“0”(R1, R4, R6, R7)= (1 0Fig. 6.

1

1

0

1

0

0)

Operation of Proposed Error-Correcting Scheme).

In this section, we concentrate the proposed error-correcting
scheme without redundancy sequences. For ease of explanation, we use Fig. 6 and explain an operation of the proposederror-correcting scheme. Here, we assume K = 8 and thetransmitted symbols = (1, 0, 1, 1, 0, 1, 0, 0).
First of all, the receiver performs the noncoherent detection
for each received block and demodulates each symbol. In thisstudy, we apply the our suboptimal detection algorithm as
the noncoherent detection. Since the transmitted signal blocks
arrive at the receiver through AWGN channel. the noise isadded to the received signal samples. Thus, the detection
error occurs depending on an amplitude of noise. Here, we
assume that decoded symbols become (1, 0, 1, 0, 0, 1, 0, 0) andthe detection error has occurred at R3 .
Next, the receiver sorts the received signal blocks based on
decoded symbols and makes two large received sequences;
Seq“1”(.) and Seq“0”(.). Figure 7 shows the received sequences which are sorted according to decoded symbols. The
reason for sorting blocks is to analyze the chaotic dynamics
of the received sequences. In the transmitting side, the chaotic
sequences are generated according to transmitted symbols formaking the transmitted signal blocks. We have assumed someconditions for an initial value in Seq 2. An initial value ischosen at random when beginning to make signal blocks andis different in each chaotic signal generator. In addition, the jth sequence is generated from the initial value which is the end
value of the former sequence with same symbol of j-th bit. If
the receiver can detects symbols and sorts blocks correctly, we
can obtain two large successive chaotic sequences. 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 thesecharacteristics of chaos for the error-correcting.
For analyzing the chaotic dynamics, the receiver apply our
suboptimal detection algorithm, i.e., the calculation of theshortest distance between the chaotic maps and two sorted
received sequences. Figure 8 shows conceivable combinationsof sorted sequences. Here, D1 (.) and D0 (.) mean the shortest
distance between the chaotic map of Symbol “1” and “0”,respectively. In addition, we define a reference value foranalyzing the chaotic dynamics as DBase . This equation is
the sum of the shortest distance between sorted sequences
according to symbols which detects as “1” and “0” and the2635

1.0E+00

chaotic map corresponding to their sequences. We also define
comparison values to analyze the chaotic dynamics as D1,0 (j)(j = 0, 1, · · · , K − 1). This equation means the shortest
distance between sorted sequence when the j-th decoded
symbol is reversed and the chaotic map corresponding to theirsequences. If the receiver can detects symbols and sorts blocks
correctly, D1,0 (0) ∼ D1,0 (K − 1) become lager values ascompared with DBase . On the other hands, if the detection
error is occurred, some of D1,0 (0) ∼ D1,0 (K − 1) become
smaller as compared with DBase . The reason for changing thevalues is also to change combinations of the chaotic dynamics. Namely, the receiver calculations the shortest distances
between some combinations and the chaotic maps and corrects
an error. For example, when the detection error has occurredat R3 as shown in Fig. 6, D1,0 (3) becomes the smallest values
as compared with DBase and other other D1,0 (j). The receivercan determine that the detection error is occurred at the 3rdreceived block.
Therefore, the likelihood of information can improve by analyzing the chaotic dynamics without redundancy sequences,and the receiver can correct an error in high accuracy. Although the error is occurred at one received block only to simplify the explanation, the proposed method is also applicablefor more than one error. In this case, after correcting one error,
the receiver sorts the received signal blocks according to thedecoded symbols again and repeats the same error-correcting
operation.

Conventional Method

BER

1.0E-01

Error-Correcting(K=16)

1.0E-02
Error-Correcting(K=64)
1.0E-03
Error-Correcting(K=32)
1.0E-04

0

2

4

6

8

10

12

14

16

18

20

Eb/N0 [dB](a) Nd = 4.
1.0E+00

Conventional Method

BER

1.0E-01

1.0E-02

Error-Correcting(K=16)

Error-Correcting(K=64)

1.0E-03
Error-Correcting(K=32)
1.0E-04

0

2

4

6

IV. S IMULATION R ESULTS AND D ISCUSSION

8

10

12

14

16

18

20

Eb/N0 [dB]To evaluate the proposed error-correcting scheme, we carryout computer simulations with following simulation conditions. In the transmitting side, we assume K = 16, 32, 64.
The parameter of the skew tent map is fixed as a = 0.05. We
use the chaotic sequence length N = 4 and 8. In this study,
we confirm Nd -dimensional space to the chaotic sequencelength (Nd = N ). Based on these conditions, we iterate thesimulation 10,000 times and calculate the average of BERs.
Figures 9(a) and (b) show the BERs versus Eb /N0 for
N = Nd = 4 and N = Nd = 8, respectively. To compare
the performance of the error-correcting method, Figs. 9(a) and(b) also show the performance of our suboptimal receiver,namely, the performance without the error-correcting. Fromthese results, we observe that the both BERs of the errorcorrecting method show gain over the system without thecoding. Significant improvements are observed for N = Nd =4.
As this reason, we consider that it is related to use skew tent
map for the chaotic signal generator. Since the skew tent map
is the one-dimensional map, a present chaotic signal influences
next signal. When a chaotic sequence per 1 bit is short, theinfluence its initial value on the last value is strong. On theother hand, when a chaotic sequence per 1 bit is large, theinfluence its initial value on the last value is weak. Therefore,
our proposed method with N = Nd = 4 easily analyzes the
chaotic dynamics of the sorted sequence as compared with
N = Nd = 8. In other words the method using the shortchaotic sequence per 1 bit is more suitable for our errorcorrecting than using long sequence. This characteristic isdifferent from the maximum-likelihood decoding. We can saythat these results are caused by unique characteristic of chaos.

(b) Nd = 8.
Fig. 9.

Simulation Results.

V. C ONCLUSIONSIn this study, we have focused on the chaotic dynamics
and proposed the error-correcting scheme without redundancy
sequences. As results, we have confirmed that the likelihood of
the received symbol increases and obtained the 2–3dB gain inerror performance of the proposed method with N = Nd = 4.
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