Author name used in this publication: Francis C. M. LauAuthor name used in this publication: Chi K. Tse2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Lau, FCM

Tam, WM

Tse, CKM

IEEE Transactions on Circuits and Systems I Regular Papers

PolyU Institutional Repository

GENERALIZED CORRELATION-DELAY-SHIFT-KEYING SCHEME FOR NONCOHERENTCHAOS-BASED COMMUNICATION SYSTEMSWai M. Tam, Francis C.M. Lau and Chi K. TseDepartment of Electronic and Information Engineering,The Hong Kong Polytechnic University, Hong Kong, ChinaEmail: tamwm@eie.polyu.edu.hk, encmlau@polyu.edu.hk & encktse@polyu.edu.hkABSTRACTIn this paper, we propose a generalized correlation-delay-shift-keying (GCDSK) scheme for noncoherent chaos-based commu-nications. In the proposed scheme, several delayed versions of achaotic signal are ﬁrst produced. Some of them will be modulatedby the binary data to be transmitted. The delayed signals will thenbe added to the original chaotic signal and transmitted. At thereceiver, a simple correlator-type detector is employed to decodethe binary symbols. The approximate bit error rate (BER) of theGCDSK scheme is derived analytically based on a Gaussian ap-proximation. Simulations are performed and compared with thenoncoherent correlation-delay-shift-keying (CDSK) and differen-tial chaos-shift-keying (DCSK) modulation schemes. The effectsof the spreading factor, length of delay and the number of delayunits on the BER are fully studied. It is found that GCDSK canachieve better BER performance than DCSK under reasonable bit-energy-to-noise-power-spectral-density ratios.1. INTRODUCTIONNoncoherent communication schemes, which do not require thereproduction of the chaotic signals at the receiving end, are morefeasible in practice [1]–[3]. The correlation delay-shift-keying(CDSK) scheme is one of the noncoherent detection schemes, whichis similar to the DCSK scheme in that a reference chaotic signal isembedded in the transmitted signal [3]. Unlike in DCSK [2], how-ever, the reference signal and the information-bearing signal arenow added together with a certain time delay in CDSK. As a con-sequence, each transmitted signal sample includes one referencesample and one information-bearing sample, and the transmittedsignal sample is never repeated. Since no individual reference sig-nal is sent, the bandwidth efﬁciency is improved. Moreover, byeliminating the switch required to perform the switching betweenthe reference chaotic signal and information-bearing signal in theDCSK system, CDSK allows a continuous operation of the trans-mitter. Also, the transmitted signal is more homogeneous and lessprone to interception. However, because the sum of two chaoticsignals is sent, more uncertainty (interference) is produced whenthe received signal correlates with its delayed version at the receiv-ing side. Therefore, the performance of CDSK is worse than thatof DCSK.In this paper, we propose a generalized CDSK (GCDSK) scheme.The transmitted signal is composed of a reference chaotic signalThis work was partly supported by The Hong Kong Polytechnic Uni-versity under the Young Professors Earmarked Research Grant (Number 1-ZE03) and under The Hong Kong Polytechnic University Research GrantG-T859.and a number of delayed chaotic signals, some of which are mod-ulated by the data being sent. Such a construction of the transmit-ted signal allows the transmission of more than one reference sig-nal and more than one information-bearing signal simultaneously.The useful signal component, as well as the interference compo-nent, will be enhanced at the receiving side. We show that withan appropriate choice of system parameters, the bit error perfor-mance of the proposed system improves over the CDSK scheme.The system also inherits the merits of the CDSK system such asbeing switchless and allowing continuous operation of the trans-mitter.2. GENERALIZEDCORRELATION-DELAY-SHIFT-KEYING (GCDSK)SCHEME2.1. Transmitter StructureWe propose a generalized CDSK (GCDSK) communication sys-tem, as shown in Fig. 1. The transmitter contains a chaotic signalgenerator and (M − 1) delay blocks. We assume that M > 2 be-cause when M = 2, the GCDSK system degenerates to the CDSKsystem. Denote the minimum delay and the lth transmitted sym-bol by L and dl ∈ {−1,+1} respectively. Assume that “+1” and“−1” are transmitted with equal probability. First a chaotic sig-nal, denoted by {xk}, is generated in the transmitter. The chaoticsignals with delays L, 3L, 5L, . . . are modulated by the data se-quence {dl}, whereas the signals with delays 2L, 4L, 6L, . . . areunmodulated. Finally, the transmitted signal is formed by addingthe original chaotic signal and all the delayed signals. Denote thespreading factor by β, i.e., β chaotic signals sk, are sent withinone bit duration. During the lth bit duration, i.e., for time k =(l− 1)β +1, (l− 1)β +2, . . . , lβ, the transmitted signal is givenbysk =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩M−22∑m=0xk−2mL + dlM−22∑m=0xk−(2m+1)L if M is evenM−12∑m=0xk−2mL + dlM−32∑m=0xk−(2m+1)L if M is odd(1)where in each case, the ﬁrst and second terms represent summationof all the unmodulated and modulated chaotic signals, respectively.2.2. Receiver StructureWe assume an additional white Gaussian noise (AWGN) channel.The received signal, denoted by rk, is given by rk = sk + ξkwhere ξk denotes the AWGN signal with zero mean and variance???????????????????????????????? ????????? ??????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:17 from IEEE Xplore.  Restrictions apply.Fig. 1. Block diagram of a generalized correlation-delay-shift-keying communication system.N0/2. Similar to CDSK, a correlator-type detector is used. Theonly difference in the correlator is that only (β − L) terms (as-suming β > L) will be added in the summation block. Althoughpart of the useful signal component will be lost by summing only(β − L) terms, the intra-signal interference component will alsobe reduced because the appearance of the bit value dl−1 will beavoided in the received signal for dl. For the lth symbol, the cor-responding output of the correlator equalsyl =lβ∑k=(l−1)β+L+1rkrk−L=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩lβ∑k=(l−1)β+L+1⎡⎢⎣⎛⎜⎝M−22∑m=0(xk−2mL + dlxk−(2m+1)L)+ ξk⎞⎟⎠×(∑ M−22m=0(xk−(2m+1)L + dlxk−(2m+2)L)+ ξk−L)]if M is evenlβ∑k=(l−1)β+L+1⎡⎢⎣⎛⎜⎝M−12∑m=0xk−2mL + dlM−32∑m=0xk−(2m+1)L + ξk⎞⎟⎠×(∑ M−12m=0 xk−(2m+1)L + dl∑ M−32m=0 xk−(2m+2)L + ξk−L)]if M is odd={λeven + µeven + νeven if M is evenλodd + µodd + νodd if M is odd(2)whereλeven =lβ∑k=(l−1)β+L+1dl⎛⎜⎝M−22∑m=1x2k−2mL +M−22∑m=0x2k−(2m+1)L⎞⎟⎠(3)µeven =lβ∑k=(l−1)β+L+1⎡⎢⎣⎛⎜⎝M−22∑m=0(xk−2mL + dlxk−(2m+1)L)⎞⎟⎠×⎛⎜⎝M−22∑m=0(xk−(2m+1)L + dlxk−(2m+2)L)⎞⎟⎠⎤⎥⎦ − λeven (4)νeven =lβ∑k=(l−1)β+L+1⎡⎢⎣M−22∑m=0(xk−2mL + dlxk−(2m+1)L)ξk−L+M−22∑m=0(xk−(2m+1)L + dlxk−(2m+2)L)ξk + ξkξk−L⎤⎥⎦ (5)λodd =lβ∑k=(l−1)β+L+1dl⎛⎜⎝M−12∑m=1x2k−2mL +M−32∑m=0x2k−(2m+1)L⎞⎟⎠ (6)µodd =lβ∑k=(l−1)β+L+1⎡⎢⎣⎛⎜⎝M−12∑m=0xk−2mL + dlM−32∑m=0xk−(2m+1)L⎞⎟⎠×⎛⎜⎝M−12∑m=0xk−(2m+1)L + dlM−32∑m=0xk−(2m+2)L⎞⎟⎠⎤⎥⎦ − λodd (7)νodd =lβ∑k=(l−1)β+L+1⎡⎢⎣⎛⎜⎝M−12∑m=0xk−2mL + dlM−32∑m=0xk−(2m+1)L⎞⎟⎠ ξk−L+⎛⎜⎝M−12∑m=0xk−(2m+1)L + dlM−32∑m=0xk−(2m+2)L⎞⎟⎠ ξk + ξkξk−L⎤⎥⎦ (8)with {λeven, µeven, νeven} and {λodd, µodd, νodd} denoting the setsof required signal, the intra-signal interference and the noise com-ponent when M is even and odd, respectively. The intra-signalinterference, similar to the inter-user interference in a multiple ac-cess system [4], originates from the correlation between the chaoticsamples and may contribute positively or negatively to the requiredsignal. Based on the value of yl, the symbol is decoded accordingto the following rule:dˆl ={+1 if yl > 0−1 if yl ≤ 0. (9)It can be observed that when L is small compared to β, the use-ful signal component in the GCDSK receiver is approximately(M − 1) times larger than that of the CDSK case. Althoughboth the intra-signal interference and the noise component increasecompared with the CDSK case, their effect can be compensated bythe increase in signal component. Therefore, with appropriate val-ues of M and L, GCDSK can be designed to outperform CDSK.????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:17 from IEEE Xplore.  Restrictions apply.2.2.1. Gaussian-Approximated BERsIn our study, we make use of the Chebyshev map of degree 2 togenerate the chaotic signal. The map is given byxk+1 = 2x2k − 1 (10)and its correlation properties have been reported previously [5, 6].Assuming that the chaotic signal {xk} is stationary, it is readilyshown that for a given transmitted symbol dl, the mean value ofthe correlator output equalsE[yl | dl] = dl(β − L)(M − 1)Ps (11)where E[.] represents the expectation operator and Ps = E[x2k]. Ifthe conditional correlator output follows a Gaussian distribution,the approximate bit error rate for the GCDSK system can then bederived analytically and is given byBERGCDSK =12Prob(yl ≤ 0 | (dl = +1))+12Prob(yl > 0 | (dl = −1))=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩14erfc((β − L)(M − 1)Ps√2var[yl | (dl = +1)])+14erfc((β − L)(M − 1)Ps√2var[yl | (dl = −1)])when L = 112erfc((β − L)(M − 1)Ps√2var[yl | (dl = +1)])when 2 ≤ L < β(12)where erfc(·) represents the complementary error function [5]. Dueto the limitation of space, the derivation of the var[yl | dl] is notdescribed in this paper.3. RESULTS AND DISCUSSIONSIn this section, we present our ﬁndings on the bit error performanceof the GCDSK system. We denote the average bit energy by Ebwhich can be readily shown equal toEb = βMPs. (13)For various average-bit-energy-to-noise-psd (Eb/N0) ratios, wesimulate the GCDSK system and record the BERs. Also, we com-pute the approximate BERs using (12). We then compare our re-sults with those derived from the CDSK and DCSK systems when-ever appropriate.3.1. Effect of delay LFirst, we investigate the effect of the delayL on the bit error perfor-mance. Figure 2 plots the BER of the CDSK system together withthat of the GCDSK system with M = 4 and 6. A spreading factorβ of 100 is used. It is observed that the bit error performance forthe GCDSK system degrades as the delay L increases. As statedin Section 2.2, the correlator in the GDCSK receiver computes thesum of (β − L) terms before deciding upon whether the receivedsymbol is a “+1” or “−1”. Hence, the correlator output becomesmore unreliable when the number of terms reduces due to an in-crease in L, thereby increasing the BER. For the CDSK system,the number of the terms used in the correlator block is ﬁxed at βand is independent of the delay L. Therefore, the bit error per-formance of the CDSK system is found to be unaffected by theFig. 2. Simulated BER versus delay L for CDSK and GCDSKsystems. β = 100.Fig. 3. Simulated BER versus M for the CDSK (M = 2) andGCDSK systems. β = 100 and L = 5.delay L. Comparing the CDSK and GCDSK systems, it can be ob-served that for the same Eb/N0, the GCDSK system outperformsthe CDSK system with delay values up to 50.3.2. Effect of the number of delay blocksNote that the number of delay blocks in the transmitter side equals(M − 1). In Fig. 3, we plot the simulated BER against M . (Notethat the CDSK system corresponds to the case where M = 2.)A spreading factor β of 100 is used. It is shown that for a ﬁxedEb/N0 value, the BER reaches an optimal value at a certain valueof M . Speciﬁcally, when M increases, the average bit energy andthe detected signal component given by (13) and (11), respectively,increase initially. Although the intra-signal interference and thenoise power also go up for a given Eb/N0, there is a net improve-ment in the signal quality initially, thereby improving the BER. As????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:17 from IEEE Xplore.  Restrictions apply.Fig. 4. Simulated and analytical BERs versus Eb/N0 for theGCDSK system with M = 6 and β = 100.the value of M is further increased beyond the optimal point, thepercentage increase in the detected signal component is overshad-owed by the degradation due to intra-signal interference and noise.Therefore, the BER starts to degrade for large values of M .3.3. Comparison of the simulated andGaussian-approximatedBERsFigure 4 compares the simulated and Gaussian-approximated BERsfor the GCDSK system. A spreading factor of 100 is used. For theGCDSK system with M = 6, the discrepancy between the simu-lated and Gaussian-approximated BERs is quite signiﬁcant whenEb/N0 is large. When we analyse the statistics of the correlatoroutput from the simulations, it is found that the conditional meansand variances match with those derived in this paper. However, thedistribution of the conditional correlator output does not follow aGaussian distribution.3.4. Comparison with DCSK systemIn Fig. 5, the simulated BERs are plotted again for the CDSK,GCDSK and DCSK systems. β = 100 is used for all systemsand a delay of L = 1 is employed for the CDSK and GCDSKsystems. It is found that the CDSK system gives the worst BERand is approximately 2 dB worse than the DCSK system. For theGCDSK system, the BERs are about the same for M = 4, 6, 8 and10. Its performance is similar to that of the DCSK system and isbetter for Eb/N0 values below 16 dB.4. CONCLUSIONSIn this paper, we develop and study in detail a generalized correlation-delay-shift-keying (GCDSK) scheme for noncoherent chaos-baseddigital communications. We also compare the bit error rate (BER)of the proposed system with two previously studied noncoher-ent chaos-based communication schemes, namely, the correlation-delay-shift-keying (CDSK) scheme and the differential chaos-shift-keying (DCSK) scheme. Results show that the CDSK schemeFig. 5. Simulated BER versus Eb/N0 for the CDSK, GCDSK andDCSK systems. β = 100 for all systems, and L = 1 for the CDSKand GCDSK systems.gives the worst BER and is approximately 2 dB worse than theDCSK scheme. For the GCDSK scheme, the BERs are about thesame as the DCSK scheme and are better for Eb/N0 values below16 dB. Furthermore, we ﬁnd that a simple Gaussian approxima-tion is not sufﬁcient to model the conditional correlator output atthe receiver end. Hence, a more accurate model should be furtherinvestigated. Finally, compared with the CDSK and DCSK sys-tems, the GCDSK system has the highest hardware requirementbecause of the additional delays and multipliers.5. REFERENCES[1] C.C. Chen and K. Yao, “Stochastic-calculus-based numericalevaluation and performance analysis of chaotic communica-tion systems,” IEEE Trans. Circuits and Systems I, vol. 47,no. 12, pp. 1663–1672, Dec. 2000.[2] G. Kolumba´n, M.P. Kennedy and L.O. Chua, “The role ofsynchronization in digital communications using chaos - PartII: Chaotic modulation and chaotic synchronization,” IEEETrans. on Circuits and Systems I, vol. 45, no. 11, pp. 1129–1140, Nov. 1998.[3] M. Sushchik, L.S. Tsimring and A.R. Volkovskii, “Perfor-mance analysis of correlation-based communication schemesutilizing chaos,” IEEE. Trans. Circuits and Systems I, vol. 47,no. 12, pp. 1684–1691, Dec. 2000.[4] W.M. Tam, F.C.M. Lau and C.K. Tse, “Analysis of bit er-ror rates for multiple access CSK and DCSK communicationsystems,” IEEE. Trans. Circuits and Systems I, vol. 50, no. 5,pp. 702–707, May 2003.[5] F.C.M. Lau and C.K. Tse, Chaos-Based Digital Communi-cation Systems, Heidelberg: Springer-Verlag, 2003.[6] T. Geisel and V. Fairen, “Statistical properties of chaosin Chebyshev maps,” Physics Letters A, vol. 105A, no. 6,pp. 263–266, 1984.????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:17 from IEEE Xplore.  Restrictions apply.

Generalized correlation-delay-shift-keying scheme for noncoherent chaos-based communication systems

Author name used in this publication: Francis C. M. LauAuthor name used in this publication: Chi K. TseRefereed conference paper2003-2004 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Generalized correlation-delay-shift-keying scheme for noncoherent chaos-based communication systems

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository

PolyU Institutional Repository