AN APPROACH TO CALCULATE THE BIT ERROR RATES OF MULTIPLE ACCESS CHAOTIC-SEQUENCE SPREAD-SPECTRUM COMMUNICATION SYSTEMS EMPLOYING MULTI-USER DETECTORS

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1 Internationa Journa of Bifurcation and Chaos, Vo. 4, No c Word Scientific Pubishing Company AN APPROACH TO CALCULATE THE BIT ERROR RATES OF MULTIPLE ACCESS CHAOTIC-SEQUENCE SPREAD-SPECTRUM COMMUNICATION SYSTEMS EMPLOYING MULTI-USER DETECTORS WAI MAN TAM, FRANCIS C. M. LAU and CHI K. TSE Department of Eectronic and Information Engineering, The Hong Kong Poytechnic University, Hong Kong, China tamwm@eie.poyu.edu.h encmau@poyu.edu.h enctse@poyu.edu.h Received September, ; Revised October, In a mutipe access spread-spectrum communication system, the bit error performance is usuay imited by the mutua interference between users. The use of muti-user detection is an effective method to reduce such interference. In this paper, we consider two inear muti-user detectors, namey, decorreating detector and minimum mean-square-error detector, appied to a mutipe access chaotic-sequence spread-spectrum communication system. An approach to cacuate the bit error rate BER for such systems is presented. The cacuated BERs are compared with those obtained using brute-force computer simuations which give the true system performance. Keywords: Chaos-based communications; mutipe access; muti-user detection; spread-spectrum communications.. Introduction A number of chaos-based communication systems have been proposed in the ast decade. They can be broady categorized into three main types, namey, chaos-based anaog moduation Kocarev et a., 99], chaos-based digita moduation Dedieu et a., 993; Koumbán et a., 998; Paritz et a., 99] and chaotic-sequence spread-spectrum digita moduation Heidari-Bateni & McGiem, 994; Paritz & Ergezinger, 994]. Being spread-spectrum systems, chaos-based communication systems have the capabiity to accommodate mutipe users. Mutipe access capabiities of digita moduation systems are investigated in Koumbán et a., 997; Jáó et a., ; Tam et a., ]. Aso, aperiodic chaotic-sequence spread-spectrum systems with mutipe users are studied by Yang and Chua 997, 998]. The system under study in this paper is a mutipe access chaotic-sequence spread-spectrum MA-CSSS system, which is essentiay a conventiona direct-sequence code-division mutipe access DS-CDMA system Peterson et a., 995] with chaotic sequences serving as the spreading codes. Different aperiodic chaotic sequences from chaotic circuits or discrete maps with different conditions are assigned to different users. Since a users are using the same spectrum at the same time, the different chaotic sequences are interfering one another. Because of the nonzero cross-correations of the chaotic sequences, mutipe access interference MAI aways exists to imit the performance of 83

2 84 W. M. Tam et a. the system when the users symbos are decoded independenty. Muti-user detection has been shown effective to combat MAI in conventiona DS-CDMA systems. The operating principe of a muti-user detector is, based on the avaiabe spreading codes, to cance the interference between users, and jointy decode the transmitted symbos. The optimum muti-user detector was first proposed based on the Viterbi agorithm Verdú, 986]. The decoding is performed so as to maximize the joint posterior distribution according to the maximum-ieihood criterion. However, the optimum muti-user receiver is not empoyed in practice because the compexity increases exponentiay with the number of users. As a resut, a cass of suboptimum detectors have been used. Suboptimum muti-user detectors can be broady cassified into inear detectors decorreating detector and minimum mean-square-error detector and subtractive interference detectors successive interference canceation detector, parae interference canceation detector and zero-forcing decision-feedbac detector. For the inear detectors, a inear transformation is performed at the receiver to mitigate the MAI. For the decorreating detector, the interference can be competey eiminated Lupas & Verdú, 989, 99]. However, the noise component of the remaining signa increases, especiay for a arge oading. In contrast to the decorreating detector, the minimum mean-squareerror detector aows residua interference without causing a significant increase in noise eve Xie et a., 99]. The appication of muti-user detection techniques to MA-CSSS systems, though an important topic, has not been studied unti recenty. In the wor of Argüeo et a. ], two subtractive interference detectors appied to a MA-CSSS system are studied by computer simuations. Their studies iustrate that muti-user detection techniques can substantiay improve the bit error rate BER of the MA-CSSS system. In this paper, we deveop anaytica techniques to cacuate the BERs of the inear muti-user detectors in a MA-CSSS system. In Sec., we present an overview of the MA-CSSS system and the operations of the inear detectors. In Sec. 3, we deveop techniques to evauate the BERs of the muti-user detectors. Brute-force simuation resuts are presented and compared with the anaytica resuts.. System Description.. Transmitter structure Consider the N-user mutipe access chaoticsequence spread-spectrum MA-CSSS communication system shown in Fig.. Define d i {, +} Fig.. Mutipe access chaotic-sequence spread-spectrum communication system with a inear muti-user detector.

3 An Approach to Cacuate the Bit Error Rates 85 as the th transmitted symbo for user i which is assumed to be equiprobabe for + and. Denote the chaotic sequence generated by the ith generator by {x i }, which is used to spread the binary symbo sequence d i. Assuming a spreading factor of γ, the transmitted signa for user i at time γ +, γ +,..., γ can be expressed as s i di x i. The overa transmitted signa of the system at time is thus given by s s i. Note that the chaotic sequences are independent of one another because they are derived from different generators. Aso, the mean vaue of each chaotic sequence is zero in order to avoid transmitting any non-information-bearing dc components. In other words, Ex i ] i,,..., N. 3 where Eψ] denotes the mean vaue of ψ... Singe-user and muti-user receivers We assume a simpe additive white Gaussian noise channe. The received signa is given by r s i + ξ 4 where ξ is an additive white Gaussian noise sampe with zero mean and variance N /. Assume that the chaotic spreading sequences are reproduced exacty at the receiver. As shown in Fig., the output of the jth correator j,,..., N, denoted by y j, equas y j γ+ γ+ d i r x j N γ+ s i + ξ x i xj + x j γ+ ξ x j. 5 Define the correation matrix R as where R,, R,, R,N, R,, R,, R,N, R R N,, R N,, R N,N, R j,i, γ+ 6 x i xj 7 denotes the correation mutua interference between the ith and jth users during the th symbo duration. Note that R is a symmetric matrix. The outputs of the correators can now be expressed as where y R d + n 8 y y y y N ] T 9 d d d d N ] T n n n n N ] T n j γ+ and T denotes matrix transposition. ξ x j... Conventiona singe-user detector For the conventiona singe-user detector, the decoded symbos, denoted by ˆd, are computed from ˆd ˆd ˆd ˆdN ] T sgny ] 3 where sgn.] represents the sign function. In other words, if y j >, + is detected for the jth user, otherwise, is decoded. Note that in a MA- CSSS communication system, the correation matrix R changes from bit to bit. Therefore, R needs to be recacuated for each bit duration.... Decorreating detector For a given d, we assume that y is Gaussian with mean R d and covariance matrix N /R Xie et a., 99]. The N-dimensiona probabiity

4 86 W. M. Tam et a. density function is then given by Proais, 995] py d πn N detr ] exp ] y R d T R N y R d 4 where det.] represents the determinant operator. The optimum soution which maximizes the probabiity of a correct decision is equivaent to minimizing the ieihood function Ωd y R d T R y R d. 5 From 5, we obtain the optimum soution where d d d R y 6 d dn ] T. 7 The decoded symbos of a decorreating detector can be evauated using ˆd sgn d ] Minimum mean-square-error MMSE detector In a communication system, both the noise and the interference components can degrade the bit error performance. The decorreating detector described in the previous section can eiminate the mutua interference between users, but the price to pay is an increase in the number of noise components, which is proportiona to the number of users. The overa system performance is imited by the resuting noise eve. A minimum mean-square-error MMSE detector taes the effects of both muti-user interference and noise into consideration during the detection process. The main idea is to find a mapping, represented by H, that minimizes the mean-squareerror, denoted by Jd, between the transmitted symbos d and the estimated d represented by d. The equation ining H and Jd is given by where Jd Ed d T d d ] 9 d H y. The mapping that minimizes the mean-square-error is readiy shown equa to Xie et a., 99] H R + N I where I is the identity matrix and H is a square matrix defined as H,, H,, H,N, H,, H,, H,N, H H N,, H N,, H N,N, The symbos are then decoded according to ˆd sgn d ]. 3 In genera, the symbos decoded using the inear muti-user detector shown in Fig. can be expressed as ˆd sgnγ y ] 4 where Γ is the inear mapping. For the conventiona singe-user, decorreating and MMSE detectors, Γ is represented by I, R and R + N I, respectivey. In a noiseess condition, the MMSE detector is equivaent to the decorreating detector DD. When the noise eve is arge, the DD and MMSE detectors have simiar performance as the conventiona singe-user detector. 3. Performance Anaysis In this section, we deveop techniques to evauate the bit error rate BER of the aforementioned muti-user systems. It is assumed that a users use the same chaotic generator but each uses a different set of initia conditions. We further assume that a users have identica BERs because they receive simiar interference. Without oss of generaity, we consider the first user in a two-user system and the jth user in an N-user system, and we derive the probabiity of error for the first symbo. For brevity, the subscripts of the variabes d j n j, R and H are omitted., dj 3.. Anaysis of the decorreating detector 3... Two-user system, ˆdj, y j, We consider the first user. It can be readiy shown that the inverse of the correation matrix R equas ] R R, R, R, R, R,. 5 R, R,

5 An Approach to Cacuate the Bit Error Rates 87 The outputs of the correators for the first and second users are given, respectivey, by y r x and y d R, + d R, + n 6 r x d R, + d R, + n. 7 The estimated d is then obtained by premutipying y by R, i.e. d R y 8 and the input to the first threshod detector equas d R, R, R, d R, R, R, Define + R, n R, n ]. 9 Q R, R, R, 3 Q R, n R, n 3 Q d R, R, R, + R, n R, n d Q + Q. 3 Using the Schwarz s inequaity and 7, it is readiy shown that Q >. Suppose a + is sent by the first user, i.e. d +. The mean and variance of Q d + can be cacuated, respectivey, by and EQ d +] EQ ] ER, ]ER, ] ER,] 33 varq d +] varq ] + varq ] + covq, Q ] 34 where varψ] denotes the variance of ψ and cova, B] represents the covariance of A and B defined as cova, B] EAB] EA]EB]. 35 It is aso readiy shown that the covariance term in 34 equas zero. Thus, 34 can be simpified to varq d +] varq ] + varq ] varr, ]varr, ] + E R, ]varr, ] + E R, ]varr, ] + varr,] ER, R, R,] ER, ]ER, ]ER,] + N ER,]ER, ] ER, R,]. 36 Since Q >, we can decode the symbo based ony on the sign of Q, i.e. when Q, is decoded, otherwise, + is recovered. Furthermore, Q d + is the sum of a arge number of random variabes. We may therefore assume that it is normay distributed. Given + has been sent, an error occurs when Q and the probabiity is given by Prob d d + ProbQ d + erfc EQ d +] varq d +] 37 where the compementary error function, erfc., is defined as erfcψ exp t dt. 38 π ψ Liewise, it can be shown that when a is sent, i.e. d, the mean and variance of Q d equa, respectivey, and EQ d ] EQ d +] 39 varq d ] varq d +]. 4 Assuming that Q d is norma, the conditiona error probabiity is evauated from Prob d > d ProbQ > d erfc EQ d ]. 4 varq d ] Schwarz s inequaity states that for two sets of M numbers given by {a m; m,,..., M} and {b m; m,,..., M}, M m a m M m b m M m a mb m. Strict inequaity hods uness a m/a n b m/b n for a vaues of m and n Briggs et a., 985].

6 88 W. M. Tam et a. The overa BER for the first user, denoted by BER, can be evauated by using 37 through 4, i.e. BER Prob d d + + Prob d > d ProbQ d + + ProbQ > d erfc EQ d +]. 4 varq d +] The vaues of the means and variances ER j,j ], ER j,j ], ER, ], ER, R,], ER, R, R, ], varr, ] and varr j,j] can be obtained by numerica simuation. The vaues are then substituted into 33, 36 and 4 to evauate the BER. We refer to this method as mixed anaysis-simuation technique. Chaotic sequences generated by the cu- Exampe. bic map Suppose a users use the cubic map x + 4x 3 3x 43 to generate the chaotic sequences and each user uses a different initia condition. The invariant probabiity density function of {x }, denoted by ρx, is given by Kohda & Tsuneda, 994] ρx π if x < x 44 otherwise and the chaotic sequences {x j ; j,,..., N} possess the foowing statistica properties see Appendix A: Ex j x j Ex j Ex j ] 45 m ] Ex j ]Ex j m ] for m 46 xj m ] for m 47 Ex j xj m x u j ] for < m < u 48 Ex j xj m x u j x j v ] for < m < u < v 49 Ex j x m j ] for m 5 for m + and m Ex j 3 x m j ] 8 for m + 5 Ex j x m j x u j ] for nonequa integers, m and u 5 Denote the average transmission power by P s where Aso, define P s Ex ] Ex ] Ex N ]. 53 Λ Ex 4 ] P s Ex 4 ] P s ExN 4 ] P s 54 which is a constant dependent ony upon the type of chaotic sequence but not on the average power. Using 45 through 54, it is readiy shown that the mean and variance of Q d + are given, respectivey, by and varq d +] EQ d +] γγ P s 55 γγ E x 4 ] + γ Ex 4 ]P s + γ + 5P 4 s ] + γγ N Ex 4 ]P s + γ P 3 s ]. 56 Substituting 55 and 56 into 4, the anaytica BER for the first user equas BER Λ erfc γγ + 4γ Λ γγ + γ+5 γγ Λ + γ + Eb N ] 57 where E b represents the average bit energy defined as E b γp s. 58

7 3... N-user system We define the inverse of the correation matrix R as P, P, P,N R P, P, P,N P P N, P N, P N,N For the jth user, using 8, 6 and 59, the estimated vaue of d j, denoted by d j, is found equa to d j d j + P j,i n i. 6 Based on the estimated d j, the symbo is decoded according to the foowing rue: { + ˆd j dj > 6 dj. For a given d j {, +}, the mean vaue of d j is given by E d j d j ] d j 6 and the variance equas Lupas & Verdú, 989; Yoon & Ness, ] see Appendix B var d j d j ] N ] N var P j,i n i E P j,i n i N EP j,j] N ] E adjr]j,j. 63 detr] where adjr] j,j represents the j, jth eement of the matrix adjr], and adjr] denotes the adjoint of R, i.e. adjr] detr]r. It shoud be noted that because R is independent of d j, so is var d j d j ]. Assuming that for a given d j, d j is norma. The BER of the jth user can thus be expressed as BER j Prob d j d j + Probd j + + Prob d j > d j Probd j Prob d j d j + + Prob d j > d j Since An Approach to Cacuate the Bit Error Rates 89 4 erfc E d j d j +] var d j d j +] + 4 erfc E d j d j ]. var d 64 j d j ] E d j d j +] E d j d j ] 65 var d j d j +] var d j d j ], 66 the BER equation can be rewritten as BER j erfc E d j d j +] var d j d j +] erfc. 67 N EP j,j ] The term EP j,j ] cannot be evauated anayticay and has to be found by numerica simuation. Then, 67 can be used to cacuate the BER vaue mixed anaysis-simuation technique. 3.. Anaysis of MMSE detector 3... Two-user system From, it can be shown that the map H minimizing the mean-squared error is equa to H 4R, R, 4R, + N R, + N R, + N Define ] R, + N 4R,. 68 4R, R, + N G 4R, R, 4R, + N R, + N R, + N. 69 Using the Schwarz s inequaity, it can be shown that R, R, R, > and hence G >. The estimated d is given by d Hy. 7

8 9 W. M. Tam et a. For the first user, the estimated d is computed from d G R, + N d R, + d R, + n 4R, d R, + d R, + n ] G 4d R, R, R, + 4R, n R, n + N d R, + d R, + N n ] Z G. 7 where Z 4d R, R, R, + 4R, n R, n + N d R, + d R, + N n. 7 d The decoded symbo ˆd is given by the sign of which in turn equas the sign of Z. For a given d d d ] T, the mean vaue of Z d equas EZ d] E4d R, R, R, + N d R, ] 4d ER, ]ER, ] 4d ER, ] + d N ER, ]. 73 Define C N R, 74 The variance of Z d is given by C N R, 75 F N n. 76 varz d]var4d Q +4Q +d C +d C +F ] 6 varq]+varc ]+varc ]+varf ] +8 covq, C ]+8d d covq, C ] +8d covq, F ]+8d covq, C ] +8d covq, C ]+8 covq, F ] +d d covc, C ]+d covc, F ] +d covc, F ] 77 where Q, Q and Q are defined as in 3 through 3. It can aso be shown see Appendix C that covq, F ] 78 covq, C ] 79 covq, C ] 8 covc, F ] 8 covc, F ] 8 covc, C ]. 83 Thus the variance of Z d can be simpified to varz d] 6varQ] + 4N varr, ] + 4N varr, ] + 4N varn ] + 6N covr, R, R,, R, ] + 6d d N covr, R, R,, R, ] + 6N covr, n R, n, n ] 6 varq] + 4N varr, ] + 4N varr, ] + N 3 ER, ] + 6N varr, ]ER, ] ER, R,] + ER, ]ER,] +6d d N ER, R, R, ] ER, 3 ] + 8N ER,]ER, ] ER, ]. 84 Since Z d is the sum of a arge number of random variabes, we can assume that it is normay distributed. The bit error rate for the first user is thus given by BER Prob d d + + Prob d > d ProbZ d + + ProbZ > d 4 ProbZ d +, d ProbZ d +, d + 4 ProbZ > d, d ProbZ > d, d

9 4 erfc EZ d +, d +] varz d +, d +] + erfc An Approach to Cacuate the Bit Error Rates 9 EZ d +, d ] varz d +, d ] ] erfc EZ d, d +] + varz d, d +] erfc EZ d, d ]. 85 varz d, d ] Simiar to the case of the decorreating detector, the vaues of the means and variances in 73 and 84 can be obtained by numerica simuations and the BER can be cacuated by the mixed anaysissimuation technique. Chaotic sequences generated by the cu- Exampe. bic map Suppose a users use the cubic map defined in 43 to generate the chaotic sequences. Given a + is sent by the first user, i.e. d +, and appying 45 through 54, the mean and variance of Z d can be shown equa to EZ d +, d ] 4γγ P s + γn P s 86 varz d +, d ] 6γγ E x 4 ] + γ Ex 4 ]P s + γ + 5Ps 4 ] + 8γγ N 3Ex 4 ]P s + γ 3P 3 s ] + 4γN Ex 4 ] + γ P s ] + γn 3 P s. 87 It can be observed that both the mean and variance of Z d do not depend on d. In other words, EZ d +, d ] EZ d +] 88 varz d +, d ] varz d +]. 89 It is aso readiy shown that when a is sent by the first user, the mean and variance are, respectivey, and EZ d, d ] EZ d ] EZ d +] 9 varz d, d ] varz d ] varz d +]. 9 Substituting 86 through 9 into 85, the anaytica BER for the first user is simpified to BER ProbZ d + ] erfc γγ +γ Eb N 8γγ Λ +γ Λ+ γ+5 +4γ γ Eb N 3Λ+γ 3 +γ 3 Eb Λ+γ N +γ 4 Eb N 3... N-user system 3 ]. 9 Based on 8, and to, the estimated vaue of d for user j can be shown equa to d j H j,i z i N H j,i R i,u d u + n i u d u u N From, we aso have H j,i R i,u + H j,i n i. 93 H j,i R i,u + N δ iu δ ju 94 where δ ju denotes the Kronecer s deta defined as { when j u δ ju 95 when j u.

10 9 W. M. Tam et a. Putting 94 into 93, we obtain d j d j N d i H j,i + H j,i n i. 96 Given a fixed transmitted symbo vector d, the mean and variance of dj are given, respectivey, by see Appendix C and E d j d] E d j N d j N ] d i H j,i d i EH j,i ]. 97 var d j d] N N ] 4 var N ] d i H j,i + var H j,i n i + cov N d i H j,i, ] H j,i n i. 98 It is readiy shown that the covariance term in 98 equas zero see Appendix C. Thus, the variance of d j d is simpified to var d j d] N N varh j,i ] 4 + u,u i + N EH j,j] N 4 N EH j,j] N 4 + N 4 covd i d u H j,i H j,u ] u,u i EH j,i] E H j,i ] d i d u covh j,i H j,u ]. 99 Assuming that d j d is norma, the BER for user j is given by BER j d Prob d j d j +, d Probd j + Prob d + Prob d j > d j, d Probd j Prob d] N Prob d j d j +, d d + Prob d j > d j, d] where the vector d denotes d d d d j d j+ d N ] T. Simiar to the case of the decorreating detector, the mean and covariance of the terms in 99 are obtained by simuation. Then they are substituted into to obtain the BER of the users. 4. Resuts and Discussions In a simuations, we assume that the cubic map defined in 43 is used by a users and each user uses a different initia condition. With the cubic map, it has been cacuated in Appendix A that Ex ].5 and Ex4 ].375. The vaue of Λ in 54 thus equas Λ Three types of resuts are reported for the case of a two-user system. They are brute-force BF simuation resuts obtained by simuating the actua system; mixed anaysis-simuation MAS resuts obtained by simuating the statistica vaues mean, variance, covariance, etc. and substituting the vaues into the anaytica BER equations; and anaytica soutions found from cacuating the statistica vaues mean, variance, covariance, etc. and substituting the vaues into the anaytica BER equations. When there are more than two users, ony the first two types of resuts are avaiabe. For comparison, the BER for a singe-user system interference free case is aso given as a reference. 4.. Decorreating detector DD 4... Two users Figure pots the BER vaues for spreading factors of and when a decorreating detector is

11 An Approach to Cacuate the Bit Error Rates 93 a b Fig.. Anaytica, mixed anaysis-simuation MAS and brute-force simuation BF BERs versus E b /N for a MA-CSSS communication system using a decorreating detector. a γ ; b γ.

12 94 W. M. Tam et a. used. The anaytica BER is obtained by substituting into 57 with appropriate E b /N vaues. Here, we ceary see that the BF simuation resuts, the MAS resuts and the anaytica soutions agree we with one another. When γ, a curves approach the singe-user performance bound N users Figure 3 pots the BF simuation resuts and the MAS resuts for four-user, eight-user and ten-user systems. Spreading factors of, 5, and are used. When the spreading factor increases, the BF simuation resuts and the MAS resuts get coser. This is because the assumption of norma distribution made in the MAS scheme becomes more reaistic as the spreading factor increases. In Fig. 4, the simuated BERs are potted for γ, 5, and when N. It is observed that the BER reduces as the spreading factor increases. This is because when the spreading factor increases, the variation in the received bit energy estimation probem quoted in Koumbán et a., 998] reduces, causing ess degradation to the system performance. a b c d Fig. 3. Mixed anaysis-simuation MAS and brute-force simuation BF BERs versus E b /N for a MA-CSSS communication system using a decorreating detector. a γ ; b γ 5; c γ ; d γ.

13 An Approach to Cacuate the Bit Error Rates 95 Fig. 4. Brute-force simuation BER versus E b /N for a MA-CSSS communication system using a decorreating detector. Number of users is. a b Fig. 5. Anaytica, mixed anaysis-simuation MAS and brute-force simuation BF BERs versus E b /N for a MA-CSSS communication system using an MMSE detector. a γ ; b γ. 4.. MMSE detector 4... Two users Figure 5 pots the BER vaues for spreading factors of and when an MMSE detector is used. The anaytica BER is found by substituting into 9. It can be observed that the BF simuation resuts, the MAS resuts and the anaytica soutions agree we with one another N users In Fig. 6, the BER curves for the BF simuation

14 96 W. M. Tam et a. a b c d Fig. 6. Mixed anaysis-simuation MAS and brute-force simuation BF BERs versus E b /N for a MA-CSSS communication system using an MMSE detector. a γ ; b γ 5; c γ ; d γ. resuts and the MAS resuts are shown for four-user, eight-user and ten-user systems. Spreading factors of, 5, and are used. The BER curves ceary indicate that the two sets of resuts are consistent. In Fig. 7, the simuated BERs are potted for γ, 5, and when N. As in the case of the decorreating detector, the BER reduces as the spreading factor increases. This is aso due to the reduction of the variation in the received bit energy as the spreading factor increases Comparison of conventiona, decorreating and MMSE detectors Figure 8 pots the simuated BER versus E b /N for different number of users when the conventiona singe-user, decorreating and MMSE detectors are used. From Fig. 8a, it can be observed that the decorreating and MMSE detectors achieve simiar performance. For a ow E b /N, say beow 5 db, the decorreating and MMSE detectors are sighty

15 An Approach to Cacuate the Bit Error Rates 97 Fig. 7. Brute-force simuation BER versus E b /N for a MA-CSSS communication system using an MMSE detector. Number of users is. a b Fig. 8. Brute-force simuation BER versus E b /N for a MA-CSSS communication system using conventiona singe-user conventiona, decorreating DD and MMSE detectors when γ. a N,, 4, ; b N 3. better than the conventiona singe-user detector. With a high E b /N, say arger than 5 db, the BER improvement of the mutiuser detectors over the singe-user detector is substantia. In particuar, for E b /N vaues arger than 6 db, the BERs for the decorreating and MMSE detectors with users are ower than those for the conventiona singe-user detector with ony two users. For N 3, the resuts are shown in Fig. 8b. Here, we observe that the MMSE detector sighty outperforms the DD. When the E b /N increases, however, the difference becomes smaer.

16 98 W. M. Tam et a. 5. Concusions In this paper, two muti-user detectors, namey, decorreating detector and minimum mean-squareerror detector, have been appied to a mutipe access spread-spectrum communication system based on chaotic sequences. Techniques for cacuating the bit error rate BER of the muti-user detectors under an additive white Gaussian noise environment have been presented. Resuts show that the mutiuser detectors can substantiay improve the BER performance of the mutipe access chaotic-sequence spread-spectrum system. Acnowedgments This wor was supported by a competitive earmared research grant PoyU 537/E funded by Hong Kong Research Grants Counci and aso by a Hong Kong Poytechnic University Research Grant. References Argüeo, F., Bugao, M. & Amor, M. ] Muti-user receivers for spread spectrum communications based on chaotic sequences, Int. J. Bifurcation and Chaos, Briggs, W., Bryan, G. H. & Waer, G. 985] The Tutoria Agebra, Vo. Hong Kong University Press, Hong Kong. Dedieu, H., Kennedy, M. P. & Haser, M. 993] Chaos shift eying: Moduation and demoduation of a chaotic carrier using sef-synchronizing Chua s circuit, IEEE Trans. Circuits Syst.-II 4, Heidari-Bateni, G. & McGiem, C. D. 994] A chaotic direct-sequence spread-spectrum communication system, IEEE Trans. Commun. 4, Jáó, Z., Kis, G. & Koumbán, G. ] Mutipe access capabiity of the FM-DCSK chaotic communications system, Proc. Int. Worshop on Noninear Dynamics of Eectronic Systems, Catania, Itay, pp Kocarev, L., Hae, K. S., Ecert, K., Chua, L. O. & Paritz, U. 99] Experimenta demonstration of secure communications via chaotic synchronization, Int. J. Bifurcation and Chaos, Kohda, T. & Tsuneda, A. 994] Even- and oddcorreation functions of chaotic Chebychev bit sequences for CDMA, Proc. IEEE Int. Symp. Spread Spectrum Technoogy and Appications, Ouu, Finand, pp Koumbán, G., Kennedy, M. P. & Kis, G. 997] Mutieve differentia chaos shift eying, Proc. Int. Worshop on Noninear Dynamics of Eectronic Systems, Moscow, Russia, pp Koumbán, G., Kennedy, M. P. & Chua, L. O. 998] The roe of synchronization in digita communications using chaos Part II: Chaotic moduation and chaotic synchronization, IEEE Trans. Circuits Syst.-I 45, 9 4. Lupas, R. & Verdú, S. 989] Linear muti-user detectors for synchronous code-division mutipe-access channes, IEEE Trans. Inform. Th. 35, Lupas, R. & Verdú, S. 99] Near-far resistance of muti-user detectors in asynchronous channe, IEEE Trans. Commun. 35, Paritz, U., Chua, L. O., Kocarev, L., Hae, K. S. & Shang, A. 99] Transmission of digita signas by chaotic synchronization, Int. J. Bifurcation and Chaos, Paritz, U. & Ergezinger, S. 994] Robust communication based on chaotic spreading sequences, Phys. Lett. A88, Peterson, R. L., Ziemer, R. E. & Borth, D. F. 995] Introduction to Spread Spectrum Communications Prentice Ha, Engewood-Ciff. Proais, J. G. 995] Digita Communications McGraw- Hi, Singapore. Tam, W. M., Lau, F. C. M., Tse, C. K. & Yip, M. M. ] An approach to cacuating the bit error rate of a coherent chaos-shift-eying digita communication system under a noisy mutiuser environment, IEEE Trans. Circuits Syst.-I 49, 3. Verdú, S. 986] Minimum probabiity of error for asynchronous Gaussian mutipe-access channes, IEEE Trans. Inform. Th. IT-3, Xie, Z., Short, R. T. & Rushforth, C. K. 99] A famiy of suboptimum detectors for coherent muti-user communications, IEEE J. Seected Areas in Commun. 8, Yang, T. & Chua, L. O. 997] Chaotic digita codedivision mutipe access CDMA communication systems, Int. J. Bifurcation and Chaos 7, Yang, T. & Chua, L. O. 998] Error performance of chaotic digita code-division mutipe access CDMA systems, Int. J. Bifurcation and Chaos 8, Yoon, S. & Ness, Y. B. ] Performance anaysis for inear mutiuser detectors of randomy spread CDMA using Gaussian approximation, IEEE J. Seected Areas in Commun., Appendix A Statistica Properties of the Chaotic Sequences Generated by the Cubic Map In this appendix, we evauate the statistica properties of the chaotic sequences {x } generated by the cubic map x + gx 4x 3 3x. A.

17 An Approach to Cacuate the Bit Error Rates 99 The invariant probabiity density function of x, denoted by ρx, equas Kohda & Tsuneda, 994] ρx π if x < x A. otherwise. We first define g x ggx g 3 x gg x.. g n x gg n x. A.3 Maing the substitution x cos φ into A.3 and appying the formua 4 cos 3 υ 3 cos υ cos 3υ to g i cos φ i,,..., n repeatedy, we obtain gcos φ 4 cos 3 φ 3 cos φ cos 3φ g cos φ ggcos φ gcos 3φ 4 cos 3 3φ 3 cos3φ cos3 φ g 3 cos φ gg cos φ gcos3 φ 4 cos 3 3 φ 3 cos3 φ cos3 3 φ.. g n cos φ gg n cos φ gcos3 n φ 4 cos 3 3 n φ 3 cos3 n φ cos3 n φ. A.4 In the foowing, we derive the statistica properties of the chaotic sequences. A symbos are defined as in Secs. and 3.. Derivation of Ex ], Ex ], varx ] and varx ] Ex ] xρxdx x π x dx varx ] Ex ] E x ] Ex ] x ρxdx x π x dx varx ] x Ex ] ρxdx A.5 A.6 x π x dx 8 A.7. Derivation of Ex x m] We consider the case where m. Without oss of generaity, we assume m + n for some positive integer n. Ex x m] x g n x ρxdx x g n x π x dx. A.8 Maing the substitution x cos φ and appying A.4, A.8 becomes Ex x m] π π π π cos φg n cos φ sin φdφ π sin φ 4π cos φg n cos φ dφ cos φ cos 3 n φdφ + cos φ + cos 3 n φ dφ + cos φ + cos 3 n φ + cos3n + φ + cos3n φ dφ

18 W. M. Tam et a. φ + sin φ + 4π 3 n sin 3n φ + 43 n + sin3n + φ + 43 n sin3n φ ] π 4. A.9 Combining with A.6, we get Ex x m] Ex ]Ex m] m. A. 3. Derivation of Ex x m ] We consider the case where m. Assume m +n for some positive integer n. Thus, Ex x m ] equas Ex x m ] π π xg n xρxdx cos φg n cos φdφ cos φ cos3 n φdφ cos3n + φ π + cos3n φ dφ π 3 n + sin3n + φ + 3 n sin3n φ. ] π A. 4. Derivation of Ex x m x u ] We consider the case where < < m < u. Assume m + κ, u + κ + µ for some positive integers κ and µ. Therefore, Ex x m x u ] π π 4π xg κ xg κ+µ xρxdx cos φg κ cos φg κ+µ cos φdφ cos φ cos3 κ φ cos3 κ+µ φdφ cos3 κ κ+µ φ] + cos3 κ + 3 κ+µ φ] + cos3 κ + 3 κ+µ φ] + cos3 κ 3 κ+µ φ]dφ 4π 3 κ κ+µ sin3κ κ+µ φ] + 3 κ + 3 κ+µ sin3κ + 3 κ+µ φ] +. 3 κ + 3 κ+µ sin3κ + 3 κ+µ φ] + 3 κ 3 κ+µ sin3κ 3 κ+µ φ] ] π A.

19 An Approach to Cacuate the Bit Error Rates 5. Derivation of Ex x m x u x v ] We consider the case where < < m < u < v. Assume m + κ, u + κ + µ, v + κ + µ + θ for some positive integers κ, µ, θ. Thus, Ex x m x u x v ] xg κ xg κ+µ xg κ+µ+θ xρxdx. Maing the subsitution x cos φ and appying the formuae in A.4, the above equation becomes A.3 Ex x m x u x v ] π π 8π. cos φg κ cos φg κ+µ cos φg κ+µ+θ cos φdφ cos φ cos3 κ φ cos3 κ+µ φ cos3 κ+µ+θ φdφ cos3 κ κ+µ + 3 κ+µ+θ φ] + cos3 κ + 3 κ+µ 3 κ+µ+θ φ] + cos3 κ κ+µ 3 κ+µ+θ φ] + cos3 κ + 3 κ+µ + 3 κ+µ+θ φ] + cos3 κ + 3 κ+µ + 3 κ+µ+θ φ] + cos3 κ 3 κ+µ 3 κ+µ+θ φ] + cos3 κ + 3 κ+µ 3 κ+µ+θ φ] + cos3 κ 3 κ+µ + 3 κ+µ+θ φ]dφ A.4 6. Derivation of Ex x m] When m, Ex x m] Ex 3 ] x 3 ρxdx x 3 ρxdx A.5 because ρx is an even function whereas x 3 is odd. Next, we consider the case where > m. Assuming m + n for some positive integer n, we obtain Ex x m] g n x xρxdx. A.6 Within the integra in A.6, both ρx and g n x are even whie x is odd. Thus, it can be concuded that Ex x m ] for > m. Finay, for < m, assuming m + u for some positive integer u, we have Ex x +u] π π π π. x g u xρxdx cos φ cos3 u φdφ + cos φ cos3 u φdφ cos3 u φ + cos3u + φ + cos3u φdφ 3 u φ sin3u φ + 3 u + φ sin3u + φ + ] π 3 u φ sin3u φ A.7

20 W. M. Tam et a. 7. Derivation of Ex 3 x ] When m, Ex 3 x m] Ex 4 ] x 4 ρxdx x 4 ρxdx 3 8. A.8 Next, we consider the case where > m. Assuming m + n for some positive integer n. Maing the substitution x cos φ and appying A.4, we have Ex 3 x m] π π 4π g n x 3 xρxdx cos 3 3 n φ cos φdφ + cos 3 n φ cos3 n + φ + cos3 n φ dφ cos3 n + φ + cos3 n φ + cos3n+ + φ + cos3n φ + cos3n+ φ + ] cos3n + φ dφ 4π n + sin3n + φ n sin3n φ 3 n+ + sin3n+ + φ + 3 n+ sin3n+ φ For < m, we assume m + u for some positive integer u. Ex 3 x m] π π 4π x 3 g u xρxdx cos 3 φ cos3 u φdφ + cosφ cos3 u + φ + cos3 u φ dφ cos3 u + φ + cos3 u φ + cos3u + 3φ + cos3u φ + cos3m + φ + ] cos3u 3φ dφ 3 3 4π 3 u + sin3u + φ + 3 u sin3u φ + 3 u + 3 sin3u + 3φ for u 8 for u. ] π + 8π cos3 u 3φdφ ] π A.9 A.

21 An Approach to Cacuate the Bit Error Rates 3 Thus, we concude that 3 for m 8 Ex 3 x m] for + m 8 otherwise. A. 8. Derivation of Ex x mx u ] We first consider the case where < m < u. Assuming m + κ and u + κ + µ for some positive integer κ and µ, we have Ex x mx u ] π π 4π x g κ xg κ+µ xρxdx cos φ cos3 κ φ cos3 κ+µ φdφ + cos φ cos3 κ + 3 κ+µ φ] + cos3 κ 3 κ+µ φ] cos3 κ + 3 κ+µ φ] + cos3 κ 3 κ+µ φ] + cos + 3κ + 3 κ+µ φ] + cos 3κ 3 κ+µ φ] + cos + 3κ 3 κ+µ φ] + ] cos 3κ + 3 κ+µ φ] dφ dφ. A. Next, consider the case where > m > u. Assuming u + κ + µ and m u + µ for some positive integers κ and µ, we have Ex x mx u ] π π 4π g κ+µ x g µ xxρxdx. cos3 κ+µ φ cos3 µ φ cos φdφ + cos 3 κ+µ φ] cos3 µ + φ] + cos3 µ φ] cos3 µ + φ] + cos3 µ φ] + cos 3κ+µ + 3 µ + φ] + cos 3κ+µ 3 µ φ] + cos 3κ+µ + 3 µ φ] + ] cos 3κ+µ 3 µ + φ] dφ dφ. A.3 Using a iewise procedure, it is readiy shown that Ex x mx u ] for nonequa integers, m and u.

22 4 W. M. Tam et a. Appendix B Derivation of Mean Reevant to the Anaysis of the Decorreating Detector In this appendix, we derive the mean of N P j,in i which is required in anayzing an N-user system using decorreating detectors. A symbos are defined as in Secs. and 3. N N ] N E P j,i n i E P j,i n i P j,u n u u E Pj,in i + E E P j,i Pj,i Eξ ]E u,u i ξ x i + ξ xi + P j,i n i P j,u n u P j,i u,u i u,u i Pj,i x i + u, u i N E Pj,i R i,i + N E Pj,j R j,j + u,u j N E P j,j P j,j R j,j + + N E,i j P j,i u,u i P j,i P j,u P j,i P j,u P j,i P j,u R u,i P j,j P j,u R u,j + u,u j P j,i R i,i + ξ x i,i j P j,u R u,j u,u i P j,u m ξ xi xu x i xu P j,i R i,i + P j,u R u,i ξ m x u m,i j u,u i P j,i P j,u R u,i N EP j,j] B. where the ast equaity comes from the fact that see Eq. 59 P j,u R u,i δ ji u B. and δ ji denotes the Kronecer s deta as defined in 95. Appendix C Derivation of Covariances and Mean Reevant to the Anaysis of the MMSE Detector In this appendix, we evauate some covariance terms which are required in the anaysis of the MMSE detector. A symbos are defined as in Secs. and 3.

23 An Approach to Cacuate the Bit Error Rates 5. Using the fact that the mean vaue of the noise sampe is zero, i.e. Eξ ], it is readiy shown that the foowing covariances are zero. covr j,j R i,i R j,i, n j ] covr i,i n j R j,i n i, R j,j ] covr i,i n j R j,i n i, R j,i ] covr j,j, n j ] covr j,i, n j ] C. C. C.3 C.4 C.5. The covariance of R j,j R i,i R j,i and R j,j is derived beow. covr j,j R i,i R j,i, R j,j ] ER j,j R i,i R j,i R j,j] ER j,j R i,i R j,i ]ER j,j] ER j,j R i,i] ER j,i R j,j] ER j,j R i,i ]ER j,j ] + ER j,i]er j,j ] varr j,j ]ER i,i ] ER j,ir j,j ] + ER j,i]er j,j ] C.6 If a users use the cubic map defined as in 43, the covariance term is simpified to covr j,j R i,i R j,i, R j,j ] γγ Ex j 4 ]Ex j ] γγ E 3 x j ]. C.7 3. The covariance of R j,j R i,i R j,i and R j,i equas covr j,j R i,i R j,i, R j,i ] ER j,j R i,i R j,i R j,i] ER j,j R i,i R j,i ]ER j,i] ER j,j R i,i R j,i ] ER 3 j,i ]. C.8 If a users use the cubic map defined as in 43, the above equation equas zero, i.e. covr j,j R i,i R j,i, R j,i ]. C.9 4. The covariance of R i,i n j R j,i n i and n j is derived beow. covr i,i n j R j,i n i, n j ] ER i,i n j R j,i n i n j ] ER i,i n j R j,i n i ]En j ] ER i,i ]En j ] ER j,i n i n j ] C. If the cubic map defined as in 43 is used by a users, the covariance term is simpified to covr i,i n j R j,i n i, n j ] N γγ E x j ]. C. 5. The covariance of R j,j and R j,i is evauated beow. covr j,j, R j,i ] ER j,j R j,i ] ER j,j ]ER j,i ]. C. 6. The covariance of N H j,in i is equa to N di H j,i and N cov d i H j,i, ] H j,i n i N E d i H j,i N E d i H j,i N ] N ] N ] H j,i n i E d i H j,i E H j,i n i N H j,i ξ x i ] N ] N E d i H j,i E H j,i ξ x i ]. E d i H j,i H j,i x i ] Eξ ] E N ] N d i H j,i EH j,i x i ]Eξ ] C.3

24 6 W. M. Tam et a. 7. The mean vaue of N H j,in i is derived beow. N N ] N E H j,i n i E H j,i n i H j,u n u Since u E Hj,i ni + E E H j,i Hj,i Eξ ]E u,u i ξ x i + ξ xi + H j,i n i H j,u n u u,u i u,u i Hj,i x i + u,u i N E Hj,i R i,i + N E Hj,j R j,j + +,i j u,u i u,u j u,u i H j,i H j,u R u,i N E H j,j H j,j R j,j + + N E u,i j H j,i H j,i H j,i H j,u H j,i H j,u H j,i H j,u R u,i H j,j H j,u R u,j + u,u j H j,i R i,i +,i j H j,u R u,j u,u i ξ x i H j,u ξ xi xu x i xu H j,i R i,i H j,u R u,i ξ x u C.4 H j,u R u,i + N δ ui δ ji, C.5 where δ ji denotes the Kronecer s deta as defined in 95, Eq. C.4 is simpified to N E h j,i n i N H E j,j N ] H j,j + N ] N E H j,i H j,i,i j N EH j,j] N 4 EHj,i]. C.6