ZERO-FORCING MULTIUSER DETECTION IN CDMA SYSTEMS USING LONG CODES

Transcription

1 ZERO-FORCING MUTIUSER DETECTION IN CDMA SYSTEMS USING ONG CODES Cássio B Ribeiro, Marcello R de Campos, and Paulo S R Diniz Electrical Engineering Program and Department of Electronics and Computer Engineering COPPE/Poli/Federal University of Rio de Janeiro PO Box 68504, , Rio de Janeiro, RJ, Brazil diniz, cassio, campos]@lpsufrjbr Abstract In this article we show that a Code Division Multiple Access (CDMA) system employing long codes can be interpreted as a Multiple Input Multiple Output system where the transmit filters are time-varying We derive conditions for existence of zero-forcing equalizers and verify through simulations that these conditions are useful guidelines for the design of appropriate solution in the presence of noise Compared to similar works in the literature, the derived conditions allow a reduction in the length of the equalizer filters, while imposing no constraints on the channel order I INTRODUCTION MultiUser Detection (MUD) for Code Division Multiple Access (CDMA) systems employing long codes, ie, codes that span duration of more than one symbol, is still an open problem ong codes are widely used in modern third-generation communications systems (eg UMTS), hence the need to develop detectors that are able to cope with the conditions presented by these commercial systems Recent works related to CDMA systems employing long codes include Minimum Mean Square Error (MMSE) equalization and interference cancellation 1] 6], and Zero Forcing (ZF) equalization relying on multiple receiver antenna and oversample at the receiver 1], 3], 7] In this article we interpret the CDMA system as a Multiple Input Multiple Output (MIMO) system, as shown in 8], where the codes play the role of the transmit filters, which are timevarying in the case of a CDMA system employing long codes We then extend the theoretical analysis in 9] to allow the application of zero-forcing (ZF) equalization to realize MUD in the downlink of CDMA systems employing long codes, assuming the receiver has an estimate of the channel model For channel estimation, see eg 10], 11] II SYSTEM DESCRIPTION In this section we describe the communications system considered throughout this article The system model is shown in Figure 1, where s m (n), and ŝ m (n) are the transmitted and received symbols for user m, respectively, m = 0,, M 1 The channel is modeled by h(k), and the transmit filters are given by f m (k, n), where the index n indicates the timevarying nature of the transmit filters This system model can represent a downlink scenario in a wireless communications system, where N is the spreading factor The output of the synthesis filter bank (precoder) is given by s m (n)f m (k nn, n), (1) where s m (n) is the symbol transmitted by the m-th user at time instant n, and f m (k, n) is the filter corresponding to user m This is a general representation for a MIMO system, but in this article we will consider only precoder filters with length up to N As a consequence, equation (1) introduces no intersymbol interference (ISI) This is particularly useful when we consider the CDMA system, since the spreading codes and the transmit filters will be seen to have a close relationship The transmitted signal passes through a linear time-invariant (TI) channel h(k), and is received together with additive Gaussian noise with zero mean v(k), ie, y(k) = x(k) + v(k) = h(l)u(k l) + v(k) (2) l= The receiver estimates the transmitted symbols for the m-th user by filtering the signal y(k) by g m (k) and then decimating the output by N, as shown in Figure 1 This process is described by ŝ m (n) = y(nn j)g m (j) (3) j= In order to obtain a compact representation in matrix form for the process, we will define the M 1 vectors s(n) = s 0 (n) s 1 (n) s (n) ] T (4) ŝ(n) = ŝ 0 (n) ŝ 1 (n) ŝ (n) ] T, (5) and the N 1 vectors u(n) = u(nn) u(nn + N 1) ] T x(n) = x(nn) x(nn + N 1) ] T v(n) = v(nn) v(nn + N 1) ] T (6) (7) (8) y(n) = y(nn) y(nn + N 1) ] T (9) Equations (1) and (3) can be rewritten as u(n) = F i (n)s(n i) (10) i=

2 v(k) s 0 (n) f 0 (k) u(k) h(k) x(k) y(k) g 0 (k) ŝ 0 (n) s 1 (n) f 1 (k) g 1 (k) ŝ 1 (n) s (n) f (k) g (k) ŝ (n) Fig 1 Communications system in time domain as a MIMO system and ŝ(n) = G j y(n j), (11) j= where the elements of the N M matrix F i (n) and of the M N matrix G i are {F i (n)} l,m = f m (in + l, n), {G i } m,l = g m (in l) m = 0,, M 1, l = 0,, N 1 (12) Define the N N matrix H l as h(ln) h(ln 1) h(ln N +1) h(ln + 1) h(ln) h(ln N) H l = h(ln +N 1) h(ln +N 2) h(ln) (13) The received signal is then given by y(n) = H l u(n l) + v(n), (14) l= where the convolution between u(n) and H l introduces intersymbol interference In order to simplify the notation we will introduce now the assumptions that will be considered in the following sections for equalization: (a) the channel is modeled as an -th order FIR filter, with h(0), h() 0; (b) the spreading factor (or block length) is greater than or equal to the number of symbols, ie, N M; (c) f m (k, n) are causal length-n FIR filters, and g m (k) are causal length-qn FIR filters, Q integer Assumption (c) implies that matrices F i (n) are zero, except for i = 0; matrices G j are zero, except for j = 0,, Q 1 Assumption (a) implies that matrices H l are zero, except for l = 0,, B, with B = N, where denotes the smallest integer greater than or equal to ( ) Based on these assumptions, we have u(n) = F 0 (n)s(n), and hence, rewrite equation (14) as y(n) = H l F 0 (n l)s(n l) + v(n) (15) et us define the NQ 1 vectors y(n), and v(n) as y(n) = y T (n Q + 1) y T (n) ] T v(n) = v T (n Q + 1) v T (n) ] T We can thus write H l F 0 (n l Q + 1)s(n l Q + 1) y(n)= H l F 0 (n l)s(n l) + v(n) = HF s(n) + v(n) (16) (17) (18) where H is QN (Q + B)N, F is (Q + B)N (Q + B)M, and s(n) is (Q + B)M 1, and given by: H B H H B H 0 0 H = (19) 0 0 H B H 0 F 0 (n Q B+1) F 0 (n Q B+2) 0 F = 0 0 F 0 (n) (20) s(n) = s T (n Q B + 1) s T (n) ] T (21) Finally, we can write the relation between the transmitted and received symbols as ŝ(n) = GHF s(n) + Gv(n) (22) where the M QN matrix G is defined as G = G Q 1 G 0 ] (23)

3 c m (k) c m (k N) c m (k 2N) 0 N 1 2N 1 3N 1 c (scr) m (k) c (scr) (k N) c m (k, 0) c m (k N, 1) c m (k 2N, 2) 0 N 1 2N 1 3N 1 Fig 2 m Representation of short+long code as a time-varying short code A Relation with CDMA Systems In a communications system employing CDMA technology, the sequence of symbols from each user is multiplied by a code at higher rate This process can be represented by the transmultiplexer model used in this article Commercial CDMA systems (eg UMTS) usually employ long codes in order to help mitigate multipath and interference A long code is a pseudo-random sequence leading to the same spreading gain as a short code, but the former is times longer, where 1 is integer In these systems, the symbols of each user are spread by a short code and then multiplied at chip rate by a long code, that is also called scrambling code et c m (k), m = 0,, M 1, be the short code pertaining to the m-th user, and c (scr) m (k), m = 0,, M 1, be the Nlength scrambling (long) code from user m The transmitted signal is given by s m (n)c m (k nn)c (scr) m (k n N) (24) where denotes the largest integer lower than ( ) This is equivalent to consider that the short code for each user is time varying, and that the variation is periodic with period N This time-varying short-code is given by c m (k nn, n) = c m (k nn)c (scr) m (k n N) (25) as illustrated in Figure 2 We can rewrite (24) as s m (n)c m (k nn, n) (26) and hence the MIMO system considered in this article can represent this long-code CDMA system if we make f m (k, n) = c m (k, n) III ZF MUTIUSER DETECTION In this section we examine the system described in the previous section in order to derive necessary conditions for ZF equalization From equation (22), ZF equalization is obtained if v(k) = 0 and GHF = 0 M (Q+B 1)M I M ] (27) The above condition is satisfied if and only if the right side of (27) belongs to the space spanned by the columns of HF, ie, 0M (Q+B 1)M I M ] T R{F T H T }, (28) where R{ } denotes the space spanned by the columns of { } et us express the above relation in terms of the null space of H, denoted as N {H} et us define the r 1 vector e r (i) as the i-th column of the r r identity matrix Thus, we can say that if there is a (Q + B)N 1 vector α such that F T α = e (Q+B)M (i + (Q + B 1)M), (29) then the condition expressed in equation (28) is satisfied if and only if α R(H T ) Instead of dealing with R(H T ), that is not easy to characterize, we will work with the nullspace of H, that has a straightforward characterization as a function of the zeros of h(k) It is possible to verify that N (H) = span{e (Q+B)N (i), i = 1,, BN 1} span{ 0 1 BN 1 1 v l v QN+ ] T l, l = 0,, 1}, (30) where v l are the roots of the polynomial h(n), and N (H) is the null space of H The first set in the right side of (30) is due to the fact that H B, defined in equation (13), has BN 1 zero columns Without impairing the analysis, let us consider BN 1 = 0, ie, the channel length is a multiple of the decimation/interpolation ratio N, and rewrite (30) as N (H)=span{ 1 v l v (Q+B)N 1 l ] T, l =0,, 1}, (31) If α R(H T ) then α is orthogonal to all vectors in N (H) Hence condition (28) is satisfied if and only if the vector α that satisfies (29) also satisfies α T ν = 0, ν N (H) (32) et us make the assumption that F 0 (n), has full column rank, n This is the usual case, since the columns of F 0 (n) are part of a pseudo-random sequence Without loss of generality we can write ] IM F 0 (n) = F(n), (33) Φ(n) where Φ(n) is N M M, and the M M matrix F(n) possesses full column rank From (20) and (33), we can rewrite (29) as diag { I M Φ T (n Q B + 1)],, I M Φ T (n)] } α= 0, fi] (34) where f i is the i-th column of (F T (n)) 1 The next step is to characterize α that satisfies equation (34) et us divide the vector α in Q + B sections of length N like α = α ] T Q+B 1 ˆα T Q+B 1 α T 0 ˆα T 0, (35)

4 where α q and ˆα q, are M 1 and (N M) 1, respectively, q = 0,, Q + B 1 From equations (34) and (35), we have { α q + Φ T 0, q = 1,, Q + B 1 (n q) ˆα q = (36) f i, q = 0 Now we need to establish restrictions on the values of Q, N M and that guarantee orthogonality between α and the vectors in the set N T (H) et us define 1 v 0 v (Q+B)N 1 0 V T =, (37) 1 v 1 v (Q+B)N 1 1 where v l are the roots of the polynomial h(n), as in equation (30) From equations (35), (36), and (37), we can rewrite (32) as where Θ = diag ˆα = V T α = V T Θ ˆα + γ = 0 1, (38) { Φ(n Q B + 1) I N M ] T, }, Φ(n) I N M ] T ˆα Q+B 1 ˆα 0, γ = V T 0 (Q+B 1)N 1 f i 0 N M 1 (39) Therefore, ZF equalization is possible if and only if there exists a vector α such that V T Θ ˆα = γ (40) Since we want to guarantee the existence of solution for (40) regardless of F(n) (and γ), the necessary and sufficient condition is rank(v T Θ) = BN 1 In order to simplify the notation, let us define the N N M matrix Ψ(n), given by Ψ(n) = Φ(n) I N M ] T, (41) and the N 1 vectors v l and ṽ l, given by vl T = ] vl 0 v N 1 l for l = 0,, 1 With these new matrices we can write v0 0N v0 T Ψ(n Q B+1) v(q+b 2)N 0 v0 T Ψ(n) (42) V T Θ= v 1 0N vt 1Ψ(n Q B+1) v(q+b 2)N 1 v 1 T Ψ(n) (43) Equation (40) admits solution if and only if rank(v T Θ) = Then, a necessary condition for existence of solution is (Q + B)(N M) Q B (44) N M In 9], 12], the authors show that if the precoder is timeinvariant, another condition that must be satisfied is N M µ, where µ is the number of congruous zeros, ie, zeros such that vl N = ρ, l = 0,, µ 1 In order to show that this condition is no longer necessary when we use time-variant precoders, let us write the first µ lines of V T Θ as v0 T Ψ(n Q B+1) v T 0 Ψ(n) (V T Θ) 1:µ,: = vµ 1Ψ(n Q B+1) T vµ 1Ψ(n) T I N M ρ I N M ρ Q+B 2 I N M Analyzing (45), it is possible to verify that if we have (45) Ψ(n δ +1) Ψ(n δ +2) Ψ(n), δ Q+B, (46) then the left side of (45) possesses at most min(µ, (N M)δ) linearly independent columns Hence, if we make (N M)δ µ it is possible that V T Θ has full row rank, implying that it is possible to achieve perfect reconstruction or, in other words, ZF equalization In order for the equalization to be possible regardless of the channel, it suffices to consider the worst case, µ =, and in that case δ is such that (N M)δ δ N M (47) Since δ Q + B, the conditions for equalization regardless of the channel are (Q + B) δ N M Q N M B (48) and Ψ(n δ + 1) Ψ(n δ + 2) Ψ(n), δ Q + B These results show that for time-variant precoders the amount of redundancy, N M, does not depend on the number of congruous zeros, allowing the system design to be independent of the channel realization Considering the application of the present analysis for CDMA systems using long codes, we conclude that ZF equalization will be possible if the system is not at full capacity, ie, if N > M, and the complexity of the receiver depends directly on the difference N M A ZF-S Equalizer In the presence of noise, the ZF equalizer may perform poorly, since the receiver may be actually amplifying the noise This problem is specially critical if the signal-to-noise ratio (SNR) is low In 9], the authors note that this problem with ZF equalizers can be avoided if the matrix G is approximated by a least squares solution, or using a pseudo-inverse, or using a different approach like adaptive filtering In all these cases, the conditions found in the previous section should be used as a guideline for proper design and dimensioning of the system IV EXPERIMENTA RESUTS In this section we present experimental results obtained via computer simulations The results consist of an average of 50 transmissions, each one comprising BPSK modulated symbols for each s m (n), m = 0,, M 1 The performance is measured in terms of bit error rate (BER) The precoder

5 consists of random codes for each user, each code following a Gaussian distribution with zero mean and unity variance The equalizer is designed as in Section III-A, using G(n) = s(n)y (n), where s(n) is the block of transmitted symbols, y(n) is defined in equation (14), and ( ) denotes the pseudoinverse The channel model is given by an FIR approximation with 14 coefficients of the following transfer function P (z) = z z z z z z 3 (49) The results shown in Figure 3 are the BER as a function of the signal to noise ratio (SNR) for N = 16 and M = 6, 9, 12, 14 Points for which BER = 0 are not shown As one could expect, the larger the redundancy, the smaller the BER, indicating that the number of users in the system must be controlled in order to achieve a target BER level BER N M=10 N M=7 N M=4 N M= SNR Fig 3 BER as a function of SNR for N = 16 and M = 6, 9, 12, 14 Figure 4 shows the BER as a function of SNR for N = 16, M = 12, and Q = 1, 2, 3, 4 Points for which BER = 0 are not shown Since from equation (48) Q = 4 is the minimum value for Q, a degradation in performance is expected as we lower Q beyond this limit This degradation occurs for Q = 2, 3 but for Q = 1 there is a considerable loss V CONCUSION In this work we derived conditions for existence of zeroforcing equalizers in communication systems employing block transmissions, in particular those using CDMA technology The extension of the results in 9] for time-variant precoders allows the application of the techniques for ZF equalization to MUD in CDMA systems using long codes Compared to related works, eg 8], the obtained relations allow a reduction in the length of the equalizer filters, and also allows transmission through channels with impulse responses longer than the block length We also conclude that ZF equalization will be possible in CDMA systems using long codes if the system is not at full capacity, ie, if N > M Even though the existence of ZF solution exists for all N and M such that N > M, BER Q=4 Q=3 Q=2 Q= SNR Fig 4 BER as a function of SNR for N = 16, M = 12, and Q = 1, 2, 3, 4 the complexity of the receiver and its performance in the presence of noise depend directly on the difference N M Hence, the conditions derived in this article shall serve as useful guidelines for the design of a communications system, allowing to compromise performance and receiver complexity REFERENCES 1] T P Krauss and M D Zoltowski, Oversampling diversity versus dual antenna diversity for chip-level equalization on CDMA downlink, in Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop, 2000, pp ] T P Krauss, W J Hillery, and M D Zoltowski, MMSE equalization for forward link in 3G CDMA: symbol-level versus chip-level, in Proceedings of the Tenth IEEE Workshop on Statistical Signal and Array Processing, 2000, pp ] M enardi, A Medles, and D T M Slock, Downlink intercell interference cancellation in WCDMA by exploiting excess codes, in Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop, 2000, pp ] K Hooli and M Juntti, Interference suppression in WCDMA downlink by symbol-level channel equalization, in EUSIPCO 2002, Toulousse, ] K Hooli, M Juntti, M J Heikkillä, P Komulainen, M atva-aho, and J illeberg, Chip-level channel equalization in W-CDMA downlink, EURASIP Journal on Applied Signal Processing, vol 8, pp 1 14, ] Vandendorpe, F Horlin, and T Sartenaer, FSE and FSDF joint detectors for long DS-CDMA codes, in Proc IEEE Conference on Acoustics Speech and Signal Processing, May ] M D Zoltowski and T P Krauss, Two-channel zero forcing equalization on CDMA forward link: trade-offs between multi-user access interference and diversity gains, in Conference Record of the Thirty- Third Asilomar Conference on Signals, Systems, and Computers, 1999, 1999, vol 2, pp ] A Scaglione, G B Giannakis, and S Barbarossa, Redundant filterbank precoders and equalizers part I: Unification and optimal designs, IEEE Transactions on Signal Processing, vol 47, no 7, pp , July ] C B Ribeiro, M R de Campos, and P S R Diniz, FIR equalizers with minimum redundancy, in Proc IEEE Conference on Acoustics, Speech and Signal Processing, April ] Tong, A van der Veen, and P Dewilde, Channel estimation for long code CDMA, in Proc ICASSP, October ] C J Escudero, U Mitra, and D T M Slock, A toeplitz displacement method for blind multipath estimation for long code DS/CDMA signals, IEEE Transactions on Signal Processing, vol 49, no 3, pp , March ] Y-P in and S-M Phoong, Minimum redundancy for ISI free FIR DMT transceivers, IEEE Transactions on Signal Processing, vol 50, no 4, pp , May 2002