NOLTA 2006
11-14 september
Bologna - ITALY

Detection of Information Symbols and Sequence Lengths
Using Suboptimal Receiver for Chaos Shift Keying
Shintaro ARAI† and Yoshifumi NISHIO††Department of Electrical and Electronics Engineering
Tokushima University, 2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Telephone: +81-88-656-7470, Fax: +81-88-656-7471
Email: arai@ee.tokushima-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—This paper proposes a new detection methodof a receiver for Noncoherent Chaos Shift Keying. In
this scheme, the receiver detects not only an information
symbol but also a chaotic sequence length which used for
Chaos Shift Keying. The detection of the chaotic sequencelength is performed by the method adapting the suboptimal receiver proposed in our previous research, and it isthe very simple method using the feature of chaos. In order
to investigate its detection, we carry out computer simulations and observe the performance.

point where the chaotic sequence length is changed. We
carry out computer simulations and investigate the performance.
2. System Overview
We consider the discrete-time binary CSK communication system, as shown in Fig. 1.
Transmitter
Noise

Receivernk

Symbol "1"

Chaotic signal
generator

Sk

Rk

Detector

DecodedsymbolSymbol "0"

1. Introduction-1

Recently, a noncoherent receiver for digital communications systems using chaos is studied actively [1]-[4]. Especially, it is attracted to develop a suboptimal noncoherent receiver having the performance similar to the optimal
noncoherent receiver.
In the previous research, we have proposed a suboptimal receiver using very simple algorithm. Our method detects symbols from the calculated values of the shortest
distance between received signals and a chaotic map [5].
Furthermore, 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 performance for the dimension Nd , which is equal to the lengthof the chaotic sequence N. In addition, we confirmed thatthe performance of this suboptimal receiver became better as N increased. Moreover, in order to use the average energy per bit effectively and to improve the bit error performance, we proposed a Chaos Shift Keying (CSK)transmitter which the chaotic sequence length changes efficiently and obtained a better BER performance than theexisting CSK communication system [7]. However, thissimulation was carried out by the case which assumed thatthe chaotic sequence length was known at the the receivingside. Hence, the receiver has to detect the chaotic sequencelength, i.e., a point where the chaotic sequence length ischanged.
In this study, based on the detection method of the information symbol in the suboptimal receiver proposed in ourprevious research, we experiment with a detection of the

Figure 1: Block diagram of discrete-time binary CSK communication system.
2.1. TransmitterIn the transmitter, a chaotic sequence is generated by achaotic map. In this study, we use a skew tent map to generate the chaotic sequence. The skew tent map is one ofsimple chaotic maps, and it is described by Eq. (1)
2xk + 1 − a(−1 ≤ xk ≤ a)



1+a


xk+1 = 
(1)



−2x
+
1
+
a

k


(a < xk ≤ 1)
1−awhere a denotes a position of the top of the skew tent map.
The information symbol is modulated by CSK using the
skew tent map and its reversal map. In other words, if
the information symbol “1” is sent, Eq. (1) is used, andif “0” is sent, the reversed function of Eq. (1) is used.
In order to transmit 1-bit information, N chaotic signals
are generated, where N is the chaotic sequence length.
Therefore, the transmitted signal is denoted by a vector
S = (S 1 S 2 · · · S N ).
In the case of the existing CSK transmitter, the chaotic
sequence length is fixed value. In this study, in order to detect the point where the chaotic sequence length is changed,
the transmitter changes the chaotic sequence length at random. Here, we consider a very simple model which799

(x0, y0, z0)

switches 2 different chaotic sequence lengths, N = 8 or
N = 12. Also, the initial value is changed for every information symbol.

v1 (X, Y, Z)
v0

2.2. Channel and NoiseIn the channel, a noise is assumed to be additive white
Gaussian noise (AWGN) and is denoted by the noise vectorn = (n1 n2 · · · nN ). Thus, the received signal block is given
by R = (R1 R2 · · · RN ) = S + n.
2.3. Receiver
The receiver recovers the transmitted signals from the received signals and demodulates the information symbol. In
this study, we use the suboptimal receiver proposed in ourprevious research [6]. Based on the detection method of
the information symbol in the suboptimal receiver, we experiment with the detection of the point where the chaotic
sequence length is switched.

(x1, y1, z1)

D(Rk, Rk+1, Rk+2)

Figure 3: Calculation of the shortest distance.
qwhere A is (x1 − x0 )2 + (y1 − y0 )2 + (z1 − z0 )2 . In addition, vector v0 is calculated from (Rk , Rk+1 , Rk+2 ) and(x0 , y0 , z0 ) by the following equation.
v0 = (Rk − x0 , Rk+1 − y0 , Rk+1 − z0 )(3)Product T in u and v0 is calculated by the following equation.

2.4. Detection Method of Information Symbol [6]T = l (Rk − x0 ) + m (Rk+1 − y0 ) + n (Rk+2 − z0 )(4)Hence, v1 can be calculated from the product of T and u.
Therefore, we can calculate the point with the shortest distance (X, Y, Z) and the shortest distance D by the following
equations.

(Rk, Rk+1, Rk+2)

1

u

Symbol "1"
0.5

Rk+2

(X, Y, Z) = (T l + x0 , T m + y0 , T n + z0 )

(5)

q

(6)

0
Symbol "0"

D=

-0.5

-1
-1

1
0.5

-0.5
Rk

0

0
0.5

-0.5

Rk+1

1 -1

Figure 2: Detection method.
The suboptimal receiver proposed by the authors calculates the shortest distance between received signals and themap in the Nd -dimensional space using Nd successive received signals (Nd : 3, 4, · · ·).
As an example, we explain the case of Nd = 3. Figure 2 shows the 3-dimensional space of the skew tent map
whose coordinates correspond to the three successive received signals (Rk , Rk+1 , Rk+2 ) where k = 1, 2, · · · , N − 2.
In order to decide which map is closer to the point (Rk ,Rk+1 , Rk+2 ) in the 3-dimensional space in Fig. 2, the shortest distance between the point and the map has to be calculated. Therefore, we calculate the shortest distance using
the scalar product of the vector.
Any two points of (x0 , y0 , z0 ) and (x1 , y1 , z1 ) are chosenfrom each straight line in the space of Fig. 2, as shownin Fig. 3. In Fig. 3, a unit vector u is calculated from(x0 , y0 , z0 ) and (x1 , y1 , z1 ) by the following equation,
x − x y − y z − z 
1
0
1
0
1
0
,
,
(2)
u = (l, m, n) =AAA(X − Rk )2 + (Y − Rk+1 )2 + (Z − Rk+2 )2Note that if the point is outside the cube, we calculate
the distance between the point and the nearest edges of themaps.
For the 3-dimensional case, there are four straight lines
in the space. Therefore, the minimum value in four distances is chosen as the shortest distance D1 for symbol “1”.
In the same way, D of symbol “0” is chosen as D0 . We calculate both of D1 and D0 for all k and find their summations
PP
D1 and D0 . Finally, we decide the decoded symbol asPPPP
1 (or 0) for D1 < D0 (or D1 > D0 ).
The calculation of the shortest distance can be extended
to Nd -dimensional space for Nd ≥ 4.
3. Detection Method of Chaotic Sequence Length (Proposed Method)
Based on the detection method of the information symbol in the suboptimal receiver, we explain the detection
method of the point where the chaotic sequence length isswitched.
Figure 4(a) shows a time series where the chaotic sequence is switched at k = 12. This sequence contains two
chaotic sequences of the information symbols “1” and “0”.
We assume that the noise is not added to this sequence.
Figure 4(b) shows the calculation results of the shortest distances for every k. Here, we use 4-dimensional space for800

|D1 - D0|

D1 D0

R(k)
1

1.2

1.2

D1
D0

1

0.5

0.8

0.8

0

0.6

0.6

0.4

0.4

-0.5

0.2

0.2

-1

2

4

6

8

10

12

14

16

18

0

20

0

2

4

6

8

10

12

14

k

(a) Time series.

Difference between D1 and D0

1

2

16

6

8

k=7
(R7, R8, R9, R10)k(b) Calculation of shortest distance.

4

10

12

14

k=11
(R11, R12, R13, R14)16k(c) Difference between D1 and D0 .

Figure 4: Connected two chaotic sequences of the information symbols “1” and “0” (switched when N is 12) .
|D1 - D0|

D1 D0

R(k)
1

1.2

1.2

D1

0.5

0.8

0.8

0

Difference between D1 and D0

1

D0

1

0.6

0.6

0.4

0.4

-0.5

0.2

0.2
0

-1

2

4

6

8

10

12

14

16

18

0

20

2

4

6

8

10

k

(a) Time series.

12

14

2

4

6

8

10

12

14

16

k

(b) Calculation of shortest distance.

k=7(R7, R8, R9, R10)

k=11
(R11, R12, R13, R14)16k(c) Difference between D1 and D0 .

Figure 6: Connected two chaotic sequences of the information symbols “1” and “1” (switched when N is 8) .
the calculation of the shortest distances. In Fig. 4(b), thesolid line and the dotted line show D1 and D0 , respectively.
Since the noise is not added, the difference between the
two lines is very large for k = 1 ∼ 9. However, the difference between the two lines becomes small for k = 10 ∼ 12.
In order to observe the difference in detail, we show theabsolute value of the difference between D1 and D0 , asFig. 4(c). We can see that the differences around the switching point (k = 12) are much smaller than other differences.
In order to expound on this cause, we use Fig. 5.
One sequence

Another sequence

1 2 ... 7 8 9 10 11 12 13 14 15 16 17 ... ... k
The shortest distance is calculatedusing mix sequence.

Figure 5: Cause of small difference between two lines
around switching point.

Figure 5 shows the switching of two chaotic sequences.
We can observe that the two chaotic sequences are mixed
for the calculation around the switching point (k = 12).
Therefore, the shortest distance around the switching pointis calculated as if the noise was added. This is the key
point of the proposed method. Namely, if the difference be-

tween D1 and D0 around the switching point becomes verysmall, it becomes easy to find the point where the chaotic
sequence length is switched. In this study, we consider avery simple model which switches 2 values (N = 8 and
N = 12). Hence, the receiver calculates 2 differences between D1 and D0 around 8 and 12, and these two values
are compared. Finally, the receiver can decide the chaotic
sequence length by which is smaller between N = 8 and
N = 12. In the case of Fig. 4(c), the difference aroundN = 12 is smaller than that around N = 8. Therefore, thechaotic sequence length is decided as N = 12. Moreover,since D1 is smaller than D0 , the information symbol is detected as “1”.
In the same way, the proposed method can detect the
chaotic sequence length and the information symbol evenwhen the same information symbol continues like “1” and“1”, as Fig. 6. From Figs. 6(b) and (c), we can find thatthe chaotic sequence length and the information symbol is
detected as N = 8 and “1”, respectively.
This detection method improves the performance when
D1 and D0 around the switching point is very small.
Namely, the initial value should be chosen such that the
distances in the Nd -dimensional space between the signal
point and the two maps, which correspond to the symbols“1” and “0”, is very small. Therefore, when choosing theinitial value of the next transmitted signal, the transmitter
calculates D1 and D0 using two values, the last value of the801previous transmitted signal and the random value. This operation is performed repeatedly. Finally, the random value
which is the closest to D1 and D0 is chosen as the initialvalue of the next transmitted signal.
4. Simulation Result
In this section, we study the performance of the proposed
method by computer simulations. The simulation conditions are as follows.
In the transmitting side, the chaotic sequence length N
is switched as N = 8 or N = 12. In the channel, noise isassumed to be only AWGN. Hence, the noise at the transmitter and the receiver are not considered. In order to calculate the shortest distance, we use the 4-dimensional and8-dimensional space on the receiving side. Based on these
conditions, the system performance is evaluated by plotting the sequence length error rate and BER against Eb /N0
when 104 bits of information are transmitted. Here, we assumed that the bit error rate (BER) is calculated in the casewhere the chaotic sequence length is detected correctly.

Sequence length error rate

1.0E+00

1.0E-01

1.0E-02
4-Dimention
8-Dimention

1.0E-03

5. Conclusions
In this study, based on the detection method of the information symbol in the suboptimal receiver proposed in ourprevious research, we experimented with a detection of thechaotic sequence length, i.e., the point where the chaotic
sequence length is changed. We have confirmed that thedetection of the information from the chaotic sequence ispossible even when the chaotic sequence length changed.
Moreover, in the previous research, we proposed the CSKtransmitter which the chaotic sequence length changes efficiently and obtained a better BER performance than theexisting CSK communication system [7]. Therefore, investigating the performance of the chaos communication system which changes the chaotic sequence length efficiently
using the proposed method in this study is our future work.

4-Dimention

References(Initial value chosen at random)

8-Dimention(Initial value chosen at random)

1.0E-04

0

5

10

15
Eb/N0 [dB]

20

25[1] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int . Journal of Bifurcation and
Chaos, vol. 10, no. 4, pp. 719-736, 2000.

30(a) Sequence length error rate.
1.0E+00[2] F. C. M. Lau and C. K. Tse, “Approximate-optimal detector for chaos communication systems,” Int. Journalof Bifurcation and Chaos, vol. 13, no. 5, pp. 13291335, 2003.

4-Dimention
8-Dimention

1.0E-01
BER

results, we can find that the performance of 8-dimension isbetter than 4-dimension. This is because the sequences are
generated from chaotic dynamics. In other words, chaotic
sequences with 8 values have more appropriate to express
the chaotic feature than 4 values. Therefore, it can be said
that the detection of chaotic sequence length is effective
by using higher-dimensional data. In addition, since theperformance of the method choosing the initial value efficiently is better than that chosen at random, we can saythat choosing the initial value of the method choosing isvery important.

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

1.0E-02

1.0E-03

1.0E-040

5

10
Eb/N0 [dB]15[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication Systems, Springer, 2003.

20[5] S. Arai and Y. Nishio, “Research on noncoherent detection for chaos shift keying,” Proc. NCSP’04, pp.
265-268, Mar. 2004.

(b) Bit error rate.
Figure 7: Simulation results.
Figures 7(a) and (b) show the sequence length error rateand BER. In order to compare the performance of choosing the initial value of the next transmitted signal, Fig. 7(a)shows the result of the simulation with the initial value ofthe chaotic sequence chosen at random together. From both

[6] S. Arai and Y. Nishio, “Research on suboptimal receivers for chaos shift keying,” Proc. NCSP’05, pp.
375-378, Mar. 2005.
[7] S. Arai and Y. Nishio, “Variable sequence lengthtransmitter for noncoherent chaos shift keying,” Proc.
NCSP’06, pp. 301-304, Mar. 2006.

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