PERFORMANCE ANALYSIS OF MULTIPLE ACCESS CHAOTIC-SEQUENCE SPREAD-SPECTRUM COMMUNICATION SYSTEMS USING PARALLEL INTERFERENCE CANCELLATION RECEIVERS

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1 Internationa Journa of Bifurcation and Chaos, Vo. 14, No. 10 (2004) c Word Scientific Pubishing Company PERFORMANCE ANALYSIS OF MULTIPLE ACCESS CHAOTIC-SEQUENCE SPREAD-SPECTRUM COMMUNICATION SYSTEMS USING PARALLEL INTERFERENCE CANCELLATION RECEIVERS WAI M. TAM, FRANCIS C. M. LAU and CHI K. TSE Department of Eectronic and Information Engineering, The Hong Kong Poytechnic University, Hong Kong, P. R. China tamwm@eie.poyu.edu.h encmau@poyu.edu.h enctse@poyu.edu.h Received August 1, 2003; Revised September 22, 2003 In this Letter, we appy combined inear detector/parae interference canceation (PIC) detectors to jointy decode symbos in a mutipe access chaotic-sequence spread-spectrum communication system. In particuar, three different types of inear detectors, namey singe-user detector, decorreating detector and minimum mean-square-error detector, are used to estimate the transmitted symbos at the first stage of the PIC detector. The technique for deriving the approximate bit error rate (BER) is described and computer simuations are performed to verify the anaytica BERs. Keywords: Chaos-based communications; mutipe access; mutiuser detection; spread-spectrum communications; parae interference canceation. 1. Introduction Spread-spectrum digita moduation schemes based on chaotic sequence was first proposed by Heidari- Bateni and McGiem [1994], and Paritz and Ergezinger [1994]. By assigning different chaotic sequences to different users, mutipe access in chaotic-sequence spread-spectrum systems can be accompished [Yang & Chua, 1997, 1998]. Since each transmitted symbo is spread by a chaotic sequence of finite ength, there exist finite nonzero crosscorreations between the spread symbos from different users. This introduces interference among the users and imits the performance of the system when the users symbos are decoded independenty. Mutiuser detection is an effective technique to reduce mutua interference between users in a mutipe access environment. Mutiuser detectors can be broady categorized into inear and noninear types. For the inear detectors, a inear transformation is performed at the receiver to mitigate the mutiuser interference [Lupas & Verdú, 1989; Lupas & Verdú, 1990; Verdú, 1986; Xie et a., 1990] whereas noninear detectors cance the interference in severa stages by subtracting the reconstructed signas from the received signas [Hui & Letaief, 1998; Xue et a., 2000; Yoon & Ness, 2002; Yoon et a., 1993]. Recenty, two noninear mutiuser detectors, namey parae interference canceation (PIC) and the successive interference canceation (SIC) detectors, have been appied to a chaotic-sequence spread-spectrum communication system [Argüeo et a., 2002]. The performance, however, has ony been evauated by computer simuations and no anaytica BER has been derived. Aso, the combined effects of the inear and noninear mutiuser detectors have not been examined. 3633

2 3634 W. M. Tam et a. Fig. 1. An N-user mutipe access chaotic-sequence spread-spectrum (MA-CSSS) communication system empoying singeuser detectors. In this Letter, we study three types of inear detectors appied in conjunction with parae interference canceation (PIC) detector in a mutipe access chaotic-sequence spread-spectrum (MA-CSSS) communication system [Tam et a., 2003]. The inear detectors incude conventiona singe-user detector, mutiuser decorreating detector (DD) and mutiuser minimum mean-squareerror (MMSE) detector [Madsen & Cho, 1999; Tam et a., 2002, 2004]. In Sec. 2, we give an overview of the MA-CSSS system. The structure of the PIC detector is described and the decision-maing mechanism is presented. The technique for deriving the approximate bit error rates (BERs) for the PIC detectors (i.e. conventiona/pic, DD/PIC and MMSE/PIC detectors) is aso shown. Finay, in Sec. 3, brute-force simuation resuts are presented and compared with the approximate BER vaues. 2. System Description 2.1. Mutipe access chaotic-sequence spread-spectrum communication system Consider an N-user mutipe access chaoticsequence spread-spectrum (MA-CSSS) communication system shown in Fig. 1. Denote the th transmitted symbo for the ith user by d (i), which assumes the vaue 1 or 1 with equa probabiity. Aso, we represent the chaotic sequence used to spread the binary symbo sequence of the ith user by {x (i) }. Assuming a spreading factor of γ, the transmitted signa for the ith user at the th symbo duration, i.e. at time = ( 1)γ 1, ( 1)γ 2,..., γ, can thus be expressed as s (i) = d(i) x (i). (1) The overa transmitted signa of the system at time is thus given by s = i=1 s (i). (2) Assuming a simpe additive white Gaussian noise (AWGN) channe, the received signa equas r = i=1 s (i) ξ (3) where ξ is an AWGN sampe with zero mean and variance N 0 /2. Assume that the chaotic spreading sequences can be reproduced exacty at the receiver. It is readiy shown that when conventiona singeuser detectors are empoyed, as in Fig. 1, the output of the jth correator (j = 1, 2,..., N), denoted by y, is equa to y = d γ (x )2 =( 1)γ1 i=1,i j γ d (i) =( 1)γ1 γ =( 1)γ1 x (i) x ξ x (4)

3 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3635 Fig. 2. A mutistage parae interference canceation (PIC) detector. Fig. 3. The nth stage of a PIC detector. ˆd, is com- and the decoded symbo, denoted by puted from ˆd = sgn[y ] (5) where sgn[ ] represents the sign function. In (4), the first term represents the desired signa, the second term denotes the inter-user interference and the third term comes from noise. It can be seen that the existence of the inter-user interference, which is nonzero, imits the performance of the system even when the noise power is sma Parae interference canceation (PIC ) detector A parae interference canceation (PIC) detector with mutipe stages is shown in Fig. 2. At the zeroth stage of the PIC detector, the transmitted symbos are first estimated using a inear detector such as a conventiona singe-user detector, decorreating detector or MMSE detector [Tam et a., 2002, 2004]. At each of the subsequent stages, the interuser interference wi be estimated and removed from the decision statistics. The structure of the nth (n 1) stage of the PIC detector is shown in Fig. 3. The transmitted symbos estimated by the previous stage are first spread by the corresponding chaotic sequences so as to approximate the transmitted signas for a users. Then, the inter-user interference is reconstructed and subtracted from the received signa for each of the users. At the nth stage, the output of the jth correator is given by y,(n) = γ =( 1)γ1 = d γ r =( 1)γ1 ˆd (i), x(i) i=1,i j (x )2 } {{ } required signa γ =( 1)γ1 i=1,i j x ( ) d (i) (i) ˆd, x (i) x } {{ } inter-user interference γ =( 1)γ1 ξ x } {{ } noise (6)

4 3636 W. M. Tam et a. and the symbo is estimated again according to the sign of y,(n), i.e.ˆd,(n) = sgn[y,(n) ]. (7) It can be easiy seen that when some symbos are correcty estimated, i.e. d (i) (i) = ˆd, for some i {1, 2,..., N}, some of the inter-user interference is eiminated and the estimation process becomes more reiabe Performance anaysis Consider the nth stage of the PIC detector. Without oss of generaity, we consider the jth user in an N-user system and we derive the probabiity of error for the first symbo, i.e. = 1. For brevity, we omit the subscript in the foowing anaysis. Define D = [D(1) D (2) D (j 1) D (j1) D (N) ]T (8) where T represents the transpose and D (i) = d(i) (i) ˆd (9) denotes the difference between the transmitted symbo of the ith user and the estimated symbo at the (n 1)th stage of the PIC detector. The input to the jth threshod detector, Eq. (6), is now rewritten as y (n) = d γ (x )2 =1 i=1,i j γ =1 D (i) γ =1 x (i) x ξ x. (10) Note that if the estimated symbo for the ith user is correct, i.e. d (i) (i) = ˆd, D(i) becomes zero and the interference from the ith user is eiminated. However, when d (i) (i) ˆd, D(i) equas ±2. Under such a condition, the interference may contribute positivey or negativey to the required signa of the jth user, depending on the signs of d (i), d and γ =1 x(i) x. Assume that the transmitted symbo is 1 for the jth user. For a given D, the mean and variance respectivey, and of y (n) can be shown equa to, E[y (n) (d = 1, D )] = γe[(x )2 ] (11) var[y (n) (d = 1, D )] = γ var[(x )2 ] γe[(x )2 ] γ γ γ =1 m=1,m i=1,i j =1 m=1,m cov[(x (D (i) )2 E[(x (i) )2 ] i=1,i j γ E[x (i) x(i) m ]E[x )2, (x m )2 ] x m ] γn 0 E[(x )2 ]/2 (12) where E[ ] and var[ ] represent the expectation and variance operators, respectivey, and cov[a, B] denotes the covariance between A and B. In the derivations of (11) and (12), we have assumed 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. Using a simiar procedure, we can derive E[y (n) (d = 1, D )] and var[y (n) (d = 1, D )]. Assuming that both y (n) (d = 1, D ) and y (n) (d = 1, D ) are norma when γ is arge, it can be readiy shown that the conditiona error probabiities of both cases are the same, i.e. Prob(y (n) 0 (d = 1, D )) = Prob(y (n) > 0 (d = 1, D = 1 2 erfc )) E[y (n) (d = 1, D )] 2var[y (n) (d = 1, D )] (13) where erfc[ ] denotes the compementary error function [Proais, 1995]. Therefore, the bit error probabiity for the j user at the nth stage of the

5 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3637 PIC detector can be computed from BER (n) = [Prob(y (n) 0 (d = 1, D )) Prob(d = 1) D Prob(y (n) > 0 (d = 1, D )) Prob(d = 1)] Prob(D ) = D 1 2 [Prob(y (n) 0 (d = 1, D )) Prob(y (n) > 0 (d = 1, D ))] Prob(D ) = D and Prob(D ) is cacuated using with Prob(y (n) 0 (d = 1, D )) Prob(D ) (14) Prob(D ) = d = [d (1) d (2) d (N) ] T (16) denoting the transmitted symbo vector. Note that in (14) and (15), there are a ot of terms to compute because a arge number of possibe combinations have to be considered. Instead of deaing with such a arge number of terms, we mae use of a simper approach to cacuate the bit error rate. Suppose that under the given condition D, at the (n 1)th stage, there are N e {0, 1,..., N 1} symbos (excuding the one for the jth user) that have been incorrecty estimated, i.e. d (i) (i) ˆd, i j. For such cases, (D(i) )2 = 4 (because D (i) = ±2) and hence (12) can be BER N 1 (n) = = u=0 N 1 u=0 Prob(D d { 1,1} N rewritten as d) Prob(d), (15) var[y (n) (d = 1, D )] = var[y (n) (d = 1, N e = γ var[(x )2 ] γ 4γN e E2 [(x γ )] γ =1 m=1,m γ )2 ] i=1,i j =1 m=1,m E[x (i) x(i) cov[(x )2, (x m ) 2 ] m ]E[x x m ] γn 0 E[(x )2 ]/2. (17) With the introduction of N e, the bit error probabiity in (14) can be expressed as Prob(y (n) 0 (d = 1, N e = u)) Prob(N e = u) 1 2 erfc E[y (n) (d = 1, N e = u)] 2 var[y (n) (d = 1, N e = u)] Prob(N e = u) (18) where E[y (n) (d = 1, N e = u)] = E[y (n) (d = 1, D )] = γe[(x )2 ]. (19) By computing the probabiity of occurrence of N e {0, 1,..., N 1} and using (17) (19), the bit error rate at the nth stage of the PIC detector, BER (n), can be found. In addition, Prob(N e =

6 3638 W. M. Tam et a. u) is ony significant when u is very sma. Therefore, not more than a few vaues of u need to be considered. 3. Resuts and Discussions Assume that a users use the cubic map [Geise & Fairen, 1984] x 1 = 4x 3 3x (20) to generate the chaotic sequences and each user uses a different initia condition. It is readiy shown that the mean vaue of each chaotic sequence is zero. Aso, the mean and variance in (19) and (17) can be simpified to, respectivey, E[y (n) (d = 1, N e = u)] = γp s (21) and var[y (n) (d = 1, N e = u)] = γ var[(x )2 ] 4γuP 2 s γn 0P s /2 (22) where P s denotes the average power of the chaotic sequence and is given by P s = E[(x (1) )2 ] = E[(x (2) )2 ] = = E[(x (N) ) 2 ]. (23) For brevity, we define Ψ = var[(x )2 ] P 2 s = var[(x )2 ] E 2 [(x )2 ]. (24) As a consequence, the bit error probabiity for the j user at the nth stage of the PIC detector can now be computed using [ BER (n) = 1 N 1 erfc 2 u=0 2Ψ γ 8u ( ) ] γ Eb Prob(N e N = u) (25) 0 where E b = γp s (26) denotes the average bit energy. Three inear detectors, namey conventiona singe-user detector, decorreating detector (DD) and minimum mean-square-error (MMSE) detector, have been appied as the first stage (Stage 0) of the PIC detector. These combined detectors are termed as, respectivey, conventiona/pic, DD/PIC and MMSE/PIC detectors. We assume that the cubic map defined as in (20) is used by a users, and each uses a different initia condition. Aso, the spreading factor (γ) is 100 and the number of users (N) is 10. We aso assume that Prob(N e = u) is sma when u is arge. Therefore, in our anaysis, we assume that Prob(N e = u) is negigibe under the foowing circumstances and the corresponding terms are negected in the computation of (25). u 3 for the conventiona/pic detector at the first stage (n = 1). u 2 for the conventiona/pic detector at the second stage and beyond. u 2 for the DD/PIC detector at the first stage and beyond. u 2 for the MMSE/PIC detector at the first stage and beyond. Detais on the cacuations of Prob(N e = u) can be found in the Appendix. Figures 4 6, respectivey, pot the simuated BERs for the conventiona/pic, DD/PIC and MMSE/PIC detectors, at various stages. The BER for a singe-user system, which is equivaent to an interference-free system, is aso given as a reference. For a the detectors under study, the BERs at Stage 1 are ower than those at Stage 0. For the conventiona/pic detector, the BER at Stage 2 is further reduced compared to that at Stage 1. The BER resuts for the conventiona/pic detector with three stages, the DD/PIC and the MMSE/PIC detectors with two stages, are cose to that of the singe-user system. When the number of stages is further increased, no improvement is observed. Overa, the resuts indicate that the parae interference canceation technique can further enhance the performance of inear mutiuser detectors. With a ow BER at Stage 0 and at subsequent stages, it is we justified that terms corresponding to arge vaue of u can be ignored in the computation of (25). Figures 7 to 9 pot the approximate BERs (using (25)) and the brute-force (BF) simuated BERs when the PIC detectors are appied. It can be observed that the approximate and the simuated resuts agree with each other for a the PIC

7 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3639 Fig. 4. Simuated BER versus E b /N 0 for the conventiona/pic detector at various stages. γ = 100 and N = 10. Fig. 5. Simuated BER versus E b /N 0 for the DD/PIC detector at various stages. γ = 100 and N = 10.

8 3640 W. M. Tam et a. Fig. 6. Simuated BER versus E b /N 0 for the MMSE/PIC detector at various stages. γ = 100 and N = 10. Fig. 7. Approximate and brute-force (BF) simuated BERs versus E b /N 0 for the conventiona/pic detector. γ = 100 and N = 10.

9 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3641 Fig. 8. Approximate and brute-force (BF) simuated BERs versus E b /N 0 for the DD/PIC detector. γ = 100 and N = 10. Fig. 9. Approximate and brute-force (BF) simuated BERs versus E b /N 0 for the MMSE/PIC detector. γ = 100 and N = 10.

10 3642 W. M. Tam et a. Fig. 10. Comparison of the conventiona/pic, DD/PIC and MMSE/PIC detectors. γ = 100 and N = 10. detectors under study. Finay, we compared the simuated BERs for the conventiona/pic, DD/PIC and MMSE/PIC detectors. The resuts shown in Fig. 10 indicate that the BERs for the DD/PIC and MMSE/PIC detectors are amost the same. At Stage 1 of the PIC detector, the performances of the DD/PIC and MMSE/PIC detectors are better than that of the conventiona/pic detector. With an additiona stage (Stage 2), the conventiona/pic can achieve the same performance as the DD/PIC and MMSE/PIC detectors. 4. Concusion In this Letter, we have appied three combined inear detector/parae interference canceation (PIC) detectors, namey conventiona singe-user detector/pic, decorreating detector/pic and minimum mean-square-error/pic detectors to a mutipe access chaotic-sequence spread-spectrum communication system. The technique for deriving the approximate BERs has been described. It is found that the approximate BERs agree with the simuation resuts. Aso, it is shown that the PIC detectors can further improve the bit error performance compared to that of the inear mutiuser detectors. With ony one or two PIC stages, the bit error performance of the PIC detectors approaches to that of a singe-user (interference free) system. Increasing the number of PIC stages further wi not enhance the bit error performance because the system is now imited by noise. Acnowedgment This wor was supported by a Hong Kong Poytechnic University Research Grant. References Argüeo, F., Bugao, M. & Amor, M. [2002] Muti-user receivers for spread spectrum communications based on chaotic sequences, Int. J. Bifurcation and Chaos 12, Geise, T. & Fairen, V. [1984] Statistica properties of chaos in Chebyshev maps, Phys. Lett. A105, Heidari-Bateni, G. & McGiem, C. D. [1994] A chaotic direct-sequence spread-spectrum communication system, IEEE Trans. Commun. 42, Hui, A. L. C. & Letaief, K. B. [1998] Successive interference canceation for mutiuser asynchronous DS/CDMA detectors in mutipath fading ins, IEEE Trans. Commun. 46, Lupas, R. & Verdú, S. [1989] Linear muti-user de-

11 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3643 tectors for synchronous code-division mutipe-access channes, IEEE Trans. Inform. Th. 35, Lupas, R. & Verdú, S. [1990] Near-far resistance of muti-user detectors in asynchronous channe, IEEE Trans. Commun. 35, Madsen, A. H. & Cho, K. S. [1999] MMSE/PIC mutiuser detector for DS/CDMA systems with inter- and intra-ce interference, IEEE Trans. Commun. 47, Paritz, U. & Ergezinger, S. [1994] Robust communication based on chaotic spreading sequences, Phys. Lett. A188, Proais, J. G. [1995] Digita Communications (McGraw- Hi, Singapore). Tam, W. M., Lau, F. C. M. & Tse, C. K. [2002] Muti-user detection techniques for mutipe access chaos-based digita communication system, Proc. Int. Symp. Noninear Theory and Appications, Xi an, China, pp Tam, W. M., Lau, F. C. M. & Tse, C. K. [2003] Performance anaysis of mutipe access chaotic-sequence spread-spectrum communication systems empoying parae interference canceation detectors, Proc. IEEE Int. Symp. Circuits and Systems, Bango, Thaiand, Vo. III, pp Tam, W. M., Lau, F. C. M. & Tse, C. K. [2004] An approach to cacuating the bit error rates of mutipe access chaotic-sequence spread-spectrum communication systems empoying muti-user detectors, Int. J. Bifurcation and Chaos 14, Verdú, S. [1986] Minimum probabiity of error for asynchronous Gaussian mutipe-access channes, IEEE Trans. Inform. Th. IT-32, Xie, Z., Short, R. T. & Rushforth, C. K. [1990] A famiy of suboptimum detectors for coherent muti-user communications, IEEE J. Se. Areas in Commun. 8, Xue, G., Weng, J. F., Ngoc, T. L. & Tahar, S. [2000] An anaytica mode for performance evauation of parae interference canceers in CDMA systems, IEEE Commun. Lett. 4, Yang, T. & Chua, L. O. [1997] Chaotic digita code-division mutipe access (CDMA) communication systems, Int. J. Bifurcation and Chaos 7, Yang, T. & Chua, L. O. [1998] Error performance of chaotic digita code-division mutipe access (CDMA) systems, Int. J. Bifurcation and Chaos 8, Yoon, Y. C., Kohno, R. & Imai, H. [1993] A spreadspectrum mutiaccess system with cochanne interference canceation for mutipath fading channes, IEEE J. Se. Areas in Commun. 11, Yoon, S. & Ness, Y. B. [2002] Performance anaysis for inear mutiuser detectors of randomy spread CDMA using Gaussian approximation, IEEE J. Se. Areas in Commun. 20, Appendix Cacuation of the Probabiity of Occurrence of N e A symbos are defined as in Sec. 2. Reca in (16) that the transmitted symbo vector d is defined as d = [d (1) d (2) d (N) ] T. (A.1) Here, we denote ˆd as the estimated symbo vector at the (n 1) stage of the PIC detector, i.e. (1) ˆd = [ ˆd ˆd (2) ˆd(N) ]T. (A.2) Aso, when some of the eements are removed from d, we denote the resutant vector by d w where w is the vector containing the indices of the eements that have been removed. For exampe, assuming i < j, d (i,j) = [d (1) d (2) d (i 1) d (i1) d (j 1) d (j1) d (N) ] T. (A.3) Simiary, we denote the resutant vector by ˆd w where w contains the indices of the eements that have been removed from ˆd. Probabiity that N e = 1 The probabiity that N e = 1 can be shown equa to [Madsen & Cho, 1999] Prob(N e = 1) = i=1,i j i=1,i j Prob(d (i) Prob(d (i) = (N 1)BER. (i) ˆd, d(i,j) (i,j) = ˆd ) ˆd (i) ) We approximate Prob(N e bound (N 1)BER (A.4) = 1) by its upper. The approximation is good when the probabiity of error of each user is sma, e.g. ess than Using the anaytica BERs of the inear detectors, namey the conventiona singeuser detector, decorreating detector and MMSE detector [Tam et a., 2002, 2004], as the BER vaues of the zeroth stage (denoted by BER ), and by

12 3644 W. M. Tam et a. appying (A.4) and (25) repeatedy, the probabiity that N e = 1 at the (n 1)th stage can be found. Probabiity that N e = 2 Using a simiar approach, the probabiity that N e = 2 is given by Prob(N e = 2) = N i=1,i j v=1,v j,v<i i=1,i j v=1,v j,v<i Prob(d (i) Prob(d (i) ˆd (i), d(v) ˆd (i), d(v) (v) ˆd, d(i,j,v) (i,j,v) = ˆd ) ˆd (v) ). In the case where a conventiona singe-user detector is used as the zeroth stage of the PIC detector, i.e. n = 1, the probabiity that N e = 2 can be evauated by Prob(N e = 2) (A.5) Prob(d (i) i=1,i j v=1,v j,v<i ˆd (i), d(v) ˆd (v) (d(i), d (v) )) Prob(d (i), d (v) ) = i=1,i j v=1,v j,v<i [Prob(y (i) 0, y(v) 0 (d(i) = 1, d (v) = 1))Prob(d (i) = 1, d (v) = 1) Prob(y (i) 0, y(v) > 0 (d(i) = 1, d (v) = 1))Prob(d (i) = 1, d (v) = 1) Prob(y (i) > 0, y(v) 0 (d(i) = 1, d (v) = 1))Prob(d (i) = 1, d (v) = 1) Prob(y (i) > 0, y(v) > 0 (d(i) = 1, d (v) = 1))Prob(d (i) = 1, d (v) = 1)] = 1 2 (N 1)(N 2) [ , y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v) ], y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v) where y (i) and y(v) represent the outputs of the ith and vth correators, respectivey, in the singe-user detectors. In (A.6), f (ψ, ω) denotes the two-dimensiona (bivariate) norma probabiity density function (pdf) of ψ and ω and is given by 1 f(ψ, ω) = 2π var[ψ]var[ω](1 ρ 2 ) { [ 1 (ψ E[ψ]) 2 exp 2(1 ρ 2 2ρ ) var[ψ] (ψ E[ψ])(ω E[ω]) var[ψ]var[ω] ]} (ω E[ω])2 var[ω] (A.6) (A.7)

13 Mutipe Access Chaotic-Sequence Spread-Spectrum Communication Systems 3645 where ρ is the correation coefficient between ψ and ω, and is defined as It is readiy shown that = = = ρ = E[ψω] E[ψ]E[ω] var[ψ]var[ω]. (A.8), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v) 0 0 Hence, Prob(N e = 2) can be simpified to Prob(N e = 2) 1 2 (N 1)(N 2) 0 0, y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v). (A.9), y(v) (d(i) = 1, d (v) = 1))dy (i) dy(v). (A.10) Simiar to the case for N e = 1, we approximate Prob ( N e = 2 ) by its upper bound given in (A.10). Probabiity that N e 3 The probabiity that N e 3 is given by Prob(N e = u) = j / u ũ, u ũ=, j / u ũ, u ũ=, u ũ j={1,2,...,n} u ũ j={1,2,...,n} Prob(d (j,u) (j,u) ˆd, d (j,ũ) (j,ũ) = ˆd ) Prob(d (j,u) (j,u) ˆd ) (A.11) where the vectors u and ũ contain the indices of the users that have made, respectivey, wrong decisions and correct decisions. Using simiar procedures as in the case N e = 2, if a conventiona singe-user detector is used as the zeroth stage, the probabiity that N e 3 can be shown equa to ( ) N 1 Prob(N e = u) u d { 1,1} u y (u) f(y d )dy (1) y (2) y (1) dy(2) dy (u) Prob(d ) (A.12) ( ) w where denotes the number of ways to choose v v items out of w. Aso, the vectors Y and d are defined as Y = [ŷ (1) ŷ (2) ŷ (u) ]T (A.13) d = [d (1) d (2) d (u) ] T. (A.14) The u-dimensiona norma pdf of Y for a given vector d is denoted by f(y d ). It is represented by f(ψ) = where 1 (2π) u det[u] [ exp 1 ] 2 (ψ E[ψ])T U 1 (ψ E[ψ]) (A.15) ψ = [ψ (1) ψ (2) ψ (u) ] T, (A.16)

14 3646 W. M. Tam et a. U 1 is the u u covariance matrix of the random variabes ψ (1), ψ (2),..., ψ (u), and det[ ] denotes the determinant operator. Probabiity that N e = 0 Since N 1 m=0 Prob(N e = m) = 1, (A.17) the probabiity that N e by Prob(N e = 0) = 0 can be evauated N 1 = 1 Prob(N e = m). m=1 (A.18)