I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 475 Performance of Chaos-Based Communication Systems Under the Influence of Coexisting Conventional Sread-Sectrum Systems Francis C. M. Lau and Chi K. Tse Abstract This brief studies the erformance of selected chaos-based systems which share their frequency bands with conventional sread-sectrum systems. Such a scenario may occur in normal ractice when chaos-based systems are introduced while the conventional systems are still in oeration. The articular chaos-based systems under study are the coherent chaos-shift-keying system and the noncoherent differential chaos-shift-keying system, and the coexisting conventional system emloys the standard direct-sequence sread-sectrum modulation scheme. Analytical exressions for the bit-error rates are derived in terms of system arameters such as sreading factor, chaotic signal ower, conventional sread-sectrum signal ower, and noise ower sectral density. Finally, comuter simulations are erformed to verify the analytical findings. Index Terms Chaos communication, chaos shift keying (CSK), coexistence, differential chaos shift keying (DCSK), differential sequence sread sectrum. I. INTRODUCTION Communication using chaos has attracted a great deal of attention from many researchers for more than a decade [] [4]. Much of the research work has focused on the basic modulation rocesses and the noise erformance assuming ideal channel conditions. Both analog [], [5] and digital [6] [9] modulation schemes have been roosed, and it has been found that digital schemes are comaratively more robust than analog schemes in the resence of noise and thus reresent a more ractical form of systems for imlementation. Direct alication of chaos to conventional direct-sequence sread-sectrum (DSSS) systems was also reorted on the code level [0], []. The basic rincile is to relace the conventional binary sreading sequences, such as m sequences or Gold sequences [2], by the chaotic sequences generated by a discrete-time nonlinear ma. The advantages of using chaotic sreading sequences are that an infinite number of sreading sequences exist and that the sread signal is less vulnerable to intercetion. Instead of alying analog chaotic sequences to sread the data symbols, Mazzini et al. roosed quantizing and eriodically reeating a slice of a chaotic time series for sreading. It was also reorted that systems using the eriodic quantized sequences have larger caacities and lower bit-error rates (BRs) than those using m sequences and Gold sequences in a multile-access environment [3], [4]. Among the various digital chaos-based communication schemes, coherent chaos-shift-keying (CSK) and noncoherent differential chaos-shift-keying (DCSK) schemes have been most thoroughly analyzed [5] [8]. Comared with chaotic-sequence sread-sectrum modulation, CSK and DCSK modulation schemes make use of analog chaotic wideband waveforms directly to reresent the binary symbols. No sreading as in traditional DSSS systems is required. Recently, the DCSK scheme has also been considered for ractical imlementation [9]. For coherent systems (e.g., coherent CSK system), the receiver is required to reroduce the chaotic carriers through a rocess called Manuscrit received December 5, 20; revised December 3, 2002. This work was suorted by the Hong Kong Polytechnic University under the Young Professors armarked Research Grant -Z03 and RGC Direct Allocation A-PD65. This aer was recommended by Associate ditor M. Di Bernardo. The authors are with the Deartment of lectronic and Information ngineering, The Hong Kong Polytechnic University, Hong Kong (email: encmlau@ olyu.edu.hk; encktse@olyu.edu.hk). Digital Object Identifier 0.09/TCSI.2003.8869 chaos synchronization, and detection is normally achieved by correlating the incoming signal with the reroduced carriers. Although ractical robust synchronization methods for chaotic signals are still not available, the study of CSK systems is imortant in that it can rovide benchmark erformance for comarison with other chaos-based communication systems. Desite the fact that their erformance is inherently inferior than coherent systems, noncoherent systems resent more ractical forms of systems because they do not require the reroduction of chaotic carriers at the receiving end. Being sread-sectrum, chaos-based communication systems are exected to erform well even in the resence of other wideband signals sharing the same bandwidth. Such a scenario may occur in normal ractice, for examle, when chaos-based systems are introduced while the conventional systems are still in oeration. This asect of erformance is imortant, though it has rarely been addressed in the literature. It is therefore of interest to robe into the error erformance of chaos-based systems in channels where other wideband communication systems coexist. Furthermore, it is useful to comare the relative erformances of coherent and noncoherent chaos-based systems and the extent to which coherent systems excel in the resence of other coexisting wideband systems. In this brief, we investigate the erformance of selected chaos-based digital systems when their bandwidths are co-occuied by a conventional sread-sectrum signal. The chaos-based systems under study are the coherent CSK and the noncoherent DCSK systems, and the coexisting system is a standard DSSS system. Analytical exressions for the BRs are derived in terms of system arameters such as sreading factor, chaotic signal ower, conventional sread-sectrum signal ower, and noise ower sectral density. Finally, comuter simulations are erformed to verify the analytical findings. II. SYSTM OVRVIW The basic roblem we wish to investigate in this brief is the erformance of chaos-based digital communication systems when the channel is under the influence of additive white Gaussian noise; wideband signal generated from a coexisting conventional sread-sectrum communication system, which shares the same frequency band as the chaos-based system under study. Fig. shows a block diagram of the system under study. In this system, two indeendent data streams, denoted by l and l, resectively, are assumed to be sent at the same data rate. At time t, denote the outut of the chaos transmitter by s(t) and the conventional DSSS signal by u(t). Assuming that noise (t) is added to the transmitted signals, the received signal consists of three comonents, namely, chaotic, conventional sread-sectrum, and additive noise. The receivers of the chaos-based system and the conventional system will attemt to recover their resective data streams. Coherent or noncoherent detection schemes may be alied in the chaos-based system receiver, deending uon the modulation method used. In Section III, we focus on the coherent CSK system and the noncoherent DCSK system [3], [8]. The coexisting conventional system is a DSSS system. Our analysis will be carried out in a discrete-time fashion, and we will develo analytical exressions for the BRs of the recovered data streams for each of the chaos-based communication schemes under the afore-described environment. III. ANALYSIS OF BIT RROR PRFORMANC A. Coherent CSK System We consider a discrete-time binary CSK communication system combined with a DSSS communication system, as shown in Fig. 2. 057-722/03$7.00 2003 I
476 I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 Fig.. Block diagram of a combined chaos-based and conventional DSSS digital communication. Fig. 2. Block diagram of a combined CSK-DSSS communication system. Also, we assume that the samling rate equals the sreading code rate of the DSSS system. In general, two sets of chaotic signal samles, denoted by f^x k g and fx k g, are roduced in the CSK transmitter by two chaos generators. If the symbol is sent, f^x k g is transmitted during a bit eriod, and if is sent, fx k g is transmitted. Further, we assume that and occur with equal robabilities. For simlicity, we consider a CSK system in which one chaos generator is used to roduce chaotic signal samles fx k g for k = ; 2;... The two ossible transmitted signal samles are f^x k = x k g and fx k = 0x k g. Suose l 2f; g is the symbol to be sent during the lth bit eriod. Define the sreading factor 2 as the number of chaotic samles used to transmit one binary symbol. During the lth bit duration, i.e., for k = 2(l 0 ), 2(l 0 ) 2;...;, the outut of the CSK transmitter is s k = l x k : () For the DSSS system, we assume that the eriod of the seudorandom sreading sequence is very long. We denote the outut ower of the system by P B. ssentially, we can model the outut of the DSSS system as a random binary sreading code b k 2f; g multilied by P B. Also, and occur with equal robabilities. Thus, for time k, the transmitted signal of the DSSS system is reresented by u k = PB b k : (2) The CSK and DSSS signals are added and corruted by an additive white Gaussian noise before arriving at the receiving end. The received signal, denoted by r k, is thus given by r k = s k PB b k k (3) where k is a Gaussian noise samle of zero mean and variance (ower sectral density) N 0 =2. For the CSK system, we assume that a correlator-tye receiver is emloyed. As shown in Fig. 3, the correlator outut for the lth bit, y l, is given by y l = = l r k x k required signal noise x 2 k PB b k x k interfering DSSS signal k x k : (4) Suose a is transmitted in the CSK system during the lth symbol duration, i.e., l =. For simlicity, we write y l j( l = ) as y l j( l = ) = A B C (5)
I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 477 Fig. 3. Block diagram of a coherent CSK receiver. where A, B, and C are the required signal, interfering DSSS signal, and noise, resectively, and are defined as A = B = PB C = The mean of y l j( l =)is [y l j( l = )] = PB x 2 k (6) b k x k (7) k x k : (8) x 2 k [b k ][x k ] [ k ][x k ] =2P s (9) where P s = [x 2 k ] denotes the average ower of the chaotic signal. The last equality holds because [b k ]=0and [ k ]=0. The mean value and the average ower of the chaotic signal can be comuted by numerical simulation, or by numerical integration if the invariant distribution function of fx k g is available. The variance of y l j( l = ) is var [y l j( l = )] = var[a] var[b] var[c] 2cov[A;B] 2cov[B;C]2cov[A; C] (0) where cov[x; Y ] is the covariance of X and Y defined as cov[x; Y ]=[XY ] 0 [X][Y ]: () It can be shown that all the covariance terms are zero and the variance terms are given by where 3 is the variance of fx 2 kg, i.e., var[a] =23 (2) var[b] =2P BP s (3) var[c] =N 0 P s (4) 3=var x 2 k : (5) In the derivation of var[a], it has been assumed that the autovariance of fx 2 kg vanishes, i.e., cov x 2 j ;x 2 k = x 2 j x 2 k 0 x 2 j x 2 k =0 forj 6= k: (6) Using (2) and (4), and because all covariance terms in (0) are zero, we may write (0) as var [y l j( l = )] = 23 2P B P s N 0 P s : (7) For the lth symbol, an error occurs when y l 0j( l = ). Since y l j( l = ) is the sum of a large number of random variables, we may assume that it follows a normal distribution when 2 is large. The error robability is thus given by Prob (y l 0j( l = )) = 2 erfc [y l j( l = )] 2var [y l j( l = )] = erfc 2P s (8) 2 43 4P BP s 2N 0P s) where erfc(.) is the comlementary error function, which is defined as erfc( ) 2 e 0 d: (9) Similarly, when l =, the corresonding error robability can be shown equal to Prob (y l > 0j( l = )) = 2 erfc 0 [y l j( l = )] 2var [y l j( l = )] = erfc 2P s : (20) 2 43 4P BP s 2N 0P s) Hence, the overall error robability of the lth transmitted symbol is BR (l) CSK = Prob( l = ) 2 Prob (y l 0j( l = )) Prob( l = ) 2 Prob (y l 0j( l = )) = 2 erfc 2P s 43 4P BP s 2N 0P s) : (2) It can be seen from (2) that BR (l) CSK is indeendent of l. Thus, the error robability of the lth transmitted symbol is the same as the BR of the system. The BR of the CSK system, denoted by BR CSK,is therefore where BR CSK = 2 erfc = 2 erfc 2P s 434P BP s 2N 0P s) 3 P P P N 2P = 2 erfc 43 2P N (22) (23) b =2P s (24) denotes the average bit energy of the CSK system. The exression given in (22) or (23) is thus the analytical BR for the noisy coherent CSK system under the interference of a DSSS signal. Note that for
478 I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 Using the same notations as defined in Section III-A, during the lth bit duration of the DCSK system, the transmitted DCSK signal can be written as x k ; for k =2(l) ; 2(l 0 ) s k = l x k0; 2;...; 2(l 0 ) for k =2(l) ; 2(l 0 ) (30) 2;...; whereas the kth transmitted signal for the DSSS system is given by The received noisy signal r k is given by u k = PBb k : (3) r k = s k u k k (32) Fig. 4. Block diagram of a noncoherent DCSK system. (a) Transmitter. (b) Receiver. fixed DSSS signal ower P B and noise ower sectral density N 0 =2, the BR can be imroved by making one or more of the following adjustments. ) Reduce the variance of fx 2 kg. 2) Increase the sreading factor 2. 3) Increase the CSK signal ower P s. xamle: Consider the case where a logistic ma is used for chaos generation. The form of the ma is x k = g(x k )=0 2x 2 k where x k 2 (; ): (25) Given that the invariant distribution function of fx k g equals [20] we obtain P s = x 2 k = (x k )= 0x ; if jx kj < 0; otherwise x 2 (x)dx = 3=var x 2 k = x 4 k 0 2 x 2 k = x 2 (x)dx = 2 (26) (27) x 4 (x)dx0 4 = 8 : (28) For the case where the logistic ma is used to generate the chaotic samles, we substitute (27) and (28) into (22) to obtain the BR, i.e., BR CSK = 2 erfc 2 4P B 2N 0 : (29) B. Noncoherent DCSK System In this section, we consider the noncoherent DCSK system. For this system, the basic modulation rocess involves dividing the bit eriod into two equal slots. The first slot carries a reference chaotic signal, and the second slot bears the information. For a binary system, the second slot is the same coy or an inverted coy of the first slot deending uon the symbol sent being or. ssentially, the detection of a DCSK signal can be accomlished by correlating the first and the second slots of the same symbol and comaring the correlator outut with a threshold. Fig. 4 shows the block diagrams of a DCSK transmitter and receiver air. As in the CSK case, we assume that the DCSK signal is interfered by the DSSS signal and corruted by additive white Gaussian noise. where the symbol k is as defined reviously in Section III-A. At the DCSK receiver, the detector essentially calculates the correlation of the corruted reference and data slots of the same symbol. We consider the outut of the correlator for the lth received bit, y l, which is given by y l = where = = l D required signal r k r k l x 2 k PBb k x k l PBb k x k P B b k b k k x k l k x k PB k b k PB k b k k k PBF l PBG P BH interfering DSSS signal J l K PB L PB M N D = F = G = H = J = K = L = M = N = noise (33) x 2 k (34) b k x k (35) b k x k (36) b k b k (37) k x k (38) k x k (39) k b k (40) k b k (4) k k : (42)
I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 479 Fig. 5. BRs versus of the coherent CSK system under the interference of a DSSS signal. Simulated BRs are lotted as oints and analytical BRs lotted as lines. (a) Sreading factor is 20. (b) Sreading factor is 50. (c) Sreading factor is 00. (d) Sreading factor is 200. The means and variances of the variables D to N can be shown equal to [D] =[x 2 k ] P s var[d] =var x 2 k 3 [F ]=0 var[f ]=P s [G] =0 var[g] =P s [H] =0 var[h] = [J] =0 [K] =0 var[j] = P N 2 var[k] = P N 2 [L] =0 var[l] = N 2 [M ]=0 var[m ]= N 2 [N ]=0 var[n ]= N 4 (43) where in the derivation of var[d], it has been assumed that the autovariance of fx 2 kg vanishes. Further, it can be readily shown that cov[; ] =0 8; 2fD; F; G; H; J; K; L; M; N : 6= g: (44) Using a likewise rocedure as in Section III-A, the means and variances of y l j( l =)and y l j( l = ) can be shown equal to [y l j( l = )] = 0 [y l j( l = )] = P s (45) var [y l j( l = )] = var [y l j( l = )] = var[d] P B var[f ]P B var[g] PBvar[H] 2 var[j] var[k] P B var[l] P B var[m ]var[n ] = 3 2P BP s P 2 B P sn 0 P B N 0 N2 0 4 : (46) Since all terms in (45) and (46) are indeendent of l, the BR of the DCSK system under the interference of a DSSS signal, denoted by BR DCSK, equals the overall error robability of the lth transmitted symbol (BR (l) DCSK ), i.e., BR DCSK = BR (l) DCSK = Prob( l = ) 2 Prob (y l 0j( l = )) Prob( l = )Prob (y l > 0j( l = )) = 2 erfc P s = 2 erfc 234P B P s 2P 2 B 2P sn 0 2P B N 0 N 2 434P 4P N N 2P 4P 2N P (47)
480 I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 Fig. 6. BRs versus of the noncoherent DCSK system under the interference of a DSSS signal. Simulated BRs are lotted as oints and analytical BRs lotted as lines. (a) Sreading factor is 20. (b) Sreading factor is 50. (c) Sreading factor is 00. (d) Sreading factor is 200. = 2 erfc 838P 8P N 2N 8P 4N (48) where b is defined as in (24). The exression given in (47) or (48) is then the analytical BR for the noisy DCSK signal under the influence of a DSSS signal. Note that for fixed DSSS signal ower P B and noise ower sectral density N 0=2, the BR can be reduced by making similar adjustments as suggested in Section III-A. xamle: Consider the case where the logistic ma is used for generating the chaotic signal samles. We substitute (27) and (28) into (47) to obtain the BR of the DCSK system, i.e., BR DCSK = 2 erfc 8P 8P N 2N 8P 4N IV. COMPUTR SIMULATIONS AND DISCUSSIONS : (49) In this section, the erformance of the chaos-based digital communication systems under the influence of a DSSS signal is studied by comuter simulations. The logistic ma described in Section III-A has been used to generate the chaotic signal samles. For comarison, we also lot in each case, the analytical BRs obtained from the exressions derived in Sections III-A and B. The relevant simulated BRs for the CSK system and DCSK systems are shown in Figs. 5 and 6. From these figures, the simulated erformance is found to be better than that from the analysis. The discreancy is due to the limited validity of the assumtion of a normal distribution for the correlator outut in the analysis [2]. For large sreading factors (e.g., 2 = 00and 200), where the assumtion of normal distribution of the conditional correlator outut holds better, we clearly see that the analytical and simulated BRs are in very good agreement. Also, at low sreading factors, the BR erformance is worse. This can be attributed to the larger variation of bit energy sent for each symbol. As a general observation, the coherent CSK system consistently erforms better than the noncoherent DCSK system under the influence of a coexisting conventional DSSS signal. As shown in Fig. 7, for the CSK system, at b =N 0 = 7 db and a sreading factor of 00, the BR degrades from about 0 03 to 0 02 when the conventional-to-chaotic-signal-ower ratio (P B=P s) increases from 0 db to 0 db. For the DCSK system, with the same increase in P B =P s, the BR now degrades from about 0 03 to 0.3 at b =N 0 =20dB. Thus, the CSK system is more tolerant to wideband interfering signals comared with the DCSK system.
I TRANSACTIONS ON CIRCUITS AND SYTMS I: FUNDAMNTAL THORY AND APPLICATIONS, VOL. 50, NO., NOVMBR 2003 48 Fig. 7. Simulated BRs versus for the coherent CSK and noncoherent DCSK systems under the interference of a DSSS signal. (a) Sreading factor is 00. (b) Sreading factor is 200. V. CONCLUSION In this brief, the erformance of chaos-based communication systems under the influence of a wideband signal generated from a coexisting conventional sread-sectrum system is investigated. The roblem is imortant technically since chaos-based systems are sread-sectrum systems which are exected to resist interfering and the kind of interference considered here, namely one being generated from another conventional sread-sectrum system such as a DSSS system, reresents a realistic (future) ractical concern when chaosbased systems need to cooerate with existing systems. For the coexisting CSK-DSSS system, coherent correlation CSK receiver has been assumed. 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