Joint Antenna Selection and Grouping in Massive MIMO Systems

Transcription

1 Joint Antenna Selection and Grouping in Massive MIMO Systems Mouncef Benmimoune, Elmahdi Driouch, Wessam Ajib Department of Computer Science, Université du Québec à Montréal, CANADA {benmimoune.moncef, driouch.elmahdi, Abstract Massive MIMO (Multi-Input Multi-Output) is considered as a promising technology for the fifth generation of wireless communication systems (5G). In this paper, we deal with the CSI feedback reduction issue when a base station (BS) equipped with a large number of antennas serves a limited number of receiver nodes disposed in several groups. This paper considers the practical case where spatial correlation exists among the transmit antennas of the BS. We propose a novel scheme that achieves a considerable reduction in CSI feedback overhead communicated by the receiver nodes to the BS. The proposed approach performs a joint antenna selection and grouping to handle the spatial correlation issue. To this end, we propose a low complexity algorithm that runs antenna selection distributively at each group of receiver nodes. We show that the proposed scheme offers enormous reduction in CSI feedback while ensuring acceptable performance in terms of achievable sum-rate and low computational complexity thanks to its greedy nature. Index Terms Massive MIMO, user grouping, antenna selection, channel correlation I. INTRODUCTION Recently, massive MIMO (Multi-Input Multi-Output) technology has attracted a significant interest in the definition of potential technologies for 5G wireless systems [1]. A massive MIMO system refers to a system where a transmitter equipped with a large number of antennas (e.g. tens or hundreds) communicates with a relatively limited number of users [2]. It was shown that the large antenna array at the transmitter can provide high degrees of freedom and thus increase significantly the system capacity [3]. Also, based on random matrix theory, it was demonstrated that massive MIMO systems can achieve the capacity gain with simple and linear signal processing methods [4]. The availability of channel state information (CSI) at the transmitter is essential to fully benefit from massive MIMO systems. This CSI could be obtained either explicitly by feedback in frequency-division duplexed (FDD) mode or implicitly by channel reciprocity in time-division duplexed (TDD) mode. Although most prior work on massive MIMO considers TDD mode, this mode faces serious challenges such as the pilot contamination problem [5] and the need for tight calibration of radio frequency (RF) chains. Therefore, since many contemporary networks are operating in FDD mode, which is generally considered to be more effective for systems with symmetric traffic and delay-sensitive applications [6], it is of great interest to explore effective approaches for obtaining CSI for such mode in massive MIMO systems. However, as CSI feedback overhead is proportional to the number of active antennas at the base station (BS), there has been much research proposing methods to reduce this overhead; two original examples are [7] and [8]. In [7], the authors propose to partition the user space into groups of users with similar transmit correlation in order to reduce the effective channel dimension and consequently reduce the CSI feedback. Based on the grouping process, other proposals have been presented in the literature. For instance, the authors in [9] propose an antenna grouping scheme that maps multiple correlated antenna elements into a single representative value using predefined patterns. On the other hand, in [8], the authors propose an efficient transmit antenna selection strategy to significantly reduce the CSI feedback. In this paper, an FDD-based massive MIMO downlink channel is considered. It also assumes the practical scenario where spatial correlation exists among the BS transmit antennas. A novel scheme that achieves a reduction in the CSI feedback overhead between the BS and the receiver nodes (RNs) is proposed. The proposed scheme performs jointly antenna selection and grouping in order to handle the spatial correlation issue. In fact, this issue, if not taken into consideration when devising the communication techniques, may result in serious sum-rate degradation. Therefore, our proposed scheme disposes the BS antennas to several groups, where each of these groups serves independently a group of RNs. In fact, by allocating a small portion of BS antennas to each group of receivers, the number of channel coefficients required to deliver towards the BS can be reduced substantially, resulting in a significant reduction in the overall feedback overhead. To this end, a low complexity algorithm that performs the antenna assignment is proposed. We show through simulations that the proposed scheme presents interesting performance in terms of the achievable sum-rate, especially in low signal-tonoise (SNR) regime, while reducing dramatically the amount of CSI feedback compared to a conventional zero-forcing beamforming (ZFBF) system. Throughout the paper, lower-case bold letters are used for vectors and upper-case bold letters for matrices; denotes the cardinality of a set or the absolute value of a scalar; I denotes the identity matrix; ( ) H and ( ) T represents the Hermitian and the transpose of a matrix, respectively and denotes the floor function. The remainder of this paper is organized as follows. The system model is given in the next section. The proposed joint /16/$ IEEE

2 antennas and small cells grouping is detailed in Section III. Numerical results are presented and discussed in Section IV, and we conclude with Section VI. II. SYSTEM MODEL We consider a single-cell downlink scenario consisting of one BS and K RNs. The BS is equipped with M antennas whereas each RN has one antenna. The term receiver node is a general term that may designate either a small base station (SBS) (such as a relay node as proposed in the Long term evolution-advanced (LTE-A) standard [10]) or a user equipment (UE). We assume that RNs are predisposed in a given number of groups based on their geographical positions (as depicted in Fig. 1). The RNs in the same group are assumed to be able to communicate directly with each other. In fact, SBSs may communicate both data and control messages using fast and reliable communication links as proposed in the coordinated multipoint (CoMP) technique in LTE-A [10]. Also, UEs can exchange such messages using deviceto-device communication techniques [11]. Since both kinds of communication techniques add delay, a tradeoff between performance gain and communication cost in terms of delay should be efficiently resolved by restricting the number of RNs in a group. Studying the impact of this delay is out of the scope of this paper and will be subject of future work. In this setup, the received signal of the k th RN can be expressed as: y k = h H k x + n k, k =1,...,K, (1) where h k C M 1 is the channel vector of the k th RN which is assumed to be a quasi-static Rayleigh fading channel, x C M 1 is the transmit signal and n k is the noise term according to an independent complex Gaussian distribution with zero mean and unit variance, i.e. CN(0,σn). 2 Considering the utilization of linear beamforming at the BS, the transmitted signal is a summation of the products formed by the desired signal and the associated beamforming vector. Hence, the received signal at the k th RN belonging to group g can be written as: y k = p k h H k v k s k + pi h H k v i s i i =k i g + pj h H k v j s j + n k, (2) g =g j g where v k C 1 K denotes the beamforming vector, s k represents the data symbol signal of the k th RN and i g denotes that RN i belongs to group g. The first term of the right-hand side in (2) represents the desired signal for the k th RN, the second term represents the intra-group interference caused by the transmissions destined to other RN in the same group, the third term represents the inter-group interference caused by the transmissions towards RNs belonging to groups other than g, which we denote in the rest of this paper by I g and the last term is the background noise that is assumed to be additive Fig. 1. Illustration of a downlink cell with three groups of RNs. white Gaussian noise (AWGN). The optimal beamforming for achieving the sum capacity in MIMO downlink channels, when there is no inter-group interference, is dirty paper coding (DPC) but its implementation is unpractical [12]. In this work, the use of linear ZFBF is assumed because of its low complexity and near optimal performance. Since the channels are spatially correlated across the transmit antenna array, the well-known Kronecker correlation model is considered [13], [14]: H =Σ 1/2 R H(iid) Σ 1/2 T, (3) where the elements of H (iid) are independent and identically distributed (i.i.d.), Σ R and Σ T denote the receive and transmit correlation matrices, respectively. To model a typical cellular downlink case, we adopt the one-ring model which is a popular geometry-based stochastic model [13]. In this model, the transmitter BS is elevated and not obstructed by local scattering, whereas RNs may be often placed near ground level and are surrounded by local scatterers. Thus, we have Σ R = I and let Σ = Σ T for notational simplicity. Let θ be the azimuth angle of the RN location, d the distance between the BS and the RN, r the radius of the scattering ring, and Δ the angle spread, which can be approximated as Δ arctan(r/d). The correlation between the channel coefficient 1 m, p M is given by (see [13] and references therein): [Σ] m,p = 1 Δ e jkt (α+θ)(u m u p) dα, (4) 2Δ Δ where k(α) = 2π λ (cos(α), sin(α))t is the vector for a planar wave impinging with angle of arrival α, λ denotes the carrier wavelength, u m and u p are the position vectors of antennas m and p, respectively.

3 By using the Karhunen-Leoeve transform, the channel vector from the BS antenna array to the k th RN can be expressed as: h k = UΛ 1/2 h (iid) k, (5) where U and Λ denote respectively a unitary matrix comprising eigenvectors of Σ and a diagonal matrix with eigenvalues of Σ. Given that the ZFBF is used and assuming perfect knowledge of CSI at the BS, the second term in (2) drops to zero. Therefore, the received signal-to-interference-plus-noise ratio (SINR) at the k th RN can be written as: γ k (h k, v k )= p k h H k v k 2 I g + σn 2, (6) where p k denotes the transmit power towards the k th RN. p k for k =1,...,K must satisfy the following inequality: K p k P, (7) k=1 where P is the total transmit power. Power allocation is performed using the well-known water-filling algorithm. Therefore, the power allocated to the k th RN must satisfy ( p k = μ I g + σn 2 ) + h H, (8) k v k 2 where (x) + is equal to max(0,x), and the water level μ is chosen to satisfy K ( μ I g + σn 2 ) + h H P. (9) k v k 2 k=1 Therefore, assuming that the inter-group interference is treated as noise, the downlink sum-rate can be given by: R sum = K log 2 (1 + γ k (h k, v k )). (10) k=1 III. JOINT ANTENNA SELECTION AND GROUPING In practice, massive MIMO based-fdd system suffers from a large CSI feedback overhead. Indeed, to enable effective downlink precoding at the BS, the RNs need to feed back a huge amount of CSI, which consumes prohibitively uplink resources. On the other hand, with spatial correlation among antennas, which is a practical issue that has to be considered in wireless communication systems design in general and in the massive MIMO context in particular, the system performance is highly degraded especially when ZFBF is used. To reduce the CSI feedback overhead and tackle the spatial correlation issue, we propose in this section a joint antennas selection and grouping algorithm. The basic idea is to divide the BS antennas into several groups. Each group of antennas serves simultaneously a given group of RNs. Therefore, the required amount of CSI to feed back to the BS is reduced substantially, since the antenna selection is performed at the Algorithm 1: Greedy Antenna Selection and Grouping Algorithm (executed at group g) Input: Channels coefficients of RNs in group g; Already selected groups of antennas A g,g <g; Number of antennas to select K g = N/G ; Initialization: A g M\ A g ; g <g t M K g ; g =g 1 while t>k g do 2 maxrate 0; 3 foreach a in A g do 4 R a = R sum (A g \{m}); 5 if R a > maxr then 6 maxr R a ; 7 a wo a; 8 end 9 end 10 A g A g \{a wo }; 11 t t 1; 12 end Output: The set of antennas selected by group g, A g. RNs side. Indeed, instead of communicating all its channel coefficients to the BS, each group of RNs feeds back only the coefficients of the selected antennas. In addition, as it will be shown in the Section IV, the effect of channel correlation is reduced especially in low SNR regime. Let G be the number of the RN groups. Each group of RNs selects a group of antennas to be served with (see Fig. 1). The BS can use N antennas out of its M transmit antennas to serve the G groups. Let A g be the group of antennas selected to serve RN group g. For the sake of fairness, we assume that each group of RNs selects an almost equal number of antennas. Furthermore, since the paper assumes no cooperation between groups, it is difficult to optimize the number of antennas per group for sum rate maximization. Therefore, we impose that A g = N/G, for all g {1,...,G 1} and A G = N (G 1) N/G. A. Antenna Selection and Grouping Algorithm In this subsection, we propose a low complexity greedy algorithm, which aims to select the best antennas by disposing them into groups that serve the predefined groups of RNs. The proposed algorithm is run distributively at each group of RNs in a successive manner. We assume that group g+1 is aware of the decisions made by the g preceding groups, and selects its preferred antennas according to these decisions. Once again, for the sake of fairness and because the order of the groups has a non negligible impact on their performance, we impose that the position of a group in this order changes from one time slot to another in a round robin manner. Hence, group g in time slot

4 t will be g+1 in time slot t+1, and group G becomes group 1. Therefore, because channel conditions change randomly from one time slot to another (according to a Rayleigh fading distribution), long term fairness is ensured. As in our previous work [15], the basic idea of the algorithm is to add iteratively the best antennas, one by one, in a greedy iterative fashion where the decision at each iteration is definitive and cannot be changed in the subsequent iterations. The steps of the proposed algorithm run at group g are presented in Algorithm 1. The algorithm takes as input the channel coefficients between the BS transmit antennas and the antennas of the RNs forming the group as well as the parameter K g = A g. The algorithm constructs the set of selected antennas A g. It starts by initializing A g to the set of available transmit antennas, i.e. A g = M\ g 1 A i where i=1 M is the set of BS antennas. In each iteration, the algorithm iterates over all the antennas in set A g and then removes from A g the antenna without which the system can provide the maximum sum-rate. The sum-rate values are computed taking into account the inter-group interference generated by the g 1 groups that already have selected their antennas. By the end of each foreach loop (lines 3-9), the proposed algorithm removes one antenna and a total of N/G antennas remains in the set A g, the set of antennas that will serve group g and for which the channel coefficient have to be fed back to the BS. B. Feedback Reduction In this subsection, we present a comparison between the feedback resources needed in a conventional massive MIMO ZFBF based system and the proposed scheme. In the former system, each RN estimate all the channel coefficients between its antenna and the BS antennas before transmitting them back to the BS through finite-rate feedback. Due to computational complexity of best codewords search the well-know limited feedback approaches, codebook techniques are poorly adapted to massive MIMO systems. Therefore, the use of large scale antenna arrays constrains the system to deal with a feedback overhead that is proportional to both the number of antennas and RNs. The total number of feedback which are sent to the BS is F conv = M K. After collecting this large number of channel coefficients, the conventional system performs an exhaustive search in order to determine the best antennas to use for transmission introducing another processing delay because of the complexity of this search. On the other hand, the proposed scheme partitions the antenna array into groups of small sizes and then applies ZFBF separately towards each group of RNs. By allocating a small portion of BS antennas to each group of RN, the number of channel coefficients required to deliver towards the BS can be reduced substantially. Accordingly, the number of feedback coefficients forwarded by each RNs group to the BS is F prop (g) =K g N/G (where K g is the number of RNs in group g) with a total of only F prop = N K/G coefficients. it is clear that the proposed scheme provides a drastically reduction of CSI feedback as it will be shown in Section IV. Average Sum Rate (bps/hz) Conventionnal ZFBF Proposed scheme SNR(dB) Fig. 2. Sum-rate performance as a function of SNR for a massive MIMO system with M =32, N =24, K =8and G =2. Furthermore, due to the greedy nature of the algorithm, the complexity of selecting and grouping the antennas is highly reduced at the cost of acceptable performance degradation. IV. NUMERICAL RESULTS In this section, several numerical simulations are performed to evaluate the sum-rate performance and feedback reduction of the proposed scheme. To this end, we consider a system consisting of one BS and several single antenna RNs. The RNs are deployed in a cell sector of 120 degrees. The presented results are averaged over 1000 correlated Rayleigh channel realizations. The other parameters are similar to [16]: θ min = 60,θ max =60, Δ min =5 and Δ max =15. We also assume that the CSI is perfectly known at each RN. In all results, we compare the proposed scheme against the conventional ZFBF system. Fig. 2 presents the performance in terms of average sumrate as a function of SNR. We assume that there are K =8 RNs in the system disposed in two groups with K 1 = K 2 =4. The BS serves the RNs using N =24antennas selected out of M =32transmit antennas. It can be seen from the figure that the proposed scheme outperforms the conventional ZFBF scheme at SNR values below 15 db. However, the two curves cross at an SNR of near 16 db. Indeed, at high SNR, the benchmark performs better than the proposed scheme when the inter-group interference becomes more severe than the effect of channel correlation. Therefore, the inter-group interference forces the sum-rate achieved by the proposed scheme to start saturating. Anyhow, the savings in CSI feedback and computational complexity obtained by the proposed scheme may favor it compared to the conventional ZFBF even for high SNR values beyond 16 db. In Fig. 3, we compare the performance in terms of average sum-rate as a function of number of RNs in the system. In this simulation, the number of RNs varies from 6 to 10. We set N =24, M =32and the SNR to 10 db. The RNs are

5 Average Sum Rate (bps/hz) Proposed scheme Conventionnal ZFBF Number of users K Feedback (Number of channel coefficients) Conventionnal ZFBF Proposed scheme Number of groups G Fig. 3. Sum-rate performance as a function of SNR for a massive MIMO system with M =32, N =24, G =2and SNR=10dB. Fig. 4. Feedback as a function of number of group for a massive MIMO system with M =32, K =8and SNR=10dB. disposed in two groups with K 1 = K 2. We observe that the proposed scheme outperforms the benchmark when K is more than 8 RNs but achieves lower performance for a small value of K. As expected, the conventional ZFBF system suffers from channel correlation, especially when K grows, since the matrix inversion impacts negatively the system sum-rate. Therefore, as the number of RNs grows large, the performance gap between the two schemes becomes more and more important, e.g. the gap attains more than 4 bps/hz for K =10(i.e. the proposed scheme is two times higher than the conventional one). Fig. 4 plots the amount of CSI feedback in terms of the number of channel coefficients transmitted to the BS as a function of the number of RNs groups G. In this simulation, the number of groups varies from 2 to 8 and the SNR is fixed to 10 db. The number of RNs is fixed to K =8with K g = 8/G, g G and K G = K g K g. The BS uses at most N =24out of its M =32transmit antennas. Since K is fixed, the amount of feedback needed by the conventional scheme is the same for all values of G. However, as the value of G increases, the proposed scheme provides a drastic CSI feedback reduction. However, as the inter-group interference impacts highly the sum-rate achieved by the proposed scheme (as noticed from Fig. 2), there is a trade-off between the sum-rate performance and the number of RNs groups. Not forgetting also that the number of users in each group has an impact on the aforementioned trade-off. V. CONCLUSION In this paper, we presented an efficient scheme to reduce the CSI feedback overhead in massive MIMO systems. The studied system is made from one base station equipped with a large antenna array, which serves receiver nodes predisposed in several distinct groups. We considered the practical case where the channels are spatially correlated assuming the wellknown one ring model. The proposed scheme is based on a low complexity greedy algorithm that performs joint antenna selection and grouping in a distributed fashion. The algorithm is run successively in each group of receiver nodes in order to select their best group of antennas. It has been shown using simulations that the proposed scheme outperforms the conventional ZFBF (taken as a benchmark) in low SNR regime. Moreover, it reduces remarkably the CSI feedback overhead compared to the same benchmark. However, this gain comes at the expense of inter-group interference, which causes performance saturation at high SNR. In future work, we will investigate the design of a beamforming scheme that can reduce or eliminate the effect of intergroup interference in order to further improve the sum-rate performance. Another important and practical issue to resolve is the impact of imperfect CSI, which must be studied carefully and taken into account for a robust beamforming design. REFERENCES [1] J. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A. Soong, and J. Zhang, What will 5g be? IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp , June [2] T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov [3] F. Rusek, D. Persson, B. K. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Signal Process. Mag., vol. 30, no. 1, pp , Jan [4] L. Lu, G. Li, A. Swindlehurst, A. Ashikhmin, and R. Zhang, An overview of massive MIMO: Benefits and challenges, IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp , Oct [5] J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, Pilot contamination and precoding in multi-cell TDD systems, IEEE Trans. Wireless Commun., vol. 10, no. 8, pp , Aug [6] P. Chan, E. Lo, R. Wang, E. Au, V. Lau, R. Cheng, W. H. Mow, R. Murch, and K. Letaief, The evolution path of 4g networks: FDD or TDD? IEEE Commun. Mag., vol. 44, no. 12, pp , Dec [7] J. Nam, J.-Y. Ahn, A. Adhikary, and G. Caire, Joint spatial division and multiplexing: Realizing massive MIMO gains with limited channel state information, in CISS 2012, Mar. 2012, pp. 1 6.

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