Energy Efficient Bidirectional Massive MIMO Relay Beamforming

Transcription

1 1 Energy Efficient idirectional Massive MIMO Relay eamforming Michal Yemini, Alessio Zappone, Eduard Jorswieck and Amir Leshem Abstract In this paper we investigate the global energy efficiency of a bidirectional amplify-and-forward relay MIMO system. It is assumed that the relay serves two end-users, each requiring a minimum target rate. Two algorithms are proposed, a sub-optimal one with lower complexity, and an optimal one with slightly higher complexity. We present numerical results that compare the two algorithms and exhibit several optimality properties concerning the global energy efficiency function. I. INTRODUCTION The advent of the next generation 5G of cellular networks has created serious sustainable growth concerns. y 00, the number of connected devices is anticipated to exceed 50 billions, and in order to support this many devices, 5G networks will have to provide 1000 times higher rates compared to present systems [1]. However, it is clear that simply scaling up the transmit powers is not a viable option, as this would require an unmanageable amount of energy. Moreover, high energy consumption leads to huge operating costs, shorter battery lifetimes, and detrimental environmental effects. Indeed, one of the requirements of 5G is to achieve the 1000x datarate increase at half of the energy consumption of today s networks [], [3]. This requires a 000x increase in the energy efficiency compared to today s levels, defined as the amount of transmitted data per Joule of consumed energy. In order to meet the 5G requirements, several candidate technologies have been proposed, among which massive multiple-output multiple-output MIMO appears to be one of the most promising. ased on the deployment of a very large number of antennas, this technique provides several advantages, including inter-user interference cancellation/mitigation, noise cancellation, efficient power allocation, and wider coverage [4] [6]. On the other hand, a well-established technology in current networks is relaying, which improves coverage and reliability, and reduces the possibility of deep fade events [7] [14]. Among the different relaying protocols, amplify-andforward AF has the attractive advantage of not requiring the relay to know the users codebooks, thus allowing faster and simpler design and deployment [15]. In particular, a digital implementation of AF has been recently proposed for the LTE-Advanced standard, and is usually referred to as layer-1 Faculty of Engineering, ar-ilan University, Ramat-Gan, Israel. CNIT and University of Cassino, Italy. Communications Laboratory TU Dresden, Germany. The work of M. Yemini, E. Jorswieck, A. Leshem was supported by the Israel-German foundation under grant 143/014. The work of A. Zappone was supported by the PRASG project, funded by the University of Cassino and Southern Lazio. relaying [16]. This AF implementation amplifies the discretetime, base-band version of the received signal, thus enabling the use of sophisticated beamforming strategies. Energy efficiency for AF relaying recently attracted a growing research interest. In [17] a pricing-based approach is employed to come up with energy-saving, distributed, power control algorithms. In contrast, a centralized approach is taken in [18], where a multi-stream MIMO system is considered. oth distributed and centralized resource allocation algorithms are developed in [19], [0], for MIMO relay systems. Relay interference neutralization is studied for MIMO relay networks in [19], [1], []. Power control in one-way and two-way relay channels is investigated in [3], while three-way relay channels are addressed in [4]. However, none of these works considers the impact of the massive MIMO technique on the energy efficiency of relay systems. In this work, we address this gap by developing energyefficient power control algorithms for a system in which two users communicate with each other via a massive AF MIMO relay. The proposed resource allocation algorithms account for the users individual quality-of-service as well by ensuring minimum rate guarantees. The problem is formulated as the maximization of the system global energy efficiency GEE, defined as the system sum-rate over the total system energy consumption. This leads to a non-convex fractional maximization problem, which is tackled by leveraging tools from fractional programming theory [5]. Two algorithms are proposed. The former provides a low-complex, but possibly suboptimal solution, whereas the latter globally solves the problem at the expense of a slightly higher complexity. Notations: Vectors and matrices appear in bold. Let a be a vector, we denote by a T the transpose vector of a and by a the conjugate of a. The conjugate transpose is denoted by a H. Finally, denotes the L norm and F denotes the Frobenius norm. A block diagonal matrix that is composed of the matrices A and is denoted by diaga,. II. PROLEM FORMULATION Consider a communication network composed of two single antenna users, u 1 and u, and a massive MIMO amplify-andforward relay with N antennas. The communication between the users takes place in two stages. First, the two users send their information to the relay; i.e. s 1 and s, respectively. The received signal at the relay is: y r = h 1 s 1 + h s + n r where n r CN 0, σ ri N, h 1, h C N 1 are i.i.d. complex Gaussian vectors with zero-mean and independent real and

2 imaginary parts, and are independent of each other and of the noise. We also denote by ν 1 and ν the mean square values of h 1,i and h,i for any i, so that it holds and E [ h1,i h,i h1,i h,i H ] ν = ν, [ h1 ] H h1 C = E = diagν1i N, νi N. h h After receiving the signal y r the relay transmits the signal: x r = W y r = W h 1 s 1 + h s + n r, with W C N N the relay amplification matrix. Then, for each i, j {1, }, i j, the received signal is y i =h T i x r +n i =h T i W h j s j +h T i W h i s i +h T i W n r +n i 1 where n i CN 0, σi are statistically independent. Denoting by P r the relay transmit power, it holds, P r W, P 1, P = E [ x H ] r x r h 1, h = W h 1 P 1 + W h P + E W n r = W h 1 P 1 + W h P + σ r W F. Letting be the communication bandwidth, the achievable rate of user i at receiver j is 1 R i j = log h T j W h i P i 1 + σj + σ r h T j W + h T, j W h j P j wherein, for each i = {1, }, j denotes the other user s index. Then, the bit-per-joule GEE of the above communication system is [5] GEEW, P 1, P = R 1 W, P 1, P + R 1 W, P 1, P, P 1 + P + P r W, P 1, P + θ wherein θ represents the static circuit power consumption in all hardware blocks of the communication system. The goal of this paper is the maximization of with respect to W, P 1, and P, subject to maximum power constraints and minimum rate constraints. However, in the considered massive MIMO setting, the following main challenges arise. The GEE is not jointly concave in the optimization variables. Moreover, N + variables are to be optimized, which is computationally prohibitive since N takes on large values if massive MIMO relaying is used. In the considered massive MIMO setting, the selfinterference terms in 1 represent a significant impairment. To show this, let us consider the simple case in which W = I N. Then, the amplification coefficient of the direct channel between user i and user j i is at best h T i h j. In contrast, the amplification coefficient of the echo channel is h T. i h i Assuming that hi, h j are zeromean statistically independent complex Gaussian vectors, 1 The factor in the denominator arises because the users transmit only in the first stage of the communication period. when N 1, by the law of large numbers it holds h T i h j = on and h T i h i = on with probability one. That is, the desired signal and the self interference factors are of the same order of N. Thus, the self-interference term h T i W h i must be suppressed. The most common self-interference suppression method in two-way relaying is to let the user cancel out the interference terms. However, this requires the users to know the optimal relay matrix W, which in turn requires global channel state information. However, in this work we assume that the users only have local channel state information and thus do not know the optimal relay matrix W. Alternatively, self-interference cancellation at the users is possible if the relay feeds back the two terms h T i W h i and h T j W h j. However, this would lead to additional feedback overhead. The coming section shows how the relay matrix W can be chosen so as to completely suppresses the self-interference terms, leading to a low-complex power control scheme, and overcoming all above issues. III. EAMFORMING AT THE RELAY Given the setup and challenges described in the previous section, this section develops a relaying protocol that we name eamform and Forward FF, and in particular Zero Forcing and Forward ZFF relay. It should be pointed out that zero forcing ZF techniques are known to be optimal in the high- SNR regime, and are used in many multi-users systems as a standard way to eliminate interference see for example the standard [6]. Finally, as it will be shown, ZF allows reducing the number of optimization variables to only three, regardless of the value of N, thereby significantly simplifying the resource allocation problem. The goal of the relay is to form a beam that amplifies the users signal, while at the same time suppressing the interference suffered by each user due to the echoes of their own amplified signal. Interestingly, the bidirectional relay interference neutralization problem can be formulated as a linear problem. Letting w = vecw, with vec the vectorization operator, we have y i = h T j h T i s j + h T i h T i s i + n T r h T i w + n i. Next, in order to eliminate the self-interference that each user suffers due to the amplified copy of its own signal, the following constraints are enforced: h T j h T i w = 1, i j 3 h T i h T i w = 0, i = 1,. 4 Note that equal right-hand-side for both users are considered in 3, for fairness reasons. Nevertheless, the results to follow can be readily extended to account for different direct channel gains, thus assigning different priorities to the two users. Also, we see that the linear system to be solved admits in general multiple solutions, since there are N variables and equations. In the following, to save power, we determine

3 3 the solution which minimizes W F = w, by solving the problem: ŵ, ˆθ 1, ˆθ = arg min w,θ w 1,θ subject to: h T i h T j w = e jθi, i j h T i h T i w = 0, i = 1,, 5 whose solution can be readily determined as follows. Let M h 1 h, h h 1, h 1 h 1, h h T, θ θ 1, θ and bθ e jθ1, e jθ, 0, 0 T. The matrix W has full row rank with probability one, thus, the matrix M MM H 1 exists with probability one. For every given θ 1, θ the optimal value of ŵ with respect to 5 is ŵθ = M H MM H 1 bθ. It is left to find the optimal θ 1, θ. Denote by M ij the entry of in the ith row and the jth column of M. Let z C and denote by z the angle of z. It can be shown that ŵθ obtains its minimum value whenever θ = θ 1 M 1. We observe that the solution of 5 depends on h 1 and h, but not on the users transmit powers, and hence it can be computed offline, and only once for each channel coherence time. Furthermore, we observe that there is no loss of generality in considering equality to one in 3 since it is always possible to scale the relay matrix as αw, with α a positive scaling factor, which can be optimized for maximal GEE. Thus, we have effectively reduced the number of optimization variables to only three, namely α, P 1, P. IV. ENERGY EFFICIENCY OPTIMIZATION Let b σ r W F, c i σ r h T i W and d i W h i. Then, plugging the ZF matrix in the GEE function yields where GEEα, P 1, P = Rα, P 1, P P 1 + P + P r α, P 1, P + θ, 6 Rα, P 1, P = R 1 α, P 1, P + R 1 α, P 1, P, 7 R i j α, P 1, P = log 1 + αp i σ j + αc j, 8 P r α, P 1, P = αp 1 d 1 + P d + b. 9 Note that the self-interference terms are suppressed in 8, and the GEE maximization problem is formulated as max GEEα, P 1, P α,p 1,P subject to : 0 P i P i,max, i {1, }, 0 αp 1 d 1 + αp d + αb P r,max R 1 α, P 1, P R 1 R 1 α, P 1, P R Observe that the numerator and denominator of the function GEEα, P 1, P are not concave/convex in α, P 1, P and neither is GEEα, P 1, P. Moreover, the third, fourth, and fifth constraints of 10 are not convex constraints in α, P 1, P. For these reasons we can neither directly employ standard convex optimization theory, nor can we directly resort to fractional programming optimization methods such as Dinkelbach s algorithm [7], as they would require a prohibitive computational complexity. Instead, in the rest of this section we develop two alternative optimization methods. The former has a very limited complexity, but does not guarantee global optimality, whereas the latter is able to determine the globally optimal solution, requiring only a slightly higher complexity. A. Alternating Maximization Algorithm We observe that if we fix α, Problem 10 becomes a pseudo-concave maximization with respect to P 1 and P. The same is true with respect to α once we fix P 1 and P. Then, a convenient approach is to resort to the alternating optimization method, solving each subproblem by means of fractional programming methods. This approach is able to monotonically increase the objective of 10 even though no further optimality claim can be made due to the presence of the last three constraints, that are coupled in the different blocks of variables. More formally, the algorithm alternatively solves the following two subproblems: 1 For fixed ˆP 1 P 1,max and ˆP P 1,max the problem max α GEEα, ˆP 1, ˆP subject to : 0 α ˆP 1 d 1 + α ˆP d + αb P r,max R 1 α, ˆP 1, ˆP R 1 R 1 α, ˆP 1, ˆP R 1 11 is a fractional programming maximization problem with linear constraints, concave numerator, and affine denominator. For fixed ˆα, the problem max GEEˆα, P 1, P P 1,P subject to : 0 P i P i,max, i {1, }, 0 ˆαP 1 d 1 + ˆαP d + ˆαb P r,max R 1 ˆα, P 1, P R 1 R 1 ˆα, P 1, P R 1 1 is also a fractional programming maximization problem with linear constraints 3, concave numerator, and affine denominator.. Line Search Algorithm Since the parameter α is a scalar, we can perform a line search over α in order to determine the global solution. In particular, for each value of α the optimal P 1 and P can be found by solving Problem 1 by means of Dinkelbach s algorithm. It is interesting to observe that the complexity of this approach can be significantly reduced by narrowing down the interval in which α can lie, that we define as [l α, u α ]. From the two rate constraints in 10 we find l α max{0, l α1, l α }, l αi R ji 1σi, i, j = 1,, j i. P j,max R ji 1c i For fixed ˆP 1 and ˆP the rate constraints can be expressed as linear constraints of α. 3 For fixed α the rate constraints can be expressed as linear constraints of ˆP 1 and ˆP.

4 GEE [Mbit/J] GEEG [Mbit/J] 4 As for the upper bound u α, lower bounds on αp and αp 1 can be found by the following inequalities log 1 + αp σ1 R 1, log 1 + αp 1 σ R 1. Plugging these lower bounds on αp 1 and αp in the constraint αp 1 d 1 + αp d + αb P r,max yields the following upper bound on α: u α1 1 [ ] P r,max R 1 1σ b d 1 R 1 1σ 1 d Prmax=8.3dm Prmax=10dm Prmax=13.4dm On the other hand, log 1 + αp R 1, αc 1 and thus another upper bound on α is u α log 1 + αp 1 R 1, αc P r,max R 1 1c d 1 + R 1 1c 1 d + b, and u α = min{u α1, u α } upper-bounds any feasible α. Remark 1. After computing the optimal α, P 1 and P, the relay notifies each user of its transmit power. Also, the self interference terms h T i W h i and h T j W h j need not be signaled to the users since they have been suppressed. V. NUMERICAL RESULTS In our numerical simulations we considered a massive bidirectional relay MIMO with bandwidth = 1MHz, and N = 50 antennas, which is in line with massive MIMO operation [8]. Each user has a maximum power constraint of 8dm, i.e., P 1,max = P,max = 8dm. Also, σ 1 = σ = σ r = 9dm, ν 1 = ν, and ν 1/σ 1 = 10d. As stated in Section II, the channel coefficients for each of the channels users to relay are generated according to an i.i.d. circularly symmetric complex Gaussian distribution. The rate constraints of both users are 4.7Mbps. Additionally, θ = 5dm. We also set the tolerance of the Dinkelbach s algorithm for α to be and for P 1, P to be Figure 1 depicts a realization of the optimal GEE, denoted by GEEG, with respect to α for several values of P r,max. It can be seen that GEEG is non-decreasing with P r,max. Moreover, GEEG is not defined for large/small α since the problem is not feasible for these values of α. Figure 1 shows that the function GEEGα may be unimodal for each value P r,max. Finally, as for the low value of α, it should be recalled that α is not directly the relay power, but rather the power scaling to be applied to obtain the desired channel gain. In Figure we evaluate the performance of the alternating algorithm presented in Section IV-A GEEA and the line search algorithm presented in Section IV- GEEG. We report average results over 500 feasible scenarios for all P r,max values in the figure, considering two initializations for the alternating algorithm: 1 the smallest ˆP1 = ˆP such that Problem 1 is feasible; the smallest ˆP1, ˆP such that ˆP 1 /c = ˆP /c 1 and 1 is feasible. The results indicate that line search outperforms the alternating optimization when the 4 A lower tolerance is used for α, since α takes lower values than P 1, P , #10-6 Fig. 1. N=50, comparison among the optimal GEE as a function of alpha for three possible values of P r,max: a 8.3dm b 10dm and c 13.4dm. initial values of ˆP1 and ˆP are not in the vicinity of the optimal solution 5. Instead, the gap between the GEEA and GEEG shrinks as the initial points get closer to the optimal ones. Finally, the saturation of GEEG can be predicted from the definition of the GEE in 6 since the denominator grows linearly while the numerator only grows logarithmically GEEA - init P1 = P GEEA - init P1=c = P=c1 GEEG Prmax [dm] Fig.. N=50, comparison between the alternating algorithm GEEA and the line search algorithm GEEG as functions of P r,max. VI. CONCLUSION This paper has considered the maximization of the energy efficiency of a bidirectional massive MIMO relay system with two end-users, subject to maximum power and minimum rate constraints. Assuming perfect CSI only at the relay, two optimization algorithms have been proposed. The former alternated between two fractional optimization problems which were solved by Dinkelbach s algorithm. The latter is guaranteed to determine the problem global solution by performing a line search over the possible relay transmit power. oth algorithms perform satisfactorily, with the performance of alternating algorithm being affected by the distance between the initialization point and the optimal solution. 5 The considered initializations are such that their distance from the optimal solution decreases as P r,max increases in the considered range.

5 5 REFERENCES [1] Ericsson White Paper, More than 50 billion connected devices, Ericsson, Tech. Rep Uen, Feb [] NGMN, Next generation mobile network alliance 5G white paper, February 015. [3] S. uzzi, C.-L. I, T. E. Klein, H. V. Poor, C. Yang, and A. Zappone, A survey of energy-efficient techniques for 5G networks and challenges ahead, IEEE Journal on Selected Areas in Communications, vol. 34, no. 4, pp. 1 14, April 016. [4] F. Rusek, D. Persson,. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO: Opportunities and challenges with very large arrays, IEEE Signal Processing Magazine, vol. 30, no. 1, pp , Jan 013. [5] J. Hoydis, S. ten rink, and M. Debbah, Massive MIMO in the UL/DL of cellular networks: How many antennas do we need? IEEE Journal on Selected Areas in Communications, vol. 31, no., pp , February 013. [6] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, Massive MIMO for next generation wireless systems, IEEE Communications Magazine, vol. 5, no., pp , February 014. [7] Y. C. Liang and R. Zhang, Optimal analogue relaying with multiantennas for physical layer network coding, in 008 IEEE International Conference on Communications, May 008, pp [8] Y. Jing and H. Jafarkhani, Network beamforming using relays with perfect channel information, IEEE Transactions on Information Theory, vol. 55, no. 6, pp , June 009. [9] A. U. T. Amah and A. Klein, Non-regenerative multi-way relaying with linear beamforming, in 009 IEEE 0th International Symposium on Personal, Indoor and Mobile Radio Communications, Sept 009, pp [10] J. H. Lee and O. S. Shin, Distributed beamforming approach to fullduplex relay in multiuser MIMO transmission, in Wireless Communications and Networking Conference Workshops WCNCW, 01 IEEE, April 01, pp [11] M. Fozooni, M. Matthaiou, S. Jin, and G. C. Alexandropoulos, Massive MIMO relaying with hybrid processing, CoRR, vol. abs/ , 015. [Online]. Available: [1] W. Zhang, M. Noori, Y. Jing, and M. Ardakani, Partial zero forcing for multi-way relay networks, in 015 IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications SPAWC, June 015, pp [13] Y. Dai and X. Dong, Power allocation for multi-pair massive MIMO two-way AF relaying with linear processing, To Appear in the IEEE Transactions on Wireless Communications, 016. [14] Z. Zhang, Z. Chen, M. Shen, and. Xia, Spectral and energy efficiency of multipair two-way full-duplex relay systems with massive MIMO, IEEE Journal on Selected Areas in Communications, vol. 34, no. 4, pp , April 016. [15] L. Sanguinetti, A. A. D Amico, and Y. Rong, A tutorial on the optimization of amplify-and-forward MIMO relay systems, IEEE Journal on Selected Areas in Communications, vol. 30, no. 8, pp , September 01. [16] M. Iwamura, H. Takahashi, and S. Nagata, Relay technology in LTEadvanced, NTT DOCOMO Technical Journal, vol. 1, no., pp. 9 36, 010. [17] S. Ren and M. van der Schaar, Pricing and distributed power control in wireless relay networks, IEEE Transactions on Signal Processing, vol. 59, no. 6, pp , June 011. [18] A. Zappone, P. Cao, and E. A. Jorswieck, Energy efficiency optimization in relay-assisted MIMO systems with perfect and statistical CSI, IEEE Transactions on Signal Processing, vol. 6, no., pp , January 014. [19] A. Zappone, E. A. Jorswieck, and S. uzzi, Energy efficiency and interference neutralization in two-hop MIMO interference channels, IEEE Transactions on Signal Processing, vol. 6, no. 4, pp , December 014. [0] A. Zappone, L. Sanguinetti, G. acci, E. A. Jorswieck, and M. Debbah, Energy-efficient power control: A look at 5G wireless technologies, IEEE Transactions on Signal Processing, vol. 64, no. 7, pp , 016. [1] D. Wu, C. Yang, T. Liu, and Z. Xiong, Feasibility conditions for interference neutralization in relay-aided interference channel, IEEE Transactions on Signal Processing, vol. 6, no. 6, pp , March 014. [] D. Wu, C. Yang, and Z. Xiong, On maximum linearly achievable degrees of freedom region of x x interference network with arbitrary antenna configurations, IEEE Transactions on Signal Processing, vol. 63, no. 14, pp , July 015. [3] C. Sun and C. Yang, Energy efficiency analysis of one-way and twoway relay systems, EURASIP Journal on Wireless Communications and Networking, vol. February, pp. 1 18, 01. [4]. Matthiesen, A. Zappone, and E. A. Jorswieck, Design of 3-way relay channels for throughput and energy efficiency, IEEE Transactions on Wireless Communications, vol. 14, no. 8, pp , August 015. [5] A. Zappone and E. Jorswieck, Energy efficiency in wireless networks via fractional programming theory, Foundations and Trends R in Communications and Information Theory, vol. 11, no. 3-4, pp , 015. [6] IEEE, IEEE Standards Association. [Online]. Available: [7] W. Dinkelbach, On nonlinear fractional programming, Management Science, vol. 13, no. 7, pp , Mar [8] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, Massive MIMO for next generation wireless systems, IEEE Communication Magazine, vol. 5, no., pp , February 014.