2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

Analysis on Chaotic Sequence with Biased Values
for Noncoherent Chaos Communication
Shintaro ARAI† , Yoshifumi NISHIO† and Takaya YAMAZATO‡†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‡EcoTopia Science Institute, Nagoya University
Furo-cho, Chikusa-ku,Nagoya, 464-8603, JAPAN,
yamazato@nuee.nagoya-u.ac.jp
Abstract—In this paper, we analyze a chaotic dynamics
generating a chaotic sequence with biased values and apply it to noncoherent chaos communications. We examine
a behavior of the chaotic dynamics by increasing the slopeof the chaotic map and investigate the invariant measure
and the correlation function. A high quality performanceof noncoherent chaos communications is obtained by controlling the distribution of the chaotic sequence with biased
values. Finally we carry out the computer simulation using
its sequence and discuss the obtained results and the future
problem for chaos communication.
1. Introduction
Recently, a digital communication system using chaos
becomes a hot topic [1]- [7]. Especially, it is attracted to develop noncoherent detection systems which does not needto recover a basis signals (unmodulated carries) at the receiver. The differential chaos shift keying (DCSK) [1] andthe optimal receiver [2] are well known as a typical noncoherent system. In addition, the correlation delay shift
keying (CDSK) [3] similar to the DCSK scheme is also regarded.
Analyzing chaotic sequence as well as its behavior is essential for improving the the performance of chaos communications. A Chaotic sequence is a series of nonperiodic signals generated from nonlinear dynamical systems. These signals are sensitive to initial conditions anddifficult to predict the behavior of the future from the pastobservational signals. Also a chaotic sequence can be generated from a simple model, such as a one-dimensional
chaotic map. In our previous research, we investigated atransmitter changing a chaotic sequence depending on an
initial value [6]. Moreover we investigated the the performance of chaos communications using the sequence withbiased values purposely [8]. As results, it could be observed that its performance was better than that of the conventional transmitter. From these results, we concluded
that the chaotic dynamics affect the performance of chaos
communications greatly. However, many subjects, such as
the behavior of the system and the change of the correlation
property, were still not solved. Then we consider that it isimportant to analyze a behavior of a chaotic sequence for
improving the performance of chaos communications.
As analysis methods to characterize chaos, there aremany measures of chaos. In this paper, we shall concentrate the invariant measure and the correlation function as
measures of chaos. The invariant measure can observe

the distribution of the value having the chaotic sequence.
The correlation function can examine the irregularity of thechaotic sequence. By calculating the invariant measure andthe correlation function, we can detail a chaotic sequence.
In this study, we analyze the chaotic map for the chaotic
sequence with biased values using the invariant measure
and the correlation function. Moreover, we control thenumber of biased values of the chaotic sequence by increasing the slope of the chaotic map and observe the behavior
of its sequence. Finally we carry out the computer simulation using its sequence and discuss the obtained results andthe future problem for chaos communication.
2. Chaotic Map With Different Slopes and its Analysis
In this section, we introduce a chaotic map with different slopes and its analysis method. Figure 1 shows the
chaotic map with 4 slopes, where this map is made from the
Bernoulli shift map well known as a typical 1-dimensional
map. Moreover it is based on the reference [7] to make thismap. Equation of this map is described by (r + 1)x − q + r
1
k
1
1



(−1 ≤ xk ≤ q1 )



q
+
1
1







(q1 < xk ≤ 0)
 (r1 − 1)xk /q1 + 1
xk+1 = 
, (1)



(r2 − 1)xk /q2 − 1(0 < xk ≤ q2 )







(1 − r2 )xk − q2 + r2
(q2 < xk ≤ 1)

1 − q2where q1 , r1 , q2 and r2 are the parameters deciding theslopes { (−1.0 < q1 < 0.0), (−1.0 < r1 < 1.0), (0.0 < q2 <1.0), (−1.0 < r2 < 1.0) } . We can change the slopes of the
map to change these parameters. In this paper, we consider
2 methods for giving the parameter. One is the parallel shiftmethod (q2 = 1 + q1 , r2 = r1 ), i.e. it is made the parallel
shift from the left side slope to the right side slope withcenter on xn = 0. Another is the point symmetry method(q2 = (−1) × q1 , r2 = (−1) × r2 ), i.e. the left side and theright side slopes are made to be the point symmetry withcenter on (xn , xn+1 ) = (0, 0). Figures 2(a) and (b) show
the chaotic maps with the two methods. As one can see,we can obtain the various maps to change the parameters.
By using these maps, we analyze the chaotic dynamics ofthis one-dimensional chaotic map family. To observe thebehavior of the chaotic dynamics using these methods, weinvestigate the invariant measure and correlation function.

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Xk+1
1

Xk+1

Xk+1

Xk+1

0.5

0.5

0.5

0

0

0

-0.5

-0.5

-0.5

1

Xk+1=Xk

0.5

r1
0

r2

1

-1
-1

-0.5

0.5

Xk

1

-1
-1

-0.5

0

0.5

Xk

1

-1
-1

-0.5

0

0.5

Xk

1

(1)
(2)
(3)
(a) Parallel shift (q2 , r2 ) = (q1 + 1, r1 )

-0.5

-1
-1

0

1

q1 -0.50

0.5 q2

Xk

1

Xk+1

Xk+1

Xk+1

0.5

0.5

0.5

0

0

0

-0.5

-0.5

-0.5

1

Figure 1: Chaotic map with 4 slopes.
2.1. Invariant measure
The invariant measure is the function deciding the iteration density of a map, namely we can observe the distribution of the value of the chaotic sequence. The invariant
measure ρ(x) is described byN
1 X
ρ(x) ≡ lim
δ(x − f i (x0 )) ,
(2)
N→∞ N
i=0
where xn+1 = f (xn ), n = 0, 1, 2, · · · , x0 is an initialvalue, δ is the delta function. If ρ(x) is the system that is
not dependent on an initial value, it is called ergodic.
2.2. Correlation function
The correlation function is the measure to calculate
the correlation between random variables at two different
points in space or time. Thus we can observe the irregularity of chaotic sequence to calculate the correlation function.
The correlation function C(m) is described by
N−1
1X
C(m) = limx̂i x̂i+m ,
(3)
N→∞ N
i=0
PN−1 iwhere x̂i = f i (x0 ) − x̄, x̄ = limN→∞ N1 i=0
f (x0 ), m isa difference with a datum point. If the correlation is calculated between random variables at two different points
in same sequence, it is called the autocorrelation function. Also if the correlation is calculated between randomvariables at two different points in different sequence, it
is called the cross-correlation function. In this paper, we
carry out the autocorrelation analysis since we are not concerned here with a multiplexing system.

1

-1
-1

-0.5

0

0.5

Xk

1

1

-1
-1

-0.5

0

0.5

Xk

1

-1
-1

-0.5

0

0.5

Xk

1

(4)
(5)
(6)
(b) Point symmetry (q2 , r2 ) = ((−1) × q1 , (−1) × r1 )Figure 2: Chaotic map with each parameter.
(1) Bernoulli shift map, (2) (q1 , r1 ) = (−0.6, −0.55),(3) (q1 , r1 ) = (−0.6, 0.35), (4) (q1 , r1 ) = (−0.1, 0.05),(5) (q1 , r1 ) = (−0.6, 0.35), (6) (q1 , r1 ) = (−0.85, 0.85)
parameters (q1 , r1 ). However, since C(m) decreases changing alternately according to m, like a (q1 , r1 ) = (−0.6, 0.35)
and (q1 , r1 ) = (−0.85, 0.85), we expect that the chaotic sequence having the similar value alternately was generated.
From these results, we can confirm that ρ(x) and C(m)of the chaotic sequence are different each by changing theparameter. Concurrently, we can also find that it is possible
to choose any ρ(x) and C(m) by changing the parameters.
However, when the chaotic map has four slopes, the deviation of the chaotic sequence is limited to one or two.
In this paper, we perform generating the chaotic sequence with any ρ(x) using the chaotic map with manyslopes. Furthermore, we also perform to generate thechaotic sequence which was excellent in the correlation
property by increasing slopes.
3. Chaotic Map with Many Different Slopes
Xk+1
1

2.3. Analysis results
Figures 3 and 4 show the numerical analysis results of
the invariant measure and the correlation function. To calculate using the computer, N of each result is assumed to106 .
In the case of the parallel shift of ρ(x) (Fig. 3(a)), we
can observe that ρ(x) shifts from −1 to 1 according to theparameters (r1 ). Namely, it can be said that the chaotic sequence with the biased value is distributed to the left or theright with center on x = 0 according to the parameters (r1 ).
However, these C(m) (Fig. 4(a)) did not almost change.
Moreover, we can also find that the correlation property
decreases as compared with the Bernoulli shift map.
In the case of the point symmetry of ρ(x) (Fig. 3(b)),
its distribution is divided into right and left with center on
x = 0 according to the parameters (q1 , r1 ). In addition, we
can observe that C(m) (Fig. 4(b)) increase according to the0(q8, r8)(q4, r4)

0.5(q2, r2)(q3, r3)(q6, r6)(q7, r7)-0.5(q1, r1)

-1
-1

-0.5

(q5, r5)

0

0.5

Xk

1

Figure 5: Chaotic map with 10 slopes.
In this section, we introduce a chaotic map with manydifferent slopes. In Sec. 2, we have observed to be different ρ(x) and C(m) of the chaotic sequence depending on
the slope of the chaotic map. However, when the chaotic
map has four slopes, the deviation of the chaotic sequenceis limited to one or two. Thus, we perform generating- 145 -

2

Bernoulli shift map

0.5(q1, r1)=(-0.60, 0.35)

1.5

2

0.6

Bernoulli shift map(q1, r1)=(-0.60, -0.55)

Type 1(q1, r1)=(-0.60, -0.55)(q1, r1)=(-0.60, 0.35)

0.4Type 2Type 3

1.5

ρ(x)

C(m)

ρ(x)

0.3

1

0.2

1

0.1

0.5

0

0.5

-0.1

0
-1

-0.5

0

0.5

x

1

-0.2

0

(a) Parallel shift.

10

20

m

30

0
-1

50

(q1, r1)=(-0.10, 0.05)(q1, r1)=(-0.85, 0.85)

0.4
C(m)

C(m)

ρ(x)

0.2
0.1

-0.5

0

x

0.5

(b) Point symmetry.

1

0.2

0

-0.1
-0.2

0.3

0.1

0

0.5

1

Type 1
Type 2Type 30.5(q1, r1)=(-0.85, 0.85)

0.3

1

0.5

0.6

(q1, r1)=(-0.60, 0.35)

0.4

0

x

(a) Invariant measure.

(q1, r1)=(-0.10, 0.05)

0.5(q1, r1)=(-0.60, 0.35)

0
-1

-0.5

(a) Parallel shift.
0.6

2

1.5

40

0

10

20

m

30

40

-0.1

50

0

10

(b) Point symmetry.

20

m

30

40

50

(b) Correlation function.

Figure 3: Invariant measure with each Figure 4: Correlation function with each Figure 7: Analysis results of Chaotic
parameter.
map with 10 slopes.
parameter.
the chaotic sequence with any distribution increasing theslopes of the chaotic map. In this paper, we generate thechaotic sequence with the biased values having 3 distribution including many values of a near (−1, 0, 1) based on
results in Sec. 2.3. Figure 5 shows the chaotic map with 10
slopes to generate the chaotic sequence with 3 distribution.
This map can change the frequency of 3 distribution, i.e.
ρ(x) of 3 distribution by changing the altitude of the slope.
To investigate the distribution by different occurrence rate,
3 types of parameters are used. These parameters are described below. Also, Figures 6(a), (b) and (c) show thechaotic map with each parameter.
• Type 1(q1 , r1 ) = (−0.75, − 0.6), (q2 , r2 ) = (−0.55, − 0.05),(q3 , r3 ) = (−0.3, 0.05), (q4 , r4 ) = (−0.2, 0.7),(q5 , r5 ) = (0.25, − 0.6), (q6 , r6 ) = (0.45, − 0.05),(q7 , r7 ) = (0.7, 0.05), (q8 , r8 ) = (0.8, 0.7)
• Type 2(q1 , r1 ) = (−0.8, − 0.6), (q2 , r2 ) = (−0.6, − 0.05),(q3 , r3 ) = (−0.4, 0.05), (q4 , r4 ) = (−0.2, 0.6),(q5 , r5 ) = (0.2, − 0.6), (q6 , r6 ) = (0.4, − 0.05),(q7 , r7 ) = (0.6, 0.05), (q8 , r8 ) = (0.8, 0.6)
• Type 3(q1 , r1 ) = (−0.8, − 0.6), (q2 , r2 ) = (−0.6, − 0.1),(q3 , r3 ) = (−0.4, 0.1), (q4 , r4 ) = (−0.2, 0.6),(q5 , r5 ) = (0.2, − 0.6), (q6 , r6 ) = (0.4, − 0.1),(q7 , r7 ) = (0.6, 0.1), (q8 , r8 ) = (0.8, 0.6)
Figure 7 (a) and (b) show the analysis results of thechaotic map with each type parameter. In Fig. 7(a), we canobserve that 3 distribution including many values of a near(−1, 0, 1). Moreover, we can also confirm the different occurrence rate of (−1, 0, 1) according to the parameters. In
Fig. 7(b), it can be observed that C(m) increase according
to the parameters as compare with Fig. 4. In addition, it canbe said that the chaotic sequence is generated without having the similar value alternately unlike Fig. 4(b). Therefore,

Xk+1

Xk+1

Xk+1

0.5

0.5

0.5

0

0

0

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1

-1
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1

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Xk

1

1

-1
-1

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0

0.5

Xk

1

-1
-1

-0.5

0

0.5

Xk

1

(a) Type 1(b) Type 2(c) Type3
Figure 6: Chaotic map with 10 slopes (each parameter).
by giving the many slopes for the chaotic map, we can control precisely the distribution of the chaotic sequence andobtain the high correlation properties.
4. Noncoherent Chaos Communication
In this section, we carry out the numerical simulationof a noncoherent chaos communication using the chaotic
map with 10 slopes. In this paper, we perform the DCSK
simulation as a noncoherent chaos communication. As areason, it is because that DCSK is one of the noncoherent
correlation-based communication systems, and its system
is very simple. First we introduce DCSK operation. Next
we explain the numerical simulation and discuss the obtained results.
Figure 8 shows the block diagram of a DCSK transmitter(a) and receiver (b). In this scheme, the transmitter out(a)xiChaotic signal
generator

Si
Delay N(b)

Delay NRi

blxi-NN

Information symbols
bl
N

ΣRR

Ri+N

i=1

i

Decoderi+N

Threshold

Figure 8: DCSK operation. (a) transmitter. (b) receiver.
puts a chaotic sequence xi followed by the same sequence- 146 -

1.0E+00

multiplied by the information symbol bl (±1). To transmit
1-bit information, N chaotic signals are generated, where N
is the chaotic sequence length. Therefore, the transmitted
signal is given byx(1 ≤ i ≤ N)
S i = bi x
(4)
(N + 1 ≤ i ≤ 2N)

Type 1
Type 2Type 3
Bernoulli shift map

BER

1.0E-01

1.0E-02l i−N

Also, the transmitted signal can be written as S =(S 1 S 2 · · · S 2N ) by vector. Since the noise n =(n1 n2 · · · n2N ) is added to the transmitted signal by
the channel, the received signal can be written as R =(R1 R2 · · · R2N ) = S + n.
On the receiving side, it is evaluated by the correlationof 2 signals, which are obtained by dividing the received
signals into two halves (length N). Thus, the output of thecorrelation can be written asN
X

Ri Ri+N .

1.0E-040

5

10
Eb/N0 [dB]1520(a) N = 16.
1.0E+00

Type 1
Type 2Type 3
Bernoulli shift map

1.0E-01
BER

C1 =1.0E-03(5)

1.0E-02i=1

The decoded symbol is decided as “+1” or “−1” dependingon C1 being larger or smaller than 0.
To generate the transmitted signal, the DCSK transmitter
needs to switch correctly by the chaotic sequence length
N. Therefore, the sophisticated switch is required, and it isregarded as the important issue to design DCSK.
5. Simulation results and Discussions
The simulation conditions are as follows. In the transmitting side, the chaotic sequence length N are 16 and 32.
Also, as the parameters deciding the slopes of the chaotic
sequence, we use 3 type parameters in Sec. 3, i.e. Type1, Type 2 and Type3. In the channel, noise is assumed
to be AWGN. Based on these conditions, the system performance is evaluated by plotting the BER against E b /N0
when 104 bits of information are transmitted.
Figure 9 plots the BERs versus E b /N0 for each type parameter. To compare the performance of the chaotic sequence with biased values, Fig. 9 shows the performanceof the conventional DCSK using the Bernoulli shift map.
From these results, we can observe that the both BERs
(N = 16, 32) improve as compare with the conventional
DCSK according to the parameters. Especially, Type 1’s
performance is the best in Fig. 9. This reason for improving its performance is considered that we can control the
distribution of the chaotic sequence with biased values sothat the correlation property improved. In Figs. 4(a) and 7,
we have showed the correlation function of the Bernoulli
shift map and the chaotic map with 10 slopes, respectively. We recognize from these figures that C(0) of thechaotic sequence with biased values is higher than that of
the Bernoulli shift map. The higher C(0) represents thatthe autocorrelation property of the proposed chaotic map
is high, namely, the chaotic sequence with biased values
become strong to interference of the noise even if it does
not give the distribution in the extreme. From these results,
it would be expected to be strong to a multipass fading.
In addition, it might be able to obtain a stable BER which
does not depend on an initial value by giving the biased
values for the chaotic sequence. Therefore, it can be said
that the chaotic sequence which changed the distribution of
biased values is very effective in the chaos communication
systems.

1.0E-03

1.0E-040

5

10
Eb/N0 [dB]1520(b) N = 32.
Figure 9: Simulation results.
6. Conclusions
In this study, we have analyzed the chaotic map for the
chaotic sequence with biased value deeply and carried out
the computer simulation using its sequence. As results, wehave obtained the higher autocorrelation property and thebetter BER performance by controlling the distribution of
the chaotic sequence with biased values, namely, the performance of the chaos communication is improved by the
chaotic sequence with biased values depending on its distribution. Consequently, it would be expected that the performance of chaos communication is improved further bycontrolling the number or frequency of its distribution forany purpose of communication.
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