Joumal of Signal ProcesSlng, Ⅵ)日3, No･ 2, pp･ 16日69, March 2009

PAPER

Suboptimal Receiver Using Shortest DistanceApproximation Method for Chaos Shift Keylng●

Shintaro Arai and Yoshifumi Nishio
Department of Electrical and Electronic Englneerlng, Tbkushima University●●

2‑1 Minami‑JosanJlma, Tokushima 770‑8506, Japan● ●E一mai1: (訂ai, nishioI◎ee･toknshima‑tl･aC･jp

Abstract ln this paper we describe the design of a receiver for use in chaos communications systems･●First, We focus on the optimal receiver proposed by Hasler and Schimmlng･ The optimal receiver calculates
the probability density function (PDF) between the received signaland the clmotic maps used on the trans一

mitting side and detects the information symbol by choosing the larger probability･ Howeverl the optimalreceiver suqersかOm high computational complexity. Also, the use of a long chaotic Sequence decreases

the performance of the optimal receiver･ Thusl it is important to develop a receiverwith performance●●

equiⅥdent or similar to that of the optimal receiver uslng a more efRcient algorithmI l･e･I a Suboptimal●receiver. In this study, We propose a new suboptimal receiver uslng a Shortest‑distance approximation

method instead of calculating the PDF. The proposed method has smaller computational complexity thanthe optimalreceiver. We performed computer simulations to show the validity of the proposed suboptimalreceiver. Additionally, to take advantage or the proposed method, we also propose a chaotic‑sequence･
length detection method. This method can detect the point at which the length of the chaotic‑sequence ofthe transmitted signalchanges. Simulations confirm that the detection of the information symbol from thechaotic sequence is possible even when the chaotic sequence length is changed.

Keywords: chaos communications, noncoherent detection, optimalreceiver, suboptimalreceiver

1. Introduction

by chaos shi氏keying (CSK), the receiver estimates

the likelihood of different chaotic maps being used
Cha加ic sequences obtained from a certain class offor modulation at the transmitter by calculating thedifference equations are nonperiodicand sensitive to

PDF between the received signaland the chaotic maps

initial conditions, and it is di氏cult to predict theirused on the transmitting side. The optimalreceiver

future behavior from past Observations. It is impor‑canthen detect the information symbol by comparlng●

features of chaos. Digital communication systems us‑

the probabilities. However, the optimalreceiver suf‑
fers from computationalcomplexity due to the large

ing chaos 【1日11] areaninteresting topic in thefieldchaotic‑sequence length. Also, the long chaotic se‑of englneerlng Chws.. Many researchers in thefield of

quence decrea5eS the performance of the optimalre‑
ceiver. Thus, it is important to develop a receiver with●●tant to develop englneerlng Systems exploiting these●●

nonlinearcircuits and systems have focused on the de一●

代lopment of noncoherent detection systems that do

not need to recovFr basis signals (unmodulated car‑performance equivalent or similarto the optimalre‑
ceiver uslng a different algorithm, i.e., a suboptimal●●

riers) at the recelVer･ Differential chaos shift key‑lng (Desk) ll]and the optimal receiver t2] are well‑

reCelVer.

The purpose of this paper is to propose a new sub‑●known typicalnoncoherent systems. In addition, cor‑

optimalreceiver uslng a Shortest‑distance approxima‑

relation delay shift keying (CDSK) [3】, which is similartion. The core problem of the optimalreceiver d&to Desk, iSalso a noncoherent system.

sign is the calculation complexity of the PDF due to

The optimalreceiver l2] performs optimaldetec‑●the large daotic‑sequence length. In other words, totion by using a probability density function (PDF).

realize a.suboptimalreceiver, it is necessary to sim‑

When the optimalreceiver receives slgnals modulated ‑plify the detectionalgorithm. Therefore, instead of

Joumal of Si9nal Processln9, Vol. 13, No. 2, March 2009161●

calculating the PDF between the received signalandthe chaotic maps, the proposed suboptimalreceiverapproximates the PDF by calculating the shortest
distance between the received signaland the chaotic
mapsand detecting the transmitted symbol･ To eval‑uate the proposed methodI we perform computer sim‑
ulati｡ns弧d observe the bit error rate (BER)･

Moreover, wealSo consider a method of detecting
the dwtic‑Sequence length using the proposed sub‑optimalreceiver･ When the chaotic sequence is used,(b) Symbol "0"(a) Symbol "1"

the transmitted energy depends on the initial value･This sometimes causes a decrea5e in the BER perfor‑

Fig･ 2

Chaos shift keying

mance. Fbrthemore, in the case of CSK) two different
symbols may generate similar sequences,and this also

decreases the performance･Asa simple method ofsolving these problemsI a CSK transmitter with a vari‑able chaotic‑sequence length) which was more effec‑

tive thanthe co.nventionalCSK, was proposed in l8]･
However, this simulation was carried out under theassumption that the血aDtic‑sequence length for euh●●

infomation symbol was known on the recelVlng Side･

2.1

Transmitterh the transmitter, a daotic sequence is generated
by a chaotic map･ In this study卜the transmitter uses

a skew tent map, a simple chaotic map) as shown in

Fig. 2(a), whid is described by Eq･ (1),Therefore) in this studyT wealso propose a method of

2xh+1‑adetecting the Sequence‑length swit血ing point using

1+α

the proposed suboptimalreceiver･ The performance

Xk+1 =of this detection method is investigatedand evaluated

CSK communication system is introduced･ The d&tection method used by the optimalreceiver isalso
focused on in this section. In Sect･ 37 the proposed

Suboptimalreceiver is described･ In Sect･ 47 we dis‑cussthe computer simulations that were performed toevaluate its performance･ In Sect･ 5, the method of de‑tecting the chwtic‑Sequence length using the proposed

suboptimalreceiverand the computer‑Simulated r&
sultSare PreSented･ Finally) we conclude this paper in(1)‑2xk+1+a(a<xh≦1)1‑αby performing a simulation･b Sect. 2, the model of the discrete‑time binary(‑1≦xk≦a)where a denotesthe position of the peak of the skew
tent map･ The transmitter can generate different●

chaotic sequences for every symbol by changlng theinitialvalue. The information symbol is modulatedby CSK.

csK is a digitalmodulation system using daos･
when thle transmitter generates the slgnalI the choice●of chaotic map depends on the transmitted symbol･ If
the information symbol I"1" is sent, Eq･ (1) is used

(Fig. 2(a)),and if "0" is sent, the reversed functionofEq. (1) is used (Fig･ 2(b))･ To transmit each bit ofSect. 6.

●

2. System Overview

we considerthe discrete‑time binary CSK commu‑
nication system Shown in Fig･ 1･ This system consists

information, N daDtic slgnalSamplesare generatedI
where N is the chaotic‑sequence length･ Therefore)
the transmitted signalis denoted by a vector S ‑(SIS2 ･･･ SN)･

of a transmitter, a channeland a receiver･ Each blockis described in detail below.

2.2

Channel and noise

we舶Sumean

additive white Gaussiannoise(AWGN) channel with a meanof zeroand a varianceof No = q2. The AWGN channel is the most com‑●

ChaoticBip81

monly used basic channel model･ Here, the noise slg‑

generator

nalis denoted bythenoisevectorn ‑ (nl n2 ･･ ･ nN)･
Thus) the received signal block is given by a ‑●(RIR2 ･･･ RN)‑S+n･
Fig･ 1 Block diagram of discrete‑time binary CSK

2.3

Receivercommnication System
The receiver recovers the transmitted signal fromthe received signaland demodulates the information

162

Journal of SignaI Processtng'VoI･ 13･ No･ 2･ March 2009

Symbol. Since we consider a noncoherent receiver, thereceiver has the chaotic map used for the modulation

Table 1 Relationship between chaotic‑sequencelength and number of subintervals

at the transmitter in its memory. However, the initialvalueand the information symbol transmitted are notPartitioned SubintervalS

known on the recelVIng Side. Before explaining the●●

proposed method, We introduce the optimal receiverand the suboptimalreceiver.

【 Optimal Receiver 】

The optimalreceiver was proposed by HaslerandSchimming l2]. This receiver selects the symbol q us‑
lng the a postem'ori probability after the slgnal block

functions. As the totalnumber of error functions is

氏 is received.

glVen aS 2N, the implementation of the optimalre‑

●

●

●

ceiver becomes di瓜cult for large N. Actually, in l2】

q〜 ‑argmaxProb(q is sent J 氏)

(2)q

it is reported that the number of calculations carriedout by the optimal receiver was reasonable only up to●

However, since the a posteTior令prObability is not easy●

to calculate, We rewrite the equation uslng Bayes'rule,

N‑15. In addition, in l2] the BER performance wasonly investigated for N‑2, 4 and 8.

i.e.,

To implement the optimal recelVlngalgorithm for●●large N, the PDF of the optimalreceiver is calculatedProb(q is sent I A)

using今numericalintegration method l51. By applyingp(RI q is sent) x IProb(q is sent)

p(a)

(3)

numerlCal integration, Eqs. (5)and (6) are rewritten
a8

where p(.) is the PDF. Here, Prob("1" is sent)andProb("0" is sent)are 1/2･ In addition, since p(A) isp(Rl "1"is sent)
上independent of q, Eq. (2) can be written as

HRm

LA3A& 2(2打q2)N/2 皇e‑∑‡ ̀Ri;.1'111'̀Z'"26･･ (7)qk argm㌍p(RI q is sent)

J=1

(4)

Asanexample, we explain the case ofN ‑ 2. Inp(R巨̀0"is sent)

this study, it is a5Sumed that the initialValue SI Of(Ri一g(i‑1)(.J))2●

e批h chaotic slgnal block S is chosen randomly in ac‑

日加1
LAB‑i 2(2打q2)N/2

cordance with the naturalinvariant probability densi‑

ぬ一(8)ties of land ‑I, where I is the function of the skew
tent map. The naturalinvariant probability densities

where 6xl ‑ 2/L, ∬l ‑ ‑1 + (k‑ 1)6xk, I(3')(α) de‑of land ‑I are both the constant 1/2 in the inter‑notes the iteration number 3'of function I, which has

Val[0, 1]･ Thus, the PDFs ofp(Rll "1"is sent)andp(RIJ "0"is sent) arealso equalto 1/2 andare limited

aninitialValue ofα (i.e. I(0)(α) ‑ α),and 9 ‑ ‑I.

In this case, the accuracy of the calculated PDF de‑pends on the parameterムSince the skew tent map

to 【0, 1】, i･e･,●

is a pleCeWise‑linear map, it has two subinterⅦユs Jl

and I2 in the whole intervalI, aB Shown in Fig. 3.

p(Rl "1"is sent)&/̲Il eWhen the chaotic‑sequence length N is decided, thedSl (5)In (Il,I2, ･ ･ ･ I2N‑I). This means the trajectories withchaotic‑sequence length N have their initialValue in a
subintervalIn. In other words, uslng numericalinte‑p(RI "0"is sent)&IIle

whole intervalI is partitioned into 2N‑1 subintervals●

dSl (6)

where q2 is the variance of the noise･ Finally the op‑

gration, the PDF has to be calculated with L > 2N‑1･Thus, the complexity of the calculation of the PDF is

exponentialin N as shown in Table 1. In other words,

althoughthe calculation of the PDF for a short length

timal receiver determines the decoded symbol to be(N ‑ 4 or 8) is possible, that for alonglength (N ‑ 16

1 (or 0) for p(RI "1"is sent) > p(Rl "0"is sent) (oror 32) is extremely difEcult. I

p(Rl "1"is sent) < p(RI "0"is sent))･

Asmentioned above, in the case of N ‑ 2, two【 Subopti‑al Receiver 】

The suboptimalreceiver is a recelVlng System that●error functions need to be computed for the PDF. In●

other words, to detemine the decoded symbol, theapproxim孔teS the operation of the optimalreceiver

optimalreceiver has tO Calculate a totalof four error

uslng a differentalgorithm. Note that the detection

Joumal of Si9nal Processln9, Vol. 13, No. 2, March 2009

●

163

'㌔
1

I

Rk = (Rh, Rh'), Rh'2)
Fig･ 5

Samples (Nd : 2,3,･･･)･
Asanexample, we explain the case for Nd ‑I

Fig･ 3Calculation of the shortest distance

Skew tent map illustrating the mapping ofsubintervals tland Z2 Onto I

3･ Figure 4 shows the 3‑dimensionalspace ofthe skew tent map whose coordinates correspond●to three successive received slgnalSamPles Rk‑(Rh,Rh+1,Rh+2) where k ‑ 1,2,･･･,N‑2･ To de‑●termine which map lS dosest to point Rk in the 3‑
dimensional space in Fig･ 41 the shortest distance be‑

tween the pointand the map has to be calculated･●

The receiver calculates the shortest distance uslng

the scalar product･ Two vectorsare required for ascalarproduct. Any two points Po ‑ (xo,yo,zo)and

Pl ‑ (xl,yl,Zl)are Chosen from each straight line in
the 3‑dimensional space, as shown in Figl 5･Using Fig･ 5, we can calculate the point P ‑(X,Y,Z) closest to Rkand the shortest distance D●●

uslng the following equations･P‑ (X,Y,Z)‑ (u･vo)u+Po

(9)

1 ･1

Fig･ 4

Proposed detection method

D‑

JIP‑RkII(X‑Rk)2 +(Y‑Rk.1)2+(Z‑Rh.2)2(10)

char批teristic of the suboptimalreceiver cannot be su‑

perior to that of the optimalreceiver･H ere

3. Shortest‑Distance Approximation Method(proposed Suboptimal Receiver)b this section, we propose a new suboptimal re‑

Pl‑Pounit vector u‖Pl ‑Poll

vo

‑

Rk‑Po

●

ceiver uslng a Shortest‑distance approximation･ As

described in Sect. 2, the drawback of the optimalreceiver is the complexity of the calculation of the

PDF for large･N･ In other wordsl tO realize a sub‑optimalreceiverl it is necessary tO Simplify the de‑tectionalgorithm･ In this studyT instead of calcu‑

latingthe PDF between the received signalandthe
血wtic map, our suboptimalreceiver approximates

the PDF by calculating the shortest. distance between
the received signaland the chaotic map then d&●tects the transmitted symbol uslng that information･Concretely, the receiver calculates the shortest dis‑Note that if Rk is outside of the cube) we calculate
the distance betw既n Rkand the nearest point thatlies on one of the edges of the map･

For the 3‑dimensionalcase, thereare four stmightlines in the space･ Therefore) the minimum of thefour distances is chosen as the shortest distance Dlfor symbol uln･ In the same way) D for symbol uOn
is chosen as D｡. The receiver calculates both Dland
D｡ forall hand their summations ∑Dland ∑Do･

Finally, we define the decoded symbol a8 1 (or 0) for∑Dl <∑Do (or∑Dl >∑Do)･
The calculation of the shortest distance canbe ex‑tance between the received signaland the map in Nd‑
dimensionalSpace using Nd Successive received signal

tended to Nd‑dimensionalSpace for Nd >̲ 4･

164

Joumal of Signa一 Processln9, VoI･ 13, No･ 2, March 2009●

4.

1.OE十00

Numerical Simulations

Tb evaluate the proposed method, we carry out

1.OE‑01●

numericalsimulations under the followlng COnditions.
On the transmitting side, 10,000 Symbols are trams一

t[的中

●

mitted uslng Chwtic sequences with different initial

1

防舵

●

Ⅶ山es. Here, the parameter of the skew tent map lS

fixed as a ‑ 0.05,and chaotic sequence lengths rang‑

ing from 4 to 32 are used. On the receiving side, BER

is recorded for various Eb/No･ To compare the per‑formances for different numbers of dimensions used

1.OE‑0410

in the calculation of the shortest distance, we use 4‑

dimensionalspace (4‑D)and 8‑dimensionalspace (8‑D). On the basis of these conditions, we repeat the20

15

Eb/No ldB](a) Optimalreceiver calculated using Eqs･ (7)and (8)simulation 1 ,000 times and calculate the average BER.

1.OE◆00

Figures 6(a)‑(C) show for BER versus Eb/No for the●

optimalreceiverand for the proposed method uslng 4‑
D and 8‑D, respectively. Here, to simulate the optimalreceiver, we apply the numericalintegration method(Eqs･ (7)and (8)) with L = 2000･ This value is chose.non the basis of 【5】. Also, Fig. 6(d) shows a comparト

son of the optimal receiver and the proposed method
when 〟 is 4 and 8. The pe血rmances of the optimalreceiverand the proposed method improve aB Eb/No
increases. However, the performance of the optimalreceiver decreases with large N, as shown in Fig. 6(a).

10■

Eb/No 【dB]

The main reason for this is that the performance of(b) Proposed method (4‑D)

the optimal receiver depends on the accuracy of thenumerical integration used to obtain the PDF. As de‑

I.OE+00

scribed in Sect. 2.3, the complexity of the calculationof the PDF is exponentialin N. Another rea50n forthe low performance when N ‑ 16and 32 is that forthese values of N, L ‑ 2000 is too low for the optimalreceiver to performanaccurate calculation. Although

1.OE‑01

PS

R l･OBO2●

the calculation accuracy canbe increa5ed by uslng alarger L, it results in increa5ed computation time andmemory usage. Figure 6(a) shows the limit of appli‑

cability for the optimalreceiver with calculation us‑●lng numerical'integration. Later, We will discuss the10

relation between Land computation cost in more de‑

Eh/No ldB]

tail. On the other hand, the proposed method can(C) Proposed method (&D)

detect infomation symbols with highaccuracy evenfor larger values of N because of the smaller computa‑

1.OE◆00

tionalcomplexity of the proposed method. Although

一■ 凵｡ ● ● ● ● ● ● 一■ ●一■ ● I ■‑in the case of the optimal receiver, the number of PDF

1.OE･O I■

calculations increases with large N, for the proposed
method, the number of dimensions used for the calcu‑

鋸lation of the shortest distance is constant. Therefore,g l･OEJ2

the computational complexity of the proposed method

Propodneth○d脚=4)... 劍

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Optindrher(〃ヒ4)0 劔…丸.̲̲̲.̲̲..一.̲.̲. ●●decreases relative to that of the optimalreceiver with

PropoBedmethodU=NJ=8)‑●

increaslng 〟.

OptindrcceiyerW=8)+
一

Fbrthermore, it canbe observed that the curvefor the proposed method corresponds to that forthe optimalreceiver when N

‑

●

●

proposed method using the shortest distance in Nd‑
dimensionalspace becomes almost idQnticalwith the
Journal ofSignaJ Processtng, Vol. 13, No. 2, March 2009 I

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Nd, aS Shown in

Fig. 6(d). Ftom this result, We consider that the●(d) Comparison of optimalreceiverand proposed
methodFig. 6

BER performance165

Table2ComputationtimeandBERperformancewithN‑Nd‑4(106symbols,Eb/No=13dB) ●

I 微

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Numberofstraightlin甲̲̲̲̲

optimalreceiver for the case that the dimension Nd isequalto the length of the chaotic sequence N･ Namely,
we have confirmed the validity of the proposed sub‑optimalreceiver a5anapprOXimation method of the
optimalreceiver･

ﾓ#Sb
繝R

3X2JV‑l=512
姪

B

2JYJ‑1=128
ｳ

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X

carried out under the assumption that the daotic‑
Sequence length for each information symbol was

known on the recelVlng Side･ We propose a method●●●uslng the proposed suboptimal receiver for detecting
the point where the chaotic‑sequence length changes･･ Finally, We investigate the computation cost of the

proposed method･ We will compare the BER perfor‑
mance with the AWGN dannel and the computation

5.1 Method of detecting daotic‑sequence length

time required to decodeal1 the symbols･ The tranS‑

The system model used to detect the switding

mitte, sends 100,000 symbols with N ‑ 4 or ‑8. Eb/No

point is assumed to be the same as the system de‑is fixed at 13 dB. Also, to compare the performance,

scribed in Sect. 2. The only difference is that the

L for the optimalreceiver is set to multiplies of 2N‑1 ･chaotic‑sequence length changes at the transmitter

These simulations are carried out using a PC with aand is detected､ by the receiver･ In this studyl We

core2Duo 2.4 GH2i CPUand 2 GB of RAM.

consider a very simple model that switdes between
2 different chaotic‑sequence lengths: N1 ‑ 8andNi ‑ 12. Figure 7(a) showsanexample of a time

Tables 2and 3 show the BER performance andthe computation time･ The BER performances ofthe optimalreceiver calculated using Eqs･ (7)and (8)and the proposed method arealmost the same whenseries where the chaotic sequence length is changed

at k ‑ 12 (i.e., Ni). The depicted sequence contains●

L = 3 × 2Nll. Thus, to obtain sufRcient calculation

accuracy of the optimalreceiver) L must be at least
3 × 2N‑1. In addition, the computation time for the●two chaotic‑sequences comprlSlng the symbols ulnand"o", respectively･ HereI it is assumed that no noise isadded to this sequence. Figure 7(b) shows the re‑optimalreceiver withthis value of L is 4‑6 times larger

sults of calculating the shortest distances for eyery k･thanthat of the proposed method･ Ftom these resultsIHere, we use 4‑D for the calculation of the shortest dis‑

we can see that a very large Ⅶ山e ofいs needed totances to explain the detection of the chaotic‑sequenceobtain good calculation accuracy of the'pDF uslng

length visually･ Actually) the detection of the chaotic

the numericalintegrationwith N ‑ 16and 32･ Fbr‑sequence is possible for a space withany number of di‑themorel it is expected that the computation cost ofmensions. In Fig. 7(b), the solid line and da血ed line●

the proposed method is lower thanthat of the opti‑

show Dland Do, respectively･ Since no noise is addedI

malreceiver uslng the numericalintegration method
for N = 16and 32. Therefore, the lower computation

the difference between the two lines is very large forcost isanadvantage of the proposed method･for 10 < k ≦ 12. To observe this difference in detail,●k < 9 and 13 < k. This difference becomes small
we show the absolute value of the difference between

5. Detection of Chaotic‑Sequence I‑ength
Using Proposed Suboptimal Receiver

Dland D. in Fig. 7(C). The differencesaround theswitching point are mud smaller than those at otherpoints.

As described in Sect. 1, we proposed a CSK trans一

mitter in which the daotic‑sequence length danges

To explain this) we use Fig･ 8, which shows theswitching between two cha加ic sequences･ It canbe

e瓜ciently [8】. However, the above simulation wasobserved that parts of both chaotic sequences are

166

Joumal of Si9nal Processln9, VoI･ 13, No･ 2, March 2009

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Diff(Nl) ‑ IDl ‑ Dolat h=Ni‑1
Diff(Ni) ‑ lDl ‑ Dolat kNi‑1(13)

The ShorteSt∋st distance is calculated

uSlng parts 0f bothSequenCeS.
●

Diff(Nl) < Diff(Ni) ･ ･ ･ Switching at NI
Diff(Nl) > Diff(Ni) ･ ‑ Switching at Ni
where Diff() means the difference. In the case Of

Fig. 8

CauSe Of small difference between two lines

around switching point

Fig･ 7(C), since Diff(12) iS Smaller thanDiff(8), theswitching point is determined to be N1 ‑ 12. More‑over, since Dl is smaller thanDo, the information
symbol is detected as "1".

used for the calculationaround the switching point(k ‑ 12). Therefore, the shortest distancearoundIn the same way, the proposed method candetect

the chaotic‑sequence lengthand the information sym‑the Switdling point is calculated as if there was noise

bol even when the same information symbol compnses

present. This is the key point of the proposed method.

both sequences, as demonstrated by the sequences

●

●

●

Namely, if there is a point at which the difference be‑comprlSlng "1"and "1'',舶Shown in Fig. 9. Ftom

tween Dland Do is very small, the chaotic sequenceFigs. 9(b)弧d 9(C), we can see that the switding point

may have switched at that point. Because we assume

and the infomation symbol are detected as NI'= 8that N1 ‑ 8and N1 ‑ 12 in this study, the receiverand "1" , respectively.

only calculates the difference between Dland Do at

Asone can see from Figs. 7(C)and 9(C), the detec‑

8 and 12, and these two values are compared. Thetion becomes easier if Diff() at the switching point

receiver candetermine the chaotic‑sequence length byis very small. In other words, by choosing a suit‑●

血oosing that with the smaller difference. These op‑

erations are described byable initialValue that makes the differencearound theswitching･point of the sequence small, the perfomanceof the chaotic‑sequence‑length detectionwill be im‑●

Joumal of Si9naI Processln9, Vol. 13, No. 2, March 2009167

1.OE+00●

proved･ Hence) We propose the followlng method forchoosing a Suitable initialValue at the switching point･
●

I ● I

al｡1 0ヒ○患uola tZOnbes

First, the transmitter chooses an initialValue at●●■●●●●■●■●●●●●● ヽ■‑●‑●‑● 凵怐怐怐怐怐怐怐怐怐｡●●●●●!'q.I:i ●1.OE‑01

一

●ヽ

･ヽ

;ヽ

ﾈ耳

ﾈ

ﾂ

randomand generates a chaotic sequence･ It then cal‑culates the differences at the possible switching pointsT

4‑Du血gprop8edtAethoJ ○fchooBineidti8lydue‑ー 劔

1.OE‑02ヽ

namely Diff(Nil)弧d Diff(Ni)･ This calculation is
8‑DuBingpr○pBedm ○fchoBinginiti8I 册thod yaJue‑ 劍

repeated M times for different initialValues chosen at
1.OE‑03

random. All the results of the calculations are stored･

4‑Dtni叩initidyalue.
cho8en8trBndOm'●●''

"

B

ﾂ

●I

劔;t

8‑DsEiniittid,8SI.u=‑.‑ 劔 白 ﾂ B

After M calculations, the transmitter chooses the ini‑
tialValue that results in the greatest difference be‑

1.OE･04

tween Diff(Ni)and Diff(Ni) fromamongall the

一LJ

05101520253

Eb/No [dB]stored results. Note that this proposed method has(a) Sequence‑length error rate

better performance for larger M,althoughthe calcu‑
lation timealso increases.
5.2

1.OE+00

Simulation results

1.OE‑01

In this section, we study the performance of theP弓method of detecting daotic‑sequence length by using

呂1･OE‑02

computer simulations･ In additionl tO COnfirm the ef‑●

fect of the proposed method of chooslng the initial

1. OE‑03

Valuey wealso carry out a simulation in whichthe ini‑
tialValue is chosen at random. The cha加ic‑sequence

1.OE‑04

lengthswit血es between N1 ‑ 8and N1 ‑ 12 atrandomand the best initialValueamong M ‑ 100
randomly selected values is chosen･ To calculate the(b) Bit error rate

Shortest distance, Nd ‑ 4 and 8 (i･e･, 4‑Dand 8‑D)打e used. Other simulation conditionsare the same as

Fig･ 10

Simulation results f♭r the detection of

the chaotic‑sequence length

those in Sect. 4. In this simulation, it is assumed that

BER is calculated after the chaotic‑sequence length isdetected correctly.

initialValue that meets the condition Dl穴ブDo is cho‑

Figures 10(a)弧d 10(b) respectively show thesequence‑length error rate a'nd BER･ Ftom both re‑sults) the performance of 8‑D is better thanthat of

4̲D. The main reason br this is that the sequences
are generated from the chwtic dynami･cs･ In otherwords, chwtic sequences with 8 Valuesare more ap‑●

propriate for expresslng Chaotic properties thanthosewith 4 Values. Therefore, the detection of chaotic‑

sen. currently, we choose the initialValue from M
random values, but if we knew the conditions for ini‑
tialValues that result in Dl穴ゴDo at sequence switch‑●lng pOintsI we can avoid the method of random sam‑pling for obtaining a suitable initial value･ In addition,

the calculation time might decrease･ Therefore,anin‑●

vestigation of the theoreticalconditions resulting ln
Dl穴〜 Do is planned･

sequence length canbe made more effective by using

higher‑dimensionaldata･ In this study,althoughwe

6.

Conclusions

have performed the simulation in the case of N1 ‑ 8

and N1 ‑ 12, we expect thatthe proposed method
candetectalso other sequence lengths･●

Next, let us observe the performance when uslng●

the proposed method of chooslng the initialValue･ We
comparethe proposed method with the result of asimulation with the initialValue chosen at random, as

showninFig･ 10(a)･ It is clearthat the performanceof the proposed method of choosing the initialValue lS
betterand that the choice of the initialValue is very
import弧七･ Nextl let us consider the conditions thatmust be met by the initialValue to ensure the easy de‑tection of the chaotic‑sequence length･ The proposed
method calculates Diff() M times with different ini‑
tialValues. When Dl矢‡ Do, Diff() becomes smallest,

namely, the sequence‑length error rate decreases iran168

we have proposed a new suboptimal receiver usinga shortest‑distance approximation･ The advantages ofthe proposed method are its very simplealgorithm
and lower cost. It wa￡ Shown by simulation that'thepe血rmance of the proposed method corresponds tothat of the optimal receiver when the dimension of

the space used Nd is equalto the length of the chaotic
sequence N･ Therefore)the validity of the proposed
suboptimalreceiver as an approximation method of
the optimalreceiver is confirmed･ However) we have
only conかmed the pe血rmance of the proposed re一

cciver up to N ‑ Nd ‑ 8･ An investigation of the
maximum value of N that canbe applied to the pro‑
posed receiver is left aB afuture study･Journa一 of Signal Processlng'VoL 13･ Not 2, March 2009

shintaro Arai

we havealSo proposed a method of detecting themaguchiI

血脈tic‑sequence length using the proposed subopti一

waS born in Ya‑

Japanl

in

1981･

He

received the B.E. and M.E. de一

malreceiver. The simulation results showed that thegrees from Tokushima University'

detection of the information symbol from the chaotic

Tbkushima, Japanl insequence is possible even when 2 different sequence

2006, reSpeCtively･ He is cu汀ently

2004

and

working towards Ph･D･ degree at thelengths are switded･
h this paper we have introduced an example of asame univerSity･ From Jan･ 2007 to
Dec. 2008, he has been also working●

at Nagoya Universityl aB a 8peCialre‑

new chaos communications system using a Suboptimalreceiver. Therefore, an important future task is tosearch student. His research intereStS include chaos,based communi‑

cation systems and visible light com‑develop other e氏cient applications using this receiver･●munication 8y8temS u81ng LED for ITS･ He iS a Student memberof the IEEE and IEICE.

Referencesll】 G. Kolumban, B･ Vizvariand W･ Schwarz and A･ Abel:

YoShifumi Nishio received
B.E., M･E･I and Ph･D･ degrees in

Differentialchaos shift keying: A robust coding for chaos

communication, Proc･ NDESI96I pp･ 87‑92I Jun･ 1996･●

electriCal[2] M. Haslerand T･ Schimming: Chaos communication over
nolSy ChannelSI Int･ J･ Bifurcation Chaos, Vol･ 10, No･ 4,●

englneenng from Keio

University, Ybkohama Japanl in●1988, 1990, and 1993I reSpeCtively･

In 1993, he joined the Department

pp. 719‑736, 2000･

of Electrical and Electronic Engi‑[3] M. SuShchik, L･ S･ T8imring and A･ R･ VolkovSkii:●

neenng at TokuShima UniversityI

performanceanalysis of correlation‑based communication

TbkuShima Japan, where he iS

schemeS utilizing chaosI IEEETranS･ Circuits SySt･ Part I,

currently an

Vbl. 47, pp 1684‑1691, 2000･

Associate Profe880r･

Ftom May 2000 he spent a year
in the Laboratory of Nonlinear[4] F. C. M. Lau and C･ K･ Tse: Approximate‑optimaldetec‑

systems (LANOS) at the Swi88

tor for chaos COmmunication sy8temSI Int･ J･ Bifurcation

FderalInstitute of Technology Lausanne (EPFL) aB a Vi8iting

ChaoS, Vol･ 13I No･ 5I pp･ 1329‑1335I 2003･

profe880r. His research interests include a.nalysisand applica‑【5】 F. C. M. Lau and C･ K･ T8e‥ On optimal detection ofnon‑

Int. I. Bifurca.Lion Chaos, Vol･ 13, No･ 67 pp･ 1587‑15977

tion of chaos in electriCalcircuits,analysis of synchroni2;ationin coupled oscillatory circuitsl development of analyzing
methods for nonlinear circuits, theory and application of2003.

cellular neural networkS, and neural network architecture･ He●●

cohercnt chaos‑Shift‑keying SlgnalS in a･ noISy environment,[61 F. C. M. Lauand C･ K･ TSe‥ Chaos‑Based DigitalCommu‑

nication Sy8temS, Springer, 2003･

was the Chair of the IEEE GAS Society TechnicalCommittee
on Nonlinea.r Circuitsand Systems (NCAS) during 2004‑2005,●

the Steenng Committee Secretary of the IEICE 氏‑arch[7] W. M. Tan, F･ C･ M･ Lauand C･ K･ Tse: General‑society of Nonlinear Theory and its Applications (NOLTA)

ized correlation‑delay‑shift‑keying SCheme for noncoherent

during 2004‑2007, and is currently the Secretary/Treasurer Of

chaos‑based communica.tion systems) IEEETranS･ Circuits

the IEEE CAS Society Shikoku Chapter･ He was an A88OCiate

Syst･ Part II Vbl･ 531 No･ 3I pp･ 712‑721, 2006･

Editor of the IEEETransactions on Circuits and SystemSlI:●[8] S. Araiand Y･ NiShio: Variable sequence length transmitter●for noncoherent chaos shift keying, I. SignalProceSSlng,
Vbl. 10, No･ 4I pp･ 255‑258I 2006･

Regular Papers during 2004‑2005I and is currently 8erVlng aS

anAs80Ciate Editor for the IEEE CAS Maga2;inel the IEEE
CAS Society Newsletterl and the RISP Joumal of Signal●

proceSSlng･ He isal80 the Editor of the IEICE FbndamentalS(91 G. Cimatti, R･ Rovattiand G･ Setti: Chaos‑based gener‑

Review and is a member of the Editorial Board oHnternational

ation of optimalSPrea･ding sequences for DS‑UWB sensor

Journalof Circuit Theory and Application8･ He is a Senior
member of the IEEE,and a member of the IEICE and the
RISP.

networkS, Proc･ NOmIO6I pp･ 583‑586I Sep･ 2006･[10] S. Araiand Y･ Nishio: Detection of informa･tion 8ymbolSand Sequence lengths using Suboptimalreceiver for cha･os
shift keying) Proc･ NOLTA7067 pp･ 799‑802, Sep･ 2006･(Received June 16, 2008; revised August 4, 2008)[11] L. E. Larson, ∫‑M･ Liu and L･ S･ TSimring: Digital

Communications Using Chaos and Nonlinear Dynamics,
Springer, 2006･

Joumal of Signa一 Processln9, Vol･ 13, No･ 2, March 2009169