Beamforming with Imperfect CSI

Transcription

1 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 007 proceedings Beamforming with Imperfect CSI Ye (Geoffrey) Li School of Electrical Computer Engineering Georgia Institute of Technology, Georgia, USA Anthony C K Soong, Yinggang Du, Jianmin Lu Huawei Technologies, Inc Huawei North American Division, Texas, USA Abstract With channel state information (CSI) at the transmitter, beamforming can be used for spatial diversity multiple spatial access Due to latency feedbac bwidth limitation, the CSI at the transmitter is often nown with some ambiguity In this paper, we develop a robust method for downlin beamforming that taes the ambiguity of the CSI into consideration It is shown by computer simulation that, compared with the existing method, the required signal-to-noise ratio (SNR) for a 1% bit-error rate () is reduced by over db for a system with 4 transmit antennas users when the variance of the CSI is -0 db The performance gain increases with the number of transmit antennas when the number of users is fixed The required SNR for a 1% is reduced by over 4 db if the the number of transmitter antennas is 8 We also study the impact of power allocation on the downlin beamforming Key Words: beamforming, partial CSI, transmitter diversity, spatial multiplexing I INTRODUCTION Multiple antenna techniques has emerged as one of the most significant enabling technologies in beyond 3G wireless communication systems In these systems, multiple antennas can be mounted at the base station for performance capacity improvement However, only a few antennas can be used at the mobile stations because of their size power consumption limitations In this case, downlin beamforming can be used for spatial diversity multiplexing provided that certain channel state information (CSI) is available at the base station In the past several years, downlin transmit beamforming has been investigated for spatial diversity multiplexing, which has been summarized in [1], [] In [], the authors have focused on the capacity improvement of downlin beamforming when different types of CSI statistics are available at the base station In [3], [4], [5], optimal downlin beamforming has been studied for users with different requirements of signal-to-interference-plus-noise ratio (SINR) when exact channel statistics are nown to the transmitter A multi-user beamforming scheme to maximize signal-to-leaage ratios (SLR) [6] has been proposed when exact CSI is available at the transmitter Due to channel estimation error, quantization Ye (Geoffrey) Li is also holding visiting chair professorship at Communications Laboratory at UESTC, Chengdu, China The wor was done when he consulted YGL Telecomm Lab, Inc on the MIMO wireless communication project supported by Huawei Technologies, Inc error (for low data-rate feedbac), delay in feedbac, the CSI or channel statistics available at the transmitter for beamforming are imperfect For transmitter with corrupted channel statistics (co-variance matrix), robust beamforming [8] optimal power allocation [7] have be proposed In this paper, we focus on robust downlin beamforming when imperfect CSI is available at the base station The rest of this paper is organized as follows After formulating robust downlin beamforming problem in Section, we present our schemes for downlin beamforming power allocation for users with only one antenna when imperfect CSI is available at the base station in Section 3 Then we present computer simulation results in Section 4 The paper is concluded in Section 5 II SYSTEM DESCRIPTION Figure 1 shows a down-lin cellular system with spatial multiplexing diversity In the system, there are N T transmit antennas at the base station K (K N T ) active users, each with one receive antenna, sharing the same frequency b by means of spatial division multiple access Without loss of generality, the fading of the channel is assumed to be flat the CSI is fed bac to the base station for beamforming Denote h i as the (complex) channel gain corresponding to the i-th transmit antenna at the base station the -th user Furthermore, we assume that h i s are (circular) complex Gaussian with zero mean unit variance are independent for different i s or s The CSI estimated at the receiver is fed bac to the base station Due to channel estimation error, quantization error (for low data rate feedbac), delay in feedbac, the CSI available at the base station for beamforming is imperfect can be expressed as ĥ i = h i + e i, e i is the error in the CSI, which is assumed to be complex Gaussian with zero mean variance σh, independent of h i, independent identically distributed (iid) for different i s s Denote ĥ = Then, ĥ 1 ĥ NT, h = h 1 h NT, e = e 1 e NT ĥ = h + e (1) /07/$ IEEE 1157

2 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 007 proceedings Base Station h1 h suser s 1 if the steering vector, u, is used Consequently, given imperfect CSI at the base station, the average power of the desired signal component of user is λ = E (ĥ e ) T u = ĥt u + σ h u (5) hk suser s 1 Similarly, the interference power at the -th user from the transmitted symbol for the n-th user can be found to be λ n = ĥt u n + σ h u n (6) Fig 1 Beamforming with partial CSI suser s 1K With imperfect CSI at the base station, a beam is formed for each user for spatial diversity Denote u = (u 1,,u NT ) T the steering vector for the -th user The transmitted signal vector at the base station can be expressed as K x = u s, () =1 s is the transmitted symbol for the -th user, which is assumed to be iid with zero mean unit variance The received signal at the -th user will be r = h T u s }{{} desired signal + l = h T u l s l } {{ } MUI + n }{{}, (3) AWGN the first term is the desired signal component, l = ht l u s l is the multi-user interference (MUI), n is the additive white Gaussian noise (AWGN) at the -th user, which is assumed to be with zero mean variance σn From the received signal in (3), the instantaneous SINR will be h T γ = u h T u (4) l + σn l = III BEAMFORMING WITH IMPERFECT CSI In this section, we discuss beamforming with imperfect CSI investigate power allocation for further performance improvement A Beamforming At the base station, only imperfect CSI, ĥi is observed The observed signal component for user will, therefore, be h T u =(ĥ e ) T u, Ideally, we should select the steering vectors, u for = 1,,K, to maximize the minimum of the observed SINR at the transmitter; that is, choose u for =1,,K to maximize { } γ(u 1,, u K ) = min 1 K λ l = λ l + σ n However, the max-min or min-max problem is usually untractable has no closed form solution The optimal solution can, therefore, only be obtained by iteration [5], which is usually computationally complicated Instead of dealing with the max-min problem, we choose each steering vector to maximize the ratio of the signal-toleaage ratio(slr), as in [6], that is, γ (u )= λ l = λ (7) l Using (5) (6), (7) can be further expressed as ĥt u + σh u γ (u ) = ( ) ĥ T + σh u l = l u = uh R su u H R iu, (8) R s = ĥ ĥt + σ hi, R i = l = ĥ l ĥt l +(K 1)σ hi It can be easily seen that both R s R i are positive definite if σh 0 Let the eigen-decomposition of R i be d R i = U 0 d 0 UH, 0 0 d K d l are all positive U is a unitary matrix Denote or v = D U H u, u = U D 1 v, 1158

3 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 007 proceedings Then d D = 0 d d K D 1 = d d 1 K d 1 γ (u )= vh D 1 UH R su D 1 v v Let γ o be the largest eigen-value of D 1 UH R su D 1 v o be the corresponding eigen-vector Then, γ (u ) reaches its maximum value, γ o, when u taes u o = U D 1 v o (9) Usually, the vector calculated from (9) is not necessarily normalized Steering vector can be obtained by normalizing u o B Optimum Power Allocation In the previous section, we have discussed steering vector optimization for each user to maximize the ratio of the desired signal power the leaage power (the overall power of interference to other users) In this section, we will discuss transmission power allocation among active users to further optimize the whole system Without loss of generality, we assume that the steering vectors obtained in the previous section has already been normalized, that is, u o =1 There are two different ways to optimize the power allocation We can minimize the total transmission power for a given SINR constraint of each user [5], [7] Alternatively, we can also maximize the SINR for a given total transmission power of all active users We will focus on the later one here Assume that the average transmission power of each user is unit Then for a system with K active users, the total transmission power will be K Let the transmission power for user be p Then K p = K p > 0 =1 As before, denote λ n be the power of interference/desired signal when the optimum steering vector is used Then the SINR for the -th user will be p λ γ (p 1,,p K )= l = p lλ l + σn It can be easily seen that we can always adjust the power distribution to increase the lower γ (p 1,,p K ) as long as they are not equal As a result, all γ (p o1,,p ok ) s must be equal for optimal power allocation Therefore, the optimal SINR, γ o, power allocation can be obtained by the following identities p o λ = γ o p ol λ l + γ o σn, (10) for =1,,K Denote l = K p o = K =1 Λ d = diag{λ 11,,λ KK }, λ 11 λ K1 Λ =, λ 1K λ KK the optimal power allocation vector p o =(p o1,,p ok ) T Then (10) can be expressed into a more compact form as ( (1 + 1 ) )Λ d Λ p o = σ γ n1, (11) o From (11), p o = σ n 1 T p o = K, 1 =(1,, 1) T }{{} K1 s ( (1 + 1 γ o )Λ d Λ) 1 1 (1) Then the optimal SNIR (γ o ) is determined by the following identity σ n1 T ((1 + 1 γ o )Λ d Λ) 1 1 = K (13) Once γ o is determined by (13), the optimal power allocation can be found from (1) When system is interference limited, that is, the signalto-noise ratio is very large, or σ n 0, the optimum power allocation approach can be simplified In that case, denote Then (11) turns into p o = Λ d p o, Λ = ΛΛ 1 d (1 + 1 γ o ) p o = Λ p o Note that Λ is a non-negative matrix From [3], [9], there is an nonnegative vector, p o, with p o 0 for =1,,K such that ρ p o = Λ p o Then the optimum power allocation will be p o = Λ 1 d p o, 1159

4 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 007 proceedings the optimum signal-to-interference ratio (SIR) in this case will be γ o = 1 ρ 1 TSS,K= Proposed,K= IV SIMULATION RESULTS In this section, we demonstrate the performance of the proposed algorithm by computer simulation In our simulation, channel gains corresponding to different pairs of transmit receive antennas, h i s, are assumed to be independent, complex (circular) Gaussian with zero mean unit variance Only imperfect CSI is available at the transmitter for downlin beamforming The transmitted symbols are independent are romly drawn from 4-QAM constellations, {± 1 ± j 1 }, each with the same probability TSS,K= Proposed,K= TSS,K= Proposed,K= TSS,K= Proposed,K= Fig 3 Performance of downlin beamforming for a system with N T =8 transmit antennas K =users, versus CSI ambiguity when SNR=0 db versus SNR when σ h = 0 db Fig Performance of downlin beamforming for a system with N T =4 transmit antennas K =users, versus CSI ambiguity when SNR=0 db versus SNR when σ h = 0 db A Beamforming with Imperfect CSI Figures -4 compare the performance of the proposed method with the one in [6] for systems with different numbers of transmit antennas users From those figures, we can see that the proposed method is more robust to CSI ambiguity In particular, from Figure, the required SNR for a 1% is reduced by over db compared with the algorithm in [6] for a system with N T =4transmit antennas, K =users, σ h = 0 db The performance gain increases with the number of transmit antennas when the number of users is fixed As we can see from Figure 3, the required SNR for a 1% is reduced by over 4 db if the number of transmit antennas is changed into N T =8in the above environment If we fix the number of transmit antennas increase the number of users, then the performance gain will be reduced, as we can see by comparing Figure 3 Figure 4 B Optimum Power Allocation To demonstrate the effectiveness of optimum power allocation, we consider shadowing in wireless channels In the simulation here, we assume a log-normal shadowing with an 8 db stard deviation The shadowing is same for different transmitter antennas at the same mobile; however, independent for different mobiles Figure 5 demonstrates the performance improvement through power allocation with imperfect CSI for a system with different numbers of transmit antennas users when σ h = 0 db From the figure, we can see significant performance improvement, especially when the SNR is high 1160

5 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 007 proceedings TSS,K=4 Proposed,K=4 without PA,K= with PA, K= TSS,K=4 Proposed,K= without PA,K=4 with PA, K= σ h (db) Fig 4 Performance of downlin beamforming for a system with N T =8 transmit antennas K =4users, versus CSI ambiguity when SNR=0 db versus SNR when σ h = 0 db V CONCLUSIONS In practical mobile communication systems, the base station can have multiple antennas while the hsets can have at most a couple of antennas The multiple antennas at the base station can be used for downlin beamforming for spatial diversity spatial division multiple access To perform downlin beamforming, CSI is usually required at the transmitter, which is usually obtained through feedbac from the receiver Due to bwidth limitation latency, the CSI at the transmitter is usually nown with some ambiguity In this paper, we have proposed a novel method for robust downlin beamforming with imperfect CSI It is shown by computer simulation that the proposed method can significantly improve the system performance compared with the existing one that does not tae CSI ambiguity into consideration in performing beamforming Therefore, the proposed method is a potential technique for future mobile communications Even through we have assumed that there is only one receive antenna at the mobile station in developing the algorithms in the paper, with minor modification, the developed algorithms can be also used for the systems with multiple receive antennas Fig 5 Impact of adaptive power allocation on a system with N T =4 transmit antennas K = users, N T = 8 transmit antennas K =4users when σ h = 10 db REFERENCES [1] Q H Spencer, C B Peel, A L Swindlehurst, M Haardt, An introduction to the multi-user MIMO downlin, IEEE Commun Mag pp 60-67, Oct 004 [] A Goldsmith, S A Jafar, N Jindal, S Vishwanath, Capacity limits of MIMO channels, IEEE J Sel Areas Commun, vol 1, no 5, June 003 [3] H Boche M Schubert, A general duality theory for uplin downlin beamforming, Proc of IEEE 55th Veh Tech Conf, pp , May 00 [4] M Schubert H Boche, Solution of the multiuser downlin beamforming problem with individual SINR constrains, IEEE Trans Veh Tech vol 53, no 1, pp 18-8, Jan 004 [5] M Schubert H Boche, Iterative multiuser uplin downlin beamforming under SINR constrains, IEEE Trans Signal Processing vol 53, no 1, pp , Jan 004 [6] A Tarighat, M Sade, A H Sayed, A multi user beamforming scheme for downlin MIMO channels based on maximizing signal-toleaage ratios, Proc of IEEE International Conf on Acoustics, Speech, Signal Processing (ICASSP 05), pp 18-3, March 005 [7] M Biguesh, S Shahbazpanahi, A B Gershman, Robust downlin power adjustment in cellular communication systems with antenna arrays at base stations, Proc of 003 4th IEEE Worshop on Signal Processing Advances in Wireless Commun, pp , June 003 [8] B K Chalise A Czylwi, Robust downlin beamforming based upon outage probabiltiy criterion, Proc of IEEE 60th Veh Tech Conf, pp , Sept 004 [9] R A Horn C R Johnson, Matrix Analysis, Cambridge University Press,