NOLTA2001 Proceedings pp. 637-640
An Approach for CSK Detection Based on Return Maps C. K. Tse, K. Y. Cheong, F. C. M. Lau and S. F. Hau Department of Electronic & Information Engineering, Hong Kong Polytechnic University, Hong Kong, China Email:
[email protected]
Abstract — Chaos-based communication can be applied advantageously only if the property of chaotic systems is suitably exploited. In this paper a simple non-coherent detection method for chaos-shiftkeying (CSK) modulation is proposed. Unlike the bit-energy-based detection which makes no specific use of any chaotic system property, the method proposed here exploits some distinguishable property of the two chaotic maps for recovering the digital message. Specifically, the proposed method exploits the difference in the return maps of the signals representing the digital symbols. Simple skew tent maps are used for illustration. The determining property of the return maps is computed using a simple linear regression algorithm. The bit-error-rate under additive white Gaussian noise is studied by computer simulations.
I.
Map 1
This work was supported by a competitive earmarked research grant (PolyU5124/01E) funded by Hong Kong Research Grants Council and also by a Hong Kong Polytechnic University Research Grant.
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mk
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e1
+
m^ k
transmitter
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Map 2
I NTRODUCTION
In a chaos-shift-keying (CSK) communication system, M digital symbols are represented by chaotic signals generated from M dynamical systems or from one system with M different parameter values [1]–[2]. In the binary case, i.e., M = 2, the transmitted signal essentially switches between two chaotic signals, which are generated from two dynamical systems or from one dynamical system having a parameter switched between two values, according to the digital symbol to be represented. Detection can take either a coherent form or a noncoherent form. In coherent detection, the receiver is required to reproduce the same chaotic signals sent by the transmitter, often through a ‘chaos synchronisation’ process which is unfortunately a fragile process [3]. Once reproduced, the digital symbols can be recovered by standard correlation detection [4]–[5]. In non-coherent detection of CSK, however, the receiver does not have to reproduce the chaotic signals. Rather, it makes use of some distinguishable property of the chaotic signals to determine the identity of the digital symbol being transmitted. The most commonly exploited distinguishable property has been the bit energy [7, 8]. However, when bit energy is chosen as the distinguishable property, detection can be accomplished easily by intruders, jeopardizing the security of the system. It has been shown in [9] for a differential CSK system that suitable exploitation of the determinism of the chaotic signals can lead to improved performance. In this paper we consider non-coherent detection of CSK, and in particular we make use of the built-in determinism of the chaotic signals for demodulation. Thus, instead of the bit energy, some built-in determin-
a
-+ ?j,
an
detector
channel
yn =xn+n receiver
Fig. 1: Block diagram of the CSK system. mk is the digital message and m ^ k is the restored message. a and b are the means of the chaotic signals an and bn . n denotes white Gaussian noise.
istic property is used to distinguish the digital symbols. In the CSK system proposed here, the chaotic signals representing the digital symbols are having the same bit energy. The demodulation method relies on detecting the deterministic property from which the symbol sent can be correctly identified. It is of interest to note that a maximum likelihood approach for non-coherent CSK, which has been recently proposed by Kisel, Dedieu and Schimming [10], achieves comparable performance with our proposed deterministic approach. However, our method enjoys simplicity of implementation. II.
T HE CSK S YSTEM
We consider a simple CSK system, in which the transmitter sends a signal consisting of chaotic signals extracted from two chaos generators. Essentially, the signal being transmitted toggles itself between the two sequences, fan g and fbng, depending upon the value of the digital message to be sent, where n is the integer index for the sequence of values generated by the chaos generators, as shown in Fig. 1. The two chaotic sequences, defined in the range [0; 1], are generated by the following skew tent maps:
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if 0 < an
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if an
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<
if bn
< (1 , yn ) ; yn+1 (~ yn; y~n+1 ) = > 1 , : (y ; y ) n
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; ;
0 1
if s^ > 0 otherwise
AND
C OMPARISON
The performance of the proposed detection method is evaluated by computer simulations. We present here the bit error rate of the system for a range of spreading factors, i.e., N = 12, 60, 100 and 200. Fig. 5 shows the plots of the bit error rate (BER) versus the usual Eb=No . The corresponding optimum performance of the non-coherent DCSK system with N = 24 is also shown for comparison [10]. This should be compared to our N = 12 case since half of the chips in DCSK are used as reference. Moreover, the DCSK doubles the bandwidth requirement. Next, we study the effect of varying the spreading factor N and the return-map parameter . From the simulation results, we may conclude that for constant Eb =No , the BER reaches a minimum for a certain spreading factor, as shown in Fig. 6 (a). Moreover, the choice of return map is also important. As shown in Fig. 6 (b), the BER can be optimized by setting at 0.5 for the case of skew tent map. Furthermore, we compare our method with the bit-energydetection method. Since our particular CSK system does not allow bit-energy-based detection because of the equal bit energy of the two chaotic signals, we deliberately perform a scaling on one of the chaotic signals so that bit energy becomes a distinguishable property for non-coherent detection. For instance, the transmitted signal defined in (3) can be modified as follows: k =0 xn = (abnn,,ab;) ; ifif m (7) mk = 1
where is a scaling factor. Note that = 1 corresponds to our proposed system (where the two chaotic signals have identical bit energy), and = 0 corresponds to a chaos-onoff-keying (COOK) system [8]. Fig. 7 compares the BER of the bit-energy-based detection and the return-map-based detection. As can be seen from the plots, our return-map-based detection outperforms the bit-energy-based detection, except for very small , i.e., for the cases approaching COOK. It should be noted that the proposed detection represents only one particular possibility of exploiting the return map features, and it may not represent the most effective algorithm for detection. Nonetheless, our purpose is to demonstrate detection possibilities by suitably exploiting the chaotic determinism. Methods based on detecting deterministic properties are still not exhausted, and further improvement is possible for non-coherent detection based on this category of methods.
(5) V.
otherwise
^ k be the message recovered. Our decision algorithm Let m can be summarized as
S IMULATION R ESULTS
(6)
where s^ is the estimated slope of the transformed return map
C ONCLUSION
In this paper we introduce an approach for demodulating CSK signals, exploitating the built-in determinism of chaotic signals. It is of the authors’ view that chaos-based communication can only be applied to the best advantage if the properties of chaos can be fruitfully exploited. We demonstrate in this paper a method using the “appearance” of the return maps as
(a)
Fig. 7: Comparison with bit-energy-based detection. R EFERENCES [1] H. Dedieu, M.P. Kennedy and M. Hasler, “Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuit,” IEEE Trans. Circ. Syst. II, vol. 40, pp. 634–642, 1993. [2] H. Yu and H. Leung, “A comparative study of different chaos based spread spectrum communication systems,” IEEE Int. Symp. Circ. Syst., pp. III-213–216, May 2001. [3] C.C. Chen and K. Yao, “Stochastic-calculus-based numerical evaluation and performance analysis of chaotic communication systems,” IEEE Trans. Circ. Syst. I, Vol. 47, No. 12, pp. 1663– 1672, Dec 2000. [4] M. Sushchik, L.S. Tsimring and A.R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circ. Syst. I, Vol. 47, No. 12, pp. 1684– 1691, Dec 2000.
(b) Fig. 6: Dependence of BER upon (a) spreading factor 0:51; (b) skew tent map parameter .
N
for
=
a distinguishing property for recovering the digital message carried by a CSK signal. We conclude this paper by reiterating that methods based on detecting deterministic properties are still not exhausted and further improvement is possible for non-coherent detection based on this category of methods.
A PPENDIX : L EAST S QUARES E STIMATE OF S LOPE The basic problem is to fit a line of the form y = sx + c to a set of data points (x1 ; y1 ), (x2 ; y2 ), ..., (xn ; yn ). The fitting objective n is to minimize the residual sum of squares, i=1 [yi (s + cxi)]2 , where s and c are the slope and y-intercept of the fitted line. The estimate of the slope s is given by the following formula:
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s^ =
n X
,X n
i=1
i=1
,
(yi , y)(xi , x )
(xi , x )2
where x and y are the means of x’s and y’s [11].
(8)
[5] M. Hasler and T. Schimming, “Chaos communication over noisy channels,” Int. J. Bifurcation and Chaos, Vol. 10, No. 4, pp. 719–735, 2000. [6] G. Kolumb´an, “Theoretical noise performance of correlatorbased chaotic communication schemes,” IEEE Trans. Circ. Syst. I, Vol. 47, No. 12, pp. 1692–1702, Dec 2000. [7] G. Kolumb´an, M.P. Kennedy and L.O. Chua, “The role of synchronisation in digital communication using chaos—part I: fundamentals of digital communications,” IEEE Trans. Circ. Syst. I, vol. 44, no. 10, pp. 927–936, Oct 1997. [8] M.P. Kennedy and G. Kolumb´an, “Digital communication using chaos,” in Controlling Chaos and Bifurcation in Engineering Systems, G. Chen (ed.), Boca Raton FL: CRC Press, pp, 477–500, 2000. [9] T. Schimming and M. Hasler, “Optimal detection of differential chaos shift keying,” IEEE Trans. Circ. Syst. I, Vol. 47, No. 12, pp. 1712–1719, Dec 2000. [10] A. Kisel, H. Dedieu and T. Schimming, “Maximum likelihood approaches for noncoherent communications with chaotic carriers,” IEEE Trans. Circ. Syst. I, Vol. 48, No. 5, pp. 533–542, May 2001. [11] P.J. Smith, Into Statistics, Second Edition, New York: Springer, 1998, Chapter 14, pp. 445–494.
