2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Research on Noncoherent Detection for 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

1. Introduction

2.1. TransmitterIn the transmitter, a chaotic sequence is generated byAn optimal noncoherent detection technique for ChaosShift Keying (CSK) digital communication system is im- a chaotic map. In this study, signals are generated fromportant for chaos-based communication [1]-[3]. How- the following skew tent map.
ever, calculation in the optimal noncoherent detection
2xn + 1 − a
becomes complicated when chaotic sequence length be(−1 ≤ xn ≤ a)

1+acomes large. Therefore, computing cost in the opti(1)xn+1 =

mal noncoherent detection becomes too large. Also,−2xn + 1 + a(a < xn ≤ 1)
whether the optimal noncoherent detection would also1−asuﬀer from a decrease in performance for long chaotic
This is expressed in Fig. 2. If the imformation symbol
sequence length is uncertain.
In this study, to improve the computing cost as keeping a good bit error rate, we propose a very simple detecXn+1
tion method for CSK which can decode even if chaotic
1
sequence length becomes long. As varying the length ofthe chaotic sequence and Eb /N0 of the channel, the bit
Symbol "0"error rate is evaluated by computer simulations.
0.5
Symbol "1"

2.

0System Model

We consider a discrete-time binary CSK communication system, as shown in Figure 1.

-0.5

Transmitter-1-1

ReceiverNoise
Symbol "1"

-0.5

00.5

nk

Chaotic signalgenerator

Sk1XnRk

Detector

DecodedsymbolFigure 2: Skew Tent Map
Symbol "0"-1“1” is sent, Formula (1) is used, and if “0” is sent, thereversed function of Formula (1) is used. In order totransmit a 1-bit signal, N chaotic signals are generated,
Figure 1: Block diagram of a discrete-time binary CSK
where N is chaotic sequence length. Therefore the transcommunication systemmitted signal is denoted by vector S = (S1 S2 · · · SN ).

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2.2. Channel and NoiseIn the channel, noise is assumed additive white Gaussian noise (AWGN) and is denoted by the noise vector
by n = (n1 n2 · · · nN ). Thus, the received signal blockis given by R = (R1 R2 · · · RN ) = S + n.

3. The shortest distance Zi is obtained than Formula(3).
22Zi =  (Xi − Rt ) + (Yi − Rt+1 ) (i = 0, 1) .
(3)
The decoded symbol is decided as “1” or “0” depending on Z1 being smaller or larger than Z0 .

2.3. Receiver
Detection method executes the following steps, asshown in Fig. 3.

3. Simulation result
Xn+1

We have developed computer programs to compute (2)and (3). Chaotic sequence length N ranging from 4 to128 are used. In simulation, 104 symbols are transmitted and the bit error rate (BER) is recorded for various
Eb /N0 values, where Eb denotes the average energy perbit. The simulation time (cpu-time) is also noted in eachcase. In addition, the optimal detection simulates withthe same condition, too-0.2

Symbol "1"

Symbol "0"-0.4(X0,Y0)-0.6

Z0(X1,Y1)Z1-0.8

3.1. Chaotic sequence
Eb /N0 change(Rk,Rk+1)
-1
-1

-0.8

-0.6

-0.4

-0.2

lengthconstantandFigures. 4(a) and 4(b) plots the BERs versus Eb /N0
for diﬀerent chaotic sequence length N . Both the detection method, when Eb /N0 value is small, the BER
performance degrades. However, when Eb /N0 value islarge, it can be observed that the BER performance ofproposed method improves than the optimal detection.

0XnFigure 3: Detection method

1. The received signals Rk and Rk+1 give a point on
the skew tent map, as (Xn , Xn+1 ) = (Rk , Rk+1 ).
3.2. Chaotic sequence length change and Eb /N0
2. From the point, we ﬁnd a point to possess the short- constant
est distance with two functions of Formula 1 and theFigures. 5(a) and 5(b) plot the BERs versus chaotic
reversed function of it, as (Xn , Xn+1 ) = (Xi , Yi ). sequence length N for diﬀerent Eb /N0 values. When
The point is obtained as Formula (2),chaotic sequence N of optimal detection becomes long,
the BER performance degrades, as shown in Fig 5(b).
RtA
even if chaotic sequence N of proposed method, as shown
Xi = 2+ Rt+1 − B
A +1 Ain Fig 5(a). However, even if N becomes long, pro(i = 0, 1) posed method doesn’t degrade than optimal detection,
2ARtas shown in Fig. 5(a). It can be said from a Fig. 5(a)+ Rt+1 − B + B
 Yi = 2
A +1 A
that the BER of the proposed method becomes very(2) small when N = 8 ∼ 16.
where A is the gradient of the two functions, B isthe intercept.

- 266 -BER

BER

1.0E+00

1.0E+00

1.0E-01
1.0E-01

N=41.0E-02

N=8Eb/N0=8dB

N=16Eb/N0=10dB
1.0E-02

N=32
1.0E-03

1.0E-04

0Eb/N0=12dB

N=64Eb/N0=14dB

N=128

Eb/N0=16dB

5101520251.0E-03

110100

Eb/N0[dB](a) Proposed method(a) Proposed method

BERBER

1.0E+00

1.0E+00

1.0E-01

1.0E-01

N=4
1.0E-02

Eb/N0=8dB

1.0E-02

N=8Eb/N0=10dB

N=16Eb/N0=12dB

N=32
1.0E-03

1000

N

Eb/N0=14dB

1.0E-03

N=64

Eb/N0=16dB

N=128

1.0E-04

1.0E-04

05

10152025

1101001000

NEb/N0[dB](b) The optimal detection(b) The optimal detection

Figure 4: BERs versus Eb /N0 for chaotic sequence length Figure 5: BERs versus chaotic sequence N for deﬀerent
Eb /N0 values for a CSK system
N 4 to 128 for a CSK system- 267 -3.3. Simulation time
In Table 1, the simulation time is tabulated against thechaotic sequence N . In both method, simulation time
also becomes long as N becomes long. When we compare
the same N , it turns out that simulation time of theproposed method is far quick. Therefore, in simulation
time, the proposed method could be said as excellent.

Table 1: Simulation time
N48163264
128

Proposed method [sec]
0.07
0.17
0.35
0.72
1.45
2.92

Optimal Detector [sec]
44.57
79.12
149.39289.66573.071130.78

4. Conclusions
In this study, to improve the computing cost as keeping a good bit error rate, we proposed a very simple detection method for CSK which can decode even if chaotic
sequence length becomes long. As a result, we obtained
a good result to even a simple method.

Acknowledgement
This work was partly supported by The Telecommunications Advancement Foundation’s Resarch Grant.

References[1] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int. Journal of Bifurcation andChaos, Vol. 10, No. 4 pp. 719-736, 2000.
[2] F. C. M. Lau and C. K. Tse, “On optimal detectionof noncoherent chaos-shift-keying signals in a noisyenvironment,” Int. journal of Bifurcation and Chaos,Vol. 13, No. 6 pp. 1587-1597, 2003.
[3] F. C. M. Lau and C. K. Tse, “Approximate-OptimalDetector For Chaos Communication Systems,” Int.
journal of Bifurcation and Chaos, Vol. 13, No. 5 pp.
1329-1335, 2003.

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