A robust sequence synchronization unit for multi-user DS ...

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A robust sequence synchronization unit for multi-userDS-CDMA chaos-based communication systems

B. Jovica,�, C.P. Unsworthb, G.S. Sandhua, S.M. Berbera

aDepartment of Electrical and Computer Engineering, University of Auckland, New ZealandbDepartment of Engineering Science, University of Auckland, New Zealand

Received 29 June 2006; received in revised form 14 January 2007; accepted 16 January 2007

Abstract

This paper demonstrates a way of achieving and maintaining sequence synchronization in multi-user direct sequence

code division multiple access (DS-CDMA) based chaotic communication systems. Synchronization is achieved and

maintained through the code acquisition and the code tracking phase, respectively. The performance of the proposed

system is evaluated in the presence of additive white Gaussian noise and interuser interferences. Throughout, a pseudo-

random binary sequence (PRBS) is used as the synchronizing pilot signal within the multi-user chaotic communication

system. In addition, the Logistic and Bernoulli chaotic maps are also used as the pilot signals in the investigation of the

code acquisition performance. The code acquisition circuit is evaluated in terms of the probability of detection and

probability of false alarm. The corresponding results demonstrate an ability to achieve initial synchronization.

Furthermore, it is shown that in terms of code acquisition, the PRBS outperforms the Logistic and Bernoulli chaotic maps.

A mathematical model of the code tracking loop is then presented. From the model, a control law for the generation of

time offset estimates is derived. The robustness of the synchronization unit is then demonstrated in terms of the bit error

rate. It has been shown that for the case of 1, 2, 3, 4, and 5 users, the bit error rate goes below the maximum acceptable

limit of 10�3 at the bit energy to noise power spectral density ratio of approximately 8, 9, 9.5, 11, and 12 dB, respectively.

Furthermore, a gradual degradation in performance, above the maximum acceptable bit error rate limit, is demonstrated

for the increasing number of users.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Synchronization; Chaos; Communications; DS-CDMA; Multi-user

1. Introduction

The synchronization of chaotic systems was firststudied by Yamada and Fujisaka in 1983 [1].However, it was not until 1990 when Pecora and

e front matter r 2007 Elsevier B.V. All rights reserved

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ing author. Tel.: +649 443 2308;

165203.

ess: [email protected] (B. Jovic).

his article as: B. Jovic, et al., A robust sequence s

n systems, Signal Process. (2007), doi:10.1016/j.sigpro.2

Carroll (PC) introduced their method of chaoticsynchronization (CS) [2] and suggested applicationto secure communications that the topic started toarouse major interest. Since then, a large number ofpapers have appeared [3–10], suggesting the appli-cation of this method of CS to chaotic communica-tions. The CS of [2] is most often established byemploying Lyapunov’s direct method [3,8,9] or byconsidering the conditional Lyapunov exponents

.

ynchronization unit for multi-user DS-CDMA chaos-based

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[11–13], leading to the design of the chaoticcommunication systems. Alternatively, the synchro-nization techniques of the traditional spread spec-trum communication systems [14–18] achievesynchronization between the transmitter and recei-ver in two distinct phases. These are called the codeacquisition and the code tracking phases [14–18].The code acquisition [14,16–18], or the initialsynchronization phase, involves determining thetime offset amidst the incoming signal and the basisfunction copy at the receiver to within a specifiedrange known as the pull-in region of the trackingloop [14–16]. Upon the successful completion of theacquisition phase, the code tracking phase startswith the fine alignment followed by the process ofmaintaining synchronization of the two signals. Dueto the mutually orthogonal properties of somechaotic signals [19–22], the synchronization techni-ques of the traditional code division multiple access(CDMA) spread spectrum communication systemshave a potential to be applied to the chaoticcommunication systems [21,23–33]. In most caseswhen evaluating the sequence synchronization ofthe chaos-based DS-CDMA systems, only the codeacquisition is analysed [21,23,24,26–32]. In [23,24],Setti et al. investigate the acquisition procedure of achaos-based DS-CDMA system and briefly discussthe possible general model for the tracking opera-tion. The tracking model of [23,24] is essentiallybased on a continuance of the acquisition procedureand it does not deal with the synchronization withinthe chip level which is required for the finealignment between the received and the despreadingsequences. It has been suggested in [23,24] that theBernoulli and the Tailed Shift chaotic maps may infact yield somewhat better performance during thecode acquisition phase than the classical spreadspectrum sequences such as m and Gold sequences.Furthermore, in [26], the authors use the Gaussianapproximation for the self-interference term toshow its effect on the acquisition performance. In[27] the moments approach is used to obtain a moreaccurate characterization of the self-interferenceterm. Throughout [23,24,26,27], the noise has notbeen included in the system in order to study theeffects of the interuser interferences on the acquisi-tion performance. However in any real communica-tion system, noise is an inevitable part of operationand is thus included here in the study of the systemperformance. In [29,31] the authors look at theacquisition performance of Markov chaotic se-quences when used as the spreading codes within a

Please cite this article as: B. Jovic, et al., A robust sequence

communication systems, Signal Process. (2007), doi:10.1016/j.sigpro.2

DS-CDMA system. It is shown in [31] that the biterror rate and code acquisition performance of theMarkov-based DS-CDMA systems are superior tothat of the independent and identically distributed(i.i.d)-based DS-CDMA systems. In [30], the dis-tribution of self-interferences of an incompletelysynchronized, to within a fraction of a chip,Markov-based DS-CDMA system is considered. Itis shown that Markov codes show promise in thisregard; however, no tracking circuit is proposed tocompletely synchronize the system. In addition, in[33], the author investigates the generation of spreadspectrum chaotic sequences via Markov chainswhose autocorrelation values always take realnumbers. Due to the reduction in the number ofunknown parameters, it is argued that the synchro-nization of such sequences is simpler than thesynchronization of Markov chain sequences whoseautocorrelation values take complex values. A morerecent advance in the synchronization of chaoticCDMA systems combines the interior penaltymethod of optimization theory and CS theory toachieve detection at the receiver [34].

Studies into the optimal spreading sequences forDS-CDMA systems have been conducted in[21–24,29–32,35–42], and it has been found that inmany instances chaotic time series are the optimalspreading sequences [21,23,24,31,32,35–42]. Forinstance, in [21,40], it has been shown thatquantized chaotic spreading codes can be generatedfor any number of users and exhibit generally betterperformance than the classical, m and Gold,sequences. Alternatively, in [35] an estimationtechnique for the minimum achievable interferencein DS-CDMA systems is proposed and used in [39]to find the autocorrelation function resulting in theminimum possible interference-to-signal ratio.Furthermore, it has been shown in [36–38] that interms of capacity, where capacity is defined as themaximum rate at which information can betransmitted without error, the suitably chosenchaotic spreading sequences outperform the classi-cal spreading sequences. Quantization of chaotictime series is recognized as one of the possiblepractical problems in the generation of the spread-ing sequences as it may affect the security and thesystem performance [20,41]. In [41] a practicalimplementation of the optimal real-valued Cheby-shev chaotic spreading sequence is investigated interms of the finite precision representation. It isshown that the bit error rate performance of a 31bit precision machine matches that of a double

synchronization unit for multi-user DS-CDMA chaos-based

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precision machine. Therefore, a 31 bit precisionmachine is sufficient for the practical implementa-tion of some chaotic sequences within DS-CDMAsystems. The digital signal processors (DSPs) are thedevices commonly used to investigate the imple-mentation of chaotic communication systems[43–48]. Under the assumption of perfect synchro-nization, the chaos-based DS-CDMA system of [19]has been investigated in [48] on a 32 bit precisionTigerSHARC DSP chip by Analog Devices. It hasbeen shown that the quantization of the Logisticmap at this precision does not affect the bit errorrate performance of the system.

Broadly speaking, chaotic communication sys-tems can be classified into those that requiresequence synchronization at the receiver, the co-herent systems, and those that do not, the non-coherent systems. However, in many cases whenstudying coherent chaotic communication systemsperfect synchronization between the spreading codeat the transmitter and its replica, or copy, at thereceiver is assumed [19,20,48,51–54]. Such analysesonly provide the benchmark performance of thesystem [51]. In [55] it was reported that the PC CSmethod of [2] is insufficiently robust for theimplementation within the practical chaotic com-munication systems. In order for the CDMA multi-user chaotic communication systems to become ofpractical and not just academic interest, robustsynchronization techniques must be developed[51,52,55,56–60]. The motivation of this work wasto develop robust synchronization technique for themulti-user DS-CDMA chaotic communication sys-tem of [19] using the traditional techniques ofsequence synchronization within the CDMA sys-tems.

In this paper, the code acquisition and trackingphase of the sequence synchronization system areimplemented within the multi-user DS-CDMAchaotic communication scheme of Parlitz andErgezinger [19]. The system is proposed andevaluated in the presence of additive white Gaussiannoise (AWGN) and the interuser interferences. Thesynchronization system proposed, utilizes a pseudo-random binary sequence (PRBS) pilot signal withinthe multi-user chaotic communication system toachieve and maintain synchronization. Under theassumption of perfect synchronization, the bench-mark performance of the system of [19] has alreadybeen investigated in the presence of noise andinteruser interferences in [19,53] as well as in theRayleigh fading channel of [54], demonstrating the

Please cite this article as: B. Jovic, et al., A robust sequence s


potentially robust nature of this system. Alsounder the assumption of perfect synchronization,the security of the system of [19] has been evaluatedin [60], demonstrating some weaknesses of thesystem to the return map and correlation functionattacks.

Section 2 presents the entire system, consisting ofthe system in [19] and the sequence synchronizationsystem proposed. The interconnections of the twosystems are explained. In Section 3, the codeacquisition circuit is presented and analysed interms of the probability of false alarm and theprobability of detection. The ability to achieveinitial synchronization in the presence of noise andinteruser interferences is demonstrated. The math-ematical model of the code tracking loop ispresented in Section 4. The control law used forthe generation of the time offset estimates is thenderived. This is followed by the investigation intothe overall noise performance of the system in termsof the bit error rate for different numbers of chaoticusers. Finally, the performance of the systemproposed is compared to the initial conditionmodulation (ICM) scheme [10] based on theprinciples of PC synchronization.

2. The chaotic communication system with the

synchronization unit

Fig. 1 shows the ‘DS-CDMA-communicationscheme based on the chaotic dynamics’ introducedin [19], with the synchronization unit proposed here.

In Fig. 1, x(t) denotes the chaotic spreadingsignals which are multiplied by the binary messagesignals m(t). The products are then summed up toproduce the signal c(t) which is transmitted throughthe channel. xp(t) denotes the PRBS which acts asthe periodic pilot signal used for synchronizationpurposes. Provided that the power of the noise inthe system is comparatively low to the power of thesignal, the synchronization unit uses the receivedsignal r(t) to generate the despreading codes whichare punctually synchronized to the spreading codesat the transmitter. In order for the spreadingwaveform generator at the receiver to producepunctual despreading codes, the initial conditionsof the spreading codes of each of the M users at thetransmitter must be available to it. The receivedsignal r(t) is then correlated with the punctualdespreading codes. For sufficiently low noise levelsin the system, the correlation value produced at theoutput of each correlator is positive if the bit 1 is


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Fig. 1. DS-CDMA chaotic communication system with the synchronization unit.

B. Jovic et al. / Signal Processing ] (]]]]) ]]]–]]]4

transmitted and negative if the bit 0 is transmitted[19]. Note that the correlator receiver of Fig. 1 hasbeen represented by integrals, rather than sums as in[19], in order to conform to the continuous timedomain which is used in this paper.

The synchronization unit of Fig. 1 is composed oftwo interconnected units, namely the acquisition orthe initial synchronization unit and the tracking unitwhich includes everything but the initial synchroni-zation unit. For the communication between thetransmitter and the receiver to take place, thesynchronization between the chaotic spreadingcodes x(t) at the transmitter and their replicas at



the receiver must be established and maintained.The synchronization is established through theacquisition or the initial synchronization unit [14,16–18] by acquiring the time offset of the receivedsignal r(t) to within a certain fraction of the chipperiod Tc. Once the synchronization has beenestablished, it is continuously maintained by thetracking unit [14–16] by ensuring that the incomingtime offset is matched by the estimated time offsetTd, as explained in Section 4. The synchronizationusing the PRBS signal as the pilot signal within thechaotic communication system is possible due to thefact that the PRBS signal and the chaotic signal


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-500 -400 -300 -200 -100 0 100 200 300 400 500

-0.2

0

0.2

0.4

0.6

0.8

1

t

Rxxp(t

)

-500 -400 -300 -200 -100 0 100 200 300 400 500

-0.2

0

0.2

0.4

0.6

0.8

1

t

Rxx(t

)

Fig. 2. (a) Cross-correlation of logistic map and PRBS time series. (b) Autocorrelation of logistic map time series.

B. Jovic et al. / Signal Processing ] (]]]]) ]]]–]]] 5

used, the logistic map [19] shown in phase space [60]in Fig. 1, are highly orthogonal as is demonstratedin Fig. 2a by the cross-correlation function with nodominant peaks. The autocorrelation function ofthe logistic map time series is presented in Fig. 2bshowing the dominant peak. The logistic map timeseries has been generated using Eq. (2.1) [19]:

X nþ1 ¼ 1� 2X 2n. (2.1)

The length of the logistic map time series used toproduce Figs. 2a and b is equal to 511 points(chips). The dynamic range of the Logistic map timeseries is confined to 71 [19]. In Figs. 2a and b, t

denotes the time delay. Also, note that the correla-tion functions have been normalized to the peak ofthe autocorrelation function.

3. The code acquisition

In this section, the first phase of the sequencesynchronization process known as the code acquisi-tion or the initial synchronization phase is presented[16,61].

3.1. Theoretical model of the system

In Fig. 3, the circuit diagram of the codeacquisition circuit is shown [61]. The followingmathematical analysis of the acquisition circuit isperformed at bandpass, as is the common practicewhen dealing with this kind of circuitry [14,16,62].

The physical description of the circuit is asfollows. The input signal r(t) is composed of theM user signals combined together and up converted,



as well as the noise component introduced by thechannel. Among the user signals is the pilot signalwhich is used for the sequence synchronizationpurposes of the communication system. The r1(t),r2(t) pair are the baseband signals. The xp(t�u) isthe copy of the pilot signal with some arbitrary timeoffset u. The time offset u can be represented asu ¼ jDTc, where j is an integer and D any valuebetween zero and one depending on the searchstrategy employed [63]. The xp(t�u) copy of thepilot signal is used to despread the r1(t), r2(t) pair.The despread signals are then integrated over theperiod of the pilot signal xp(t) equal to NTc units oftime. The decision variables Z1j and Z2j are squaredand summed to produce the decision variable Zj,which is used to decide whether the time offset hasor has not been acquired by comparing it to thepredetermined threshold. If the time offset has notbeen acquired, the despreading pilot signal is shiftedby further DTc and the new decision variableproduced. The procedure is repeated until theapproximate time offset (to within 7DTc), isdetermined. The reason of having two branches isto eliminate the influence of the carrier componentfrom the decision making [16,61]. It is assumed thatthe clock and carrier synchronization between thetransmitter and the receiver has already beenachieved, and is maintained throughout the acquisi-tion procedure, so that the system has the knowl-edge of where the chips start and end. Thisassumption is used in most cases when evaluatingthe performance of binary modulation techniques[10,64]. In the bandpass case the received signal ofFig. 1, r(t), is assumed to be of the up converted


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2 cos(ωct)

2 sin(ωct)−

LPF

LPF

Apxp(t−υ)

Apxp(t−υ)

X1(t)

X2(t)

Z1 j

Z2 j

r(t) Zj

( )2

()2

r1(t)

r2(t)

Fig. 3. Code acquisition circuit.


form given by Eq. (3.1):

rðtÞ ¼XMi¼p

Aixiðt� ZÞmiðt� ZÞffiffiffi2p

cosðoctþ jÞ þ nðtÞ,

(3.1)

where nðtÞ ¼ffiffiffi2p

nIðtÞ cosðoctÞ �ffiffiffi2p

nQðtÞ sinðoctÞ.In Eq. (3.1), Ai represents the amplitude of the

transmitted signals, xi the spreading waveforms andmi the information signals, with Z denoting somearbitrary time offset of the received signal. The limitof the sum M denotes the M users of the system,with p corresponding to an extra user (p ¼ 0), thatis, the pilot signal, as illustrated in Fig. 3. The termsnI(t) and nQ(t) denote the in phase and quadraturecomponents of the noise signal n(t). The angularfrequency of the carrier is denoted by oc and itsphase by j.

Keeping in mind that Z and u are some arbitrarytime offsets with respect to each other, let trepresent the overall time offset between thereceived signal and the despreading replicas at thereceiver. Also note that it is so chosen that mp(t) ¼ 1for all time [16]. It is then readily verifiable that thedecision variables Z1j and Z2j can be expressed byEqs. (3.2) and (3.3), respectively, as shown in[16,61]:

Z1j ¼ Ap cosðjÞTRjðtÞ þN1j , (3.2)

Z2j ¼ Ap sinðjÞTRjðtÞ þN2j. (3.3)

In Eqs. (3.2) and (3.3), R(t) denotes the autocorre-lation function of the periodic waveforms [16,61],where the pilot signal period T ¼ NTc. In general,N1j (N2j) is composed of the white Gaussian noiseterm (g) and the interferences term (I).



3.2. Theoretical upper bound on the probability of

detection

In this subsection, the theoretical expression forthe upper bound probability of correctly acquiringthe time offset between the received signal and thedespreading replicas at the receiver is given. Thestatistical properties of N1 are now briefly analysed.Noting that the phase of noise relative to the pilotsignal is arbitrary, it is readily verifiable that thesecond term of (3.2) can be expressed by Eq. (3.4)[16,61]:

N1j ¼ gj þ I j

¼

Z jNTc

ðj�1ÞNT c

nIðtÞxpðt� tÞdt

þ

Z jNTc

ðj�1ÞNT c

XMi¼1iap

AixiðtÞmiðtÞ

8<:

9=;

� cosðjÞxpðt� tÞdt. ð3:4Þ

The mean value of the white Gaussian noise term gis equal to zero, and its variance can be expressed byEq. (3.5) [14,16]

Var½g� ¼Z NT c

0

Z NT c

0

1

NT c

Z NT c

0

nIðtÞnIðuÞdt

� ��xpðt� tÞxpðu� tÞdtdu

¼

Z NT c

0

Z NT c

0

1

2N0Bdðt� uÞxpðt� tÞ

�xpðu� tÞdtdu

¼1

2N0B

Z NTc

0

x2pðt� tÞdt ¼

1

2N0BNT c.

ð3:5Þ

In Eq. (3.5), B denotes the bandwidth of theintermediate frequency filter (not explicitly shown),


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Fig. 4. Output Zj of the code acquisition circuit of Fig. 3.


so that the noise component is the baseband whiteGaussian noise process with two-sided powerspectral densities N0/2 over the frequency range|f|oB/2 [14]. Therefore, g is a Gaussian randomvariable of zero mean and N0BNTc/2 variance, andcan be represented by Eq. (3.6):

g ¼ G 0; 12

NoBNTc

� �. (3.6)

The interference term, I, is expected to always beclose to zero due to the orthogonal relationshipamong the chaotic interferences and the PRBS pilotsignal, as demonstrated in Fig. 2a, with certainvariance not equal to zero. With this in mind,Eq. (3.7) is assumed to hold [16]:

I ¼

Z jNT c

ðj�1ÞNTc

XMi¼1iap

AixiðtÞmiðtÞ

8<:

9=; cosðjÞxpðt� tÞdtffi 0.

(3.7)

Therefore, N1 is the Gaussian random variable ofzero mean and variance N 00BNTc=2, where N 00denotes the effective noise power spectral densitythat is due to both the receiver noise and theinterferences [16,61]. Variance N 00BNT c=2 thusincludes both the variance of g, and I terms. OverallEq. (3.8) holds:

N1 ¼ G 0; 12

N 00BNTc

� �. (3.8)

It is then readily verifiable that the upper bound onthe probability of detection can be expressed by theMarcum’s Q-function as shown in Eq. (3.9) [16,61]:

PDðj ¼ 1ÞpQ 2

ffiffiffiffiffiffiffiffiffiffiNEc

N0B

r;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 lnðPFðj ¼ 1ÞÞ

p� �.

(3.9)

In (3.9), PF stands for the probability of false alarmgiven by PFðj ¼ 1Þ ¼ e�bT=N 00BNT c , where bT denotesthe false alarm threshold level.

3.3. Empirical evaluation of the probability of false

alarm and the probability of detection

A way of obtaining the empirical expressions forthe probability of false alarm and the probability ofdetection is now briefly presented [61]. Assume thatat a certain level of noise in the system the output ofthe acquisition circuit is as given in Fig. 4.

Then at this certain level of noise the decisionvariable Zj will exceed the threshold value six timesand in any of those times synchronization will bedeclared. However in only one of those six times



(denoted by a circle in Fig. 4) the incoming signalr(t) and the basis function xp(t�u) will actually be insynchronism, while in the other five times (denotedby the crosses) the two will not be in synchronism.Following this example it can then be shown usingthe principles of conditional probability that theprobability of false alarm and the probability ofdetection can also be written as Eqs. (3.10) and(3.11), respectively [61]:

PFðj ¼ 1Þ ¼ limm!1

Xm

n¼1

kn

mðSn � 1Þ

� �, (3.10)

PDðj ¼ 1Þ ¼ limm!1

Xm

n¼1

PDðj ¼ 1Þnm

� �, (3.11)

where PDðj ¼ 1Þn 2 f0; 1g.In (3.10) and (3.11), m represents the number of

synchronization bits (periods) processed. The letterk represents the number of crosses in Fig. 4, that is,the number of decision variables Zj exceeding thethreshold value bT when in fact they should not. S

represents the total number of decision variables, Zj.The detailed derivation of (3.10) and (3.11) ispresented in [61].

3.4. Theoretical and numerical simulation results

The theoretical and empirical performance of thecode acquisition circuit of Fig. 3 is now examined interms of the probability of detection and theprobability of false alarm. In particular, the systemperformance is examined when the ratio of the chipenergy to the noise power spectral density, Ec/N0, isequal to �15 dB [16], and the chaotic interferencesand the period of the synchronizing pilot signal


007.01.014



vary. It has been found that the theoretical upperbound on the probability of detection (3.9)matches the empirical upper bound on the prob-ability of detection when B ¼ 12/7. Note that theempirical upper bound on the probability ofdetection is obtained by simply eliminatingchaotic users from the system and processing onlythe pilot signal. Fig. 5a shows the theoretical upperbound on the probability of detection, (3.9),when Ec/N0 ¼ �15 dB, N ¼ 255 and B ¼ 12/7,followed by the corresponding no interferenceempirical curve. The subsequent empiricalcurves associated with the increasing number ofusers in the system, demonstrate the expecteddegradation in the system performance with theincreasing level of interference. For instance, onecan see from Fig. 5a that an integrationtime equivalent to 255 chips is required to achievea detection probability of approximately 94%while maintaining a false alarm probabilityof 5% when Ec/N0 ¼ �15 dB and the total inter-ference is equivalent to an interference encoun-tered within a 5-user system. On the other handfor a 20-user system, when Ec/N0 ¼ �15 dBand N ¼ 255, one is only able to achieve a dete-ction probability of approximately 75.5%while maintaining the same false alarm probabilityof 5%.

By increasing the length of the pilot signal fromN ¼ 255 to 383 chips and N ¼ 511 chips, whilekeeping Ec/N0, B and the interferences unaltered,the results shown by Figs. 5b and c are obtained,respectively. From Figs. 5b and c one can see thatby increasing the integration time of the integratorsof Fig. 3, the effect of the noise and the interferencesis reduced resulting in a higher probability ofdetection. However, increasing the integration timeinevitably increases the overall initial synchroniza-tion time [16]. Therefore, there is a trade off betweenthe time it takes to search the possible pilot timeoffsets and the reliability of acquiring the correcttime offset. Thus, the choice of the particularintegration time will depend on the nature of theapplication. Note that although the cross-correla-tion between the chaotic signal generated by thelogistic map and the PRBS pilot signal is very low(Fig. 2a), the variance caused by the interuserinterferences (Eq. (3.7)) cannot be ignored in theanalytical model, especially for the case when thenumber of users is large.

The acquisition performance of the chaoticmaps, in particular the Bernoulli shift map [49],



has been investigated in a noiseless environment in[23,24]. In order to evaluate and compare theacquisition performance of the PRBS pilotsignal in a chaotic DS-CDMA system, Figs. 5dand e show the results obtained when the Logisticand Bernoulli chaotic map sequences are used as thepilot signal, respectively. It can be seen from Fig. 5dthat for Ec/N0 ¼ �15 dB and N ¼ 255 the Logisticmap pilot signal exhibits virtually the sameperformance as the PRBS pilot signal for the firstfive users. However, when the number of usersincreases to 10, 15 and 20 the performance of thesystem with the PRBS pilot signal is better.Furthermore, it can be seen from Fig. 5e that whenthe Bernoulli chaotic sequence is used as the pilotsignal, in a Logistic-map-based DS-CDMA system,the acquisition performance deteriorates by a non-negligible margin for any number of users in thesystem. Therefore, the acquisition performance ofthe Logistic-map-based DS-CDMA system in anoisy environment is better when the PRBS is usedas the pilot signal than the Logistic and Bernoullichaotic maps.

4. The code tracking

In this section the second phase of the sequencesynchronization process known as the code trackingphase is presented. Once the initial synchronizationcircuit of Fig. 1 has established the correct timeoffset to within the pull-in region of the trackingcircuit, the tracking circuit is able to take over thesynchronization process. Note that the pull-inregion of the tracking circuit is defined as the rangeof the time offset error that can be successfullycorrected by it [63]. The function of the trackingcircuit is to fine align the approximate time offsetacquired between the received and despreadingsequences and to maintain the synchronizationfrom this point onward [14–16,63]. In this section,the code tracking loop with a pull-in region of half achip length is considered. Therefore, to this end it isassumed that the search parameter D of the initialsynchronization circuit of Fig. 3 is equal to 1/2. Thisensures that the acquired time offset of Fig. 1, Ta, isaccurate to within half a chip length of the exacttime offset enabling the tracking circuit to correctthe inaccuracy and maintain the correct time offset.Thus, we redefine the incoming time offset Z, ofSection 3 as Td, indicating that the acquisition phasehas been finished and that the synchronization unitnow has the approximate knowledge of the correct


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-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

0.7

0.75

0.8

0.85

0.9

0.95

1

Pf

Pd

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

0.7

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0.8

0.85

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0.95

1

Pf

Pd

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

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1

Pf

Pd

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

0.7

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0.95

1

Pf

Pd

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

0.7

0.75

0.8

0.85

0.9

0.95

1

Pf

Pd

a c

b d

e

Fig. 5. (a) The probability of detection vs. the probability of false alarm for Ec/N0 ¼ �15 dB, N ¼ 255 and varying levels of interference

when the PRBS is used as the pilot signal. (b) The probability of detection vs. the probability of false alarm for Ec/N0 ¼ �15 dB, N ¼ 383

and varying levels of interference when the PRBS is used as the pilot signal. The close up is shown in the lower right-hand corner. (c) The

probability of detection vs. the probability of false alarm for Ec/N0 ¼ �15 dB, N ¼ 511 and varying levels of interference when the PRBS

is used as the pilot signal. The close up is shown in the lower right-hand corner. (d) The probability of detection vs. the probability of false

alarm for Ec/N0 ¼ �15 dB, N ¼ 255 and varying levels of interference when the Logistic map is used as the pilot signal. The close up is

shown in the lower right-hand corner. (e) The probability of detection vs. the probability of false alarm for Ec/N0 ¼ �15 dB, N ¼ 255 and

varying levels of interference when the Bernoulli map is used as the pilot signal. The close up is shown in the lower right-hand corner.


Please cite this article as: B. Jovic, et al., A robust sequence synchronization unit for multi-user DS-CDMA chaos-based

communication systems, Signal Process. (2007), doi:10.1016/j.sigpro.2007.01.014



time offset to within half a chip length:

�T c=2p Ta � Tdð ÞpT c=2. (4.1)

4.1. Theoretical model of the system

The tracking circuit examined here is knownas the delay lock loop (DLL) [14–16] and itincludes the entire synchronization unit of Fig. 1except for the initial synchronization unit. Theultimate function of the synchronization unit of Fig.1 is to produce the punctual codes for despreadingthe received signal r(t). This is achieved bycorrelating the early and late replicas of the pilotsignal by the received signal r(t), subtracting theirdifference, and ensuring that the resulting errorsignal e(t) is constantly forced to zero. In Fig. 1,VCO stands for the ‘Voltage-controlled oscillator’whose function is to increase or decrease the clockfrequency depending on the current value of e(t)[14]. The term d of Fig. 1 is defined as thenormalized difference among the incoming timeoffset Td of r(t) signal and the tracking circuit timeoffset estimate Td, that is, d ¼ ðTd � TdÞ=T c. Theloop filter of Fig. 1, is essentially an averagingintegrator, integrating over the PRBS pilot signalperiod:

eðtÞ ¼1

NT c

Z NTc=2

�NTc=2�ðt; dÞdt. (4.2)

Provided that one is already synchronized to withinhalf a chip period, after successful acquisition, it isnow shown how a punctual time offset Td, whichmatches the received signal time offset Td, isobtained at discrete time instances. With a correctestimate of Td, the receiver is able to accuratelydespread the received signal. The following mathe-matical analysis is performed at baseband and isbased on the circuit of Fig. 1. Therefore, thereceived baseband signal r(t) can now be repre-sented by Eq. (4.3):

rðtÞ ¼ cðtÞ þ nðtÞ ¼XMi¼p

Aixiðt� TdÞmiðt� TdÞ þ nðtÞ.

(4.3)

The received signal r(t) is composed of thetransmitted signal c(t) and the AWGN componentn(t). The transmitted signal c(t) is in turn composedof the mixture of signals of different users as well asthe pilot signal.



Keeping in mind that mp(t) ¼ 1 for all time [16], itis then readily verifiable that Eq. (4.4) holds:

eðtÞ ¼A2

p

NT c

Z NTc=2

�NT c=2xpðt� TdÞxp t� Td �

T c

2

� �dt

�A2

p

NT c

Z NT c=2

�NT c=2xpðt� TdÞxp t� Td þ

T c

2

� �dt

þ1

NT c

Z NT c=2

�NTc=2neðtÞdt, ð4:4Þ

where

neðtÞ ¼XM

i¼1iap

ApAixiðt� TdÞmiðt� TdÞ

� xp t� Td �Tc

2

� �� xp t� Td þ

Tc

2

� ��

þ ApnðtÞ xp t� Td �Tc

2

� ��

�xp t� Td þT c

2

� ��.

Eq. (4.4) can be rewritten in the form of Eq. (4.5):

eðtÞ ¼A2

p

NT c

Z NTc=2

�NT c=2xpðtÞxp tþ Td � Td �

T c

2

� �dt

�A2

p

NT c

Z NT c=2

�NT c=2xpðtÞxp tþ Td � Td þ

T c

2

� �dt

þ1

NT c

Z NT c=2

�NTc=2neðtÞdt. ð4:5Þ

Note that the autocorrelation function for theperiodic waveforms is defined by Eq. (4.6) [14]:

RxpðtÞ ¼

1

TCxpðtÞ ¼

1

T

Z T

0

xpðtÞxpðtþ tÞdt. (4.6)

In terms of CxpðtÞ, (4.5) can also be expressed by

Eq. (4.7):

eðtÞ ¼A2

p

NT cCxp

d�1

2

� �T c

� �� Cxp

dþ1

2

� �T c

� ��

þ1

NT c

Z NT c=2

�NTc=2neðtÞdt. ð4:7Þ

Fig. 6 shows the plots of Cxpððd� 1

2ÞT cÞ and

Cxpððdþ 1

2ÞTcÞ functions, and their difference, as

well as the plot of CxpðdT cÞ.

Assuming that the PRBS pilot signal is trackedover its entire period, the gradient in the linear


007.01.014


ARTICLE IN PRESS

Fig. 6. Plot of the early, late, and on-time PRBS correlation

functions.


region is then expressed by Eq. (4.8)

m ¼A2

pðN þ 1ÞTc

T c=2¼ 2A2

pðN þ 1Þ. (4.8)

Therefore in the linear region, the operation of thedelay lock tracking loop is governed by Eq. (4.9)

Cxpd� 1

2

� �Tc

� �� Cxp

dþ 12

� �Tc

� �¼ mdTc ¼ 2A2

pðN þ 1Þ Td � Td

� �. ð4:9Þ

Rearranging (4.9) to make Td the subject of theformula, Eq. (4.10) is obtained:

Td ¼ Td �Cxp

d� 12

� �T c

� �� Cxp

dþ 12

� �Tc

� �2A2

pðN þ 1Þ.

(4.10)

The numerator of the second term of (4.10) isdetermined by the DLL, and according to thaterror, Td, which is the estimate of the time offset Td,is determined. Although Td and Td are not shownexplicitly to be time varying, they are [14]. In orderto implement Eq. (4.10) digitally, Td and Td must berepresented as time variables. Assuming that theDLL executes a cycle, that is, calculates new valuesof Cxp

ððd� 12ÞT cÞ and Cxp

ððdþ 12ÞT cÞ every Tc

seconds, Eq. (4.10) can be re-represented by Eq.(4.11)

Td nT cð Þ ¼ Td nT cð Þ �C2 nTcð Þ � C1ðnTcÞ

2A2pðN þ 1Þ

, (4.11)

whereC1 ¼ Cxp

dþ 12

� �Tc

� �and C2 ¼ Cxp

d� 12

� �Tc

� �.

Eq. (4.11) cannot be implemented in practicesince it requires the knowledge of the time offset Td

in order to calculate the estimate Td of thattime offset in the same time instant. Under the



assumption that the time offset has been acquiredsuccessfully to within half a chip period, as denotedby (4.1), every new subsequent value of the timeoffset estimate can then be calculated based on itsprevious estimate in the following manner. Assum-ing that at the moment of the tracking phase start-up Ta ¼ Td, and substituting it into (4.11), yieldsEq. (4.12):

TdðT cÞ � Ta ¼ �C2 � C1

2A2pðN þ 1Þ

. (4.12)

Provided that indeed at the tracking phase start-upTa ¼ Td, Td(Tc) takes the value of Ta since thenumerator of the right-hand side of Eq. (4.12) goesto zero (refer to Fig. 6). If however Ta 6¼Td, when itwas assumed that Ta ¼ Td, then Td (Tc) takes on theactual value of Td at the start-up of the trackingphase, as the right-hand side of Eq. (4.12) generatesthe difference among the acquired and the actualtime offset: Ta�Td, so that (4.12) takes the form ofTdðT cÞ � Ta ¼ �ðTa � TdÞ, resulting in TdðT cÞ

¼ Td.With this thought in mind, (4.12) can be rewritten

as Eq. (4.13):

TdðnT c þ T cÞ ¼ TdðnT cÞ �C2ðnTcÞ � C1ðnTcÞ

2A2pðN þ 1Þ

,

(4.13)

where the initial condition is set as: Tdð0Þ ¼ Ta.Figs. 7a and b demonstrate the operation of the

tracking loop model developed at no noise and nointerferences present. The tracking loop was set toexecute 50 cycles, with the incoming time offset Td

varied for the first 35 cycles and set to a constantvalue, equal to the one of the previous cycle,thereafter. The figures demonstrate the optimalperformance of the tracking loop governed by thecontrol law of Eq. (4.13). In this particular case thepilot period has been set equal to 511Tc secondswith Tc represented by 8 time units. Choosing thesimulation parameters in this way allows one toobserve the ability of the tracking loop to activelytrack the changes in the incoming time offset for thefirst 35 cycles. Furthermore, when the time offsetstabilizes for the following 15 cycles, the trackingloop also stabilizes its estimate at this particularvalue, as demonstrated in Figs. 7a and b.

In order for the tracking loop to remain opera-tional and thus ensure the transfer of data betweenthe transmitter and the receiver of Fig. 1, the range


007.01.014


ARTICLE IN PRESS

Pilot signal period

Information bitduration

Pilot signal

Information signal

Transmitted signal Transmission start

Fig. 8. General transmission structure of the signals.

0 10 20 30 40 50-15

-10

-5

0

5

10

15

nTc

Am

plit

ude

Td

0 10 20 30 40 50

-5000

0

5000

nTc

C2(n

Tc)

- C

1(n

Tc)

a b

−Td

Fig. 7. (a) Plot of Td(nTc) and �TdðnT cÞ vs. nTc. (b) Plot of C2(nTc)�C1(nTc) vs. nTc.


of Eq. (4.14) must be satisfied at all times:

�T c=2p Td nT cð Þ � Td nT cð Þ� �

pT c=2. (4.14)

Equivalently, in terms of (4.13), the range ofEq. (4.15) must be satisfied at all times:

�T c=2p Td nT c þ T cð Þ � Td nTcð Þ� �

pT c=2. (4.15)

With the range of Eqs. (4.14) and (4.15) notsatisfied, the tracking circuit of Fig. 1 will no longerbe able to track the incoming time offset and theconnection among the transmitter and the receiverwill inevitably be lost. In this case the time offsetwill need to be re-acquired by the initial synchro-nization unit, as outlined in Section 3, before thesuccessful data transfer can take place again.

4.2. Performance evaluation of the system with noise

and interuser interferences

In this section, the performance of the system,highlighted in Fig. 1, is examined under theinfluence of AWGN and interuser interferencesduring its tracking mode of operation. The perfor-mance is evaluated for different numbers of chaoticusers with bit error rate curves [63] for the specifiedrange of the bit energy to noise power spectraldensity ratio (Eb/N0). The spreading factor of 73chips has been used to represent a single informa-tion bit transmitted. Tracking is conducted over thesynchronization period of the pilot signal which hasbeen chosen to be 511 chips long, that is, seven timesthe duration of the information bit. The generaltransmission structure of the signals is plotted inFig. 8. The code acquisition is required only at the



beginning of the transmission, and when the systemis no longer able to track.

The empirical BER curves for the system of Fig. 1are presented in Fig. 9 for 1–5, 10, 15 and 20 chaoticusers on top of the system’s PRBS pilot signal. Theincoming time offset Td has been uniformly variedwithin the boundaries of Eq. (4.14).

Note that with the perfect synchronizationassumed, the theoretical bit error rate curves ofthe system of [19] have been shown to be governedby Eq. (4.16) [53]:

BER ¼1

2erfc

2OLþ

2ðM � 1Þ

Lþ

Eb

No

� ��1" #�1=20@

1A,

(4.16)

where erfc denotes the complementary error func-tion [53], and O is defined as the variance of the


007.01.014


ARTICLE IN PRESS

-5 0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

BE

R

4

1 user 23

5

10

15

20

Fig. 9. The empirical BER curves of the system of Fig. 1 (marked curves), with Td varied within the limits of (4.14). The corresponding

theoretical curves with perfect synchronization assumed are shown by unmarked curves.


chaotic signal squared divided by the square of theaverage power of the same chaotic signal, and isexpressed by Eq. (4.17):

O ¼var x2

1

P2s

¼var x2

2

P2s

¼ � � � ¼var x2

M

P2s

. (4.17)

In addition to the empirical BER curves of thesystem of Fig. 1, Fig. 9 also presents the perfectsynchronization theoretical BER curves obtained byevaluating Eq. (4.16) for 1–5, 10, 15 and 20 chaoticusers without the system’s pilot signal. Note thatthese theoretical BER curves should be used as aguide only since Eq. (4.16) is somewhat inaccurate[53], especially at the low values of BER.

From Fig. 9 it can be observed that withoutassuming perfect synchronization, the noise perfor-mance of the system introduced by Parlitz andErgezinger [19] degrades by approximately 1–2 dBfor the single-user case. For a given Eb/N0 the singleuser plus the pilot signal exhibit the best perfor-mance due to the lowest interference at the receiverwhich subsequently causes the tracking unit of Fig.1 to generate least error in the time offset estimates.As the number of users increases, the interuserinterference inevitably increases, causing furtherdegradation in the performance of the trackingunit, which in turn further degrades the bit error



rate. With the decreasing levels of noise, that is withincreasing Eb/N0, the interuser interference dom-inates, causing the constant bit error rate character-ized by the flattening of the BER curves of Fig. 9.By assuming that the highest acceptable level ofBER equals 10�3 [50], it can be observed from Fig. 9that the Eb/N0 ratio for which the system perfor-mance is satisfactory for the case of 1, 2, 3, 4, and 5users is equal to approximately 8, 9, 9.5, 11 and12 dB, respectively. In the case of 10, 15 and 20 usersthe BER curves flatten before reaching the BERlevel of 10�3. This is unacceptable in practice.However, as seen from Fig. 9, in the case of 10, 15and 20 users, even the perfect synchronization BERcurves exceed the BER of 10�3. A possible methodto improve the performance in this case would be touse the filters specially designed for the chaotic timeseries [65]. The clock synchronization between thetransmitter and the receiver is assumed, as is in mostcases when evaluating the performance of binarymodulation techniques [10,15,64].

In Fig. 10, the BER curves for Td varied withinand beyond the boundaries of Eq. (4.14) are plotted.With the boundaries of Eq. (4.14) violated, thetracking loop operates outside the linear region ofFig. 6 and can no longer estimate the incoming timeoffset Td. As seen from Fig. 10 this results in thesignificant increase in the bit error rate for a given


007.01.014


ARTICLE IN PRESS

-5 0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

BE

R20 15

10

5

321 user

4

Fig. 10. The empirical BER curves of the system of Fig. 1 (marked curves), with Td varied within and beyond the limits of (4.14). The

corresponding theoretical curves with perfect synchronization assumed are shown by unmarked curves.


Eb/N0. In the case of Fig. 10 it has been assumedthat the time offset is immediately reacquired sothat the tracking loop can accurately execute thesubsequent cycles, provided that (4.14) is nowsatisfied.

4.3. Comparison and discussion

In order to evaluate and compare the perfor-mance of the system of Fig. 1, the BER curves forthe binary phase shift keying (BPSK) and therecently proposed CS-based communications tech-nique of ICM [10,65], have been produced in Fig. 11alongside the single-user curves of Figs. 9 and 10. Inthis work, the ICM scheme [10] is of interest as it isbased on a different form of synchronizationstrategy [2] used within chaotic communicationsystems.

From Fig. 11, it can be observed that the single-user chaotic DS-CDMA system of Fig. 1 outper-forms the ICM single-user communications schemebased on the principles of CS [10]. Therefore, thesynchronization scheme proposed and investigatedhere has been shown to be more robust to noisethan the PC CS-based ICM communication scheme,which in turn has been shown to be one of the morerobust PC CS communication schemes [10,65].



5. Conclusions

In this paper chaotic carriers have been embeddedwithin a practical multi-user DS-CDMA chaoticcommunication system and its performance evalu-ated in the presence of noise and interuserinterferences. The mutually orthogonal propertiesbetween the chaotic time series produced by thelogistic map and the PRBS pilot signal have enabledthe traditional ideas of the multi-user CDMAsequence synchronization process to be utilizedwithin the multi-user chaos-based DS-CDMAsystem. Both phases of the sequence synchroniza-tion process, namely the code acquisition and thecode tracking, have been proposed and investigated.

The code acquisition phase has been evaluated interms of the probability of detection and theprobability of false alarm at the chip energy tonoise power spectral density ratio of �15 dB for thethree different pilot signals and varying number ofchaotic users in the system. The theoretical upperbound on the probability of detection has beenderived and compared to the empirically determinedresults with the chaotic interferences present. Thesubsequent empirical curves associated with theincreasing number of users in the system havedemonstrated the expected degradation in thesystem performance with the increasing level of


007.01.014


ARTICLE IN PRESS

-5 0 5 10 15 20 2510-7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/No (dB)

BE

R

BPSK Chaotic, 1 user

Fig. 11. The BER curves: (a) the solid lines are for the theoretical BPSK and chaotic DS-CDMA system of [19] with the perfect

synchronization assumed; (b) the inverted triangles are for the system of Fig. 1 with Td varied within the boundaries of (4.14); (c) the

asterisks are for the system of Fig. 1 with Td varied within and beyond the boundaries of (4.14); (d) the solid squares are for the CS ICM-

based system of [10].


interference. In addition the expected increase of theprobability of detection, with the increase in theintegration time, has been demonstrated. Further-more, it has been shown that the best codeacquisition performance is achieved when the PRBSis used as the pilot signal as compared to theLogistic and Bernoulli chaotic maps.

The mathematical model for the investigation ofthe code tracking loop has been presented and usedto derive the control law used for the generation ofthe time offset estimates. The performance of theproposed code tracking circuit has been primarilyevaluated in terms of the bit error rate for varyinglevels of the chaotic interuser interferences, that is,for different numbers of chaotic users in the system.It has been shown that the system is reasonablyrobust to noise as compared to the performanceunder the assumption of perfect synchronization.The overall BER performance degradation for amulti-user system is characterized by the flatteningof the BER curves at low levels of noise due to theprevailing effects of the interuser interferences.

Finally, it has been demonstrated that the DS-CDMA chaotic communication system implement-ing the proposed sequence synchronization scheme,with a single user in the system, in general exhibits



better noise performance in terms of the bit errorrate than the PC CS-based communication techni-ques.

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