2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Variable Sequence Length Transmitter for Noncoherent Chaos Shift Keying
Shintaro ARAI† and Yoshifumi NISHIO††Department of Electrical and Electronics Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {arai, nishio}@ee.tokushima-u.ac.jp

Abstract

TransmitterThis paper proposes a new transmitter for Noncoherent Chaos
Shift Keying (CSK). In this scheme, the new transmitter canuse a different chaotic sequence length N determined by aninitial value. The performance of the communication systemusing the new transmitter is studied by computer simulation.
In addition, we compare its and an exiting CSK communication system.

Receiver

Noise

nk

Signal "1"

Chaotic signal
generator

Sk

Rk

Detector

DecodedsignalSignal "0"-1

Figure 1: Block diagram of a discrete-time binary CSK communication system.

1. Introduction

2.1. Transmitter
Recently, a noncoherent receiver for digital communicaIn the transmitter, a chaotic sequence is generated by ations systems using chaos is studied actively [1]-[4]. Espechaotic map. In this study, we use a skew tent map to gencially, it is attracted to develop suboptimal noncoherent reerate the chaotic sequence.
ceiver having the performance similar to the optimal nonco2.1.1. Skew Tent Map
herent receiver.
In the previous research, we have proposed the suboptiXk+1
mal receiver using a very simple algorithm. Our method1
Xk+1=Xk
detects symbols from the calculated values of the shortest distance between received signals and chaotic map [5].
0.5Furthermore, we extended this concept to the distance in
Nd -dimensional space using Nd successive received signals(Nd : 3, 4, · · ·) [6]. As a result, we obtained the best per0
formance for the dimension Nd is equal to the length of thechaotic sequence N . In addition, the performance of this sub-0.5
optimal receiver becomes better as N increases. Also, theaverage energy per bit becomes large.
In this study, in order to use the average energy per bit-1-1-0.5
0a0.5
1Xk
effectively and improve the bit error performance, we proFigure 2: Skew Tent Map.
pose a Chaos Shift Keying (CSK) transmitter using variablesequence length. We carry out computer simulations and inThe skew tent map is shown in Fig. 2. This map is one ofvestigate the performance of the proposed method.
simple chaotic maps, and it is described by Eq. (1)2xk + 1 − a
2. System Overview(−1 ≤ xk ≤ a)
1+axk+1 =(1)First, we explain a discrete-time binary CSK communica
−2xk + 1 + ation system with an existing transmitter, as shown in Fig. 1.
(a < xk ≤ 1)
1−aThis system consists of a transmitter, channel and a receiver.
where a denotes a position of a top of the skew tent map.
Detail of each block is described below.

- 301 -2.1.2. Chaos Shift Keying
Xk+1(Rk, Rk+1, Rk+2)

1

1

Symbol "1"

Symbol "0"

0.5

0.5

Rk+2

Symbol "1"00

Symbol "0"
-0.5
-0.5

-1-1
-1
-1

-0.5

00.5

Xk

10.5

-0.5

1RkFigure 3: Chaos Shift Keying.

00-0.50.5

Rk+1

1 -1

Figure 4: Detection method.

CSK is a digital modulation system using chaos. When thetransmitter generates the signals, we use chaotic sequences
generated by different chaotic maps depending on the valueof an information symbol. In this study, we use the skew tent
map and its reversal map, as shown in Fig. 3. If the information symbol “1” is sent, Eq. (1) is used, and if “0” is sent,
the reversed function of Eq. (1) is used. In order to transmit a1-bit information, N chaotic signals are generated, where N
is chaotic sequence length. Therefore the transmitted signal
is denoted by a vector S = (S1 S2 · · · SN ).

closer to the point (Rk , Rk+1 , Rk+2 ) of three successive received signal in the 3-dimensional space in Fig. 4, the shortest
distance between the point and the map has to be calculated.
In this method, we calculate the shortest distance using thescalar product of the vector.
(x0, y0, z1)

v1 (X, Y, Z)
v0

2.2. Channel and NoiseIn the channel, noise is assumed to be additive white Gaussian noise (AWGN) and is denoted by the noise vector n =(n1 n2 · · · nN ). Thus, the received signal block is given by
R = (R1 R2 · · · RN ) = S + n.
2.3. Receiver
The receiver detects the transmitted signals from received
signals and demodulates the information symbol. In the detection methods, there are coherent detection methods thatrecord the initial value of chaotic sequence at the receiver andnoncoherent detection methods that do not record one. In thisstudy, we use the suboptimal receiver proposed in previousresearch [6].

u(x1, y1, z1)

D(Rk, Rk+1, Rk+2)
Figure 5: Calculation of the shortest distance.

Any two points of (x0 , y0 , z0 ) and (x1 , y1 , z1 ) are chosenfrom the straight line in the space of Fig. 4, as shown in Fig.
5. In Fig. 5, a unit vector u is calculated from (x0 , y0 , z0 ) and(x1 , y1 , z1 ) by the following equation,x − x  y  − y  z  − z u = (l, m, n) =(2),,AAAq222where A is (x1 − x0 ) + (y1 − y0 ) + (z1 − z0 ) . In
addition, vector v0 is calculated by (Rk , Rk+1 , Rk+2 ) and(x0 , y0 , z0 ) from the following equation.

2.3.1. Our Suboptimal Receiverv0 = (Rk − x , Rk+ − y , Rk+ − z )The suboptimal receiver proposed by the authors calculates
the shortest distance between received signals and the map
in the Nd -dimensional space using Nd successive receivedsignals (Nd : 3, 4, · · ·).
For 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 (Rk , Rk+1 , Rk+2 ). In order to decide which map is(3)Product T in u and v0 is calculated from the following equation.
T = l (Rk − x0 ) + m (Rk+1 − y0 ) + n (Rk+2 − z0 ) (4)So, v1 can be calculated from the product of T and u.
Therefore, we can calculate coordinates of the shortest distance (X, Y, Z) and the shortest distance D from the following equation.

- 302 -(X, Y, Z) = (T l + x0 , T m + y0 , T n + z0 )q
D = (X − Rk )2 + (Y − Rk+1 )2 + (Z − Rk+2 )20.025(5)N(Nd) = 8
N(Nd) = 4

0.02

(6)
BER

0.015
Note that if the point is outside the cube, we calculate thedistance between the point and the nearest edges of the maps.
0.01
For the 3-dimensional case, there are four straight lines inthe space. So, the minimum value in four distances is decided0.005
as the shortest distance D1 for symbol “1”. In the same way,
D of symbol “0” is decided as D0 . We calculate
0P both of020406080
100120DandDforallkandfindtheirsummations
D1 and0Section
P1thePdecoded symbol as 1 (or 0) for Figure 6: Basis on proposed transmitter (average of results
P D0 . Finally,
P we decideP
D1 < D0 (or D1 >D0 ).
for Eb /N0 = [10, 20][dB]).
In this system, calculation of the shortest distance above 4dimension also possible by using this method. However, wecan not draw a high-dimensional space like 3-dimension.
3.2. Variable Sequence Length Transmitter

3. Proposed Transmitter
Variable Sequence Length Transmitter
Initial value (determinded from last value)

3.1. Basis on Proposed Transmitter
As described above, the successive received signals are required in order to calculate the shortest distance. the bit error
rate (BER) increases for some successive received signals. In
order to expound on this cause, we carried out the following
computer simulation.
On the transmitting side, −1 to 1 is divided into 128. In addition, a chaotic sequence is generated by an initial value determined at random from one section divided into 128. Here,one section is fixed until transmitting 104 bits is finished. On
the receiving side, the number of dimensions (Nd ) using thesuboptimal receiver uses the chaotic sequence length N in
the transmitter. For example, when N is 4, the receiver uses
Nd = 4.
Based on the above conditions, the BER is recorded forvarious the average-bit-energy-to-noise-spectral-density ratio(Eb /N0 ) when N is 4 or 8. Moreover, this simulation is carried out for all sections (0 ∼ 127).
Figure 6 plots the BERs versus sections for N (Nd ) = 4, 8(an average of results for Eb /N0 = [10, 20][dB]). From thisfigure, we can find that BERs of N =4 and 8 different by eachsection. In other words, when the initial value is determined,
the optimal calculation method of the shortest distance is decided by each section.
For example, we can find that the BER performance ofN =4 is better than N =8 in the 1st section. However, it canbe also observed that the BER performance of N =8 is better than N =4 in the 10th section. Therefore, it can be said
that the performance of the CSK communication system ischanged by any initial value of the chaotic sequence.
From this result, we devised new transmitter that thechaotic sequence length N is changed by the initial value.
Namely, we propose a Variable Sequence Length Transmitter. We explain about this transmitter in next subsection.

Determinationof Section

Sequence LengthControl

Chaotic signal
generaterSk

Information symbol"+1" or "-1"Figure 7: Proposed transmitter.
Figure 7 shows the block diagram of Variable Sequence
Length Transmitter. This system consists of three blocks.
First, any section is determined from the initial value by 1stblock. Next, the optimal sequence length is decided by any
section in 2nd block. Finally, chaotic sequence with optimal
sequence length is generated from 3rd block and modulatedby CSK. Moreover, next initial value is decided from Last
value of chaotic sequence.
One merit of this system is to improve the BER performance as described above. Additionally, this system has othermerit. This system can use the average energy per bit effectively by changing the chaotic sequence length. So, the transmitting efficiency improves compared with the existing CSK
communications system.
4. Simulation Result
In this section, we study the performance of the CSK communication system using the proposed transmitter by computer simulations. The simulation conditions are as follows.
In the proposed transmitter, chaotic sequence length N
uses 4 or 8 and changes by any section. In the channel,
noise is assumed only AWGN. So, noise at transmitter andreceiver is not considered. In the receiving side, the suboptimal receiver proposed by the authors is used. And in orderto calculating the shortest distance, 4 or 8 dimensional space- 303 -is changed by the chaotic sequence length N of received signals. In this study, the chaotic sequence length N of the received signal assumes that it can determine by the receiving
side.
Based on the above conditions, the system performance isshown by plotting the BER against Eb /N0 when 104 symbols
are transmitted.
1.0E+00

Using proposed transmitter
4-dimension

BER

1.0E-01

1.0E-02

length on the transmitting side is unknown with the receivingside. Hence, it can be said that developing the receiver thatcan detect the chaotic sequence length N is very important.
5. Conclusions
In this study, in order to use the average energy per bit effectively and improve the bit error performance, we proposed
the CSK transmitter using variable sequence length. As a results, a better BER performance is achieved compared withthat obtained from the existing CSK transmitter.
On the basis of the result obtained in this study, developing the receiver that can detect the chaotic sequence length
decided the transmitting side is our future work.

1.0E-03

References
1.0E-04

05

1015[1] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int . Journal of Bifurcations andChaos, vol. 10, no. 4, pp. 719-736, 2000.

20

Eb/N0 [dB](a) Comparison of 4-dimension
with the proposed transmitter.
1.0E+00[2] F. C. M. Lau and C. K. Tse, “Approximate-optimal detector for chaos communication systems,” Int. Journal ofBifurcations and Chaos, vol. 13, no. 5, pp. 1329-1335,
2003.

Using proposed transmitter
8-dimension

BER

1.0E-01[3] F. C. M. Lau and C. K. Tse, “On optimal detection ofnoncoherent chaos-shift-keying signals in a noisy environment,” Int. Journal of Bifurcations and Chaos, vol. 13,
no. 6, pp. 1587-1597, 2003.

1.0E-02

1.0E-03

1.0E-04[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication Systems, Springer, 2003.
05101520[5] S. Arai and Y. Nishio, “Research on Noncoherent Detection for Chaos Shift Keying,” Proc. NCSP’04, pp. 265268, Mar. 2004.

Eb/N0 [dB](b) Comparison of 8-dimension
with the proposed transmitter.

[6] S. Arai and Y. Nishio, “Research on Suboptimal Receivers for Chaos Shift Keying,” Proc. NCSP’05, pp. 375378, Mar. 2005.

Figure 8: Simulation results.
Figures 8(a) and 8(b) show the simulation results for theproposed transmitter. For comparison, the results of 4dimension (Fig. 8(a)) and 8-dimension (Fig. 8(b)) are alsoshown.
As expected, the performance of the proposed transmitter is better than that of 4-dimension. Moreover, it can beobserved that the curves of the proposed transmitter and 8dimension are almost the same, as shown in Fig. 8(b).
Therefore, we can find that changing the chaotic sequencelength N by the initial value is effective in CSK communication system. However, in this study, we assumed that thechaotic sequence length N of the received signal can determine with the receiving side. In fact, changed sequence- 304 -