2007 IEEE Workshop on Nonlinear Circuit Networks
NCN'07, Tokushima, Japan, Dec 7--8, 2007

Improvement of Error Performancefor 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

II. S YSTEM OVERVIEW

Abstract— This paper proposes the error-correcting for noncoherent chaos communications. To improve the error performance,
we focus attention on the successive chaotic sequence basedon the chaotic dynamics. Concretely, error performance of anoncoherent receiver improves by analyzing the chaotic dynamics
of the successive received sequence. As results of computer
simulations, we confirm about 3 dB gain in BER performancecompared to the conventional suboptimal receiver.

Transmitter
Symbol "1"

Chaotic signal
generator 1
Symbol "0"- 54 -Sj

Rj

Detector

DecodedsymbolChaotic signal
generator 0I. I NTRODUCTION
Recently, research on digital communications systems using
chaos becomes a hot topic [1]− [3]. Especially, it is attracted
to develop noncoherent detection systems which do not needto recover basis signals (unmodulated carries) at the receiver.
In this study, we focus attention on the optimal receiver which
is one of typical noncoherent systems [1]. The optimal receiver
performs an optimal detection by using the probability density
function (PDF) between the received signals and the samechaotic map of the transmitting side. However, the optimalreceiver suffers from a computational complexity due to thelarge chaotic sequence length. Thus, it is important to develop
a receiver with performance equivalent to the optimal receiverusing different algorithms, i.e., a suboptimal receiver.
In our previous research, we proposed the suboptimal receiver using the shortest distance approximation [4]. Insteadof calculating the PDF, the proposed suboptimal receiverapproximates the PDFs by calculating the shortest distance
between the received signals and the chaotic map. As resultsof the computer simulations, we confirmed the validity of theproposed suboptimal receiver as an approximation method of
the optimal receiver.
In this study, to improve the error performance of the suboptimal receiver more, we propose an error-correcting method forthe suboptimal receiver. Note that the detection characteristicof the suboptimal receiver is not superior to the optimalreceiver. To solve this problem, we focus attention on thesuccessive chaotic sequence based on the chaotic dynamics.
Concretely, 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. As
results of computer simulations, we confirm about 3 dB gain
in BER performance compared to the conventional suboptimalreceiver.

ReceiverAWGN

Fig. 1.

Block diagram of discrete-time binary CSK communication system.

We consider the discrete-time binary CSK communication
system, as shown in Fig. 1. In the transmitter, a chaotic
sequence is generated by a chaotic map. In this study, weuse a skew tent map which is one of simple chaotic maps, asdescribed by Eq. (1)2xi + 1 − a(−1 ≤ xi ≤ a)
1+a(1)
xi+1 =

−2xi + 1 + a(a < xi ≤ 1)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, and if “0” is sent, the reversed
function of Eq. (1) is used. To transmit a 1-bit information, N
chaotic signals are generated, where N is the chaotic sequencelength.
In this study, to perform the error-correcting by the receiver,information are transmitted by a packet which consists of adata part and a redundancy part, as shown in Fig. 2, where
Sj is the transmitted signal vector, j = (0, 1, · · · , K − 1), K
is total bit per packet, Seq “1” and Seq “0” are the chaotic
sequence. Note that the transmitted signal vector Sj is different
for each symbol.
[When the symbol “1” is sent]
Sj = (xj f (1) (xj ) · · · f (i) (xj ) · · · f (N −1) (xj ))(2)[When the symbol “0” is sent]
Sj = (yj g (1) (yj ) · · · g (i) (yj ) · · · g (N −1) (yj ))(3)

S0 S1 ...

Sj... SK-2 SK-1 Seq Seq"1" "0"Data(K bits N)

Fig. 2.

v0Fig. 3.

P = (X, Y, Z) = (u · v0 ) u + P0

Redundancy(2N or 4N)

D

Packet Format.
P = (X, Y, Z)

P0=(x0, y0, z0)

Using Fig. 3, we can calculate P = (X, Y, Z) and the shortest
distance D by the following equations.

D(4)
(5)

= ||P − R||

whereuP1 − P 0||P1 − P0 ||= R − P0 .

Unit vector u =P1= (x1, y1, z1)v0

R = (Rj, l, Rj, l+1, Rj, l+2)
Calculation of the shortest distance.

where f (i) and g (i) are the function of the skew tent map
for symbol “1” and “0”, i is the iteration of f or g, xj
or yj denotes the initial value of the jth symbol = “1”or “0” respectively. When K bits data are transmitted, thelength of the data part becomes K × N . The redundancy
is the sequence which has the known information symbol,
and its length is different depending on the error-correcting
method. In addition, the initial value of the chaotic sequenceis also different in each chaotic signal generator and is updated
each packet, namely, the chaotic sequence generated by eachgenerator in the packet is a successive sequence based on thechaotic dynamics. As an example, let us assume the jth and(j + 3)th symbols are “1” (i.e. the (j + 1)th and (j + 2)thsymbols are “0”). In this case, xj+3 becomes f (N ) (xj ). In thesame way, yj+2 becomes g (N ) (yj+1 ). Moreover, Seq “1” andSeq “0” are also a successive sequence based on the chaoticdynamics.
The channel distorts the signal and corrupts it by noise.
In this study, noise of the channel is assumed to be theadditive white Gaussian noise (AWGN). Thus, the received
signals block is given by Rj = (Rj,0 Rj,1 · · · Rj,N −1 ) =Sj + AW GN .
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 thechaotic 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.
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. In this case, the receiver calculates the shortest distance
between R and the chaotic map using the scalar product ofthe vector, where R is the three successive received signals
R = (Rj,l , Rj,l+1 , Rj,l+2 ), Rj,l denotes the jth symbol andlth signal of the received signals (l = 0 1 · · · N − 3). Any
two points of P0 = (x0 , y0 , z0 ) and P1 = (x1 , y1 , z1 ) arechosen from each straight line in the space, as shown in Fig. 3.

(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 ischosen as the shortest distance D1 for symbol “1”. In the sameway, D of symbol “0” is chosen as D0 . We calculate
P both ofDandDfor
alll
and
find
theirsummations
D1 and0
P1D.
Finally,
wedecide
thedecodedsymbolas1(or0) for0PPPP
D1 < D0 (orD1 > D0 ).
The calculation of the shortest distance can be extended toNd -dimensional space for Nd ≥ 4.
III. P ROPOSED E RROR -C ORRECTING M ETHOD
Our proposed error-correcting method calculates the shortest
distance by the double or triple of the sequence length N
which used for the modulation in the transmitter, i.e. thedouble or triple dimension (Nd = 2N or 3N ). Figure 4(a)shows the image of the operation of the error-correcting. To
simplify the explanation, we assume the redundancy length
to be 2N . The redundancy is the sequence which has the...

...

j-1

jj+1

j+2

SS

Seq
"1"

Seq
"0"

NNNN

K-1
K-2
"1" or "0" "1" or "0"

DataRedundancy(a)ifj-1

min D0 ( Sjj+2

K-1"1" or "0"

Seq
"0"jS

K-1
K-2
"1" or "0" "1" or "0") +D0 ( S)

= (j-1, j) = (0, 0)
j-1

min D0 ( SK-2
"1" or "0"

jj+1

K-1
"1" or "0"

Seq
"1"

j+2
Seq"0") +D1 ( S)

= (j-1, j) = (0, 1)
j-1

min D1 ( SK-2"1" or "0"

jj+2

K-1
"1" or "0"

Seq
"0"

j+1
Seq"1") +D0 ( S)

= (j-1, j) = (1, 0)
j-1

min D1 ( Sjj+1

K-1"1" or "0"

Seq
"1"jS

K-2
K-1
"1" or "0" "1" or "0") +D0 ( S)

= (j-1, j) = (1, 1)(b)Fig. 4.

- 55 -(6)Proposed Error-Correcting Method (Nd = 2N ).

IV. S IMULATION R ESULTS
To evaluate the proposed error-correcting method, we carryout computer simulations with following simulation conditions. In the transmitting side, the packet consists of the data
part with K = 64 bits and the redundancy part with 2N or 4N .
In this case, each coding rate becomes 0.96 or 0.94. Here, theparameter of the skew tent map is fixed as a = 0.05, and we
use the chaotic sequence length N = 3 and 4. To compare theperformance of Nd -dimensional space, we use Nd = 2N and
Nd = 3N . Based on these conditions, we iterate the simulation
10,000 times and calculate the average of BERs.
Figures 5(a) and (b) show the BERs versus Eb /N0 for
N = 3 and N = 4, respectively. To compare the performanceof the error-correcting method, Figs. 5(a) and (b) also show theperformance of our suboptimal receiver without the coding.
From these results, we observe that the both BERs of theerror-correcting method show gain over the system without
the coding. Significant improvements are observed for N = 4
with Nd = 3N -dimensional. In this case, since the receivercalculate the shortest distance using Nd = 12-dimensional
space, the computation complexity also increase. As ourreceiver calculates the likelihood of the received symbols not
only by the shortest distance between the received signals
and the chaotic map but also analyze the chaotic dynamics
of the successive symbols, the likelihood of the received symbol increase and the performance improvement is achieved.
Moreover, although the cording rate is very high, the proposed
method can well correct the error. Therefore, we can say thatthe efficiency of the proposed error-coding is excellent.
However, we also confirm that the curve of the proposed
method of N = 3 with Nd = 3N -dimensional space corresponds that with Nd = 3N -dimensional, as shown in Fig. 5(a).

- 56 -

1.0E+00

Error-corecting method (Nd=3N)
Error-corecting method (Nd=2N)
Conventional method(w/o coding)

BER

1.0E-01

1.0E-02

1.0E-03

1.0E-04

05

10

Eb/N0[dB]

1520(a) N = 3.
1.0E+00

Error-corecting method (Nd=3N)
Error-corecting method (Nd=2N)
Conventional method(w/o coding)

1.0E-01

BER

known information, “1” and “0”. Here, let us expect that thej = (K − 1)th bit is “1” or “0”. Since the (K − 1)th bit
is “1” or “0” and the redundancy has “1” or “0” certainly,
the (K − 1)th bit sequence also has the relation with eitherof two types of the sequence based on the chaotic dynamics.
Moreover the (K − 2)th sequence also has the relation witheither of three types of the sequence. Thus we can consider
the j and j − 1th bit as the four pattern, as shown in Fig 4(b),where D1 (.) and D0 (.) mean the shortest distance between thechaotic map of Symbol “1” and “0”, respectively. Since thesequence connected two types becomes the double sequencelength, we can calculate the shortest distance using Nd = 2N dimensional space. We calculate each distance like the patternof Fig 4(b). Finally, we choose the pattern of the shortest
distance out of the four patterns, and the (K − 1)th bit
is decided accurately. By using this determined symbol for
detection of the next bit, these operation can be shifted and it
can demodulate to all information. Therefore, the likelihood ofinformation can improve by expecting the combination of thesequence based on the chaotic dynamics, and we can detect
information in high accuracy. In this case, the coding rateK.
becomes K+2In the same way, we can calculate the shortest distance using
Nd = 3N -dimensional space to perform the error-correcting.
In this case, the redundancy length is 4N , the combination of
the sequence becomes the eight patterns and the coding rate
K.
becomes K+4

1.0E-02

1.0E-03

1.0E-04

05

10

Eb/N0[dB]

1520(b) N = 4.
Fig. 5.

Simulation results.

We have not clearly understood the reason why two curves
correspond, yet. We would like to investigate and analyze thechaotic sequence and its behavior in detail when our errorcorrecting method is used.
V. C ONCLUSIONSIn this study, to improve the error performance of the suboptimal receiver more, we have proposed the error-correcting
method using the features of chaos for the suboptimal receiver.
As results, we have confirmed that the likelihood of thereceived symbol increases and obtained the 3dB gain in error
performance.
ACKNOWLEDGMENTS
This research is supported by the fund of Open ResearchCenter Project from MEXT of Japanese Government (20072011).
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