Complexity reduced zero-forcing beamforming in massive MIMO systems

Complexity reduced zero-forcing beamforming in massive MIMO systems Chan-Sic Par, Yong-Su Byun, Aman Miesso Boiye and Yong-Hwan Lee School of Electrical Engineering and INMC Seoul National University Kwana P.O. Box 34, Seoul 151-600 Korea Email: {parcs7,ysbyun,amanmiesso}@ttl.snu.ac.r, ylee@snu.ac.r Abstract Data gathering is one of the most popular applications in multi-hop wireless sensor networs. Since resources are limited, it is important to efficiently allocate the resource Massive multi-input multi-output (m-mimo) systems can provide a high degree of freedom in signal transmission, enabling to simultaneously serve a number of users with high transmissin capacity. Conventional zero-forcing beamforming (ZFBF) techniques can transmit multi-user signal while completely canceling out interbeam interference. However, they may have implementation difficulty when applied to m-mimo systems mainly due to hugh processing complexity. In this paper, we design a complexity reduced ZFBF scheme by means of sequential interference cancellation. We first determine the beam weight according to the use of conventional maximum ratio transmission () scheme and calculate the corresponding interbeam interference. We calculate so-called an interference cancellation vector by sequentially cancelling out a predetermined number of interference sources in an order of the strongest interference. Finally, we determine the beam weight by adding the interference cancellation vector to the beam weight. The number of interbeam interferences to be cancelled out can be pre-determined taing into consideration of the processing complexity and required performance. As the number of interbeam interferences to be cancelled out increases, the performance of the proposed scheme approaches to that of ZFBF. The numerical and simulation results show that the proposed scheme can achieve about 90% capacity of ZFBF while requiring 2 7% processing complexity of ZFBF in various operating environments with the use of 32 128 transmit antennas. 1 Index terms - massive MIMO, spatial multiplexing scheme, zero-forcing beamforming, maximum ratio transmission, interbeam interference, processing complexity I. INTRODUCTION Wireless data traffic demand has dramatically increased with the activation of smart phones and vitalization of wireless multimedia service in cellular networs. CISCO expected that legacy cellular technologies cannot support future demand along with the use of increased number of mobileconnected devices [1]. To meet the demand, research wors have been investigated in various aspects including the increase of spectrum efficiency, spectrum extension and networ density. Employment of massive multiple-input multipleoutput (m-mimo) systems has received great attention as a ey technique for the increase of spectral efficiency in next 1 This research was funded by the MSIP(Ministry of Science, ICT & Future Planning), Korea in the ICT R&D Program 2013. generation wireless communication systems. Furthermore, the m-mimo technique can provide high spectrum efficiency with low energy consumption, maing it quite applicable to green communication systems [2]-[4]. M-MIMO systems can provide a high degree of freedom, enabling to simultaneously serve a number of users with high transmission capacity without the increase of time-frequency resources. However, conventional m-mimo techniques may require for high computational complexity, hampering commercial deployment. Maximum ratio transmission () and zero-forcing beamforming (ZFBF) are widely employed in multi-user MIMO systems. The technique maximizes desired channel quality without high implementation complexity, but it may not provide capacity as much as the ZFBF mainly due to the interbeam interference. It may still suffer from the interference problem even with appropriate user selection [8]-[10]. The ZFBF technique can cancel out the interbeam interference, but it may require very high computational complexity with the use of a large number of antennas [5]. The implementation complexity can be reduced through iterative QR-decomposition (QRD) [12]. The QRD based ZFBF considers full multiplexing order for the user selection. However, it may require an additional combining operation when the multiplexing order is not full. As a consequence, the QRD based ZFBF may require implementation complexity higher than conventional ZFBF since the multiplexing order is not full in m-mimo systems. It is of great concern to reduce the implementation complexity to deploy m-mimo systems in commercial wireless networs. In this paper, we consider complex reduced beamforming in m-mimo systems. We consider the cancellation of interbeam interference by taing into consideration of the tradeoff between the implementation complexity and performance. We can assume that the interference between beams is not equal without the loss of generality. To reduce the implementation complexity, we consider the cancellation of some strong interference. To this end, we first generate multiuser beam weight using conventional technique and then calculate the amount of interference between beams. We only remove a small number of strong interference by means of sequential interference cancellation. Thus, we can employ m-mimo systems with affordable performance and

implementation complexity as well. We show that as the number of interference to be cancelled out, the proposed scheme provides performance approaching to that of ZFBF. The rest of this paper is organized as follows. Section II describes the system model in consideration. Section III describes the proposed multi-user beamforming scheme. Section IV verifies the performance of the proposed scheme by computer simulation. Finally, Section V concludes this paper. II. SYSTEM MODEL For ease of description, we consider the downlin transmission in a multi-user multi-antenna wireless communication system, where a base station (BS) equipped with N T antennas simultaneously serves K users with a single antenna in a user set Ω K. The decription can easily be extented to MU-MIMO environments. We assume that the BS has perfect channel information of all users. When the BS transmits signal to K users in parallel by means of spatial multiplexing, the received signal of user can be represented as y = α h H w s + K l=1,l α h H w l s l + n (1) where α denotes the path loss from the BS to user ; h denote the (N T 1) channel vector from the BS to user, w and s respectively denote the (N T 1) beam-weight vector and the signal to user, n denotes e zero-mean complex circular-symmetric additive white Gaussian noise (AWGN) of user, and the superscript H denotes the complex-transpose of a matrix. III. COMPLEXITY-REDUCED ZFBF SCHEME and ZFBF have been employed for multi-user signal transmission mainly due to the implementation simplicity and good performance, respectively. The beamforming weight of and ZFBF can be determined by, respectively, : F = H = f 1 f 2 f f K (2) ZFBF : F ZF = H ( H H H ) 1 = f1 f 2 f f K (3) where H denotes the N T K channel matrix between the BS and all users, comprising channel vectors h. Finally, the beam-weight can be determined by w = f / f after normalization. The beamforming scheme is simple to implement, but it cannot provide high performance mainly due to the interbeam interference. The ZFBF scheme can be free from interbeam interference, but it requires high computational complexity [5], maing it quite impractical in m-mimo environments. In practice, the is preferred to the ZFBF mainly due to its robusteness and low computational complexity [6]. The complexity of ZFBF can be somewhat reduced using an iterative QR-decomposition (QRD) process to find null space of a complex matrix [7, 12]. However, the QRD based ZFBF can reduce the complexity when the multiplexing order is full. It may require an additional combining operation when the multiplexing order is not full, yielding complexity higher than conventional ZFBF. As a consequence, it may not be applicable to m-mimo, where the multiplexing order is usually lower than the antenna size. We consider to reduce the computational complexity of multi-user beamforming while significantly reducing the interbeam interference. To this end, we consider the use of beamforming vector w determined by the, which may suffer from interbeam interference. To alleviate the interbeam interference effect taing into consideration of implementation complexity, we consider the cancellation of N strong interferences among K 1 interbeam interferences. The number N (< K) can be pre-determined taing into consideration of the tradeoff between the implementation complexity and required performance. The amount of interbeam interference from beam of user to beam of user n can be represented as I,n = α n h H n w 2 where w denotes the beam weight of user, determined by the. Then, the total amount of interbeam interference from beam of user to other (K 1) users can be represented as I = I,n (5) n Ω K, n To maximally reduce the interbeam interference for a given N, we consider sequential interference cancellation from the strongest one. For a given N, we can form a set of users interfered by user, defined by (4) χ,n = {x,1, x,2,, x,n } (6) where x,j ( Ω K ) denotes the user index whose interbeam interference from w are sorted in a descending order (i.e., I,x,1 I,x,2 I,x,N ). We construct an (N + 1) demensional user index vector Z,N of user defined by Z,N = {, χ,n } = {z,1, z,2,, z, }. (7) Let w (z,n) be the interference cancellation vector that can remove the interference from w to user z,n ( Z,N ). We can calculate w (z,n) by finding a vector orthogonal to the channel direction of other users. It can be shown that w (z,n) can be calculated as w (z,n) = h z,n i=1,i n ( h H z,n e z,i ) Hez,i (8) where e z,i is the unit vector which can be obtatined recursively as algorithm 1. The corresponding weight to interference cancellation vector w (z,n) can be determined as β (z,n) = hh z,n w h H z,n w (z,n) (9)

Algorithm 1 Orthonormal vector design for proposed scheme Initialization 1: i := 2 2: e z,1 = w 3: repeat 4: e z,i = h z,i i j=1,j n h z,i i j=1,j n 5: i := i + 1 6: until i = N + 1 ( h H z,i e z,j ) Hez,j (h H z,i e z,j ) Hez,j, (i n) We can generate a new beam weight that does not yield interbeam interference to users in χ,n. The new beam weight vector of user can be obtained by adding the weighted interference cancellation vector defined by w = n=2 β (z,n) w (z,n) (10) to the initial beam weight w. After normalization, we can finally determine the beam weight vector w of user, represented as w = w + w w + w (11) By applying the above process to all users in Ω K, we can newly generate the multi-beam weight. Note that when N = K 1, we can get beam weight corresponding to the ZFBF. We calculate the computation complexity assuming the use of floating-point operation (FLOP) [11]. For example, for (N 1) vectors a and b, (M N) matrix A, and (N L) matrix B, the inner product a H b requires 6N multiplcations and 2(N 1) summations, and matrix product AB requires 6MNL multiplcations and 2ML(N 1) summations. It can easily be shown that the complexity of conventional, ZFBF and QRD based ZFBF beamforming scheme can be represented as, respectively, ψ = K (6N T + 2 (N T 1) + 6N T ) ; (12) ) ψ ZF BF = K{ (24 (K 1) N 2 T + 48(K 1) 2 N T +54(K 1) 3 + 6N T } ψ QRD = 8 (3N T 1) + K 1 (i + 1) { 8N T [N T (i 1)] i=2 +4 [3 (N T (i 1)) 1] +8N T [N T (i 1)] (N T i) } (13) (14) Note that when K < N T, the QRD based ZFBF requires an additional computational complexity of K(N T K + 2) (6N T + 2 (N T 1) + 6N T ). With the use of beam weight determined by the as the initial weight, it can be shown that the proposed scheme requires computational complexity represented as Π prop { = K[(6N T + 2 (N T 1) + 6N T ) } +N (l 1) (G + D) D + B + 6N T + 2N T l=2 +D ], (N 1) where G and B denote the computational complexity for calculating e z,i and the weight of interference cancellation vector, equal to 2N T + (6N T + 2(N T 1) + 6N T ) and 2(6N T + 2(N T 1)) + 2N T, respectively, and D denotes the computational complexity for the normalization, equal to 6N T + 2(N T 1) + 6N T. IV. PERFORMANCE EVALUATION The performance of the proposed schemes is evaluated by computer simulation. For the evaluation, we assume that the channel can be represented as H = H 0 R 1/2, where H 0 is a zero mean i.i.d. complex Gaussian channel matrix with unit variance [13], [14], and the channel correlation has a form of exponential magnitude, i.e., the (i, j)-th component of R is represented as [R] i,j = ρe jϕ i j. Here, the correlation coefficient is ρe jϕ with ϕ being uniformly distributed over [0, 2π] and ρ = 0.7, stated otherwise [15]-[17]. We assume that the channel has flat fading with the simplified path model in [18]. We also evaluate the performance of, ZFBF and QRD based ZFBF schemes for performance comparison. Fig. 1 and 2 depict the spectral efficiency and the complexity of the four schemes according to the interference cancellation number N, when 8 users signals are spatially multiplexed at an average SNR of 10dB in Rayleigh fading channel. It can be seen the QRD based ZFBF have highest complexity due to the matrix inversion operation and additional combining computation. It can also be seen that as the number N increases, the performance of the proposed scheme approaches to that of ZFBF at an expense of increasing computational complexity. It can also be seen that the proposed scheme can Spectral Efficiency[bps/Hz] 31 29 28 27 26 24 23 22 0 1 2 3 4 5 6 7 Interference cancellation number Fig. 1. Spectral efficiency according to N. (15)

10 8 10 9 10 8 0 1 2 3 4 5 6 7 Interference cancellation number 10 3 40 50 60 70 80 90 100 110 120 Number of transmit antennas Fig. 2. Complexity according to N. Fig. 4. Complexity according to the transmit antenna size. Spectral Efficiency[bps/Hz] 50 45 40 35 20 15 10 40 50 60 70 80 90 100 110 120 Number of transmit antennas proposed scheme with N = 4 provides about 90% capacity of ZFBF, while requiring computational complexity equal to about 7% that of conventional ZFBF. V. CONCLUSION We have considered multi-user beamforming with low computational complexity in m-mimo environments. The proposed scheme sequentially cancel out the interbeam interference in an order of strongest one. Thus, it can compromize the trade-off issue between the computational complexity and the performance. The numerical and simulation results show that the proposed scheme can significantly reduce the computational complexity, while providing performance similar to the ZFBF in practical m-mimo environments. Fig. 3. Spectral efficiency according to the transmit antenna size. provide almost 90% of ZFBF by only cancelling out three major interference terms (i.e., N = 3), while requiring only 4.5% computational complexity of ZFBF. Fig. 3 and 4 depict the spectral efficiency and the computational complexity according to the antenna size, when 8 users signals are spatially multiplexed using 32, 64, 96 and 128 antennas at an average SNR of 10dB and the number N is set to 3. It can be seen that the proposed scheme can achieve capacity higher than 90 % of ZFBF, while only requiring 2 7 % computational complexity of ZFBF. The reduction of the computational complexity increases as the number of antennas increases. Fig. 5 and 6 depict the the spectral efficiency and the computational complexity according to the multiplexing order, respectively, when the number of antennas is 64 and the number N is set to an integer less than 50% of the multiplexing order. It can be seen that as the multiplexing order increases, the total capacity and the computational complexity increase. It can also be seen that when the multiplexing order is 10, the REFERENCES [1] CISCO Whitepaper, CISCO Visual Networing Index: Global Mobile Data Traffic Forecast Update, 2012-2017, 6 Feb 2013 [2] T. L. Marzetta, How much training is required for multiuser MIMO?, in Proc. of Asilomar Conf. on Sign., Syst. and Computers, 2006, pp.359363. [3] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, Energy and spectral efficiency of very large multiuser MIMO systems, IEEE Trans. Comm., 2012, available: http://arxiv.org/abs/1112.3810. [4] T. L. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Comm., pp. 1-6, Nov. 2006. [5] Q. Spencer, A. Swindlehurst, and M. Haardt, Zero-forcing methods for downlin spatial multiplexing in multiusermimo channels, IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461471, Feb. 2004. [6] H. Yang and T. L. Marzetta, Performance of conjugate and zero-forcing beamforming in large scale antenna systems, IEEE J. Selected Areas in Commun., vol. 31, no. 2, pp. 172-179, Feb. 2013. [7] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. Baltimore, MD: The John Hopins Univ. Press, 1996. [8] Zuang Shenet al., Low complexity user selection algorithm for multiuser MIMO systems with Bloc Diagonalzation, IEEE Trans. Signal Process., vol 54, no 9, Sep. 2006. [9] G. Dimic and N. Sidiropoulos, On downlin beamforming with greedy user selection: Performance analysis and a simple new algorithm, IEEE Trans. Signal Process., vol. 53, no. 10, pp. 38573868, Oct. 2005. [10] Z. Youtuan, T. Zhihua, and Z. Jinang, An improved norm-based user selection algorithm for multiuser MIMO systems with bloc diagonalization, in Proc. VTC 2007-Fall, Oct. 2007, pp. 601605.

Spectral Efficiency[bps/Hz] 35 20 press, 2005. 15 4 5 6 7 8 9 10 Multiplexing order Fig. 5. Spectral efficiency according to the multiplexing order. 10 8 10 3 4 5 6 7 8 9 10 Multiplexing order Fig. 6. Complexity according to the multiplexing order. [11] Raphel Hunger, Floating point operations in Matrix-vector calculus, Techneische universitat Munchen associate institute for signal processing, Technical report, 2007. [12] Le-Nam Tranet al., Iterative precoder Design and User Scheduling for Bloc-Diagonalized systems, IEEE Trans. Signal Process., vol 60, no 7, July 2012. [13] C.-N. Chuah, D. Tse, J. Kahn, and R. Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 637650, Mar. 2002. [14] J. Kermoal, L. Schumacher, K. Pedersen, P. Mogensen, and F. Frederisen, A stochastic MIMO radio channel model with experimental validation, IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp.12111226, Aug. 2002. [15] S. Loya, Channel capacity of MIMO architecture using the exponential correlation matrix,, IEEE Commun. Lett., vol. 5, no. 9, pp.369371, Sep. 2001. [16] C. Martin and B. Ottersten, Asymptotic eigenvalue distributions andcapacity for MIMO channels under correlated fading, IEEE Trans.Wireless Commun., vol. 3, no. 4, pp. 13501359, Jul. 2004. [17] K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson, and M. Beach, Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements, in Proc. IEEE GLOBECOM, Nov. 2001, vol. 1, pp. 156160. [18] Andrea Goldsmith, Wireless Communications, Cambridge university