2008 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 5-6, 2008

Suboptimal Receiver using Chaotic
Sequences 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{arai, 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— We investigate and evaluate a suboptimal receiverusing chaotic sequences with biased values. In our previousresearch, we investigated a differential chaos shift keying (DCSK)
using chaotic sequences with biased values purposely and conﬁrmed its better performance. However, our previous study only
performed the computer simulation of DCSK using these chaoticsequences. In this study, we focus on the suboptimal receiverand evaluate its performance with chaotic sequences with biased
values.

Transmitternk

Chaotic signal
generator

SkRk

DecodedsymbolDetector

Symbol"1" or "-1"Fig. 1.

Block diagram of discrete-time binary CSK communication system.
Xk+1

Xk+1

1

1

Xk+1=Xk

I. I NTRODUCTION

Symbol "1"

0.5

0.5

Research on digital communications systems using chaosbecomes a hot topic [1]− [5]. Especially, it is attracted todevelop noncoherent detection systems which do not need torecover basis signals (unmodulated carriers) at the receiver.
DCSK [1] and an optimal receiver [2] are well-known as
typical noncoherent systems.
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 non-periodic signals
generated from nonlinear dynamical systems. These signals
are sensitive to initial conditions and difﬁcult to predict thebehavior of the future from the past observational signal. Also
the chaotic sequence can be generated from a simple model,such as a one-dimensional chaotic map. These characteristics
are advantages of using the chaotic sequence for communication systems. Thus, we consider that the utilization ofcharacteristics of the chaotic sequence is important to improve
the performance of chaos communication systems. In ourprevious research, we investigated the performance of DCSK
using chaotic sequences with biased values purposely [7]. Asresults, it could be observed that its performance was better
than that of the conventional method. However, our previous
study only performed the simulation of DCSK using thesesequences. Namely, we need to investigate chaotic sequenceswith biased values in other chaos communication systems.
In this study, we focus on a suboptimal receiver as oneof chaos communication systems. The suboptimal receiverachieves the performance equivalent or similar to the optimalreceiver using different algorithms. We apply the chaotic
sequence with biased values to the suboptimal receiver andevaluate its performance.

Suboptimal Receiver

AWGN

R=(Rk, Rk+1)

00-0.5-0.5

Symbol "-1"-1-1-0.5

Fig. 2.

0a0.5

1

Xk

Skew tent map.

-1-1

Fig. 3.
slopes.

-0.5

00.5

1

Xk

Chaotic maps with someII. S YSTEM OVERVIEW
We consider a discrete-time binary CSK communication
system, as shown in Fig. 1. This system consists of a transmitter, a channel and a suboptimal receiver.
In the transmitter, a chaotic sequence is generated by achaotic map. To generate the chaotic sequence with biased
values, we give some slopes to the skew tent map (Fig 2)
which is well-known as a typical one-dimensional map andchange slopes of map as shown in Fig 3. Figure 3 is describedby (r +1)x −q +r1
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)k2
q2 −a
 −(r2 +1)xk +q2 +r2(q2 < xk ≤ 1)
1−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. An analysis of this map is taken up in Sec. III. The transmitter generate
different chaotic sequences for every symbol by changing the- 49 -P = (X, Y)

P0=(x0, y0)v0DuwhereP1= (x1, y1)

unit vector uv0

R = (Rk, Rk+1)
Fig. 4.

Calculation of the shortest distanceinitial value. The information symbol is modulated by Chaos
Shift Keying (CSK) which is one of digital modulation systemusing chaos. If the information symbol “1” is sent, the solid
map of Fig 3 is used, and if “−1” is sent, the dashed mapof Fig 3 is used. To transmit a 1-bit information, N chaotic
signal samples are generated, where N is chaotic sequencelength. Therefore the transmitted signal is denoted by a vector
S = (S1 S2 · · · SN ).
In the channel, we assume the additive white Gaussian noise(AWGN) channel with a mean of zero and variance of N0 =σ 2 . 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.
The suboptimal receiver achieves the performance equivalent or similar to the optimal receiver using different algorithms. The receiver has the chaotic map used for themodulation at the transmitter memorized. In this study, we
use the suboptimal receiver proposed by the authors [6]. Our
suboptimal receiver calculate shortest distances between thereceived signal and the chaotic maps and performs detectionof the symbol.
Here, we introduce the operation of our suboptimal receiver.
The receiver calculates the shortest distance between received
signal and the maps in the Nd -dimensional space using Nd
successive received signal samples (Nd : 2, 3, · · ·). As an
example, we explain the case of Nd = 2. In this case,we consider two successive received signal samples R =(Rk , Rk+1 ) as coordinate of chaotic map (Fig 3) , wherek = 1, 2, · · · , N − 1. To decide which map is closer to thepoint R in Fig. 3, the shortest distance between the pointand the map has to be calculated. Therefore, the receiver cancalculate the shortest distance using the scalar product of thevector.
Any two points P0 = (x0 , y0 ) and P1 = (x1 , y1 ) arechosen from each straight line in the space of Fig. 3, as shownin 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)

P1 − P0||P1 − P0 ||= R − P0 .
=(4)
(5)

Note that if the point is outside of the cube, we calculate thedistance between the point and the nearest edges of the maps.
In the case of using Fig 3, there are eight straight lines inthe space. Therefore, the minimum value from four distances
is chosen as the shortest distance D1 for symbol “1”. In
the same way, D of symbol “−1” is chosen as D−1 . thereceiver calculates∑both D1 ∑and D−1 for all k and ﬁnds
their summations
D1 andD−1 . ∑Finally, we∑decide thedecodedsymbolas1(or−1)
for
D1 <D−1 (or∑∑D1 >D−1 ).
The calculation of the shortest distance can be extended toNd -dimensional space for Nd ≥ 3.
III. A NALYSIS OF C HAOTIC M AP WITH S OME SLOPESIn this section, we analyze chaotic maps for the chaotic
sequence with biased values. To analyze the chaotic map inthis study, we calculate the invariant density ρ(x). ρ(x) is
the function deciding the iteration density of a map, namely
we can observe the distribution of the value of the chaotic
sequence. ρ(x) is described by1 ∑δ(x − f i (x0 )) ,N →∞ Ni=0N

ρ(x) ≡ lim(6)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 notdependent 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 density; The ﬁrst is the skew tent map as the chaotic
map without biased values, as shown in Fig. 5(a); The second
is the chaotic map with (q1 , r1 ) = (−0.3, 0.3), (q2 , r2 ) =(0.3, −0.3), as shown in Fig. 5(b); The third is the chaotic
map with (q1 , r1 ) = (−0.6, 0.6), (q2 , r2 ) = (0.6, −0.6), as
shown in Fig. 5(c). In this study, we label the second and thethird as Type 1 and Type 2, respectively.
Figures 6(a), (b) and (c) show the invariant densities of theskew tent map, Type 1 and Type 2, respectively. First of all, weobserve the invariant density of the skew tent map (Fig. 5(a)).
From Fig 6(a), ρ(x) is constant. 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 thatthe skew tent map is ergodic. Second, we focus on the invariant
density of Type 1 (Fig 5(b)). As one can see, ρ(x) increases asx approaches −1. Thus, the chaotic sequence generated fromType 1 includes many negative values. Finally, we observe
the invariant density of Type 2 (Fig 5(c)). This distribution isdivided into right and left with center on x = 0 according to(q1 , r1 ). Especially, ρ(x) increases as x approaches −1 or 1.
Namely, the chaotic sequence generated from Type 2 includes
many values around 1 and −1.
As described above, we become available the chaotic sequence with the desired biased values by changing the slopeof the chaotic map.

- 50 -

Xk+1

Xk+1

0.5

0.5

001

Xk+1

1

1

(-0.6, 0.6)

0.5(-0.3, 0.3)

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

(a)

00.5

-1
-1

1

Xk

-0.5

(b)

00.5

1

Xk

(c)

Fig. 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

-0.5

0x

0.5

(a)

1

0-1

-0.5

0x

0.5

0-1

1-0.5

(b)
Fig. 6.

0x

0.5

1(c)

Invariant densities: (a) Skew tent map, (b) Type 1, (c) Type 2.

IV. S IMULATION R ESULTS
In this section, we carry out the computer simulation of oursuboptimal receiver using Figs 5(a), (b) and (c). 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 isﬁxed as a = 0.05, As the chaotic sequence length, we set
N = 256. On the receiving side, BER is recorded for various
Eb /N0 . Also, to calculate the shortest distance, the receiverapplies two-dimensional space (i.e., Nd = 2).
Figure 7 plots the BER versus Eb /N0 for each chaoticmap. To compare the performance, we perform the simulationof the skew tent map without biased values, i.e., as theconventional method. We can observe that the BER of Type 1
is the best performance in Fig 7. Since the chaotic sequence
generated from Type 1 includes many negative values, thechaotic sequence with values which are biased toward one side
might be suitable for our suboptimal receiver. From results,
it can be said that the chaotic sequence which changed thedistribution of biased values is effective for the suboptimalreceiver. Therefore, chaotic sequences with biased values are
effective not only DCSK but also the suboptimal receiver.
V. C ONCLUSIONSIn this study, we have investigated the performance of thesuboptimal receiver using chaotic sequences with biased values. As results, we have obtained the better BER performanceby controlling the distribution of the chaotic sequence withbiased values, Moreover, we have conﬁrmed that the chaotic
sequence with biased values is effective not only DCSK butalso the suboptimal receiver.

1.0E+00

1.0E-01

BER0-1

1.0E-02Skew tent map
Type 1
Type 2

1.0E-03

1.0E-04

0510152025

Eb/N0 [dB]Fig. 7.

BER performance (N = 256).

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