THE TRADEOFF BETWEEN DIVERSITY GAIN AND INTERFERENCE SUPPRESSION VIA BEAMFORMING IN

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1 THE TRADEOFF BETWEEN DIVERSITY GAIN AND INTERFERENCE SUPPRESSION VIA BEAMFORMING IN A CDMA SYSTEM Yan Zhang, Laurence B. Mistein, and Pau H. Siege Department of ECE, University of Caifornia, San Diego La Joa, CA ABSTRACT In this paper, the upink of an asynchronous Direct Sequence Code Division Mutipe Access (DS-CDMA) system with mutipe antennas at both the transmitter and the receiver is considered. We anayze the system performance over a spatiay correated Rayeigh fading channe with mutipe access interference (MAI). Assuming perfect channe knowedge avaiabe at the transmitter, Maxima Ratio Transmission (MRT) is empoyed to weight the transmitted signa optimay in terms of combating signa fading. Further, adaptive beamforming reception is adopted to suppress MAI and aso to combat the fading. We examine the effect of varying the number of transmit and receive antennas on the diversity gain and interference suppression. INTRODUCTION By utiizing antenna arrays at both the transmitter as we as the receiver, the imitations of the radio channe may be reduced and the data rates increased. This triggered the deveopment of space-time codes that empoy both the spatia and the tempora dimensions to achieve a significant portion of the channe capacity cacuated in [1]. Common to the space-time coding schemes [2], [3] is that they do not expoit channe knowedge at the transmitter. Channe information, if it is avaiabe, shoud of course be utiized to maximize the performance. Beamforming is one soution when there is a dominant direction-of-arriva (DOA) for the signa-of-interest. For a transmit array, the channe information is used to focus as much energy in the direction of the receiver as possibe. For a receive array, the gain of the antenna is maximized in the direction of the path with This work was partiay sponsored by the Center for Wireess Communications at UCSD, by the UC Discovery Program of the State of Caifornia, and by the TRW Foundation. the strongest power. Compared with the space-time coding schemes, beamforming is preferred in terms of compexity. Furthermore, in a mutipe access channe, where K u users each with L T transmit antennas try to communicate with a common receiver with L R receive antennas, beamforming is not ony sufficient but aso necessary for achieving the so-caed sum capacity of mutipe access channes if the number of users is much arger than the number of receive antennas [4]. This atter condition generay hods in a CDMA system. In this paper, the upink of an asynchronous DS-CDMA system with mutipe antennas at both the transmitter and the receiver is considered. Assuming perfect channe knowedge avaiabe at the transmitter, Maxima Ratio Transmission (MRT) is empoyed to weight the transmitted signa optimay in terms of combating signa fading. Adaptive beamforming reception is adopted to suppress MAI and aso to combat the fading. We anayze the system performance over a spatiay correated Rayeigh fading channe with mutipe access interference (MAI), and evauate the antenna array performance with joint fading reduction and MAI suppression. The detaied organization of the paper is as foows. The system mode and channe mode used in the study are described in Section II. Section III presents the anaysis of system performance, and is foowed by some numerica resuts and discussions in Section IV. Finay, concusions are drawn in Section V. SYSTEM MODEL We describe a system mode expoiting mutipe antennas in a singe ce CDMA system. Assume that both the mobies and the base station use an antenna array to transmit and receive signas, where each mobie has an antenna array, of size L T, used for maxima ratio transmission [5], and the base station has an antenna array, of size L R, used for adaptive beamforming reception. 1 of 6

2 For the bock diagram shown in Figure 1, the transmitted signa of user k is given by s k (t) P k c (k) n h(t nt c ) (1) u (k) n/n s v k exp [j(ωt + θ k )] where u (k) i is the ith data symbo of user k, v k is a transmission weight vector for user k, ω is the carrier frequency, θ k is a random carrier phase associated with user k and is uniformy distributed over [0, 2π), the spreading sequences of the interfering users, c n (k), k 2,, K, are assumed to be i.i.d. random variabes taking vaues ±1 with equa probabiity, whie that of the desired user, c (1) n, is taken to be deterministic, h(t) is the impuse response of the baseband chip wave-shaping fiter, and 1/T c is the chip rate of a band-imited DS-CDMA system. We aso define x(t) F 1 H(f) 2 and assume that x(t) satisfies the Nyquist criterion, i.e., x(nt c ) δ(n). The processing gain is defined as N s T s /T c, and is taken to be much smaer than the period of the spreading sequence, where T s is the symbo duration. The channe mode is taken to be a sowy varying Rayeigh fading channe, with transfer function ξ (k) α (k) exp(jβ (k) ), for r 1,, L R and 1,, L T, where is the index for the transmit antennas and { r is the index for the receive antennas. We assume that α (k) } { and β (k) } are statisticay independent for different users, { and that α (k) } { and β (k) } are, respectivey, i.i.d Rayeigh random variabes with a unit second moment, and uniform random variabes over [0, 2π) for different transmit antennas. However, the array gain and the phase of the different eements in the receive antenna array might be correated, where the correation is determined by parameters such as direction of arriva φ (k), anguar spread (k), spacing between neighboring receive antennas D r and the waveength of the carrier signa λ, as shown in Fig. 3. where Rs I (i, j) and RsQ (i, j) are given by ( ) Rs I (i, j) J 2πDr i j J 2n (3) λ n1 ( ) 2πDr i j cos(2nφ (k) )sinc(2n (k) ) λ and Rs Q (i, j) 2 n0 ( ) 2πDr i j J 2n+1 λ sin((2n + 1)φ (k) )sinc((2n + 1) (k) ) respectivey, for 1,, L T, and where the J n s are Besse functions of integer order. When this correation is high, the signas at the antennas tend to fade at the same time, and the diversity benefit of antenna arrays against fading is significanty reduced. On the other hand, because independent fading is not required for interference suppression, antenna arrays can suppress interference even with compete correation. Thus, we need to evauate the antenna array performance with joint fading reduction and interference suppression. We define a channe matrix H k by putting the channe gain of each transmit and receive antenna pair into a matrix of size L R L T. That is to say, the i, jth entry in H k is ξ (k) i,j. Thus, the received signa vector in the antenna array is obtained as K r(t) H k s k (t τ k ) + n w (t) (5) k1 where τ k is an arbitrary time deay uniformy distributed over [0, T s ], and n w (t) is the AWGN vector added to the receive antenna array such that each of its eements is a zero-mean compex Gaussian random process with twosided spectra density η 0. An asynchronous DS-CDMA system is assumed, but the receiver is synchronized to the desired transmission, say that of user 1; thus, we assume that the power and deay of the desired signa are, respectivey, P 1 1 and τ 1 0, without oss of generaity. (4) Specificay, using the mode in Fig. 3, a cosed-form spatia correation formua is given by [6], { E (ζ (k) i, )(ζ(k) j, ) } Rs( (k), φ (k), D r, λ) (2) ] [Rs I (i, j) + jrsq (i, j) PERFORMANCE ANALYSIS We evauate the performance of the first user. Perfect carrier, code, and bit synchronization are assumed in the receiver side, as shown in Fig 2. The output of the correator during the ith symbo interva, z 1 (i), obtained by summing 2 of 6

3 the corresponding despread N s chip-matched fiter output, is given by z 1 (i) 1 in N s n (i 1)N s c (1) n (6) [(r(t) exp( jwt)) h( t)] tn T c 1 in G 1 v N 1 s c (1) n n (i 1)N s u (1) n/n s c(1) n h((n n)t c + τ)h(τ)dτ + I 1 (i) + N 1 (i) u (1) i G 1 v 1 + I 1 (i) + N 1 (i) where represents convoution, G k H k exp(jθ k ) for k 1,, K. S 1 (i) u (1) i G 1 v 1 is the signa component for the desired user, N 1 (i) 1 in N s n(i 1)N s c (1) n (7) {[n w (t) exp( jwt)] h( t)} tntc is the component due to therma noise, and K in Pk I 1 (i) c (1) n N k2 s n (i 1)N s u (k) n/n s c(k) n K k2 x((n Mn)T c τ k )G k v k Pk N s R k,1 (i)g k v k is the mutipe access interference. In (8), R k,1 (i) in c (1) n (i 1)N s (8) n (9) µ (k) n x((n n)t c τ k ) is the cross-correation function of the spreading signa between user k and user 1 during the ith symbo interva. Here we define µ n (k) u (k) n/n s c(k) n, absorbing u (k) n/n s into c(k) n, since both are random variabes taking vaues of ±1 with equa probabiity. By the Liapounoff version of the centra imit theorem, I 1 (i) can be modeed as an asymptoticay compex Gaussian vector as ong as the foowing condition is satisfied [8]: x(nt c τ) < for a τ, where 3 of 6 0 τ < T c. The correator outputs from each receive antenna are combined with the beamforming vector w 1 [w 1,1,, w 1,LR ] T to produce an estimate of the transmitted symbo of the desired user. The estimated data symbo can be represented as û 1,i w H 1 z 1(i) S 1,i + I 1,i + N 1,i. (10) Now we proceed to determine the optimum transmit and receive weight vectors v 1 and w 1 respectivey, for the desired user. Since the MAI, I 1 (i), can be modeed as an asymptoticay zero-mean compex Gaussian vector, and is independent of the AWGN vector N 1 (i), the conditiona SINR, γ i, of the estimated data, û 1,i, conditioned on G 1, is given by γ i S 1,i 2 V ar(n 1,i ) + V ar(i 1,i ) (11) w H 1 G 1v 1 v H { 1 GH 1 w 1 } w H 1 E N 1 (i)n H 1 (i) + I 1(i)I H 1 (i) w 1 w H 1 G 2 1v 1 [ w H η0 1 N s I LR + K P k k2 N s R I (0)(V k ) H R (k) ] I V k w 1 Refer to [8] for the detaied derivations of the covariance matrices for N 1 (i) and I 1 (i). Since mobie 1 has no knowedge of the other users transmit weight vectors {v k }, an ad hoc criterion of generating its own transmit weight vector v 1 is to maximize the effective received signa power P (1) r w H 1 G 1 v 1 2 ( (G1 ) H w 1 2 ) ( v 1 2) (12) where equaity hods if and ony if v 1 c 1 (G 1 ) H w 1, and c 1 is a constant for normaization. If the transmitter has perfect knowedge of channe state information G 1 can be maximized by setting v 1 c 1 (G 1 ) H w 1. This concept for determining the transmit weight vector is known as maxima ratio transmission. Subject to the power constraint v 1 2 1, the transmit weight vector is given by and the corresponding receive weight vector w 1, P (1) r v 1 (G 1) H w 1 (G 1 ) H w 1. (13) Now the goa is to choose the beamforming weight vector w 1 which maximizes the received power for the desired user. Subject to the normaization constraint, the optimum

4 receive weight vector w 1 is obtained as { (w1 ŵ 1 arg max ) H G 1 (G 1 ) H 2 } w 1. (14) w Therefore, the receive weight vector ŵ 1 is the principa eigenvector of G 1 (G 1 ) H, and the received power is the corresponding eigenvaue, i.e., the maximum eigenvaue λ of G 1 (G 1 ) H. NUMBERICAL RESULTS AND DISCUSSIONS We assume that the fading seen by each transmit antenna is independent, since L T is usuay a sma number. At the receiver, L R receive antennas are depoyed for adaptive beamforming reception, where L R can be a arge number so that the fading experienced by each receive antenna might be correated. L T independent transmit antennas and L R independent receive antennas give L T L R order spatia diversity gain. So fixing the vaue of L T L R fixes the maximum diversity order achievabe by the system. When the fading is, in fact, correated, the diversity gain from the receive antenna array is reduced. However, independent fading is not required for interference suppression, since correated receive antennas can sti be used for MAI suppression. Suppose we fix the product L T L R, just for the sake of having a frame of reference for a tradeoff. Then increasing L T wi increase the diversity gain against fading whie sacrificing some of the receive antenna array s capabiity of MAI suppression. This wi be iustrated beow. We assume the use of a raised-cosine fiter characteristic, with rooff factor α 0.5, for puse shaping. We further assume the processing gain to be fixed at N s 64. Since it is difficut to anayticay derive the pdf of the instantaneous SNR, f γ (γ), we cannot obtain a cosed-form expression for the BER. To circumvent this probem, a Monte-Caro simuation is carried out. After one miion trias, the SNR distribution of the combined outputs at the receiver is accumuated and f γ (γ) is numericay determined. The SNR vaue γ for each combined output is appied to the conditiona bit error probabiity of a BPSK system, φ( 2γ), and the average BER is cacuated by integrating P e 0 φ( 2γ)f γ (γ)dγ. In Fig. 4, we consider a DS-CDMA system with 10 users, where the interference power is og-normay distributed with a 3dB standard deviation. The average BER versus E b /η 0 with different sets of parameters is shown in the figure. With L T L R fixed to be 8, we find that the system empoying 2 transmit antennas and 4 receive antennas is better than one empoying 1 transmit antenna and 8 receive antennas. At a BER of 10 3, there is neary a 5dB enhancement by setting L T 2 and L R 4. This is primariy due to the two-fod diversity gain from the two transmit antennas with independent fading. Note that since the tota ength of the receive array is fixed at a vaue such that the mutipe receive antennas experience correated fading, the resuting effective diversity order achieved by the 8-antenna array is ess than twice that achieved by the 4-antenna array, athough the MAI suppression capabiity is enhanced with more receive antennas. We further evauate the system performance with a more severe near-far probem, i.e, interference power is og-normay distributed with a 10dB standard deviation. Compared to the system with better power contro, the BER performance of both of the above systems degrades. It is observed that the degradation is more significant for the L T 2 and L R 4 system than it is for the L T 1 and L R 8 system. The system s abiity to suppress MAI is augmented by using more receive antennas, whie sacrificing some diversity gain from the transmit antennas, and in the presence of a arge amount of MAI, this interference suppression is more important than the diversity gain against fading. In Fig. 5, we pot the BER performance curves for systems with Ku 20. Compared with the curves potted in Fig. 4 for systems with Ku 10, there is smaer degradation when L R 8 receive antennas are empoyed. However, the degradation is much more conspicuous when ony L R 4 receive antennas are used. Since MAI becomes more dominant compared to the fading when the number of interferers increases, and the system with 8 antennas is more capabe of MAI suppression, the performance gap between the two systems decreases. Furthermore, for the system with 20 users and/or 10dB standard deviation, the performance curves of L T 2, L R 4 and L T 1, L R 8 are quite cose. CONCLUSION In this paper, we proposed a DS-CDMA system empoying mutipe antennas at both the mobie and the base station. Maxima ratio transmission and adaptive beamforming reception are used to achieve the maximum received power for the desired user in a mutipe access channe with correated Rayeigh fading. The conditiona SNR is anayticay 4 of 6

5 derived and the average BER is investigated via simuation. We examined the effect of varying the number of transmit and receive antennas on both the diversity gain against fading and the MAI suppression. The benefit of using ony a singe transmit antenna is easier impementation in a sma mobie unit. However, when the number of active users is stabe and/or accurate power contro is maintained, using a arger number of independent transmit antennas is preferred. u (k) Beamformer v k d (k) 1 d k) c(k) n c(k) n I.M I.M H(f) H(f) exp[j(wt+ θ k )] exp[j(wt+ θ k )] Tx 1 Tx REFERENCES d(k) Lt c n (k) I.M H(f) exp[j(wt+ θ k )] Tx Lt [1] T. Marzetta, B. Hochwad, Capacity of a mobie mutipe-antenna communication ink in Rayeigh fat-fading, IEEE Trans. Inform. Theory, vo. 45, no. 1, pp , Jan [2] G. J. Foschini, Layered space-time architecture for wireess communication in a fading environment when using mutipe antennas, Be Labs Technica Journa, Vo. 1, pp , Autumn 1996 [3] V. Tarokh, N. Seshadri, and A. R. Caderbank, Space-time codes for high data rate wireess communication: Performance criterion and code construction, IEEE Trans. Inform. Theory, vo. 44, no. 2, pp , March [4] W. Rhee, J. M. Cioffi, On the asymptotic optimaity of beamforming in muti-antenna Gaussian mutipe access channes, Proc. Gobecom 01, San Antonio, TX, Nov [5] Titus K. Y. Lo, Maximum Ratio Transmission, IEEE Trans. Commun., vo. 47, no. 10, Oct [6] J. Saz and J. H. Winters, Effect of fading correation on adaptive arrays in digita wireess communications, Proc. IEEE ICC, pp , November [7] B.D.Van Veen and K.M.Buckey, Beamforming: a versatie approach to spatia fitering, IEEE ASSP Mag., pp. 4-24, Apri [8] Yan Zhang, Performance of combined beamforming and spacetime coding in a CDMA system, Ph.D. dissertation, University of Caifornia, San Diego. Fig. 1. r(t) n w(t) Fig. 2. Transmitter for MRT in a CDMA system exp[jw 1t] exp[jw Mt] chip matched fiter chip matched fiter y 1 y M c n (1) c n (1) N 1 n0 N 1 n0 z1 zm Adaptive fiter w 1 (k) Adaptive fiter w M (k) Receiver with adaptive beamforming in a CDMA system u (k) Mobie φ D 1 2 L R 1 L R Fig. 3. Wireess environment where a signas from a transmit antenna arrive at the BS within ± of ange Φ 5 of 6

6 10 1 std3db std10db 10 2 Ku10,Lt1,Lr8 BER 10 3 Ku10,Lt1,Lr8 Ku10,Lt2,Lr Ku10,Lt2,Lr SNR Fig. 4. BER versus E b /η 0 with K u 10 and L T L R std3db std10db Ku20,Lt1,Lr Ku20,Lt2,Lr4 Ku20,Lt1,Lr8 BER 10 3 Ku20,Lt2,Lr SNR Fig. 5. BER versus E b /η 0 with K u 20 and L T L R 8 6 of 6