Author name used in this publication: Chi K. TseRefereed conference paper2006-2007 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Chen, H

Feng, J

Tse, CKM

Abstract—A general noncoherent chaos-shift-keying (CSK) communication system with adjustable weights is proposed in this paper. The performance of the system under additive white Gaussian noise (AWGN) and multipath channel conditions are evaluated. Analytical expressions of the bit error rates are derived. The performance of the system is compared with that of the DCSK system and existing noncoherent CSK systems. The results show that the general CSK system can achieve the same performance as that of the DCSK system. A detailed analysis is presented to show that the DCSK system is the optimal form of such noncoherent systems. I

Hongbin Chen

Jiuchao Feng

Chi K. Tse

CiteSeerX

A General Noncoherent Chaos-Shift-Keying Communication System and its Performance Analysis

PolyU Institutional Repository

A General Noncoherent Chaos-Shift-KeyingCommunication System and its PerformanceAnalysisHongbin Chen1, Jiuchao Feng1,2 and Chi K. Tse31School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China2Faculty of Electronic and Information Engineering, Southwest University, Chongqing 400715, China3Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, ChinaEmail: chbscut@163.com, fengjc@scut.edu.cn & encktse@polyu.edu.hkAbstract—A general noncoherent chaos-shift-keying (CSK)communication system with adjustable weights is proposed inthis paper. The performance of the system under additive whiteGaussian noise (AWGN) and multipath channel conditions areevaluated. Analytical expressions of the bit error rates arederived. The performance of the system is compared with thatof the DCSK system and existing noncoherent CSK systems. Theresults show that the general CSK system can achieve the sameperformance as that of the DCSK system. A detailed analysis ispresented to show that the DCSK system is the optimal form ofsuch noncoherent systems.I. INTRODUCTIONThe discovery by Pecora and Caroll [1], [2] that two chaoticsystems can synchronize has triggered some interest in thedevelopment of chaos-based communication systems in thepast decade [3]–[5]. The early few proposed chaos-basedcommunication schemes were all coherent systems whichwere based upon the idealized assumption that two or morechaotic systems can be synchronized. In digital implementa-tions, the chaos shift keying (CSK) has been most widelystudied and analyzed [6], [7]. However, as the assumptionof the synchronizability of chaotic systems has been foundunrealistic or invalid under practical channel conditions, re-search direction has shifted towards noncoherent schemes. Thedifferential chaos shift keying (DCSK) has been the mostwidely studied noncoherent scheme which does not requirethe synchronization of chaotic systems [8]. Furthermore, withfrequency modulation (FM), the FM-DCSK scheme attainsconstant bit energy and hence represents a practical form ofdigital chaos-based communication scheme [9], [10].In the past few years, research interest in the developmentof alternative noncoherent detection schemes for digital chaos-based communication systems has grown rapidly. Despitebeing inferior to coherent schemes in terms of bit-error per-formance, noncoherent schemes still inherit the advantages ofcarrying information using chaotic signals and being simple. Anumber of modulation schemes have been proposed, e.g., themaximum likelihood scheme [11], the tracker-based scheme[12], [13], the extended Kalman filter (EKF) scheme withchannel equalization [14], the return-map-based scheme [15],the generalized maximum likelihood scheme for the detectionof frequency modulated differential chaos shift keying (FM-DCSK) signals [10], optimal [16], [17] and suboptimal [18]noncoherent detection method for CSK, and so on. In thispaper, we consider a general weighted noncoherent CSKcommunication system and evaluate its performance in ad-ditive white Gaussian noise (AWGN) and multipath channelenvironments.II. GENERAL WEIGHTED CSK SYSTEMWe begin with a formal description of the system to bestudied in this paper. Motivated by the fact that the DCSKsystem is a well-studied noncoherent chaos communicationsystem and weighting is a commonly used technique incommunication system design, we design the general weightednoncoherent CSK system following the concept of DCSK. Fig.1 shows the block diagram of the proposed noncoherent CSKcommunication system. The chaotic transmitter may simplyemploy a one-dimensional chaotic map, such as the logisticmap:xn+1 = 1− λx2n, −1 < xn < 1, (1)where 0 < λ ≤ 2. When 1.4 ≤ λ ≤ 2, the map is chaotic.Suppose the spreading factor of the system is 2N . Thechaotic signal x is generated by choosing λ = 2 and anappropriate initial condition, and then iterating over N steps.In our proposed scheme, x can be used to generate carriersy(1) and y(2), which are given byy(1)i ={w11xi, i = 1, · · · , Nw21xi−N , i = N + 1, · · · , 2N, (2)andy(2)i ={w12xi, i = 1, · · · , Nw22xi−N , i = N + 1, · · · , 2N, (3)where w11, w21, w12, w22 are non-zero adjustable weights (Inorder to obtain low BER, the weights should satisfy: w11w21 >0, w12w22 < 0, see (10), (12), (15)). When w11 = 1, w21 =1, w12 = 1, w22 = −1, the proposed system reduces tothe DCSK system in [8], [19]. Considering the lth symbol.Suppose the transmitted symbol is denoted by b l, which is24661-4244-0921-7/07 $25.00 © 2007 IEEE.Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 14, 2008 at 21:50 from IEEE Xplore.  Restrictions apply.ChaosgeneratorxnDelayN( )y1γw21Delay blplγ Multipathchannel hqlγvlγCorrelatorDecisionclbl~( )y2γNw11w12w22Fig. 1. Block diagram of the noncoherent CSK communication system.either 1 or 0 which occurs with equal probabilities. Duringthe lth symbol duration, the transmitted signal isplγ = bly(1)γ + (1 − bl)y(2)γ , γ = 1, 2, · · · , 2N. (4)In the receiver, a correlation-based detection is used, asshown in Fig. 1. The signal received by the receiver is givenbyqlγ = h(plγ) + vlγ , l = 1, 2, · · · ,M, (5)where v is AWGN with mean equal to zero and varianceN0/2,and h(.) is the multipath channel impulse response which canbe represented byh(plγ) =L∑j=1αjplγ−j+1, γ = 1, 2, · · · , 2N, (6)where α = [α1, α2, · · · , αL] is the normalized path gainvector, L is the number of paths in the channel [20]. Here,we consider the case where the path gains are time-invariant.The decision variable is the output of the correlator, which isgiven bycl =N∑β=1qlβqlβ+N . (7)Then, the lth decoded symbol is determined according to thefollowing rule:b˜l ={1, if cl ≥ 00, if cl < 0. (8)In the next section, we will derive the approximate bit errorrates (BERs) of the proposed noncoherent CSK system overmultipath channels.III. PERFORMANCE ANALYSISIn the following analysis, we assume that 0 < L  N .According to (2) through (7), we obtaincl =N∑β=1L∑k=1{α2k[(bl)2w11w21 + (1− bl)2w12w22]+αkbl(1− bl)(w11w22 + w12w21)}x2β−k+1+N∑β=1L∑k=1αk{vlβ [blw21 + (1 − bl)w22]+vlβ+N [blw11 + (1− bl)w12]}xβ−k+1+N∑β=1vlβvlβ+N= A+B + C, (9)whereA =N∑β=1L∑k=1{α2k[(bl)2w11w21 + (1 − bl)2w12w22]+αkbl(1− bl)(w11w22 + w12w21)}x2β−k+1,B =N∑β=1L∑k=1αk{vlβ[blw21 + (1− bl)w22]+vlβ+N [blw11 + (1 − bl)w12]}xβ−k+1,C =N∑β=1vlβvlβ+N .Assuming that “1” is transmitted (i.e., bl = 1), for large N ,the following statistics are easily obtained:E {A|bl = 1} = w11w21(α21 + α22 + · · ·+ α2L)·NE {x2n},E {B|bl = 1} = E {C|bl = 1} = 0,var{A|bl = 1} = w211w221(α41 + α42 + · · ·+ α4L)·Nvar{x2n},var{B|bl = 1} = (α21 + α22 + · · ·+ α2L)·(w221 + w211)NN0E{x2n}/2,var{C|bl = 1} = NN20/4,cov{A,B|bl = 1} = cov{B,C|bl = 1}= cov{A,C|bl = 1} = 0where E [·] and var[·] are the expectation and variance opera-tors, respectively, and cov[X,Y ] denotes the covariance of Xand Y . Then, we haveE {cl|bl = 1} = w11w21(α21 + α22 + · · ·+ α2L)·NE {x2n}, (10)2467Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 14, 2008 at 21:50 from IEEE Xplore.  Restrictions apply.var{cl|bl = 1} = w211w221(α41 + α42 + · · ·+ α4L)·Nvar{x2n}+ (α21 + α22 + · · ·+ α2L)·(w221 + w211)NN0E {x2n}/2+ NN20 /4. (11)The case of sending a symbol “0” may be computed in alikewise fashion, i.e.,E {cl|bl = 0} = w12w22(α21 + α22 + · · ·+ α2L)·NE {x2n}, (12)var{cl|bl = 0} = w212w222(α41 + α42 + · · ·+ α4L)·Nvar{x2n}+ (α21 + α22 + · · ·+ α2L)·(w222 + w212)NN0E {x2n}/2+ NN20 /4. (13)For the logistic map, we havevar{x2n} = 1/8 and E {x2n} = 1/2. (14)If y(1) is transmitted, the bit energy is E1 = (w211 +w221)NE {x2n}; else if y(2) is transmitted, the bit energy isE0 = (w212+w222)NE {x2n}. We define S1 = α41+α42+· · ·+α4Land S2 = α21+α22+ · · ·+α2L. According to (10) through (14),and assuming that cl follows a normal distribution under thegiven conditions, the conditional BER may be computed asBERc =12Prob(cl < 0|bl = 1)+12Prob(cl ≥ 0|bl = 0)=14[erfc(E {cl|(bl = 1)}√2var{cl|(bl = 1)})+erfc( −E {cl|(bl = 0)}√2var{cl|(bl = 0)})]=14{erfc([S1S22N+(w211 + w221)2N0w211w221S2E1+N(w211 + w221)2N202S22w211w221E21]−1/2)+erfc([S1S22N+(w212 + w222)2N0w212w222S2E0+N(w212 + w222)2N202S22w212w222E20]−1/2)}=14{erfc([S1S22N+W (w211 + w221)N02w211w221S2Eb+W 2NN208S22w211w221E2b]−1/2)+erfc([S1S22N+W (w212 + w222)N02w212w222S2Eb+W 2NN208S22w212w222E2b]−1/2)}, (15)where W = w211 + w221 + w212 + w222, Eb is the average bitenergy and is represented byEb = (E1 + E0)/2 = WNE {x2n}/2. (16)A. Comparison with DCSK SystemIn [19], BER for the DCSK system under multipath channelconditions is given byBERd =12erfc([S1S22N+4N0S2Eb+2NN20S22E2b]−1/2). (17)when w11 = 1, w21 = 1, w12 = 1, w22 = −1, the noncoherentCSK system is equal to the DCSK system, so they haveidentical multipath performance. For other weights, from (15)we can further derive that if w211 = w221 = w212 = w222 = 1and w11w21 > 0, w12w22 < 0, the two systems have identicalmultipath performance. However, If the weights do not takethese values, performance of the noncoherent CSK systemdegrades.B. Comparison with Existing Noncoherent CSK systemsThe AWGN performance of other noncoherent CSK systemsproposed in [11], [13], [15], which do not possess any adaptivealgorithm like those in [21], are all inferior to that of theDCSK system. Our proposed system can achieve the sameAWGN performance as that of the DCSK system, so ithas better AWGN performance than those noncoherent CSKsystems, such as those in [11], [13], [15].IV. SIMULATION RESULTSWe consider two different multipath channel models [20]:(I) α = [1, 0.45, −0.22], (II) α = [1, 0.7, −0.3, 0.5, −0.1].Numerical calculation and computer simulation of the BERsof the proposed system are performed for two cases.Case 1: The weights are w11 = 1, w21 = 1, w12 = 1, w22 =−1. The calculated and simulated BERs are ploted forN = 50in Fig. 2.Case 2: The weights are w11 = 1, w21 = 1, w12 = −1, w22 =1. The calculated and simulated BERs are ploted for N = 50in Fig. 3.Numerical calculation and computer simulation of the BERsof the DCSK system is also performed. The calculated andsimulated BERs are ploted for N = 50 in Fig. 4. It can beseen from Fig. 2–Fig. 4 that the simulated results are consistentwith numerical results and the results are almost the same.V. CONCLUSIONThe performance of the proposed noncoherent CSK systemis studied and compared with that of the DCSK systemand existing noncoherent CSK systems. The results showthat the proposed system can achieve the same multipathperformance as that of the DCSK system, and is of betterAWGN performance than some existing noncoherent CSKsystems. This formally proves that the performance of suchnoncoherent CSK schemes is bounded under that of the DCSKsystem. Possible performance enhancement may be obtainedthrough the use of adaptive algorithms [21].2468Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 14, 2008 at 21:50 from IEEE Xplore.  Restrictions apply.0 2 4 6 8 10 12 14 1610-410-310-210-1100Eb/N0(dB)BERchannel I, numericalchannel I, simulatedchannel II, numericalchannel II, simulatedFig. 2. BER performance of the proposed system (Case 1).0 2 4 6 8 10 12 14 1610-410-310-210-1100Eb/N0(dB)BERchannel I, numericalchannel I, simulatedchannel II, numericalchannel II, simulatedFig. 3. BER performance of the proposed system (Case 2).ACKNOWLEDGMENTThis paper was supported by National Natural Science Foundationof China (60572025), Program Foundation for New Century ExcellentTalents in China University (NCET-04-0813), Key Project Foundationof the Education Ministry of China (105137), and Natural ScienceFoundation of Guangdong Province, China (04205783).REFERENCES[1] L. M. Pecora and T. L. Caroll, “Synchronization in chaotic systems,”Phys. Rev. Lett., Vol. 64, pp. 821–824, 1990.[2] T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEETrans. Circ. Syst. I, Vol. 38, pp. 453–456, 1991.[3] A. Abel and W. Schwarz, “Chaos communications—principles, schemes,and system design,” Proc. IEEE, vol. 90, pages 691-710, 2002.[4] M. Hasler and T. Schimming, “Optimal and suboptimal chaos receivers,”Proc. IEEE, Vol. 90, pp. 733–746, 2002.[5] G. Kolumba´n, M. P. Kennedy, Z. Ja´ko and G. Kis, “Chaotic communi-cations with correlator receivers: Theory and performance limits,” Proc.IEEE, Vol. 90, pp. 711–732, 2002.0 2 4 6 8 10 12 14 1610-410-310-210-1100Eb/N0(dB)BERchannel I, numericalchannel I, simulatedchannel II, numericalchannel II, simulatedFig. 4. BER performance of the DCSK system.[6] H. Dedieu, M. P. Kennedy and M. Hasler, “Chaos shift keying: Mod-ulation and demodulation of a chaotic carrier using self-synchronizingChua’s circuits,” IEEE Trans. Circ. Syst. II, Vol. 40, pp. 634–642, 1993.[7] F. C. M. Lau and C. K. Tse, Chaos-based Digital CommunicationSystems, Heidelberg, Germany, Springer Verlag, 2003.[8] G. Kolumba´n, G. K. Vizvari, W. Schwarz and A. Abel, “Differentialchaos shift keying: A robust coding for chaos communication,” Proc.Int. Workshop on Nonl. Dynamics of Electron. Syst., pp. 87–92, 1996.[9] G. Kolumba´n, G. Kis, Z. Ja´ko´ and M. P. Kennedy, “FM-DCSK: Arobust modulation scheme for chaotic communications,” IEICE Trans.Fundamentals of Electronics, Communications & Computer Sciences,Vol. E81-A, pp. 1798–1802, 1998.[10] G. Kolumba´n, G. Kis, F. C. M. Lau and C. K. Tse, “Optimumnoncoherent FM-DCSK detector: Application of chaotic GML decisionrule,” IEEE Int. Symp. Circ. Syst., pp. 597–600, 2004.[11] A. Kisel, H. Dedieu and T. Schimming, “Maximum likelihood ap-proaches for noncoherent communications with chaotic carriers,” IEEETrans. Circ. Syst. I, Vol. 48, pp. 533–542, 2001.[12] J. Feng and C. K. Tse, “On-line adaptive chaotic demodulator based onradial-basis-function neural networks,” Phys. Rev. E, Vol. 63, pp. 1–10,2001.[13] J. Feng, C. K. Tse and F. C. M. Lau, “A chaos tracker applied to non-coherent detection in chaos-based digital communication systems,” Proc.IEEE Int. Symp. Circ. Syst., pp. 795–798, 2001.[14] Z. Zhu and H. Leung, “Combined demodulation with adaptive blind-channel equalization for chaotic modulation communication systems,”IEEE Trans. Circ. Syst. I, Vol. 49, pp. 1811–1820, 2002.[15] C. K. Tse, F. C. M. Lau, K. Y. Cheong and S. F. Hau, “Return-map-based approaches for noncoherent detection in chaotic digitalcommunications,” IEEE Trans. Circ. Syst. I, Vol. 49, pp. 1495–9, 2002.[16] M. Hasler and T. Schimming, “Chaos communication over noisy chan-nels,” Int. J. Bifurcation and Chaos, Vol. 10, pp. 719–735, 2000.[17] F. C. M. Lau and C. K. Tse, “On optimal detection of noncoherentchaos-shift-keying signals in a noisy environment,” Int. J. Bifurcationand Chaos, Vol. 13, pp. 1587–1597, 2003.[18] S. Arai and Y. Nishio, “Variable sequence length transmitter for nonco-herent chaos shift keying,” Proc. Int. Workshop on Nonlinear Circuitsand Signal Processing, pp. 301–304, 2006.[19] Y. Xia, C. K. Tse and F. C. M. Lau, “Performance of differential chaos-shift-keying digital communication systems over a multipath fadingchannel with delay spread,” IEEE Trans. Circ. Syst. II, Vol. 51, pp. 680–684, 2004.[20] Z. Zhu and H. Leung, “Adaptive blind equalization for chaotic commu-nication systems using extended-Kalman filter,” IEEE Trans. Circ. Syst.I, Vol. 48, pp. 979–989, 2001.[21] W. M. Tam, F. C. M. Lau and C. K. Tse, “Generalized correlation-delay-shift-keying scheme for noncoherent chaos-based communicationsystems,” IEEE Trans. Circ. Syst. I, Vol. 53, pp. 712–721, 2006.2469Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 14, 2008 at 21:50 from IEEE Xplore.  Restrictions apply.

A general noncoherent chaos-shift-keying communication system and its performance analysis

http://repository.lib.polyu.edu.hk/jspui/bitstream/10397/738/1/chaos-shift_07.pdf

A general noncoherent chaos-shift-keying communication system and its performance analysis

Abstract

Similar works

Full text

Available Versions

CiteSeerX

PolyU Institutional Repository