Research Article Design and Performance Analysis of an Adaptive Receiver for Multicarrier DS-CDMA

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1 Hindawi Publishing Cororation EURASIP Journal on Wireless Communications and Networking Volume 2007, Article ID 0462, 0 ages doi:055/2007/0462 Research Article Design and Performance Analysis of an Adative Receiver for Multicarrier DS-CDMA Huahui Wang, Kai Yen, Kay Wee Ang, and Yong Huat Chew Institute for Infocomm Research (I 2 R) (A STAR), Agency for Science, Technology and Research, 2 Heng Mui Keng Terrace, Singaore 963 Received 26 January 2006; Revised 22 January 2007; Acceted 2 May 2007 Recommended by Lee Swindlehurst An adative arallel interference cancelation (APIC) scheme is roosed for the multicarrier direct seuence code division multile access (MC-DS-CDMA) system Freuency diversity inherent in the MC system is exloited through maximal ratio combining, and an adative least mean suare algorithm is used to estimate the multile access interference Theoretical analysis on the biterror rate (BER) of the APIC receiver is resented Under a unified signal model, the conventional PIC (CPIC) is shown to be a secial case of the APIC Hence the BER derivation for the APIC is also alicable to the CPIC The erformance and the accuracy of the theoretical results are examined via simulations under different design arameters, which show that the APIC outerforms the CPIC receiver rovided that the adative arameters are roerly selected Coyright 2007 Huahui Wang et al This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited INTRODUCTION Future generations of broadband wireless mobile communication systems are exected to suort various services over a multitude of channels encountered in indoor, oen rural, suburban, and urban environments, while maintaining the reuired uality of service (QoS) [, 2] In order to meet these demands, the signal should be flexibly designed such that it is caable of adating to these communication conditions In the existing direct-seuence code-division multile access (DS-CDMA) systems, the sread sectrum (SS) modulation is exloited in mitigating various roblems encountered in different communication media Recently, the multicarrier (MC) techniue has become an imortant alternative for achieving this goal [3, 4] A number of MC-CDMA schemes have been roosed in the literature [5 9] Their erformances are analyzed and comared with that of single carrier (SC) DS-CDMA systems in freuency-selective Rayleigh fading channels [9 3] MC systems are advantageous due to their robustness in combating the freuency selectivity in broadband channels The underlying reason is the integration of an orthogonal freuency division multilexing (OFDM) overlay, which can be designed such that each subcarrier undergoes flat fading and hence reduces the severe intersymbol interference (ISI) encountered in SC-DS-CDMA systems On the other hand, the critical issue for MC-CDMA systems remains to be the imrovement of the system caacity in multiuser communications, in which the multile access interference (MAI) becomes the major caacity-limiting factor Much attention has been given to the erformance analysis of the MC systems based on the single user detection (SUD) strategy, see [4, 4, 5], for examle, where the MAI is simly treated as thermal noise Significant imrovement can be achieved when multiuser detection (MUD) techniues are emloyed to jointly detect all the users signals [6, 7] Most MUD algorithms originally roosed for SC-DS-CDMA are alicable to MC-CDMA systems, where interest in this area has been mainly focused on the erformance analysis of these various techniues It is well known that the rohibitive comlexity of the otimal multiuser detection necessitates subotimal solutions having lower comlexity A large volume of subotimal MUD algorithms have been roosed in the literature, and two different aroaches emerge, namely, adative filtering [8 20] and interference cancelation (IC) [2 23]Much more research has been dedicated to the latter rimarily due to a simler analysis tractability There are two main varieties of IC schemes, namely, serial IC (SIC) and arallel IC (PIC) In SIC, MAI is estimated and subtracted from the received signal seuentially Adatation of SIC to MC systems can be found in[2, 22], where the system erformances are also

2 2 EURASIP Journal on Wireless Communications and Networking c (t) cos(ω,t + ϕ,) d 2 Identical-bit stream Serial to arallel converter M c (t) c (t) c (t) cos(ω,2t + ϕ,2) cos(ω,mt + ϕ,m) cos(ω P,t + ϕ P,) Parallelbit branch d P 2 M c (t) cos(ω P,2t + ϕ P,2) c (t) cos(ω P,Mt + ϕ P,M) Figure : Illustration of signal transmission for user k analyzed PIC aears to be more attractive in the case when high seed detection is referred, since the cancelation of the interference is erformed in arallel However, the otential gain from PIC deends on the recise estimate of the MAI A artial PIC is roosed in [24] tomitigate theeffectof unreliablemaiestimation Motivated by [24], a hybrid aroach comrising of the PIC and an adative techniue is roosed for SC-DS-CDMA in [25] In this aer, an adative arallel interference cancelation (APIC) scheme is roosed for the MC-DS-CDMA system, in which the freuency diversity inherent in the MC system is exloited through maximal ratio combining (MRC) The contribution of the aer is twofold Firstly, the adative signal rocessing as well as the IC techniue is designed for the MC system, and the conditions under which the algorithms are able to function roerly are investigated Secondly, instead of simly imlementing a heuristic algorithm, we erform a thorough analysis on the system erformance and obtain a simle closed form exression for the bit-error rate (BER) of the APIC receiver Furthermore, under the unified signal model, we show that the theoretical result derived for the APIC is also alicable to the conventional PIC (CPIC), as long as the adative ste-size is set to zero The accuracy of the BER derivation is validated by comuter simulations The results showed a significant erformance imrovement of the APIC over the CPIC receiver The organization of this aer is as follows Section 2 introduces the MC-DS-CDMA system model Section 3 highlights the structure of the APIC receiver with re- and ost-mrc combining Section 4 analyzes the erformance of the receiver and derives the corresonding closed form BER exression Numerical results and discussions are resented in Section 5 We conclude the aer in Section 6 2 SYSTEM MODEL The structure of the transmitter for user k is shown in Figure AblockofP incoming bits is first serial-to-arallel converted into P so-called arallel-bit branches The bit on each arallel-bit branch, denoted as d, =, 2,, P, is then relicated into M streams referred to as identical-bit streams, as shown in Figure These M P bit streams are sread by the same user-secific seudorandom sreading seuence c (t), and modulated on subcarriers that are orthogonal to each other In order to ensure indeendent fading and hence achieving freuency diversity, the M P subcarriers are assigned in such a way that the freuency searation between all identical-bit subcarriers is maximized, as illustrated in Figure 2, where the identical-bit subcarriers f,m, m =, 2,, M, corresonding to data d are searated by a distance of P/T c for two neighboring subcarriers, for examle, f,m and f,m+ In this aer, the channel is assumed to be a slowvarying, freuency-selective Rayleigh fading channel with a delay sread of T m Since the sread sectrum system can resolve multiath signals with delay larger than one chi duration, for an SC-DS-CDMA system with a chi duration of T c, the number of resolvable aths L is given by L = T m /T c +,

3 Huahui Wang et al 3 f, f P, f,2 f,m f P,m f,m f P,M 2/T c (MP +)/T c Figure 2: Sectrum of the transmitted signal where x is themaximum integer less than or eual to xassuming a assband null-to-null bandwidth, the transmission bandwidth for the SC-DS-CDMA is 2/T c Maintaining this bandwidth, if the chi duration on each subcarrier of the MC-DS-CDMA system is T c, then the following condition shouldbesatisfied(referto Figure 2): 2 MP + = (2) T c T c From and(2), the number of resolvable aths L on each subcarrier of the MC-DS-CDMA system is L = T m /T c + = 2(L )/(MP +) + It is easy to show that when P and M are chosen to satisfy [4] MP 2(L ), (3) then L = and each subcarrier exeriences a flat fading channel In this case, the comlex channel gain for the th subcarrier of user k can bedefined as ζ (t) = α (t)ex [ jβ (t) ], (4) where α (t) is a Rayleigh-distributed stochastic rocess with unit second moment and β (t) is uniformly distributed over 0 and 2π It is assumed that the channel gain ζ (t) is indeendent and identically distributed (iid) for different values of k and This is a slight simlification over a real channel which would be correlated in freuency, but tyically the difference in erformance between a correlated and uncorrelated channel model is small, excet that the correlation is noticeable [2, 26] Furthermore, we investigate synchronous MC-DS-CDMA systems with BPSK modulation to considerably simlify the exosition and analysis Synchronous systems are becoming more of ractical interest since uasisynchronous aroach has been roosed for satellite and microcell alications [27] In [28], an ulink synchronous CDMA system is investigated, where users signals are assumed aligned at the base station In this aer, we consider a similar ulink synchronous MC-DS-CDMA system and study its erformance Assuming that the system consists of K number of users and that all the users emloy the same transmitter structure of Figure, the received signal at the base station can be written as P M 2Pk ( ) r(t) = M d Tb t Tb c (t) = m= (5) ζ (t)cos ( ) ω t + φ + n(t), where P k is the ower of the kth user, d {, +} is the kth user s th arallel-bit data, T b is the transmission interval for each block of data, τ (t) is defined as the rectangular ulse waveform with unit amlitude and duration τ, ω and φ are the freuency and random hase of the th subcarrier, resectively, and c (t) is the sreading seuence of user k, which is given by c (t) = n= c (n) Tc ( t ntc ), (6) where c (n) is the nth chi of the long sreading seuence for user k Suose each symbol interval contains N chis, we conduct normalization in each symbol eriod such that (l+)n n=ln [c (n)] 2 =, l Z N is called the rocessing gain Since we have assumed that the channel is slowly fading, the channel gain ζ (t) will remain constant for one transmission interval Hence the function of time in ζ (t) will be omitted hereafter The arameter = +(m )P is the subcarrier index corresonding to the th arallel-bit branch and the mth identical-bit stream The variable n(t) in(5) is the additive white Gaussian noise (AWGN) with zero mean and one-sided ower sectral density of N 0 3 ADAPTIVE RECEIVER STRUCTURE The structure of the roosed adative receiver is illustrated in Figure 3, which can be functionally divided into three arts: the re- and ost-mrc combining, the adative MAI estimation, and the arallel IC 3 Initial stage: MF with MRC (MF-MRC) Referring to Figure 3, after the received signal is down converted to its euivalent baseband signal and assed through the fast Fourier transform (FFT) block, the signal can be groued into P sets, where each set consists of M identicalbit streams For descrition simlicity, we only consider the rocessing of the th branch in the seuel For the th subcarrier where = +(m )P, the coherently detected signal corresonding to the nth chi, r (n), is given by (n+)tc r (n) = r(t)cos ( ) ω t + φ dt nt c (7) where = v (n) = v (n)+η t (n), 0 n N, Pk 2M d c (n)ζ, (8) (n+)tc η t (n) = n(t)cos ( ) ω t + φ dt (9) nt c It is easy to show that η t (n) is a zero mean Gaussian random variable with variance σ 2 η = N 0 /2

4 4 EURASIP Journal on Wireless Communications and Networking r(t) FFT r, r,m r, r,m r P, r P,M u, u (K), MF MF u,m u (K),M MRC MRC d d (K) Reference signal s, s,m Signal reconstruction s (K), s (K),M Signal reconstruction Reference signal MAI estimate Weights adjustment MAI estimate Weights adjustment w, w (K), w,m w (K),M PIC PIC x, x (K), x,m x (K),M MF MF MRC MRC d d (K) Pre-combining MAI reconstruction PIC stage Postcombining Figure 3: Adative receiver design The signal r (n) is then assed to the chi-rate matched filter (MF) bank, as deicted in Figure 3 Theoututcorresonding to the nth chi of the kth user is given by u (n) = r (n)c (n), 0 n N (0) Assuming erfect channel estimation, the M oututs corresonding to the identical-bit streams are combined together using the MRC coefficients g = [ζ ], where [x] is the comlex conjugate of x If the user of interest is user, the tentative decision is then given by d d { [ N M = sign R n=0 m= 32 MAI estimation stage d u (n)g ]} () After the initial decision is obtained, it is relicated into M branches to reconstruct the MAI, as shown in Figure 3 Similar to the transmitter, each of the M coiesissreadby the corresonding user s sreading code c (n) andattenuated by the channel gain ζ Hence the regenerated signal of the nth chi corresonding to the th carrier is given by s (n) = Pk 2M d c (n)ζ, m M (2) The regenerated signals of all the users are multilied by their corresonding adative weights w (n) and summed together to roduce an estimate r (n) of the signal r (n), which is written as r (n) = s (n)w (n), 0 n N (3) The difference between r (n) and r (n) constitutes the MAI estimation error Based on this error we define the cost function of the adative algorithm as ε = E [ e (n) 2] = E [ r (n) r (n) 2], (4) where E[ ] is the statistical exectation oerator and e (n) = r (n) r (n) is the error of the MAI estimation In order to minimize the cost, the weights w (n) are adjusted at the chi rate according to the normalized LMS algorithm [29]: w (n +)= w (n)+ μ s (n) Ki= [ s (i) (n) ] [ 2 e (n) ], (5) whereμ denotes the ste-size At the end of one transmission interval, the weight w (N ) is determined and it is used by the next stage to assist in the interference cancelation, as deicted in Figure 3 33 Cancelation with MRC stage: PIC-MRC At this stage, w (N ) is used to weight the inut signal s (n) over the entire transmission interval Subtracting the weighted MAI, the cleaner signal for user is given by where x v (n) = r (n) k=2 v (n), (6) (n) = s (n)w (N ) (7) The signal x (n) is then assed to the MF bank and the M identical-bit streams are combined via MRC The final decision is then obtained according to { ]} d = sign [ N M R n=0 m= x (n)c (n)g (8)

5 Huahui Wang et al 5 A multistage APIC receiver can be realized by reeating the rocess from (2)to(8) 4 PERFORMANCE ANALYSIS 4 BER of the MF-MRC receiver In Figure 3, if we only consider the first stage, then the structure becomes a conventional MF with MRC combining, which we will refer to as the MF-MRC receiver in the following context The soft outut of the MF-MRC receiver for user isgivenby Z = D + n t + I MAI, (9) where u (n) wasdefinedin(0) The desired signal D is given by M N D = v (n)c (n) [ ζ ] m= n=0 (20) P M [ ] = 2M d α 2, m= and the noise term n t is given by { M N n t =R η t (n)c (n)α ex [ jβ ] } (2) m= n=0 n t is a zero mean Gaussian random variable and its variance is given by σ 2 n = N 0 4 M m= [ ] α 2 (22) The term I MAI in (9) is the MAI which can be written into two arts: I MAI = I (s) MAI + I (d) MAI, (23) where I (d) MAI is the interference from the other users on different subcarriers, which simly vanishes in synchronous case [4] I (s) MAI is the interference from the other users on the same subcarrier, which is given by { I (s) M Tb MAI =R m= 0 2Pk ( ) M d Tb t Tb c (t) k=2 ζ cos ( ) ω t+φ c (t)cos ( )[ ω t+φ ζ M N Pk = 2M d c (n)c (n) m= k=2 n=0 cos ( β β ) α α } ] dt (24) The term I (s) MAI is commonly aroximated as a zero mean Gaussian random variable Under the assumtion that α is known at the receiver and the channel is normalized with E[(α ) 2 ] =, the variance of I (s) MAI is given by Var [ I (s) ] K P k MAI = 4MN k=2 M m= [ ] α 2 (25) Let γ = M m= [α ] 2, and assuming that a bit is transmitted, then the error robability conditioned on γ is given by P[e γ] = ( [ E Z 2 erfc ] ) 2Var [ Z ] = ( ) 2 erfc P /(2M)γ 2 ( Var [ ] ) I MAI + σ 2 n (26) Assuming that any bit can be sent via any of the P arallel branches with eual robability, the final BER of the MF- MRC receiver (the initial stage of the APIC receiver) can be written as P ini [e] = P P = 0 P [ e γ ] (γ)dγ, (27) where (γ) is the robability density function of γ and is given by [30] (γ) = (M ) γm e γ (28) 42 BER of the conventional PIC receiver In Figure 3, when there is no adative rocess involved, or euivalently by setting the weight w (N ) = in the MAI estimation stage, the receiver reduces to a conventional PIC receiver (CPIC) For CPIC, the IC is erformed by subtracting the estimated signals of the interfering users from the reference signal r (n), which forms a cleaner signal x (n)as given by x (n) = r (n) k=2 s (n), (29) where s (n) is the regenerated signal defined in (2) The outut signal after the MRC combining is given by { Ẑ M N =R x (n)c (n) [ } ζ ] m= n=0 (30) = D + n t + I MAI The desired signal D and the noise term n t above are identical with the corresonding terms in (20)and(2) The new MAI term is given by M N I MAI = m= k=2 n=0 Pk 2M ( d ) d c (n) c (n)cos ( β β ) α α (3)

6 6 EURASIP Journal on Wireless Communications and Networking If the BER of the initial stage, P ini [e], is available, then we have [3] P [ d P [ d = d = d E [( d d ] = Pini [e], d ] = Pini [e], (32) ) 2 ] d = 4Pini [e] (33) From (25)and(33), the variance of I MAI can be written as Var [ I MAI ] = 4Pini [e]var [ I MAI] (s) (34) The corresonding BER can then be obtained by using (26) and (27), with Var[I MAI ] relaced by Var[I MAI] An alternative derivation of the BER of the CPIC receiver can be obtained by regarding the CPIC as a secial case of APIC, as shown below 43 BER of the adative PIC receiver Let Δr (n) be the difference between r (n) and the comosite estimated signal K v (n), that is, r (n) = v (n)+δr (n) (35) Comaring with (7), the following relations are satisfied: v (n)+δr (n) = v (n)+η t (n), (36) Δr (n) = Δv (n)+η t (n), (37) where Δv (n) = v (n) v (n), by which the term x (n) in (29)canberewrittenas x (n) = r (n) v (n)+ v (n) = Δr (n)+ v (n) = Δr (n) Δv (n)+v (n) (38) From (38), the soft outut of the PIC-MRC stage is given by { Z M N =R x m= n=0 (n)c (n) [ } ζ ] M N = v (n)c (n) [ ζ m= n=0 + M N m= n=0 = D + I ] [ Δr (n) Δv (n) ] c (n) [ ζ ] (39) The first term D is the desired signal, which is identical to the corresonding term in (20) The second term I is the interference, which is aroximated as a zero mean Gaussian random variable From (37), with the assumtion that Δv (n) is an iid random variable, we have E [ ( Δr (n) ) 2 ] = K E [ ( Δv (n) ) 2 ] + σ 2 η, (40) E [ ( Δr (n) Δv (n) ) 2 ] = (K )E [ ( Δv (n) ) 2 ] + ση, 2 (4) where σ 2 η is the variance of η t (n)in(9) Substituting (40) into (4)gives E [ ( Δr (n) Δv (n) ) 2 ] = (K )E [ ( Δr (n) ) 2 ] + ση 2 K (42) Therefore, the variance of the interference term is given by Var[I] = M m= [ α,0] 2 (K )E [ ( Δr (n) ) 2 ] + ση 2, (43) 2K where E[(Δr (n)) 2 ] is the mean suare error (MSE) of the MAI estimation and it can be aroximated using the following result Proosition Assume that the source data is iid with E[d d (l) ] = δ k,l Furthermore, assume that ower control is ideal such that all users signals have the same ower level at the receiver If the ste-size is roerly selected such that the misadjustment of the LMS algorithm is less than 0%, then the MSE of the MAI estimation can be aroximated as ( MSE + μk ){ [ ( K 2Pini [e] ) 2] } + ση 2, 2MN 2MN (44) where μ is the ste-size, K, M, N are the number of users in the system, the identical-bit streams and the rocessing gain, resectively, P ini [e] is defined in (27) and ση 2 = N 0 /2 Proof Refer to the aendix By aroximating E[(Δr (n)) 2 ]in(43) using the MSE in (44), the corresonding BER of the APIC receiver can then be established by using (26)and(27)withVar[Z ] relaced by Var[I] Remark Other than the theoretical aroximation, a more accurate value of the MSE can be determined with the aid of comuter simulations It is easy to show that if the adative ste-size of the algorithm is fixed at 0 and the initial weights are set at, the APIC reduces to the CPIC Under these settings, if the MSE of the CPIC is available through comuter simulations, the BER exression originally derived for the APIC can also be alied to the CPIC The justification of the analysis will be illustrated in the next section

7 Huahui Wang et al BER 0 2 BER Number of users (K) SNR (db) APIC-analytic, with theoretical MSE APIC-analytic, with simulated MSE APIC-simulation CPIC-analytic CPIC-simulation APIC-analytic APIC-simulation Figure 4: Theoretical and simulation results of the one-stage APIC receiver, N = 32, SNR = 20 db Figure 5: Analytical and simulation results of the one-stage CPIC and APIC receivers, N = 32, K = 30 APIC-analytic uses the simulated MSE 5 NUMERICAL RESULTS AND DISCUSSIONS In this section, the erformance of the APIC receiver for the MC-DS-CDMA system is studied through numerical results The channel is freuency-selective with L = 3 The number of identical-bit streams M and arallel-bit branches P should be chosen to satisfy (3) in order to guarantee flat fading on each subcarrier M is referred to as the reetition deth in [2] which bears the tradeoff between the maximum number of users suortable and the achievable freuency diversity, given a fixed number of subcarriers In the simulations we set M = 2 to reflect a certain level of diversity gain The selection of P,as long as it satisfies(3), does not affect the erformance much although it is constrained by the total available bandwidth as well as the system comlexity Considering that the ractical cell secific scrambling codes could destroy the orthogonality between users sreading codes, we utilize random codes as the sreading codes in the simulations In Figure 4, the analytical and simulation results of the APIC receiver are resented The dashed curve is numerically calculated using (26), (27), (28), and (43), where the MSE of the MAI estimation is obtained from the aroximation of (44) The ste-size and the initial weight for the APIC receiver are μ = 0 andw =, resectively There is a small discreancy between the theoretical and the simulation results, due to the aroximation of the MSE However, it can be seen that if the MSE is obtained from the simulations, the resultant BER calculated from the euations is very close to the statistic BER obtained from the Monte Carlo simulations Under the same settings, Figure 5 illustrates the analytical and simulation results of both the APIC and CPIC receivers in terms of BER versus SNR The derivations of both receivers are justified through the agreement of the analytical and simulation results For the adative receiver, the ste-size μ lays an imortant role in system erformance In the seuel, simulations are conducted to investigate the effects of the ste-size and the initial weights on the erformance of the APIC receiver It is shown in Figure 6 that the lowest BER is achieved when the initial weight is w 0 = atμ = 03 Note that when the initial weight is w 0 = and the ste-size is μ = 0, the APIC reduces to the CPIC The horizontal line in the figure reresents the erformance of the CPIC receiver, and we can see that the APIC outerforms the CPIC for μ (0, 05) Under the same simulation settings, the BER erformances of the one- and two-stage APIC receiver versus the ste-size μ are resented in Figure 7 The initial weight has been fixed at It is shown that the one-stage APIC receiver achieves its best erformance at μ = 03 However, for the two-stage APIC receiver (with the ste-size of the first stage being μ = 03), the best erformance is at the oint where μ 2 = 0 Hence the APIC reduces to the CPIC at the second stage (horizontal line) The underlying reason lies in the fact that, for both the APIC and the CPIC, the MAI estimation has been reliable enough after the first stage rocessing Hence the APIC does not have sueriority over the CPIC at the second stage However, when the SNR is low or the system load is heavy such that the first stage cannot erfectly handle the MAI estimation errors, a small nonzero ste-size for the second stage guarantees advantage of the APIC over the CPIC This is verified in Figure 8, where for the APIC, the first stage adots a ste-size of μ = 03andμ 2 = 005 for the second stage Furthermore, the influence of the choice of the initial weight w 0 is shown in Figure 9, where the MSE erformances

8 8 EURASIP Journal on Wireless Communications and Networking BER 0 2 BER 0 3 Performance of CPIC Ste-size μ Number of users (K) APIC, w = 0 APIC, w = 5 APIC, w = 05 APIC, w = stage-cpic stage-apic 2stage-CPIC 2stage-APIC Figure 6: BER erformance of one stage APIC receiver with different initial weights as a function of the ste-size, N = 32, SNR = 20 db, K = 30 Figure 8: BER erformances of the CPIC versus the APIC N = 32, SNR = 20 db For the APIC, μ = 03, μ 2 = BER 0 3 Performance of CPIC, first stage MSE w 0 = 0 w 0 = 05 w 0 = Performance of CPIC, second stage Ste-size μ APIC, first stage with initial weight w 0 = APIC, second stage with initial weight w 0 = andμ = Iterations (n) Figure 7: BER erformance of the first and second stage of the APIC as a function of the ste-size Figure 9: Convergence comarison for different initial weights N = 32, K = 30, SNR = 20 db, μ = 03 with different initial weights are resented It is shown that when the initial weight is randomly chosen, it takes a while for the algorithm to converge Conseuently, when the rocessing gain of the system is small, the algorithm converges across multile symbols A very natural choice of the initial weight for the roosed scheme, however, is w 0 = The idea comes with the obvious fact that by choosing w 0 =, the algorithm starts from the CPIC, which constitutes a stationary starting oint 6 CONCLUSIONS In this aer, we designed an adative receiver for the multicarrier DS-CDMA system over Rayleigh fading channels and evaluated the erformance of the system A closed form exression of the BER is originally derived for the APIC receiver, and it was shown that the derivation of the BER for the CPIC receiver can be unified under the same framework Simulation results are rovided to verify the theoretical

9 Huahui Wang et al 9 derivations The effect of the design arameters of the APIC receiver, such as the adative ste-size and the initial weights, are investigated It is shown that with the aroriate selection of these arameters, the APIC outerforms the CPIC APPENDIX At the MAI estimation stage, the regenerated signal of the nth chi corresonding to the th carrier given in (2)isrewritten here for convenience of reference: s (n) = Pk 2M Stack K users signals in one vector as d c (n)ζ, m M (A) s(n) = [ s (n), s (2) (n),, s (K) (n) ] T, (A2) such that the K K autocorrelation matrix is defined as The eual ower and the iid source data assumtions further lead to the following result: E [ r (n) 2 ] = E [ K = v Pk E[ = K 2MN + σ2 η, 2] (n)+η t (n) 2M d 2] c (n)ζ + σ η 2 (A9) where ση 2 = N 0/2 From (A), (A6) and the exression of v (n)in(8), we can easily calculate the kth comonent of as E [ s (n)r (n) ] = 2MN E[ d d ] = 2P ini[e] 2MN (A0) R = E [ s(n) s T (n) ] (A3) Hence, the cross-correlation vector is given by Under ideal ower control, the channel is statistically identical for all users The average ower received at the base station for users can be assumed to be eual By normalizing E[P k ζ 2 ] =, for all k =, 2,, K, and incororating the iid source data assumtion made in Section 4, we then have the following result: 0 0 2MN 0 R = 2MN MN = 2MN I K, (A4) where I K is the K K unit matrix The inverse of the matrix R is given by R = 2MNI K As in (7), the reference signal is given by r (n) = (A5) v (n)+η t (n), 0 n N, (A6) thus we can define the K cross-correlation vector = 2P ini[e] 2MN [ ] } {{ } K The MMSE in (A8) can then be written as ε min = K[ ( 2P ini [e] ) 2] 2MN T (A) + σ 2 η (A2) When the LMS algorithm [29] is utilized for adative signal rocessing, the mean-suare error (MSE) of the estimation can be searated into two terms as MSE = ε min + ε excess, (A3) where ε excess is the excess MSE which is roortional to ε min, that is, ε excess = λ ε min (A4) Assuming that the ste-size μ is roerly selected such that the misadjustment of the LMS algorithm is less than 0%, that is, λ 0, we have λ = μ tr[r] μk μ tr[r] = μ tr[r] 2MN (A5) = E [ s(n)r (n) ] (A7) Finally, the exression of the MSE is given by It is then easy to obtain the minimum mean-suare error (MMSE) of the MAI estimation as [29] ε min = E [ r (n) 2] H R (A8) MSE = ( + λ) ε min ( = + μk ){ [ ( K 2Pini [e] ) 2] } + ση 2 2MN 2MN (A6)

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