Analysis of Bit Error Rates for Multiple Access CSK and DCSK Communication Systems

Transcription

1 7 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY The cases of H () (z) and H () (z) are trivial because we can tae =in Step 4), and write H () (z)= H () (z)= (z ) + (z ) + 4 z + 5 z + z 8 6 z 8 z + + :9 z +: z + + :8 z +:79 : 8 In both cases, positive realizations of diension 6 can be given as follows: c = ( :675 :57 95) A = 64 :79 :7 b = 4 c = ( :54 :54 94) A = b = 64 :79 :7 5 : A positive realization of the original H(z) is then possible to construct as in [4] or [8]. V. COCLUSIO In this brief, we provided a general finite step procedure for checing the nonnegativity of the ipulse-response sequence of H(z), which answers an open proble raised in []. For priitive transfer functions a new ethod of positive realization was proposed by reducing the pole order of the doinant pole. RFRCS [] B. D. O. Anderson, M. Deistler, L. Farina, and L. Benvenuti, onnegative realization of a linear syste with nonnegative ipulse response, I Trans. Circuits Syst. I, vol. 4, pp. 4 4, Feb [] L. Benvenuti, L. Farina, and B. D. O. Anderson, Filtering through a cobination of positive filters, I Trans. Circuits Syst. I, vol. 46, pp. 4 44, Dec [] L. Benvenuti and L. Farina, An exaple of how positivity ay force realizations of large diensions, Syst. Contr. Lett., vol. 6, no. 4, pp. 6 66, 999. [4] L. Farina, On the existence of a positive realization, Syst. Contr. Lett., vol. 8, no. 4, pp. 9 6, 996. [5] L. Farina and S. Rinaldi, Positive Linear Systes: Theory and Applications. ew Yor: Wiley,. [6] K.-H. Foerster and B. agy, onnegative realizations of atrix transfer functions, Linear Algebra Appl., vol., no., pp. 7 9,. [7] C. Hadjicostis, Bounds on the size of inial nonnegative realizations for discrete-tie LTI systes, Syst. Contr. Lett., vol. 7, no., pp. 9 4, 999. [8] T. Kitano and H. Maeda, Positive realization of discrete-tie syste by geoetric approach, I Trans. Circuits Syst. I, vol. 45, pp. 8, Mar [9] Y. Ohta, H. Maeda, and S. Kodaa, Reachability, observability and realizability of continuous-tie positive systes, SIAM J. Control Opti., vol., no., pp. 7 8, 984. Analysis of Bit rror Rates for Multiple Access CSK and DCSK Counication Systes W. M. Ta, Francis C. M. Lau, and Chi K. Tse Abstract In this paper, ultiple-access techniques for use with chaos shift eying (CSK) and differential CSK under a noisy condition are analyzed using a discrete-tie approach. A ixed analysis-siulation technique is developed to calculate the bit error rates (BRs). When certain assuptions are ade, closed-for analytical expressions of the BRs are found. Index Ters Bit error rate (BR), chaos counication, chaos shift eying (CSK), differential chaos shift eying (DCSK), ultiple access. I. ITRODUCTIO Chaos-based spread-spectru counication techniques have eerged rapidly during the last decade, and uch research effort has been devoted to the developent of analog and digital chaos-based counication systes that can achieve perforance coparable to the existing counication systes. Typically, in a digital chaos-based counication syste, digital sybols are apped to nonperiodic chaotic basis functions [] [4]. Multiple access is an iportant requireent for spread-spectru counication. For chaos-based systes, only a few ultiple-access schees have been proposed. For instance, a ethod based on ultiplexing chaotic signals has been proposed by Carroll and Pecora [5], and soe chaos-based approaches for generating spreading codes have been applied to conventional CDMA systes [6] [8]. Furtherore, ultiple access using differential chaos shift eying (DCSK) has been introduced by Kennedy et al. [] and [9], and the ultiple-access capability of frequency-odulated DCSK (FM-DCSK) has been studied by Jáo et al. []. Recently, an alternative ultiple-access technique for use with DCSK has been proposed by Lau et al. []. Only a preliinary study of the perforance of this schee under a noise-free environent has been given. Manuscript received October, ; revised January 9,. This wor is supported in part by The Hong Kong Polytechnic University and in part by the Hong Kong Research Grants Council under Grant PolyU57/. This paper was recoended by Associate ditor G. Setti. The authors are with the Departent of lectronic and Inforation ngineering, The Hong Kong Polytechnic University, Hong Kong, China (eail: taw@eie.polyu.edu.h; enclau@polyu.edu.h; enctse@polyu.edu.h). Digital Object Identifier.9/TCSI /$7. I

2 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY 7 Fig.. MA-CSK counication syste. In this paper, we investigate in depth, using a discrete-tie approach, a CSK syste with ultiple users, as well as the ultiple-access DCSK (MA-DCSK) syste proposed by Lau et al. []. It is assued that the aplitude of the carrier is odulated by the chaotic signals (aplitude odulation) and we analyze the syste using the equivalent baseband odel [], []. A ixed analysis-siulation (MAS) technique is developed to calculate the bit error rates (BRs). When certain assuptions hold, analytical expressions for the BRs can also be obtained. In Sections II and III, the ultiple-access CSK (MA-CSK) and MA-DCSK systes are described and the corresponding BR equations are derived. Results found by the MAS ethod and the analytical ethod are copared with siulations in Section IV. II. AALYSIS OF MA-CSK SYSTMS A. Transitter Structure We consider a MA-CSK counication syste with users, as shown in Fig.. A discrete-tie approach will be adopted in the following analysis. In the transitter of the ith user, a pair of chaotic sequences, denoted by f^ g and fx(i) g, are generated by a chaotic ap with different initial conditions. We also assue that the ean value of each of the chaotic sequences is zero in order to avoid transitting any dc coponent which is a waste of power. Denote the th transitted sybol for the ith user by d (i) f; +g and assue that and + occur with equal probabilities for all users. Define the spreading factor as the nuber of chaotic saples used to transit one binary sybol. During the th bit duration, i.e., for tie =() +; ( ) +;...;, the output of the transitter of user i, v (i),is v (i) = ^x(i) if d (i) =+ if d (i) =. The overall transitted signal of the whole syste at tie, denoted by v, is derived by suing the signals of all users, i.e., v = () i= v(i). B. Receiver Structure The received signal, denoted by r, is given by r = v + where denotes the additive white Gaussian noise with zero ean and variance (power spectral density) =. Assue that synchronized versions of the chaotic signals f^x g and fx g can be reproduced at the receiver. The detection essentially involves correlating the incoing signal with the locally regenerated chaotic signals and sapling the outputs of the correlators at the end of each sybol duration. The th decoded sybol for the jth user, denoted by ~, is deterined according to following rule: ~ = + if z = if z = =()+ =()+ r ^x =()+ r ^x =()+ r x > r x. C. Derivation of BR Consider the jth user. Without loss of generality, we consider the probability of error for the first sybol. For brevity, the subscripts of the variables d, d ~, ^c, c, and z are oitted. If + is transitted for user j, i.e., = +and v = ^x, it is readily shown that the ean value of z j( = +) equals z =+ = ^x Also, the variance of z j( = +) is given by var z =+ = var ^x + ^x + = l= 6=l + ^x + cov = l= 6=l ^x ^x l + i= = l= i6=j 6=l i= i6=j ^ ^x(i) l ^ ^x ^x l + ^x ^x () : () ; ^x l : (4)

3 74 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY Liewise, we can find z j( = ) and confir that z = = z =+ (5) var z = = var z =+ : (6) Since z j( = +) and z j( = ) are the su of a large nuber of rando variables, we assue that both of the are norally distributed. The BR for user j can thus be coputed fro BR = Prob z =+ + Prob z > = = erfc z =+ var [z j( = +) ] (7) where erfc(:) is the copleentary error function []. ) MAS Technique: To find the BR, we can first attain the values of the ters (expected values, variances and covariances) on the right-hand side of () and () by nuerical siulations. Then, the values of [z j( = +)] and var[z j( = +)] can be coputed and substituted into (7) to get the BR. Since both nuerical siulation and analytical ethod are involved in obtaining the BR, we refer to it as the MAS technique. ) Analytical BRs: To ensure that all users are treated equally, we assue that all users transit with equal average power P s. Further, if both the autovariance of f(^x ) g and the autocorrelation of f^x g vanish, i.e., cov ^x ; ^x l the analytical BR for user j can be reduced to BR CSK-I = erfc 9 = ; 6= l (8) ^x ^x l = ; 6= l (9) ( ) + + b () Fig.. Transission schee for the MA-DCSK counication syste. Fig. shows an MA-DCSK counication syste in a discrete-tie ode. In the transitter of the ith user, a chaotic ap is used to generate a chaotic sequence f g with zero ean. The chaotic aps for different users are different in general. Assue that chaotic saples are sent in each slot (spreading factor = ). Consider the transitted signal of user i during the th tie slot. Denote the output of the transitter by y (i). If the slot is a reference saple slot, y (i) =. If the slot corresponds to a data saple slot sending a binary sybol +, y (i) = i. Otherwise, if the slot corresponds to a data saple slot sending a binary sybol,. Thus, for the th tie slot of user i, we define a(i) as y (i) = x(i) i a (i) = +; if y(i) ; if y (i) = x(i) l = x(i) l () where l = for reference saples, and l = i for data saples. Therefore, the transitted signal of user i during the th tie slot can be represented by where 9 = var[(^x ) ]=Ps = var[(^x ) ]= [(^x ) ], and b = P s represents the average bit energy. Finally, if all users eploy the sae ap that generate sequences satisfying (8) and (9), the values 9 will becoe identical and all users will have the sae BR. III. AALYSIS OF MA-DCSK SYSTMS A. Frae Structure of the Transitted Signal In a MA-DCSK syste, to avoid excessive interference, and hence, isdetection, the separation between the reference and data saples ust be different for different users. A ultiple-access schee has been proposed by Lau et al. [] where the separation between the reference and data saples differs for different users, as illustrated in Fig.. Suppose there are users in the syste. Define a superfrae as the iniu nuber of slots within which all users transit an integral nuber of fraes. Within one superfrae, denote the nuber of fraes sent by user i by f (i). Fro now on, we restrict our discussion to one superfrae which is assued to contain s slots, where () () s = f =( )f =... = f () = f (). y (i) =a(i) l ; =( ) +; ( ) +;...; : () The overall transitted signal at tie, denoted by y, equals y = i= y(i). B. Receiver Structure At tie, the received signal, r, is given by r = y +.For each user, the signal received during a reference saple slot will correlate with the signal at the corresponding data saple slot. Depending on whether the output is larger or saller than the threshold, a + or is decoded. In Fig., a bloc diagra of a DCSK correlator receiver is shown. Define the uth sub-state of the syste at the th slot by A ~ (u), i.e., ~A (u) = a () a ()... a (u) a (u+)... a () T : () Consider the jth user and the received signal during the th tie slot. Suppose the slot corresponds to a reference saple slot for user j, i.e., = +. These reference saples will correlate with the received a

4 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY 75 Fig.. MA-DCSK counication syste. saples j slots later, i.e., the (+j)th slot. The output of the correlator, denoted by z j( A ~ ; A ~ +j ;a =+;a +j ), is given by z = = ~A ; A ~ +j ;a =+;a +j =()+ u= y (u) u= v= =() u= =()+ v= =()+ =()+ + v= a (u) a (v) +j x(u) l a (u) x (u) l +j a (v) +j x(v) +jl y (v) +j + +j x (v) +jl +j (4) where l i =(i =; ;...;) for reference saples, and l i = i for data saples. The decoded sybol corresponding to this pair of tie slots, denoted by d ~, is deterined according to the following rule: ~ = +; if z ; if z ~A ; A ~ +j ;a =+;a +j > ~A ; A ~ +j ;a =+;a +j. (5) C. Derivation of BR Denote the binary sybol transitted by the jth user in the ( + j)th slot by d, i.e., a +j = d. Using a liewise procedure as in Section II, it is readily shown that the average BR for user j, BR, is (for details, refer to [4]) BR = s S ~A U ~A U where U f; +g and S is the set of slot nubers corresponding to the reference slots of user j over one superfrae. Define B( A ~ ; A ~ +j ) as the average probability of occurrence of ) over all reference slots, i.e., ( ~ A ; ~ A +j B ~ A ; ~ A +j = s S Prob ~ A ; ~ A +j : (7) Thus, (6) can be rewritten as BR = ~A U ~A U ~z erfc var ~z + ~z erfc var ~z =+ =+ = = B ~ A ; ~ A +j : (8) ) MAS Technique: As in the MA-CSK case, we find out the required expected values, variances and covariances by nuerical siulations. The values of [~z j = +], [~z j = ], var[~z j = +] and var[~z j = ] are then coputed and put into (8) to obtain the BR. ) Analytical BR: As in Section II-C, we assue that all users transit with equal average power P s. Assue that the chaotic sequence generated by user i, f g, satisfies the following conditions: ~z erfc var ~z + ~z erfc var ~z =+ =+ = = Prob ~ A ; ~ A +j (6) cov ; x (i) n = x (i) n n = ; 6= n (9) x(i) n = ; 6= n () x(i) + x(i) + x(i) + = 8 ; ; I + and <<<: ()

5 76 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY TABL I STATISTICAL PROPRTIS OF TH ORMALIZD CHAOTIC SQUCS Then, (8) can be readily siplified to BR DCSKI = erfc 9 ( + ) +4 b + b () Fig. 4. BR versus in a three-user MA-CSK syste. Cubic ap is used and =. where 9 = var[(x ) ]=P s = var[(x ) ]= [(x ) ] and b = P s denotes the average bit energy. In addition to the statistical properties given in (9) to (), if all users eploy the sae ap, the values of 9 are now identical and all users will have the sae bit error perforance. IV. RSULTS AD DISCUSSIOS The following aps are used in our siulations. 5) Logistic ap: x + =x 6) Cubic ap: x + =4x x 7) Bernoulli-shift ap: x + = :x +; when x < :x ; when x > : In the siulations, the chaotic sequences are noralized before transission. The statistical properties of the noralized sequences are shown in Table I. Based on the invariant distribution of the logistic ap and the cubic ap [5], it is readily shown that the chaotic sequences generated fro these two aps satisfy the statistical properties assued in (8), (9), and (9) to (). Also, due to the shortage of space, we only present the results in which all chaotic signals are generated by the sae ap but with different initial conditions. (For other related results, please refer to Lau and Tse [4].) A. MA-CSK Syste In the first siulation, the cubic ap is used with a spreading factor of. In Fig. 4, the BRs are plotted against b = for a three-user syste. It can be seen that both the MAS and the analytical results are in good agreeent with the brute-force (BF) siulated results. In Fig. 5, we present the BF siulated BRs against b = for the cases corresponding to different choices of chaotic aps. It is found that the chaotic sequences generated by the Bernoulli-shift ap produce a higher BR, while the BRs of the users using cubic ap and logistic ap are the sae. One reason is that the value of var[x ] is larger for the noralized chaotic sequences generated fro the Bernoulli-shift ap copared with that of the cubic ap and logistic ap (see Table I). B. MA-DCSK Syste We consider the results for a three-user MA-DCSK syste in which all chaotic sequences are generated fro the cubic ap with a spreading factor of ( = ). As shown in Fig. 6, all ethods give consistent results. ext, we copare the BF siulated BRs for different choices of chaotic aps. Fig. 7 shows the results when Fig. 5. Siulated BR versus in a three-user MA-CSK syste. Cubic ap, Bernoulli-shift ap and logistic ap are used, respectively. =. Fig. 6. BR versus in a three-user DCSK syste. Cubic ap is used and =. Bernoulli-shift ap, cubic ap and logistic ap are used, respectively. It can be observed that using Bernoulli-shift ap produces a higher BR copared with the other two cases. One possible reason is that the value of var[x ] is larger for the Bernoulli-shift

6 I TRASACTIOS O CIRCUITS AD SYSTMS I: FUDAMTAL THORY AD APPLICATIOS, VOL. 5, O. 5, MAY 77 Fig. 7. Siulated BR versus in a three-user DCSK syste. Bernoulli-shift ap, cubic ap and logistic ap are used, respectively. =. Fig. 8. Analytical BR versus spreading factor( ) under different in a three-user DCSK syste. Logistic ap is used. [] M. P. Kennedy, G. Kolubán, G. Kis, G. Jáo, and Z. Jáo, Recent advances in counicating with chaos, in Proc. ISACS 98, Montery, CA, 998, pp [] M. P. Kennedy and G. Kolubán, Digital counication using chaos, in Controlling Chaos and Bifurcation in ngineering Systes, G. Chen, d. Boca Raton, FL: CRC,, pp [4] G. Kis, Z. Jáo, M. P. Kennedy, and G. Kolubán, Chaotic counications without synchronization, in Proc. 6th I Conf. Telecounications, dinburgh, U.K., Mar. 998, pp [5] T. L. Carroll and L. M. Pecora, Using ultiple attractor chaotic systes for counication, Chaos, vol. 9, pp , 999. [6] G. Mazzini, G. Setti, and R. Rovatti, Chaotic coplex spreading sequences for asynchronous DS-CDMA. Part I: syste odeling and results, I Trans. Circuits Syst. I, vol. 44, pp , Oct [7], Chaotic coplex spreading sequences for asynchronous DS-CDMA. Part II: soe theoretical perforance bounds, I Trans. Circuits Syst. I, vol. 45, pp , Apr [8] T. Yang and L. O. Chua, Chaotic digital code-division ultiple access counication systes, Int. J. Bifurcation Chaos., vol. 7, pp , 997. [9] G. Kolubán, M. P. Kennedy, and G. Kis, Multilevel differential chaos shift eying, in Proc. 5th Int. Specialist Worshop on onlinear Dynaics of lectronics Systes DS 97, Moscow, Russia, June 997, pp [] Z. Jáo, G. Kis, and G. Kolubáan, Multiple access capability of the FM-DCSK chaotic counications syste, in Proc. DS, Catania, Italy,, pp [] F. C. M. Lau, M. M. Yip, C. K. Tse, and S. F. Hau, A ultiple access technique for differential chaos shift eying, I Trans. Circuits Syst. I, vol. 49, pp. 96 4, Jan.. [] M. Hasler and T. Schiing, Chaos counication over noisy channels, Int. J. Bifurcation Chaos, vol., pp ,. [] J. G. Proais and M. Salehi, Counications Systes ngineering. nglewood Cliffs: Prentice Hall, 994. [4] F. C. M. Lau and C. K. Tse, Chaos-Based Digital Counication Systes: Operation, Analysis and valuation, st ed. Heidelberg, J: Springer-Verlag,. [5] T. Kohda and A. Tsuneda, ven- and odd-correlation functions of chaotic Chebychev bit sequences for CDMA, in Proc. I Int. Syp. Spread Spectru Technology and Applications, Oulu, Finland, 994, pp [6] G. Kolubán et al.. The theory and ipleentation of a robust chaotic digital counications syste. Univ. California, San Diego, CA. [Online]. Available: ap. Finally, assuing the logistic ap is used,we plot the analytical BR against the spreading factor () under different b = for a three-user syste. Fig. 8 shows that when the spreading factor increases initially, the BR iproves. After the spreading factor has reached an optiu value, increasing the spreading factor further will deteriorate the BR. V. COCLUSIO In this paper, two ultiple-access techniques for use with CSK and DCSK under a noisy condition are analyzed thoroughly. An MAS technique and an analytical ethod are developed to derive the BRs. The techniques developed are applicable to other ultiple-access schees. Finally, the DCSK schee is nown to be suboptial in the aplitude odulated version. An extension of the present nuerical analysis to the FM-DCSK schee and the ultiple-access FM-DCSK schee based on Walsh functions [6] would also be worth studying. RFRCS [] M. P. Kennedy, Chaotic counications: fro chaotic synchronization to FM-DCSK, in Proc. 6th Int. Specialist Worshop on onlinear Dynaics of lectronics Systes (DS 98), Budapest, Hungary, July 998, pp. 4.