Joumal ofSignal Processlng'Wl･ 10, No･ 4, pp･ 255‑258, July 2006
●SELECTED PAPER

Variable･Sequence･LengthTransmitter br Noncoherent Chaos Shift Keyingshintaro Arai and Yoshifumi Nishio●Department of Electrical and Electronics Englneerlng

Tokushima University● ●2‑ 1 Minami‑Josaqlma, Tokushima 770‑8506･ Japan
phone:+8 1‑88‑656‑7470, Fax :+8 1 ‑88‑656‑747 1

Email : (arai, nishio) @ee･tokushima‑u･ac･jp

A bstract

Tram smi tter

Inthis paper we propose a new transmitter for noncoherent●chaos shi允keylng･ In this scheme, the new transmitter canuse a different chaotic sequence length determined by an ini‑tial value･ The performance of the coTTmunication system●

uslng the new tran･smitter is studied by computer simulation･
I●●●●■●●一●●一一●●●●●●■一■●一一●■●●Figure 1 : Block diagram of discrete‑time binary CSK com‑

1. Introduction

munication system

Recently, the use of a noncoherent receiver for digital com‑●munications systems uslng Chaos has been studied activelyll ]‑[4]. In particular, a suboptimal noncoherent receiver hav‑●●

2.1 TransmitterIn the transmitter, a chaotic sequence is generated by alng a Similar pe血rmance to the optlmal noncoherent receiverchaotic map･ In this study, We use a skew tent map to generate

has been developed.
●In our previous research, we proposed a suboptlmal re‑ceiver using a very simple algorithm･ Our method de‑the chaotic sequence･

2.1.1 Skew tent map

tects symbols from the calculated values of the shortest
distance between received signals and a chaotic map 【51･Xh+1

Furthermore, wp extended this concept to the distance in●Nd‑dimensional space uslng Nd Successive received signals(Nd : 3,4, ･ ･ ･) [6]･Asaresult, we obtained the best perfor‑
mance for the dimension Nd, Which was equal to the lengthof the chaotic sequence N･ In addition･ we confirmed thatthe pe仙‑ance of this suboptimal receiver improved as 〟increased.

In this study, to usethe average energy per bit effectively

and to improve bit errorperformance･ we propose a chaos
shift keying (CSK) transmitter using a variable‑sequence‑･1length･ We carry out computer simulations and investigatethe performance of the proposed method･･0150aO･5xAIFigure 2: Skew tent map●●The skew tent map lS Shown in Fig･ 2･ This map lS a Simple

2. System Overview

chaotic map and is described byFirst, we explain the discrete‑time binary CSK communi‑●

2xk+1‑a●cation system uslng an eXistlng transmitter, as shown in Fig･

1. This system consists of a transmitter, a channel and a re‑ceiver. Details of each bl∝k are described below.

1+α
Xk+1 =(‑1≦xk≦a)(1)‑2xk+1+a1‑α(a<xk≦1)

where a denotesthe position of the top of the skew tent map･●Journal of Signal Processlng, VoL 10･ No･ 4, July 2006

255

2.1.2 Chaos shift keying･1･0.500.5

1Xh1 ･1●

Figure 3: Chaos shift keylng
Figure 4: Detection meth∝1●

csK is a digital modulation system uslng Chaos･ When the●transmitter generates slgnals, We use the chaotic sequences

the map has to be calculated･ Therefore,'We calculate thegenerated by different chaotic maps depending on the value

shortest distance uslng the scalar product of the vector･●of the information symbol･ In this study, we use the skew
tent m叩and its reversal m叩･ aS Shown in Fig･ 3･ Ifthe

information symbol "1" is sent, Eq･ (1) is used, and if "0" issent, the reversed function ofEq. (I) is used･ To transmit l‑●

bit infbmation, 〟 chaotic slgnals are generated, where 〟 isthe chaotic ,sequence length･ Therefore･ the transmitted signal

isdenotedbyavectorS ‑ (SI S2 ･･･ SN)･(Rh, Rh'), Rh'2)

2.2.Channel and noise

Figure 5: Calculation of shortest distanceIn the channel, a noise is assumed to be additive white

Gaussian noise (AWGN) and is denoted by the noise vec‑t?rn ‑ (nl n2 ･･･ nN)･ Thus･ thereceived signal block iseach straight line in the space ofFig･ 4, as shown in Fig･ 5･

glVenbyR‑(RIR2 ･･･ RN)‑S+n･In Fig. 5, the unit vector u is calculatedfrom (xo, yo, zo) andTwo points (xo,yo,zo) and (xl,yl,Zl)are ■chosen from(xl,yl,Zl) using

2.3 Receiver and our suboptimal receiver
The receiver recovers the transmitted signals from the re‑

Xl‑Xo yl‑yo Zl Zou‑(l,m,n)‑A

ceived signals and demodulates仙e infbmation symbol･ In'A') (2,A

the detection methods, thereare coherent detection methods
that record the initial value of the chaotic sequence at the re‑where A isceiver and noncoherent detection methods that do not record

dition, the vector vo IS Calculated from (Rh,Rk+1,one. In仙is study, we use a noncoherent detection method us‑and (xo,yo, zo) using●(xl ‑I.)2+(yl ‑y｡)2+(zl ‑Z｡)2.Inad,
●

Rk+2)

■

lng the suboptlmal receiver proposed in our previous researchvo ‑ (Rk ‑X｡,Rk+1 ‑y｡,Rk+1 ‑Z｡)

【6】.

(3)

●

Our proposed suboptlmal r∝eiver calculates the short一●●

est distance between received signals and the map ln Nd‑
dimensional space uslng Nd Successive received slgnals●

The scalar product r of u and vo is calculated uslng●

T‑l(Rk‑Xo)+m(Rk+1‑yO)+n(Rk+2‑Zo) (4)(Nd:3,4,･･･)･
Asan example, we explain the case of Nd ‑ 3･ Fig‑ure 4 shows the 3‑dimensional space uslng the skew tentHence, vI Can be calculated fromthe scalarproduct of T

map whose coordinates correspond to the three successive

and u. Therefore, we can calculate the polnt With the shortest●

received signals (Rk, Rk+1, Rk+2)･ To decide which map
is closest to the point (Rk, Rk+1, Rk+2) in the 3‑dimensional●

distance (X, Y, Z) and the shortest distance D using●

space in Fig･ 4, the shortest distance between the polnt and(X,Y,Z) ‑ (Tl+xo,Tm+yo,Tn+zo)●256(5)

JournaJ of Signal Processlng, VoI. 10, Not 4, July 2006

D=(X‑Rk)2+(Y‑Rk.1)2+(Z‑Rk.2)2 (6)0.045

Note that if the point is outside the cube, We calculate the0.04

distance between the point and the nearest edge of the maps･0.035

For the 3‑dimensional case, thereare four straight lines inI. r･ ･ ;, l ‑.･ ･㌧ ■..p .∵･

0.03

the space･ Therefore, the minimum value of the four dis‑

tances is chosen as the shortest distance Dl for symbol "1"･

1宅0.025

In the same way,the D of symbol HO" is chosen as Do･ We●●1司0.02

calculate both Dl and Do for all k and find their surrmations

0.015∑ Dl and ∑ Do. Finally, We determine the decoded symbol

0.01

asl (orO)for∑Dl < ∑Do(or∑Dl > ∑Do)･
The calculation of the shortest distance can be extended to0.005

Nd‑dimensional space for Nd ≧ 4.
0

3. Proposed Method
Figure 6: Basis of proposed transmitter (Eb/No = 1 3ldB])

3.1 Basis of proposed transmitter

L･; ‑.･･.IAs described above, successive received. signals are re‑First, one s∝tion is detemined by the initial value of仙equired in order to calculate the shortest distance･ However,血e bit e汀α rate (石ER) increases when some successive re‑ceived signals are used･ To expound on this case, We carriedchaotic sequence in the lst block･ Next, a better sequence●length is decided uslng the results shown in Fig･ 6 in the 2nd
block･ Finally, the chaotic sequence with the better sequence●out the followlng COmPuter Simulation･On the transmitting side, the interval of the skew tent mapl‑1, 1] is divided into 128 sections･ We choose one section
and selected 104 initial points from this section at random･

length is generated in the 3rd block and used forthe CSKmodulation. Moreover, the next initial value is decided fromthe last value of the chaotic sequence.

Using the chaotic sequences starting fromthese initial points･
We transmit 104 bits of information. The chaotic sequence●

Initialvalue (determined舟om last Value)●length is触ed as N ‑ 4 or N ‑ 8･ On the recelVlng Side,the suboptimal receiver proposed by the authors is used torecover the information. To calculate the shortest distance,
the number of dimensions is負xed as Nd ‑ 4.

Figure 6 shows an example of the simulation results of theBER versus the section for N ‑ 4 and N ‑ 8 when Eb/No
I‑+1… or …‑1●●

is点xed as 13ldB], where Eb is the average energyper bit･
From this figure, we can触d that the BER strongly depends

on the section selected. Furthermore, the BER for N ‑ 4 isbetter than that for N = 8 for some sections, whereas N ‑ 8is better than that for N ‑ 4 for others. For example, we cansee that the BER ofN = 4 is better than that for N ‑ 8 forFigure 7: Proposed transmitter

4. Simulation Result

the 6伽h section. However, the BER of 〟 ‑ 8 is better thanthat for N = 4 for the 120th section. This means that theperfomance of the CSK corrmunication system is changed
depending on the initial value of the chaotic sequence･

From this result, We devise a new transmitter in which thechaotic sequence length 〟 is changed by the initial valueof the chaotic sequence･ Namely, we propose a variable‑sequence‑length transmitter･ We explain this transmitter inln this section, we study the pe血mance of仙e CSK com‑●munication system uslng the proposed transmitter by com‑puter simulation･ The simulation conditions are as fbllows･
In the proposed transmitter, the chaotic sequence length 〟is 4 or 8. In the channel, noise is assumed to be only AWGN.
Hence, the noises at the transmitter and receiverare not con‑●●sidered. On the recelVlng Side, the suboptimal receiver pro‑
posed by the authors is used･ To calculate the shortest dis‑

the next subsection.

tance, we use 4‑dimensional space. In this study, We assume

3.2 Variable･sequence･Iength transmitter
Figure 7 shows the block diagram of the proposed trans‑mitter･ This system consists of three blocks･■Journal of Signal Processlng, VoI･ 10, No･ 4, July 2006

that the chaotic sequence length 〟 is known on the receiv‑lng Side･ On the basis of these conditions,the systemperfor‑●mance is evaluated by plotting BER against Eb/No when 104
bits of infわrmation are transmitted.

257

I‑1一 ‑ 一‑I ,･‑I ‑､･.J ‑ 1■J‑I l･ ‑ I1‑ ▼I･1‑ ‑ 4･lJI‑･一ter BER performance has been achieved･ However･ we as‑
1.OE+00

sumed that the chaotic sequence length is known on the re‑●●celVlng Side･ Hence, we have to develop a receiver that candetect the chaotic sequence length in our future research･
1.OE‑01

l宅Refe ren ces

岩1･OE‑02【1] M. Hasler and T. Schiming: Chaos communication●

over noISy Channels, Int ･ Joumal Bifurcation and Chaos･

1.OE‑03Ⅵ)I. 10, No･ 4, pp･ 719‑736, 2000･

1.OE‑04【2】 F. C. M. Lau and C. K. Tse: Approximate‑optimal detec‑tor f♭r chaos communication systems, Int･ Joumal B血ト(a) Comparison with N ‑ 4cation and Chaos, Ⅵ)I. 13, No･ 5, pp･ 1329‑1335, 2003･【31 F. C. M. Lau and C. K. Tse: On optimal det∝tion ofnon‑

1.OE+00●●●

coherent chaos‑sh机‑keylng Slgnals in a noISy enViron一
ment, Int. Joumal B血mation and Chaos, Ⅵ)1･ 13, No･ 6,pp･ 1587‑1597, 2003･

1.OE‑01【4] F. C. M. Lau and C. K. Tse: Chaos‑Based Digital Com‑●munication Systems, Spnnger, 2003･

P弓

Eu ･OE‑02[5] S.血ai and Y. Nishio: Research on noncoherent detec‑●tion for chaos shift keylng, Proc･ NCSP'04, pp･ 265‑268･
1.OE‑03

Mar. 2(X)4.
【6】 S. Arai and Y. Nishio: Research on suboptimal receivers●for chaos shift keylng, Proc･ NCSP'05, pp･ 375‑378･ Mar･

1.OE‑04

2005.
(b) Comparison with 〟 ‑ 8

Figure 8: Simulation results

Figures 8(a) and 8(b) show the simulation results of the

proposed method･ To comparetheperformance of the pro‑
posed system withthefixed‑length transmitter･ the perfor‑

mances ofN = 4 and N ‑ 8 are shown together in Figs. 8(a)and 8(b), respectively. From the results, we血d that the per‑formance of the proposed transmitter is similar to or better
than that of the点xed‑length transmitter for both N ‑ 4 andN ‑ 8 for any Eb/No. In other words, the proposed trans‑mitter has a goodperformance for N ‑ 4 for small Eb/No
and a goodperformance for N ‑ 8 for large Eb/No･ Hence･
we can conclude山at the proposed method uses the averageenergyper bit effectively depending on noise level･

5. Conclusions

In this study, to usethe average energy per bit effectivelyand improve bit errorperfomance･ we have proposed a CSKtransmitter uslng Variable sequence length･Asa result･ a bet‑●

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Joumal of Signa一 Processlng, VoL 10, No･ 4, Ju一y 2006