2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Research on Suboptimal Receivers 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 byRecently, digital communication systems using chaos
are studied actively [1]-[5]. In particular, a noncoherent a chaotic map. In this study, we use a skew tent map toreceiver is hard to design while information could be re- generate the chaotic sequence.
covered even if the sequence is unknown at the receiver. 2.1.1. Skew Tent Map
Therefore, it is important to study the noncoherent receiver. The optimal detection proposed by Schimming
Xk+1
1and Hasler is well known as a noncoherent receiver [1][2].
Xk+1=Xk
However, if a chaotic sequence length N becomes long,calculation becomes very complicated and the signal de0.5
tection becomes diﬃcult.
We have proposed a method of detecting symbols fromthe calculation value of the shortest distance between
0received signals and chaotic map [6]. In this study, we
propose a method improving this noncoherent receiver.
-0.5
At ﬁrst, we show the method of deciding the symbol byusing 3 received signals. We also show the method of
deciding the symbol by using 4 received signals. Com-1
-1
-0.5
0a0.5
1
puter simulations are performed for these two methods,
Xkand the bit error rates (BER) are evaluated.
2.

Figure 2: Skew Tent Map.

System overview

The skew tent map is shown in Fig. 2. This map is
We consider a discrete-time binary CSK communica- one of simple chaotic maps, and it is described by Eq. (1)tion system, as shown in Fig. 1. This system consists of⎧
2xk + 1 − a⎪a transmitter, channel and a receiver. We explain about⎪(−1 ≤ xk ≤ a)⎪⎨1+aeach block.
xk+1 =(1)⎪⎪−2xk + 1 + a⎪⎩(a < xk ≤ 1)
Transmitter1−aReceiver
Noise

Signal "1"

where a denotes a position of a top of the skew tent map.

nk

Chaotic signal
generator

Sk

RkDetector

Decodedsignal

2.1.2. Chaos Shift Keying

Signal "0"

CSK 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. In
Figure 1: Block diagram of a discrete-time binary CSK this study, we use the skew tent map and its reversal
communication system.
map, as shown in Fig. 3. If the information symbol “1”-1- 375 -Xk+1
1(Rk, Rk+1, Rk+2)

1

Symbol "0"

Symbol "1"
0.5

0.5
Symbol "1"

Rk+2

00
Symbol "0"

-0.5
-0.5

-1
-1
-1
-1

1
0.5

-0.5
-0.5

0

0.5

001

Rk

Xk

-0.5

0.5

Rk+1

1 -1

Figure 4: Three-dimensional space of (Rk , Rk+1 , Rk+2 ).

Figure 3: Chaos Shift Keying.

is sent, Eq. (1) is used, and if “0” is sent, the reversed 3.1. Calculation of the shortest distancefunction of Eq. (1) is used. In order to transmit a 1-bit
In order to decide which map is closer to the point
information, N chaotic signals are generated, where N(Rk , Rk+1 , Rk+2 ) of three successive received signal inis chaotic sequence length. Therefore the transmitted
the3-dimensional space in Fig. 4, the shortest distancesignal is denoted by a vector S = (S1 S2 · · · SN ).
between the point and the map has to be calculated.
2.2. Channel and Noise(x0, y0, z1) v1 (X, Y, Z)
uIn the channel, noise is assumed to be additive white(x1, y1, z1)
Gaussian noise (AWGN) and is denoted by the noisevector n = (n1 n2 · · · nN ). Thus, the received signal
D
block is given by R = (R1 R2 · · · RN ) = S + n.

v0

2.3. Receiver(Rk, Rk+1, Rk+2)
The receiver detects the transmitted signals from received signals and demodulates the information symbol.
Figure 5: Calculation of shortest distance.
In the detection methods, there are coherent detection
methods that record the initial value of chaotic sequence
Calculating the shortest distance in 3-dimensionalat the receiver and noncoherent detection methods thatspace is more diﬃcult than the case of 2-dimensional
do not record one. In this study, we propose a noncoplane. So, we calculate the shortest distance using theherent receiver.
scalar product of the vector.
At ﬁrst, any two points of (x0 , y0 , z0 ) and (x1 , y1 , z1 )3. Noncoherent Receiverare chosen from the straight line in the space of Fig. 4,
as shown in Fig. 5. In Fig. 5, a unit vector u is calWe have proposed a method of detecting symbols from culated from (x , y , z ) and (x , y , z ) by the following
0 0 0
1 1 1
the calculation value of the shortest distance between a equation,
point (Rk , Rk+1 ), which is given by the two successive received signals, and the skew tent map on the 2-dimensionx1 − x0 y1 − y0 z1 − z0,,u = (l, m, n) =(2)plane [6].
AAAIn this study, we extend this concept to the distancein 3-dimensional space using three successive received
222
where A is (x1 − x0 ) + (y1 − y0 ) + (z1 − z0 ) . In
signals. Figure 4 shows the 3-dimensional space whose
coordinates correspond to the three successive received addition, vector v0 is calculated by (Rk , Rk+1 , Rk+2 ),R). In the ﬁgure, the relationships and (x0 , y0 , z0 ) from the following equation.
signals (R , Rkk+1

k+2of the received signals made by using the skew tent map
and its reversal map are also drawn.

- 376 -v0 = (Rk − x0 , Rk+1 − y0 , Rk+1 − z0 )(3)

Product T in u and v0 is calculated from the following increases, as shown in Fig. 6(a). However, when N beequation.
comes large, the BER performance is similar for all of
the three diﬀerent dimensions, as shown in Figs. 6(b)T = l (Rk − x0 ) + m (Rk+1 − y0 ) + n (Rk+2 − z0 ) (4) and (c).
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 thefollowing equation.

1.0E+00

2-D
3-D
4-D
BPSK

1.0E-01

BER(X, Y, Z) = (T l + x0 , T m + y0 , T n + z0 )(5)222
D = (X − Rk ) + (Y − Rk+1 ) + (Z − Rk+2 ) (6)1.0E-02

1.0E-03

Note that if the point is outside the cube, we calculate
the distance between the point and the nearest edges ofthe maps.

1.0E-0405

10Eb/N0[dB]

1520(a) N = 4.

3.2. Decision of symbol
The shortest distance is calculated for both the skew
tent map and its reversal map. For the 3-dimensionalcase, there are four straight lines in the space for symbol“1”. So, the minimum value in four distances is decided
as the shortest distance D1 for symbol “1”. In the sameway, D of symbol “0” is decided as D0 . We calculate
and D0 for all k and ﬁnd their summations
both of D1
D1 and Dsymbol
0 . Finally, we decide the decoded
as 1 (or 0) forD1 < D0 (orD1 >D0 ).

1.0E+00

2-D
3-D4-D
BPSK

BER

1.0E-01

1.0E-02

1.0E-033.3. Four-dimensional case

1.0E-04

05

10
Eb/N0[dB]

1520(b) N = 8.

This method based on the shortest distance can beextended to 4-dimensional space. In this case, the four
successive received signals (Rk , Rk+1 , Rk+2 , Rk+3 ) areused for the calculation. The calculation method using
the vector is similar to the case of the 3-dimensional
space.

1.0E+00

2-D
3-D
4-D
BPSKBER

1.0E-01

4. Simulation result
The simulation conditions are as follows.

1.0E-02

1.0E-034• 10 symbols are transmitted.
1.0E-04• Chaotic sequence length N are 4, 8 and 16.

510Eb/N0[dB]

1520(c) N = 16.

• In the channel, noise is assumed only AWGN.
Noise at transmitters and receivers is not considered.
In the simulation, we recorded the BER when Eb /N0 waschanged from 1 into 20, where Eb denotes the averageenergy per bit. For comparison, the result using thedistance on 2-dimensional plane is also shown.
Figures 6(a), (b) and (c) show the BERs versus Eb /N0for N are 4, 8 and 16, respectively. It can be observedthat the BER performance improves as the dimension

0Figure 6: BER performance.
Figures 7(a) and (b) show the BERs for the 4dimensional method and the optimal detection for N
are 4 and 8. When N is 4, the two curves are almost thesame. However, when N is 8, the BER performanceof the optimal detection is better than the proposed
method.

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From this result, we consider that our method using
References
the shortest distance in Nd -dimensional space becomesalmost identical with the optimal detection for the case [1] M. Hasler and T. Schimming, “Statistically motivated detection methods for chaos shift keying,”that the dimension Nd is equal to the length of theProc. NOLTA’99, vol. 2, no. 2, pp. 569-572, 1999.
chaotic sequence N . We believe that our method canbe extended for larger dimensions such as Nd = 16 or [2] M. Hasler and T. Schimming, “Chaos communication
Nd = 64. However, it is diﬃcult to compare those resultsover noisy channels,” Int . Journal of Bifurcations
with the optimal detection, because it is too diﬃcult toand Chaos, vol. 10, no. 4, pp. 719-736, 2000.
perform the optical detection for larger N . Investigating
the relationship between the two detection methods is [3] Francis C. M. Lau and Chi K. Tse, “Approximateour important future research.
Optimal detector for chaos communication systems,”
Int. Journal of Bifurcations and Chaos, vol. 13, no.
5, pp. 1329-1335, 2003.
1.0E+00
4-D

Optimal[4] Francis C. M. Lau and Chi K. Tse, “On optimal detection of noncoherent chaos-shift-keying signals in anoisy environment,” Int. Journal of Bifurcations andChaos, vol. 13, no. 6, pp. 1587-1597, 2003.

BER

1.0E-01

1.0E-02[5] Francis. C. M. Lau and Chi. K. Tse, Chaos-Based
Digital Communication Systems, Springer, 2003.
1.0E-03

1.0E-04

0510Eb/N0[dB]

15[6] S. Arai and Y. Nishio, “Research on Noncoherent
Detection for Chaos Shift Keying,” Proc. NCSP’04,
pp. 265-268, Mar. 2004.

20(a) N = 4.
1.0E+00

4-D
OptimalBER

1.0E-01

1.0E-02

1.0E-03

1.0E-04

0510Eb/N0[dB]

1520(b) N = 8.
Figure 7: Comparison between the optimal detection
and the proposed method.
5. Conclusions
In this study, we proposed the method of detecting
symbol from the calculation value of the shortest distance between successive received signals and 3 or 4 dimensional space.
On the basis of the result obtained in this study, whenchaotic sequence length is Nd , detecting the shortest distance of Nd -dimensional space is our future work.

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