Performance Analysis of Suboptimal ReceiverUsing Shortest Distance Approximation Method
for Chaos Shift Keying
Shintaro ARAI† and Yoshifumi NISHIO†† Department of Electrical and Electronic Engineering, Tokushima University

2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN{arai, nishio}@ee.tokushima-u.ac.jp
Transmitter

I. I NTRODUCTION
Research on digital communications systems using chaos
becomes a hot topic [1]− [8]. Especially, it is attracted todevelop noncoherent detection systems which do not need torecover basis signals (unmodulated carries) at the receiver. In
this study, we focus attention on the optimal receiver which isone of typical noncoherent systems [2]. The optimal receiver
performs an optimal detection by using the probability density
function (PDF) between the received signals and the samechaotic map of the transmitting side. However, the optimalreceiver suffers from a computational complexity due to thelarge chaotic sequence length. Thus, it is important to develop
a receiver with performance equivalent to the optimal receiverusing different algorithms, i.e., a suboptimal receiver.
In our previous research, we proposed the suboptimal receiver using the shortest distance approximation [9]. Insteadof calculating the PDF, the proposed suboptimal receiverapproximates the PDFs by calculating the shortest distance
between the received signals and the chaotic map. As resultsof the computer simulations, we conﬁrmed the validity ofthe proposed suboptimal receiver as an approximation methodof the optimal receiver. However, our previous study did not
sufﬁciently investigate the performance and the computational
cost of the suboptimal receiver.
In this study, we observe computational costs of the optimal and our suboptimal receiver to evaluate the accuracy of
calculation and the processing time.
II. S YSTEM OVERVIEW
We consider the discrete-time binary CSK communication
system, as shown in Fig. 1. This system consists of a transmitter, a channel and a receiver.
In the transmitter, a chaotic sequence is generated by achaotic map. In this study, the transmitter uses a skew tent
map which is one of simple chaotic maps, as shown in Fig.
2(a), and it is described by Eq. (1)
2xk + 1 − a(−1 ≤ xk ≤ a)
1+a
xk+1 =(1)
−2xk + 1 + a(a < xk ≤ 1)
1−aReceiver

Noise

nk

Symbol "1"

Chaotic signal
generator

Rk

Sk

Detector

DecodedsymbolSymbol "0"-1

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

Xk+1

Xk+1
1

1

Xk+1=Xk

Xk+1=Xk

0.5

0.5

0

0

-0.5

-0.5

-1
-1

-0.5

0a

0.5

1

Xk

(a) Symbol “1”Fig. 2.

-1
-1

-0.5

0a

0.5

1

Xk

(b) Symbol “0”Chaos Shift Keying.

where a denotes a position of the top of the skew tent map.
The transmitter can generate the different chaotic sequence
for every symbol by changing an initial value of chaos. The
information symbol is modulated by CSK.
CSK is a digital modulation system using chaos. Whenthe transmitter generates the signals, it is used that chaotic
sequences generated by different chaotic maps depending on
the value of an information symbol. If the information symbol“1” is sent, Eq. (1) is used (Fig. 2(a)), and if “0” is sent, thereversed function of Eq. (1) is used (Fig. 2(b)). To transmit a
1-bit information, N chaotic signals are generated, where N
is chaotic sequence length. Therefore the transmitted signal isdenoted by a vector S = (S1 S2 · · · SN ).
In the channel, we assume the additive white Gaussian noise(AWGN) with a distribution of mean zero and variance N0 =σ 2 . AWGN channel is well known as the most popular andbasic channel model. Here, the noise signals is denoted by thenoise vector n = (n1 n2 · · · nN ). Thus, the received signals
block is given by R = (R1 R2 · · · RN ) = S + n.
The receiver recovers the transmitted signals from the re--3-

ceived signals and demodulates the information symbol. Since
we consider a noncoherent receiver, the receiver memorizes the
chaotic map used for the modulation at the transmitter. However, the receiver never knows the initial value of chaos andthe information symbol in the transmitter. Before explaining
our receiving method, we introduce the optimal receiver andthe suboptimal receiver.

R = (Rk, Rk+1, Rk+2)

1

Symbol "1"
0.5

Rk+2

0
Symbol "0"-0.5

A. Optimal Receiver
The optimal receiver was proposed by Hasler and Schimming [2]. This receiver selects the symbol q using the aposteriori probability when the received signals block R is
received.
q̃ = arg max Prob(q is sent | R)
q

1

Rk

Fig. 3.

0
0.5

-0.5

Rk+1

1 -1

Proposed detection method.

P = (X, Y, Z)

v0

where p(.) is the probability density function (PDF). Here,Prob(“1” is sent) and Prob(“0” is sent) are 1/2. In addition,
since p(R) is independent of q, Eq. (2) can be written as

D

u

P1
= (x1, y1, z1)

R = (Rk, Rk+1, Rk+2)
Fig. 4.

q0

P0=(x0, y0, z0)p(R| q is sent) × Prob(q is sent)
, (3)
p(R)

q̃ = arg max p(R| q is sent) .

0.5

-0.5

(2)

However, since the a posteriori probability is not convenient
to calculate, we convert them using Bayes’ rule, i.e.
Prob(q is sent | R) =-1
-1

Calculation of the shortest distance.

(4)

As an example, we explain the case of N = 2. In thisthe numerical integration method [5]. By applying numerical
study, it is assumed that the initial value S1 of each chaotic
integration, Eqs. (5) and (6) are rewritten bysignal block S is chosen randomly in accordance with thenatural invariant probability density of f and −f , where f isp(R| “1”is sent)∑N
the function of the skew tent map. In addition, since the natural
L(Ri −f (i−1) (xl ))2
∑
i=1
1
−
δxl
invariant probability density of f and −f is the constant
2σ 2
= lim
(7)
e
L→∞ 2(2πσ 2 )N/2
1/2 in the interval [0, 1], the PDF p(R| “+1”is sent) andl=1
p(R| “0”is sent) are also equal to 1/2 and limited to [0, 1],i.e.
p(R| “0”is sent)∫ +1∑N(R −S )2 +(R2 −f (S1 ))2
1
− 1 1
L
2
(Ri −g (i−1) (xl ))22σ∑p(R| “1”is sent) =e
dS1 (5)
i=1
1
2
−
δxl
2πσ −1
2σ 2
= lim
(8)
e
L→∞ 2(2πσ 2 )N/2
l=1
1
2πσ 2

∫ +1

(R1 −S1 )2 +(R2 +f (S1 ))2
2σ 2

(j)
dS1 (6) where δxl = 2/L, xl = −1 + (k − 1)δxk , f (α) denotes
the iteration of the function f and j times with the initial−1
condition α (i.e. f (0) (α) = α), g = −f . As one can see,where σ 2 is the variance of noise. Finally the optimal receiver the calculation accuracy of the PDF by using the numericaldecides the decoded symbol as 1 (or 0) for p(R| “1”is sent) > integration method depends on the parameter L. Although thep(R| “0”is sent) (or p(R| “1”is sent) < p(R| “0”is sent)).
calculation accuracy improves as L increases, the calculation
As mentioned above, two error functions need to be com- time also increases.
puted for two PDFs in the case of N = 2. In other words,
to decide the decoded symbol, the optimal receiver has toB. Suboptimal Receivercalculate a total of four PDFs. Because the total number ofN
the PDFs is given as 2 , the implementation of the optimal
The suboptimal receiver is a receiving system to implement
receiver becomes quite difﬁcult for large N .
the optimal receiver using a different algorithm from the
To implement the optimal receiving algorithm with large optimal receiver. Note that the detection characteristic of theN , the PDF of the optimal receiver is calculated by using suboptimal receiver can not be superior to the optimal receiver.

p(R| “0”is sent) =

e−

-4-

1.0E+00

III. S HORTEST D ISTANCE A PPROXIMATION M ETHOD

P = (X, Y, Z) = (u · v0 ) u + P0
D

(9)

= ||P − R||
√
2
2
2
=
(X − Rk ) + (Y − Rk+1 ) + (Z − Rk+2 )
(10)

where
Unit vector u =v0=

P1 − P0||P1 − P0 ||R − P0 .

(11)
(12)

Note that if the point is outside the cube, we calculate thedistance between the point and the nearest edges of the maps.
For the 3-dimensional case, there are four straight lines inthe space. Therefore, the minimum value in four distances ischosen as the shortest distance D1 for symbol “1”. In the sameway, D of symbol “0” is chosen as D0 . the receiver calculates∑
both∑of D1 and D0 for all k and ﬁnd their summations
D1
and ∑ D0 . Finally,
wedecidethedecodedsymbol
as
1
(or
0)
∑
∑
∑
for
D1 < D0 (orD1 > D0 ).
The calculation of the shortest distance can be extended to
Nd -dimensional space for Nd ≥ 4.
IV. I NVESTIGATION OF C OMPUTING C OST
To investigate computational costs of the optimal and our
suboptimal receiver, we carry out computer simulations with

1.0E-01

BER

In this section, we explain our suboptimal receiver using ashortest distance approximation [9]. As described in Sec. 2,
the core of the optimal receiver’s problems is the complexityof the calculation of the PDF due to large N . In other words,
to realize a suboptimal receiver, it is necessary to simplify thedetection algorithm. In our receiving method, instead of calculating the PDF between received signals and the chaotic map,
our suboptimal receiver approximates the PDF by calculating ashortest distance between the received signals and the chaotic
map and performs a detection of information. Concretely,
the receiver calculates the shortest distance between received
signals and the map in the Nd -dimensional space using Nd
successive received signals (Nd : 2, 3, · · ·).
As an example, we explain the case of Nd = 3. Figure 3
shows the 3-dimensional space of the skew tent map whose
coordinates correspond to the three successive received signals
R = (Rk , Rk+1 , Rk+2 ) where k = 1, 2, · · · , N − 2. To decide
which map is closer to the point R in the 3-dimensional spacein Fig. 3, the shortest distance between the point and the map
has to be calculated. Therefore, the receiver can calculate theshortest distance using the scalar product of the vector. Any
two points of P0 = (x0 , y0 , z0 ) and P1 = (x1 , y1 , z1 ) arechosen from each straight line in the space of Fig. 3, as shownin Fig. 4.
Using Fig. 4, we can calculate the point with the shortest
distance P = (X, Y, Z) and the shortest distance D by thefollowing equations.

1.0E-02
Our Suboptimal Receiver (N=Nd=4)
Optimal Receiver (N=4)

1.0E-03

Our Suboptimal Receiver (N=Nd=8)
Optimal Receiver (N=8)

1.0E-040

5

10

Eb/N0 [dB]15

20

Fig. 5. BER performance (Comparison of optimal receiver and our suboptimal receiver).

following simulation conditions. On the transmitting side,
100,000 symbols are transmitted using chaotic sequences withdifferent initial values. Here, the parameter of the skew tent
map is ﬁxed as a = 0.05. Also, as the chaotic sequencelength, N = 4 and 8 are used. The receiver calculates a BER
performance under AWGN and a computational time spent todecide all decoded symbols. To compare the performances ofdifferent dimensional spaces to calculate the shortest distance,
we use 4-dimensional space (4-D) and 8-dimensional space(8-D). Also, to compare the performance, the optimal receiveris changed L from 2N −1 and is carried out the simulation.
These simulations are carried out using PC implemented CPU:
Core2Duo 2.4GHz, RAM2GB.
First of all, to conﬁrm the validity of the our suboptimalreceiver as an approximation method of the optimal receiver,
we show the BER comparison of the optimal and our suboptimal receiver in Fig 5. Here, to simulate the optimal receiver,
we apply the numerical integration method (Eqs. (7) and (8))
with L = 2000. It can be observed that the curve of the oursuboptimal receiver corresponds that of the optimal receiver
when N = Nd -dimension. From this result, we consider that
our method using the shortest distance in Nd -dimensional
space becomes almost identical with the optimal receiver forthe case that the dimension Nd is equal to the length of thechaotic sequence N .
Next, we compare a computational cost of our method withthat of the optimal receiver. Tables I and II show the BER
performance and the actual computational time. Here, Eb /N0
is ﬁxed as 13dB. From these tables, we can see that the
BER performances of the optimal receiver and our method
are almost the same when L = 3 × 2N −1 . Thus, to obtainthe calculation accuracy of the optimal receiver, the minimum
required intervals of L needs 3 × 2N −1 . In addition, thecomputational time of the optimal receiver with the minimum
required interval spends 4 ∼ 6 times of that of our method.
From these results, we expect that the computational cost of
our method is lower than that of the optimal receiver using
the numerical integration method, even if the accuracy of the-5-TABLE IC OMPUTATIONAL T IME AND BER PERFORMANCE WITH N = Nd = 4 (106 SYMBOLS , Eb /N0 = 13D B)

Computational Time [sec]BER

2N −1 = 8
1.04
1.18 × 10−1

Optimal ReceiverL
2 × 2N −1 = 16
3 × 2N −1 = 32
1.72
3.23
2.82 × 10−2
1.61 × 10−2

Our Suboptimal Receiver
Number of Straight Line
2Nd −1 = 8
0.74
1.57 × 10−2

TABLE IIC OMPUTATIONAL T IME AND BER PERFORMANCE WITH N = Nd = 8 (106 SYMBOLS , Eb /N0 = 13D B)

Computational Time [sec]BER

2N −1 = 128
21.06
1.74 × 10−2

Optimal ReceiverL
2 × 2N −1 = 256
3 × 2N −1 = 512
44.23
90.85
1.26 × 10−2
1.15 × 10−2

optimal receiver with N = 16 and 32 is obtained by increasing
the computer performance. Therefore, it can be said that thelower computational cost is advantage of our method.
V. C ONCLUSIONSIn this study, to evaluate the accuracy of calculation andthe processing time, we have observed computational costsof the optimal and our suboptimal receiver As the simulation
results, we have conﬁrmed that the lower computational costis advantage of our method.
R EFERENCES[1] G. Kolumbán, B. Vizvári, W. Schwarz, and A. Abel, “Differential chaos
shift keying: A robust coding for chaos communication,” Proc. NDES’96,
pp. 87-92, Jun. 1996.
[2] M. Hasler and T. Schimming, “Chaos communication over noisy channels,” Int . Journal of Bifurcation and Chaos, vol. 10, no. 4, pp. 719-736,
2000.
[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, pp 1684-1691, Dec. 2000.
[4] F. C. M. Lau and C. K. Tse, “Approximate-optimal detector for chaos
communication systems,” Int. Journal of Bifurcation and Chaos, vol. 13,
no. 5, pp. 1329-1335, 2003.
[5] F. C. M. Lau and C. K. Tse, “On optimal detection of noncoherent chaosshift-keying signals in a noisy environment,” Int. Journal of Bifurcation
and Chaos, vol. 13, no. 6, pp. 1587-1597, 2003.
[6] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Communication
Systems, Springer, 2003.
[7] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalized correlation-delayShift-Keying Scheme for Noncoherent Chaos-Based Communication Systems,” IEEE Trans. Circuits and Systems Part I, vol. 53, no. 3, pp. 712721, Mar. 2006.
[8] L. E. Larson, J-M. Liu, L. S. Tsimring, Digital Communications Using
Chaos and Nonlinear Dynamics, Springer, 2006.
[9] 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.

-6-

Our Suboptimal Receiver
Number of Straight Line
2Nd −1 = 128
14.28
1.15 × 10−2