Investigation of Noncoherent Detection
using Chaotic Sequence with Biased Values
Shintaro ARAI† , Yoshifumi NISHIO† and Takaya YAMAZATO‡†Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN‡EcoTopia Science Institute, Nagoya University
Furo-cho, Chikusa-ku,Nagoya, 464-8603, JAPAN
Email: {arai, nishio}@ee.tokushima-u.ac.jp, yamazato@nuee.nagoya-u.ac.jp
Abstract
We investigate the chaotic sequence with biased values
in chaos-based communication systems. In our previous research, we investigated the the performance of chaos communications using the sequence with biased values purposely. As results, we concluded that the chaotic dynamics
affect the performance of chaos communications greatly.
However, our previous study only performed the computer
simulation the DCSK system using the chaotic sequencewith biased values. In this study, we focus on the suboptimal receiver as one of chaos communication systems andobserve its performance with the chaotic sequence with biased values.

In this study, we focus on the suboptimal receiver as oneof chaos communication systems. The suboptimal receiveris a detecting system that approximates the working of theoptimal receiver using a different algorithm. We apply thechaotic sequence with biased values to the suboptimal receiver and investigate its performance.
2. System Overview
We consider the discrete-time binary CSK communication system, as shown in Fig. 1. This system consists of atransmitter, a channel and a receiver. Detail of each blockis described below.
Transmitter

1. Introduction

AWGN

Suboptimal Receivernk

Research on digital communications systems using
chaos becomes a hot topic [1]− [6]. Especially, it is attracted to develop noncoherent detection systems which do
not need to recover basis signals (unmodulated carries) atthe receiver. The differential chaos shift keying (DCSK) [1]and the optimal receiver [2] are well known as typical noncoherent systems. In addition, the correlation delay shiftkeying (CDSK) [3] similar to the DCSK scheme is also a
noncoherent system.
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 thethe performance of chaos communications using the sequence with biased values purposely [8] [9]. As results,
it could be observed that its performance was better thanthat of the conventional transmitter. From these results,
we concluded that the chaotic dynamics affect the performance of chaos communications greatly. However, ourprevious study only performed the computer simulation theDCSK system using the chaotic sequence with biased values. Namely, we need to investigate the chaotic sequencewith biased values in other chaos communication systems.

Chaotic signal
generator

SkRkDetector

DecodedsymbolSymbol"1" or "-1"Figure 1: Block diagram of discrete-time binary CSK communication system.
2.1. TransmitterIn the transmitter, a chaotic sequence is generated bya chaotic map. Since we consider the chaotic sequencewith biased values, the chaotic maps of this study has someslopes as shown in Fig. 2. This map is made from the skew
tent map well known as a typical 1-dimensional map. Also,
this map is described by (r +1)x −q +r
1
11k
(−1 ≤ xk ≤ q1 )
q1 +1k −q1 +ar1
(q1 < xk ≤ a)
 (1−r1 )xa−q1, (1)xk+1 =
(r2 −1)xk +q2 −ar2(a<x≤q)k2q2 −a −(r2 +1)xk +q2 +r2 (q < x ≤ 1)2
k1−q2
where a denotes a position of the top of the skew tent map,
q1 , r1 , q2 and r2 are the parameters deciding the slopes {(−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- 221 -Rk+1

map to change these parameters. An analysis of this map
is taken up in Sec. 3.The transmitter can generate different
chaotic sequences for every symbol by changing the initialvalue. The information symbol is modulated by CSK.

1

Symbol "1"
0.5

R=(Rk, Rk+1)

Xk+1
10

Xk+1=Xk

0.5-0.5(q2, r2)(q1, r1)
0-1
-1

00.5

1

Rk

Figure 3: Detection method of our suboptimal receiver

-0.5

-1
-1

Symbol "-1"
-0.5

-0.5

0a0.5

1

P = (X, Y)

P0=(x0, y0)

Xk

Figure 2: Chaotic map with 4 slopes.

Dv0

CSK is a digital modulation system using chaos. Whenthe transmitter generates the signal, it is used that choice ofchaotic map depends on the transmitted symbol. If the information symbol “1” is sent, Eq. (1) is used, and if “0” issent, the reversed function of Eq. (1) is used. To transmit a
1-bit information, N chaotic signal samples are generated,
where N is chaotic sequence length. Therefore the transmitted signal is denoted by a vector S = (S1 S2 · · · SN ).
2.2. Channel and Noise
We assume the additive white Gaussian noise (AWGN)
channel with a mean of zero and variance of N0 = σ 2 .
AWGN channel is well known as the most popular and basic channel model. Here, the noise signal 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. Suboptimal Receiver
The suboptimal receiver is a detecting system that approximates the working of the optimal receiver using a different algorithm. The receiver has the chaotic map used
for the modulation at the transmitter memorized. In thisstudy, we use the suboptimal receiver proposed by the authors [7]. Our suboptimal receiver calculate shortest distances between the received signal and the chaotic maps
and performs detection of the symbol. Our suboptimal receiver calculates shortest distances between the received
signal and the chaotic maps and performs detection of thesymbol.
Here, we introduce the operation of our suboptimal receiver. The receiver calculates the shortest distance between received signal and the map in the Nd -dimensional
space using Nd successive received signal samples (Nd :2, 3, · · ·). As an example, we explain the case of Nd =2. Figure 3 shows the 3-dimensional space of the skew

uP1= (x1, y1)

R = (Rk, Rk+1)

Figure 4: Calculation of the shortest distancetent map whose coordinates correspond to the three successive received signal samples R = (Rk , Rk+1 ) wherek = 1, 2, · · · , N − 1. To decide which map is closer tothe point R in the 3-dimensional space in Fig. 3, the shortest distance between the point and the map has to be calculated. Therefore, the receiver can calculate the shortest
distance using the scalar product of the vector.
Any two points P0 = (x0 , y0 ) and P1 = (x1 , y1 ) arechosen from each straight line in the space of Fig. 3, asshown in Fig. 4.
Using Fig. 4, we can calculate the point P = (X, Y )closest to R and the shortest distance D using the followingequations.
P = (X, Y ) = (u · v0 ) u + P0

D(2)

= ||P − R||√22(X − Rk ) + (Y − Rk+1 )=(3)whereUnit vector uv0

P1 − P0||P1 − P0 ||= R − P0 .

=(4)
(5)

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

- 222 -Xk+1

Xk+1

0.5

0.5

001

Xk+1

1

10.5

(-0.3, 0.3)(-0.6, 0.6)

0(0.3, -0.3)

-0.5

-0.5

-0.5
(0.6, -0.6)

-1
-1

-0.5

00.5

-1
-1

1

Xk

-0.5

0(a)

0.5

-1
-1

1

Xk

-0.5

00.5

1

Xk

(c)

(b)

Figure 5: Chaotic Maps: (a) Skew Tent Map, (b) Type 1 (q1 , r1 ) = (−0.3, 0.3), (q2 , r2 ) = (0.3, −0.3), (c) Type 2(q1 , r1 ) = (−0.6, 0.6), (q2 , r2 ) = (0.6, −0.6).

0.8

0.8

0.8

0.6

0.6

0.6
ρ(x)

1

ρ(x)

1

ρ(x)

1

0.4

0.4

0.4

0.2

0.2

0.2

0-1-0.5

0x

0.5

1

0-1

-0.5

0x

(a)

0.5

10-1-0.5

(b)

0x

0.5

1(c)

Figure 6: Invariant Measures: (a) Skew Tent Map, (b) Type 1, (c) Type 2.

In this case, there are eight straight lines in the space.
Therefore, the minimum value from four distances is chosen as the shortest distance D1 for symbol “1”. In the sameway, D of symbol “0” is chosen as D0 . the receiver calculates
1 and D0 for all k and ﬁnd their summations∑ both D∑
D1 andD0 . Finally,symbol∑∑ we decide∑ the decoded∑
as 1 (or 0) for D1 < D0 (or D1 > D0 ).
The calculation of the shortest distance can be extended
to Nd -dimensional space for Nd ≥ 3.
3. Analysis of Chaotic Map with Some slopes
In this section, we analyze chaotic maps for the chaotic
sequence with biased values. Here, we apply the invariant
measure to analyze the chaotic map. The invariant measure is the function deciding the iteration density of a map,
namely we can observe the distribution of the value of thechaotic sequence. The invariant measure ρ(x) is describedbyN
1 ∑δ(x − f i (x0 )) ,(6)
ρ(x) ≡ lim
N →∞ Ni=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 isnot dependent on an initial value, it is called ergodic. In
this study, to calculate using the computer, N is assumed
to 106 .

We use three types of chaotic maps for calculating theinvariant measure; The ﬁrst is the skew tent map as thechaotic map without biased values (Fig. 5(a)); The second
is the chaotic map with (q1 , r1 ) = (−0.3, 0.3), (q2 , r2 ) =(0.3, −0.3) (Fig. 5(b)); The third is the chaotic map with(q1 , r1 ) = (−0.6, 0.6), (q2 , r2 ) = (0.6, −0.6) (Fig. 5(c))
In this study, we label the second and the third as Type 1
and Type 2, respectively. To calculate using the computer,
N of each result is assumed to 106 .
Figures 6(a), (b) and (c) show the invariant measures ofFig 5. Now, let us observe each invariant measure.
First of all, we observe the skew tent map. From Fig 6(a),ρ(x) is constant, namely, In other words, the distributionof the chaotic sequence generated from the skew tent map
is not dependent on an initial value, namely, we can conﬁrm that the skew tent map is ergodic. Second, we focus
on Type 1 (Fig 6(b)). As one can see, ρ(x) increases asx approaches −1. Thus, we can say that the chaotic sequence with the biased value is distributed to the left side.
Finally, we observe Type 2 (Fig 6(c)). This distribution isdivided into right and left with center on x = 0 according
to (q1 , r1 ). Especially, ρ(x) increases as x approaches −1or 1. Namely, we can say that the chaotic sequence generated from Fig 6(c) has many values of surrounding 1 and−1.
From these analysis, by changing the slope of the chaotic
map, we become available the chaotic sequence with the- 223 -1.0E+00

5. Conclusions

BER

1.0E-01

1.0E-02
Skew Tent Map
Type 1: (q1, r1)=(-0.3, 0.3)
Type 2: (q1, r1)=(-0.6, 0.6)

1.0E-03

1.0E-040510
15
Eb/N0[dB]2025

In this study, we have investigated the suboptimal receiver by changing the some slopes. As results, we haveobtained the better BER performance by controlling thedistribution of the chaotic sequence with biased values,Moreover, we have conﬁrmed that the chaotic sequencewith biased values is effective not only DCSK but also thesuboptimal receiver. Since the simulation with Nd = 2dimensional space has been performed for calculating theshortest distance, investigating of the chaotic sequence withbiased values is our future work in the case of our suboptimal receiver with Nd ≥ 3.
References(a)1.0E+00[1] G. Kolumbán, B. Vizvári, W. Schwarz, and A. Abel,“Differential chaos shift keying: A robust coding forchaos communication,” Proc. NDES’96, pp. 87-92,
Jun. 1996.

BER

1.0E-01

1.0E-02

Skew Tent Map
Type 1: (q1, r1)=(-0.3, 0.3)
Type 2: (q1, r1)=(-0.6, 0.6)

1.0E-03

1.0E-04[2] M. Hasler and T. Schimming, “Chaos communication
over noisy channels,” Int . Journal of Bifurcation andChaos, vol. 10, no. 4, pp. 719-736, Apr. 2000.

05

10
15
Eb/N0[dB]

2025[3] M. Sushchik, L. S. Tsimring and A. R. Volkovskii,“Performance Analysis of Correlation-Based Communication Schemes Utilizing Chaos,” IEEE Trans. Circuits and Systems Part I, vol. 47, no. 12, pp. 16841691, Dec. 2000.
[4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication Systems, Springer, 2003.

(b)
Figure 7: BER: (a) N = 64, (b) N = 128.

[5] L. E. Larson, J-M. Liu, L. S. Tsimring, Digital Communications Using Chaos and Nonlinear Dynamics,Springer, 2006.

desired biased values.
4. Computer Simulation
In this section, we carry out the computer simulation ofour suboptimal receiver using Figs 5. The simulation conditions are as follows. On the transmitting side, 10,000
symbols are transmitted using chaotic sequences with different initial values. Here, the parameter of the skew tent
map is ﬁxed as a = 0.05, As the chaotic sequence length,
we set N = 64and128. On the receiving side, BER isrecorded for various Eb /N0 . Also, to calculate the shortest
distance, the receiver applies 2-dimensional space (2-D).
Figures 6(a) and (b) plot the BER versus Eb /N0 foreach type parameter. To compare the performance, we perform the simulation of the skew tent map without biasedvalues, i.e., as the conventional method. We can observethat the performance of Type 1 from Figs 6(a) and (b).
From these results, it can be said that the chaotic sequencewhich changed the distribution of biased values is effective
for the suboptimal receiver. Therefore, we can sat that thechaotic sequence with biased values is effective not only
DCSK but also the suboptimal receiver.

[6] G. Cimatti, R. Rovatti and G. Setti, “Chaos-BasedSpreading in DS-UWB Sensor Networks Increases
Available Bit Rate,” IEEE Trans. Circuits and Systems
Part I, vol. 54, no. 6, pp. 1327-1339, Jun. 2007.
[7] S. Arai and Y. Nishio, “Detection of Information Symbols and Sequence Lengths Using Suboptimal Receiver for Chaos Shift Keying,” Proc NOLTA’06, pp.
799-802, Sep. 2006.
[8] Shintaro ARAI and Yoshifumi NISHIO, “Research onDifferential Chaos Shift Keying Changing Deviation ofChaotic Sequence,”Proceedings of RISP International
Workshop on Nonlinear Circuits and Signal Processing(NCSP’07), pp. 349-352, Mar. 2007.
[9] S. Arai, Y. Nishio and T. Yamazato, “Analysis on
Chaotic Sequence with Biased Values for Noncoherent Chaos Communication,” Proc NOLTA’07, pp. 144147, Sep. 2007.

- 224 -