LINEAR DETECTORS FOR MULTI-USER MIMO SYSTEMS WITH CORRELATED SPATIAL DIVERSITY

Transcription

1 LINEAR DETECTORS FOR MULTI-USER MIMO SYSTEMS WITH CORRELATED SPATIAL DIVERSITY Laura Cottateucci, Raf R. Müer, and Mérouane Debbah Ist. of Teecommunications Research Dep. of Eectronics and Teecommunications Mobie Communications University of South Austraia Norwegian Univ. of Science and Technoogy Institut Eurecom Mawson Lakes Bouevard 749 Trondheim 2229 Route des Cretes B.P. 93 Adeaide, Austraia Norway Sophia Antipois, France ABSTRACT A mutiuser CDMA system with both the transmitting and the receiving sites equipped with mutipe antenna eements is considered. The mutiuser MIMO channe is correated at the transmitting and the receiving sites. Mutistage detectors achieving near-inear MMSE performance with a compexity order per bit inear in the number of users are proposed. The arge system performance is anayzed in a genera framework incuding any mutiuser detector that admits a mutistage representation. The performance of this arge cass of detectors is independent of the channe correation at the transmitters. It depends on the direction of the channe gain vector of the user of interest if the channe gains are correated.. INTRODUCTION The semina works in [] and [2] on mutipe antenna eements at the transmitter and the receiver show a huge increase in throughput of this point-to-point channe, referred to aso as mutipe input mutipe output (MIMO) system. These promising resuts motivated the introduction of mutipe antenna eements in the standardization of third generation systems based on code division mutipe access (CDMA), e.g. UMTS. The beneficia effects of spacia diversity, eventuay obtained by mutipe antenna eements at a singe base station, on code division mutipe access (CDMA) systems have been investigated in [3]. Modeing the spreading matrices as random matrices and focusing on inear minimum mean square error (MMSE) detectors, Hany and Tse [3] found a very simpe reation between the degrees of freedom introduced by spatia diversity (L receiving antennas) and the degree of freedom in frequency given by spread spectrum techniques (spreading factor N), when the channe gains are independent and identicay distributed. The muti-antenna system behaves ike a system with a singe receive antenna but with spreading factor mutipied by the number of receiving antennas, and the received power of each user being the sum of the received powers at the individua antennas. This behaviour is known as resource pooing effect. It shows the possibiity to trade bandwidth (spreading factor) with antennas and viceversa according to the pecuiarity of the communications system. The interchangeabiity between degrees of freedom in frequency and space suggests the idea of treating the two effects in the same way performing antenna array processing and mutiuser detection jointy. Joint processing outperforms techniques that try to expoit separatey the degrees of freedom in space and frequency significanty [4]. However, the optima agorithms for this task are known for their prohibitive compexity. The inear MMSE detector has been proposed as a suboptima approach abe to attain good performance with a substantia reduction in compexity. However, when appied to arge CDMA systems, i.e. systems with arge spreading sequences and arge number of users, its compexity is sti very demanding for rea-time impementations. With the aim of finding a good trade-off between compexity and performance, aso in the chaenging scenario of arge CDMA systems with random spreading, inear mutistage detectors with universa weights have been proposed in [4, 5]. They are obtained as asymptotic approximation of the mutistage Wiener fiter (MSWF) [6]. These mutistage detectors consist of a projector onto a ryov subspace and a subsequent fiter using universa weights as fiter coefficients instead of taiored weights depending on the transmitted spreading sequences. The design of universa weights benefits from the asymptotic sef averaging properties of random matrices and reduces the computationay demanding part of the detector into a computation of a poynomia depending on the statistica properties of random matrices via few essentia system parameters. The assumption of independent channe gains underies the design of universa weights in both works. Thanks to the additiona feature of detecting jointy a active users, the mutistage detectors proposed in [4, 7] achieve near-lmmse performance with the same compexity order per bit as the singe user matched fiter, aso in the upink. In fact, by processing jointy a users, most of projection computations becomes identica and the compexity drops by a factor of. In [4] an asymptotic approximation of the poynomia expansion detectors [8] is aso proposed. In this work we generaize the resuts in [4] to a synchronous CDMA system with correated spatia diversity and/or ine of sight components. We refer to the asymptotic approximation of the MSWF detector as detector Type J-I to underine the joint projection of the received signa for a users and the asymptotic individua optimization of the fiter coefficients for each user. The asymptotic approximation of the poynomia expansion detector is referred to as Type J-J detector to emphasize the joint optimization of the fiter coefficients. The design of the universa weights reies on (i) the convergence of the diagona eements of the system correation For a detaied discussion on the convergence rate of the mutistage detector performance to the LMMSE performance the interested reader is referred to [9, 0].

2 matrix R and of its positive powers when the system dimensions go to infinity with constant ratio, for detector Type J-I, (ii) the convergence of the traces of R and its powers for detector Type J-J. To compute the diagona eements of R m, m Z + or the trace of R m we propose a recursive agorithm for the genera case and a simpified version for the correated Rayeigh fading channes. As shown in [7], the knowedge of the asymptotic diagona eements of R m enabes the asymptotic anaysis of any inear mutiuser detector that can be expressed as inear mutistage detector with projection in the same ryov subspace. A part from the MSWFs and the poynomia expansion detectors, this cass of detectors incudes the parae interference canceers (PIC), the weighted PICs, the matched fiter, and, asymptoticay as the number of stages goes to infinity 2 the inear MMSE detectors. The arge system anaysis shows that the asymptotic performance of this arge cass of detectors is independent of the correation of the channe gains at the transmitters. In contrast to the case of a system with a singe receive antenna, the mutiuser efficiency does not characterize univocay the system performance and varies from user to user according to the direction of the vector of the channe gains. This property has the foowing impication. Whie the MSWFs and the poynomia expansion detectors are equivaent for synchronous CDMA with singe antennas [7] or mutipe receiving antennas with independent and identicay distributed channe gains, in case of perfect power contro, the MSWFs outperform the poynomia expansion detectors aso in case of perfect power contro if the channe gains are correated. 2. SYSTEM MODEL We consider a CDMA system with spreading factor N and users. Each user empoys a transmit antenna array with N T eements sending independent data streams through each of the eements. Thus, we may speak of a system with = N T virtua users. The signa is received by L receive antennas. These antennas can be part of an array or can be paced at different ocations, but processed jointy. The baseband discrete-time system mode, as the channe is fat fading and the system is synchronous, is given by y = Hb + n () where y is the NL-dimensiona vector of received signas, b is the -dimensiona vector of transmitted symbos, and n is discrete-time, circuary symmetric compex-vaued additive white Gaussian noise with zero mean and variance σ 2. The infuence of spreading and fading is described by the NL matrix H = L = (SDΛ ) e (2) where S is the N spreading matrix whose k th coumn is the spreading sequence of the k th virtua user. The diagona square matrix D C contains the transmitted ampitudes of a virtua users such that its k th diagona eement d k is the ampitude of the signa transmitted by the virtua user indexed by k. The diagona matrices Λ,Λ 2,...,Λ L C 2 Note that the convergence to the inear MMSE performance is very fast [9], exponentia in the number of stages [0]. take into account the effect of the fat fading channe. The k-th diagona eement of Λ is the channe gain between the transmitting antenna eement of the k th virtua user and the th receive antenna and wi be denoted by λ k in the foowing. The channe gains can be, in genera, correated and contain ine of sight components as in Rice channes. e is the L-dimensiona unit coumn vector whose eements are zero except the th that equas, i.e. e = (δ j ) L j=. In order to simpify notation, it wi be hepfu in the foowing to define the L-dimensiona vectors k = d k [λ k,λ 2k,...,λ Lk ] T, k =,..., and the diagona square matrices L = DΛ, =,...,L. Let us consider the empirica joint distribution function of the random variabes (,k, 2,k,... L,k ), k =,..., F () L,L 2,...L L () = k= ( k ) (3) where ( ) is the L-dimensiona indicator function, i.e. (x) = L i= (x i) with (x) = for x 0 and (x) = 0 esewhere. In the asymptotic design and anaysis carried out in this work, we assume that the sequence of the empirica joint distribution functions {F () L,L 2,...L L ()} converges weaky with probabiity to a imit distribution function F L,L 2,...L L () with bounded support. In the foowing, the spreading matrix is modeed as a random matrix whose eements are independent and identicay distributed (i.i.d.) with zero mean and variance N. Moreover, we assume the transmitted symbos to be uncorreated random variabes with zero mean and unit variance, i.e. E{bb H } = I. For carity sake, we adopt the foowing notation: β = N is the system oad; h k denotes the k th coumn of H; T = HH H ; R = H H H. 3. MULTISTAGE DETECTION The design of mutistage detectors with universa weights for mutiuser MIMO systems with correated spatia diversity foows aong the design of mutistage detectors for synchronous CDMA systems with singe antenna in [7]. The mutistage detectors Type J-I perform the projection onto the ryov subspace χ M,k (H) = span(t m h k ) jointy for a users and the subsequent fitering individuay for each user. The Type J-I detector for user k is defined as bk = w k,m h H k Tm y (4) where M < is an integer and w k,m are the universa weights. The universa weights w k,m are obtained as w k,m = im w k,m(n) N β N, where w k,m (N) are the taiored fiter coefficients minimizing the mean square error (MSE) E{ b k w k,m(n)h H k Tm y 2 }. The taiored weight

3 w k,m (N) is the (m + ) st eement of the vector w k (N) given by w k (N) = Ξ k (N)ξ k (N) where Ξ k (N) = ( (R i+ j ) kk + σ 2 (R i+ j ) kk )i, j=...m and ξ k (N) = ( (R j ) ) kk j=...m. The matrix form of Type J-I detector for the joint projection is given by b = W m R m H H y where W m is a diagona matrix whose k th diagona eement coincides with w k,m. Type J-J detectors perform the projection onto χ M,k (H) and subsequenty fiter a projections with the same fiter coefficients. They are defined as b pe = w m R m H H y where the scaar w m are the universa weights of Type J-J detectors. The universa weights are obtained as w m = im w m (N) N β N, where w m (N) are the taiored fiter coefficients minimizing the MSE E{ b w m(n)r m H H y 2 }. The taiored weight w m (N) is the (m + ) st eement of the vector w(n) given by w k (N) = Ξ (N)ξ (N) with Ξ(N) = ( trace(r i+ j ) + σ 2 trace(r i+ j ) ) i, j=...m and ξ (N) = ( trace(r j ) ) j=...m. The design of universa weights reduces to the computation of the asymptotic vaues R m kk, = im =βn (R m ) kk for Type J-I detectors and to the computation of m m R = im =βn trace(r m ), the asymptotic eigenvaues moments of R, for Type J-J detectors. The foowing theorem shows that (R m ) kk converges amost surey to a deterministic vaue conditionay on k. Theorem Let S be an N compex matrix with random i.i.d. zero mean entries with variance E{ s i j 2 } = N, and im N E{N 3 s i j 6 } < +. Let k be the vector of the received ampitudes of the virtua user k. Let us assume that, amost surey, the empirica joint distribution of, 2,... converges to some imiting joint distribution F (, 2,..., L ) with bounded support as. L, =,...,L, is a diagona matrix whose k th eement coincides with the th component of k, i.e. (L ) kk = ( ) k. Define H = L = SL e and assume that the spectra radius of the matrix R = H H H is upper bounded. Then, as N, with N β and L fixed, the diagona eements of the matrix Rm corresponding to the virtua user k, with given fading ampitude k, converges with probabiity to the deterministic vaue R m ( k ) a.s. = im =βn (Rm ) kk with R m () determined by the foowing recursion m R m () = m T m = g(t m,) = H T m. g(t m s,)r s () βe{r m s () H }T s The recursion is initiaized by R 0 () = and T 0 = I L. The proof is in []. This theorem yieds the foowing coroary to compute m m R. Coroary Let S, H, R, and k be defined as in Theorem. Let the assumptions of Theorem be satisfied. Then, the asymptotic eigenvaue moments of the matrix R are given by m m R = E{R m ()} where R m () is obtained by the recursion in Theorem and the expectation is taken over the imiting joint distribution F (, 2,..., L ) defined in Theorem. Theorem and Coroary yied a simpe agorithm for the computation of R m () and m m R, m Z+. Agorithm st step Let ρ 0 () = and µ 0 = I. th step Define u () = H µ. Define v () = ρ () H and write it as a poynomia in the monomias r...r L L,s... s L L. Define m (r,...,r L,s,...,s L ) = E{ L = r s } and repace a monomias L = r s in v () by the corresponding m (r,...,r L,s,...,s L ). Assign the resut to V. Set ρ () = u s ()ρ s () µ = βv s µ s. Assign ρ () to R (). Write ρ () as a poynomia in,... L,,... L and repace a monomias L = r s in ρ () by the correspondent moments m (r,...,r L,s,...,s L ) and assign the resut to m R. If the channes at the receiving site are independent, the previous agorithm simpifies since the matrix T s, s Z +, is diagona. If the coefficients are asymptoticay independent and identicay distributed as in the micro-diversity scenario anayzed in [3] the imiting diagona eements of the matrix R and the eigenvaue moments m R can be derived from Agorithm in [7] for synchronous singe receiving antenna systems by repacing (i) β with β = LN ; (ii) The received energy of user k at a singe antenna, by the tota received energy of user k at a antennas, H ;

4 (iii) The moments of the received energy at a singe antenna by the moments of the tota received energy at a antennas E{ H s }. This resut can be obtained directy from Theorem in [3] as proposed in [4] or, aternativey, from Agorithm noting that V s is proportiona to the identity matrix and R () is a function of H. In practice, fading ampitudes are often compex Gaussian distributed and correated 3 and their imiting joint distribution is given as f () = π L detc exp ( H C ). (5) In the absence of power contro, i.e. D = I, C is the correation matrix { of} the fading at the receiving side with entries r i j = E λ i λ j. Consider the eigenvaue decomposition C = MΨM H with Ψ = diag(ψ,...,ψ L ) and the change of variabes g = M H and g k = [g k,...,g Lk ] T = M H k creating statisticay independent components in the random vector g. Then, substituting = Mg, k = Mg k and taking into account that g,g 2...g L, the components of g, are independent compex Gaussian variabes with variances ψ,ψ 2,...ψ L, Agorithm can be simpified as foows: Agorithm 2 st step Let ρ 0 (g) = and µ 0, =, for =,...L. th step Define u n, (g) = L = µ n, g 2. Define v n, (g) = ρ n (g) g 2, =,...L and write them as poynomias in the monomias L = g 2r. Define m (r,...,r L ) g = L = E{ g 2r } and repace a monomias L = g 2r in v n, (g), =,...L by the corresponding m (r,...,r L ) g. Assign the resut to V n,, =,...L, respectivey. Set n ρ n (g) = n µ n, = u n s (g)ρ s (g) βv n s, µ s,. Assign ρ n (M H ) to R n (). Write ρ n (g) as a poynomia in the monomias L = g 2r and repace a monomias L = g 2r in ρ n (g) by the correspondent moments m (r,...,r L ) g and assign the resut to m n R. 4. PERFORMANCE ANALYSIS The asymptotic signa-to-interference and noise ratio (SINR) of user k at the output of a mutistage detector with weighting vector w k is given by SINR k = w H k w k Hξ k ξ T k w k ( Ξk ξ k ξ T ) k wk 3 Rayeigh fading vioates the demand for a distribution with bounded support in Theorem. However, it can be approximated arbitrary cosey by a distribution with bounded support. Thus, from an engineering perspective, we need not worry about that fact. SINR db SNR=30 db SNR=5 db SNR=0 db SNR=5 db SNR=0 db u Figure : Output SINR in decibe of a Type J-I detector with M = 4 versus the coefficient u of the inear combination v = uv MAX + ( u)v MIN for severa vaues of the input SNR and correated Rayeigh fading (soid ines) or independent and identicay distributed fading (dashed ines). where Ξ k = im =βn Ξ k (N) and ξ k = im =βn ξ k (N). It speciaizes for the poynomia expansion detector to SINR pe,k = ξ T Ξ Ξ k Ξ ξ ( ξ Ξ ξ ) 2 with Ξ = im =βn Ξ(N) and ξ = im =βn ξ (N). The asymptotic SINR at the output of a MSWF is given by SINR MSWF,k = ξ T k Ξ k ξ k ξ T. k Ξ k ξ k 5. NUMERICAL RESULTS The numerica resuts presented in this work were obtained using L = 3 receiving antennas at the base station and assuming a system oad β = 2. The channe was fat Rayeigh fading with imiting joint distribution (5) and correation matrix [ ] C = In case of correated fading channe, the output SINR of a inear MMSE depends on the direction of the channe gain vector of the user of interest [2]. For correated Rayeigh fading the performance is maximum or minimum when the channe gain vector is parae to some of the eigenvectors of the correation matrix C [2]. The same property hods aso for Type J-I and Type J-J detectors, as verified numericay. Let us denote by MAX and MIN the eigenvectors of C corresponding to the maximum and minimum eigenvaues, respectivey. Figure shows the performance of a Type J-I detector with M = 4 as the channe gain vector span the subspace { MIN, MAX }, i.e. it is a inear combination = umax +( u) MIN. The soid ines pot the output SINR as a function of u, the coefficient of the inear combination, for different vaues of the input SNR. The performance is

5 SINR db Poynomia expansion detectors versus MSWF u=0 u=0.5 u= SNR db Figure 2: Asymptotic output SINR in decibe of poynomia expansion detectors (dashed ines) and MSWF (soid ines) with M = 4 versus SNR for severa coefficients u of the inear combination v = uv MAX + ( u)v MIN. maximum when the channe gain vector is parae to MIN and minimum when the the channe gain vector is parae to MAX. The gap between maximum and minimum SINR increases as the input SNR increases. The dashed ines iustrate the performance of the same detector for a mutiuser MIMO system with independent Rayeigh fading for the sake of comparison. For independent Rayeigh fading, the SINR does not depend on the direction of the channe gain vectors and it has an intermediate vaue between the maximum and the minimum SINR for a correated fading channe. In Figure 2 the asymptotic output SINR of poynomia expansion detectors or Type J-J detectors (dashed ines) and of MSWF or Type J-I detectors (soid ines) is potted as a function of the input SNR for three different channe gain vectors (u = 0,0.5,) and perfect power contro. In case of singe receive antenna or mutipe antennas with independent and identicay distributed fading gains, the MSWF and the poynomia expansion detectors are equivaent if perfect power contro is performed [4, 7]. On the contrary, for correated fading channes, even in case of perfect power contro, the MSWF outperforms the correspondent poynomia expansion detector. The gap between the performance of the two detectors increases as the input SNR increases and/or u decreases. 6. CONCLUSIONS In this contribution we propose two mutistage detectors for CDMA systems with random spreading and spatia diversity in the genera case as the channe gains are correated and with ine of sight components. Type J-I detector achieves near inear MMSE performance with the same compexity order per bit as the matched fiter. The resuts presented incude as specia cases the resuts in [4] derived there under the constraints of independence of the channe gains and uniformy distributed phases. A framework for the asymptotic performance anaysis of any mutistage detector with projection onto the ryov subspace χ M,k (H) is aso provided. It is shown that the correation at the transmitting sides does not affect the performance of the arge cass of inear mutiuser detectors under investigation, whie the system is sensitive to the correation at the receiving site. The performance depends on the direction of the channe gain vector of the user of interest. In this scenario, the MSWF outperforms the correspondent poynomia expansion detector aso in case of perfect power contro. 7. ACNOWLEDGEMENTS This work was supported by the French-Austraian Science and Technoogy Programme (FAST). REFERENCES [] G. Foschini and M. Gans, On imits of wireess communications in a fading environment when using mutipe antennas, Wireess Persona Communications, vo. 6, pp , 998. [2] I. E. Teatar, Capacity of muti antenna Gaussian channes, European Transactions on Teecommunications, vo. 0, no. 6, pp , Nov./Dec [3] S. V. Hany and D. N. Tse, Resource pooing and effective bandwidth in CDMA networks with mutiuser receivers and spatia diversity, IEEE Transactions on Information Theory, vo. 47, no. 4, pp , May 200. [4] R. R. Müer and L. Cottateucci, Joint antenna combining and mutiuser detection, in Smart antennas in Europe. State-of-art, ser. EURASIP book series on Appied Signa Processing, aiser, Ed. Hindawi Pubishing Cooperation, [5] L. Li, A. Tuino, and S. Verdú, Design of reduced-rank MMSE mutiuser detectors using random matrix methods, IEEE Transactions on Information Theory, vo. 50, no. 6, pp , June [6] J. S. Godstein and I. S. Reed, Reduced rank adaptive fitering, IEEE Transactions on Signa Processing, vo. 42, no. 2, pp , Feb [7] L. Cottateucci and R. R. Müer, A systematic approach to mutistage detectors in mutipath fading channes, IEEE Transactions on Information Theory, vo. 5, no. 9, pp , Sept [8] S. Moshavi, E. G. anterakis, and D. L. Schiing, Mutistage inear receivers for DS CDMA systems, Internationa Journa of Wireess Information Networks, vo. 3, no., pp. 7, Jan [9] M. Honig and W. Xiao, Performance of reduced rank inear interference suppression, IEEE Transactions on Information Theory, vo. 47, no. 5, pp , Juy 200. [0] P. Loubaton and W. Hachem, Asymptotic anaysis of reduced rank Wiener fiters, in Proc. of IEEE Information Theory Workshop (ITW), Paris, France, Apr. 2003, pp [] L. Cottateucci, Low compexity mutiuser detectors with random spreading, Ph.D. dissertation, TU Wien, Vienna, Austria, Mar [2] L. Cottateucci and R. R. Müer, A generaized resource pooing resut for correated antennas with appications to asynchronous CDMA, in Proc. of Internationa Symposium on Information Theory and its Appications (ISITA), Parma, Itay, Oct