Minimizing Transmit Power for Cooperative ... - Semantic Scholar
The 10th Annual IEEE CCNC- Wireless Communications Track
Minimizing Transmit Power for Cooperative Multicell System with Massive MIMO Jinkyu Kang and Joonhyuk Kang
Namjeong Lee, Byung Moo Lee and Jongho Bang
Department of Electrical Engineering Korea Advanced Institute of Science and Technology (KAIST) Daejeon, South Korea Email: [email protected] and [email protected]
Samsung Advanced Institute of Technology (SAIT) Samsung Electronics, Yoinin-si, Gyeoggi-do, 446-712, Korea Email: {namjeong.lee, bm37.lee and jh0278.bang}@samsung.com
Abstract—We consider the problem of designing transmit beamformer and power for downlink cooperative base-station (BS) system with a large antenna arrays. Since the design of the beamforming vector at the transmitter requires high computational complexity, in a large antenna arrays, we utilize the zero-forcing transmit beamformer, which is the simplest form and the optimal performance in a large antenna arrays. Therefore, this paper focuses on the design of power allocation with fixed transmit beamformer for minimizing the transmit power while meeting target signal-to-interference-and-noise-ratio (SINR) of each user and power constraints. We consider two scenarios according to the power constraints of cooperative BSs. One scenario is the sum power constraint on the cooperative basestations. In this case, the cooperative BSs share the total available transmit power. However, each BS exists a maximum available transmit power in practical implementations. Thus, we consider a more realistic per BS power constraints. We proposed the solution strategies for both scenarios: For the sum power constraint case, a simple intuitive solution, where the power is allocated without regard to the power constraint until the SINR constraints is satisfied, is presented. For the per BS power constraints case, we use the properties of a large antenna arrays to find the solution of closed form. We also demonstrate, via numerical simulation, the performance of proposed strategy is convergent to the optimal performance which is achieved by using the iterative algorithm.
I. I NTRODUCTION Conventional wireless networks are designed with a cellular architecture in which BSs of different cells communicate with their individual mobiles independently. In conventional cellular systems, signal processing is independently performed on each cell and inter-cell interference is treated as noise. As a result, conventional network can be considerably improved by designing joint signal processing for minimizing or removing inter-cell interference. Cooperation and coordination of BSs for multicell downlink system have been recently dealt in the literature [1]-[5]. Cooperative BSs fully cooperate both the channel state information (CSI) and data for being able to perfectly control the intercell interference from other cell [4]. In other words, they are available at all BSs, who share their antennas together to serve the users jointly. In multicell system, new algorithms are developed through many papers for meeting various proposed performance objectives (e.g. power minimization under given
978-1-4673-3133-3/13/$31.00 ©2013 IEEE
SINR constraints [1], [2], or minimum SINR maximization [5]). This paper considers the total transmit power minimization of various objectives. The power minimization problem for a single cell multi-user multiple input multiple output (MuMIMO) system with a single antenna at each user is researched in [6]. In [4], the optimization problem of cooperative BS system is to minimize the transmitted power subject to same SINR requirements of all users and same power constraints of all BSs. Although above explained cooperative BS system can lead to considerable decreasing the transmit power, it is unlikely that they will be able to carry the exponentially growing the performance gains. Due to this reason, the massive MIMO system which has the large number of antennas is introduced in [7]. In [7], it is particularly interesting that with an infinite number of antennas, the transmit power can be made extremely small and the received signal-to-noise-ratio (SNR) of each user becomes infinite. Due to the large number of antennas, however, the design of the beamforming vector at the transmitter requires high computational complexity. Hence, we utilize the zero-forcing (ZF) beamformer that is the simplest form and becomes optimal in the high SNR [8]. In this paper, we focus on the design of power allocation with fixed transmit beamformer for minimizing the transmit power while meeting target SINR of each user and power constraints in cooperative massive MIMO system. We consider two scenarios according to the power constraints of cooperative BS. In first scenario, the total transmit power of cooperative BSs should be smaller than or equal to the sum power constraint on the cooperative BSs [9], [10]. That is, the cooperative BSs share the transmit power constraint. Since each BS exists a maximum available transmit power in practical implementations, however, the cooperative BSs do not share the total transmit power constraint. Thus, we consider a more realistic per BS power constraints [3]. We proposed the solution strategies for both scenarios: For the sum power constraint case, a simple intuitive solution, where the power is allocated without regard to power constraint until the SINR constraints is satisfied, is presented. For the per BS power constraints case, we use the properties of a large antenna
438
2
Through the received signal yk , the SINR for kth user can be expressed as 2 √ 2 hkj wkj pkj j=1 , ∀k. (4) SINRk = 2 2 K √ 2 hkj wk j pk j + σn k =1,k =k j=1 Fig. 1.
B. Transmit Beamforming
Cooperative BS system with massive Mu-MIMO
arrays to find the solution of closed form. The remainder of the paper is organized as follows. Section II contains the system model and ZF transmit beamforming. In section III, we consider the power minimization problem with the sum power constraint in cooperative massive MIMO system. The total power minimization with the per BS power constraints is treated in Section IV. Section V provides numerical results. Concluding remarks are made in Section VI. II. S YSTEM M ODEL A. Channel Model Consider a cooperative BS system with Nt (note that Nt 1) transmit antennas at each of J BSs and a single receive antenna at each of K active users as illustrated in Figure 1. We assume that the channels between each BS and user are flat fading and independent. Let hkj ∈ C1×Nt denote the channel vector between the jth BS and the kth user whose element is complex Gaussian random variable with zero mean and unit variance. Then, the received signal of the kth user is given by yk =
J
hkj xj + nk ,
(1)
j=1
where nk is the additive white Gaussian noise with variance σn2 and xj ∈ CNt ×1 denotes transmit signal of jth BS, which consists of the linear beamformer and data symbol. In the cooperative BS system, the BSs fully cooperate: both CSI and data are available at all transmitters, who pool their antennas together to serve the users jointly. That is, the transmit beamforming for multiuser downlink is employed at each base-station. Let us suppose sk and wkj ∈ CNt ×1 as the data symbol for kth user with sk = 1 and the beamformer performed at the jth BS for kth user with wkj = 1, respectively. Then, the transmit signal xj is of the form xj =
K
√
wkj pkj sk ,
(2)
k=1
where pkj is the transmission power at the jth BS for kth user. The received signal at the kth user denotes a summation of the intended signal, intra-cell interference and noise: yk =
J j=1
√ hkj wkj pkj sk +
K
J
k =1,k =k j=1
√ hkj wk j pk j sk +nk (3)
We consider the beamforming vectors wkj and power allocation pkj for cooperative BS system in a large antenna arrays. In a large antenna arrays, the received SNR is proportional to the number of transmit antenna [7]. Although massive MIMO system has high received SNR, the design of the beamforming vector at the transmitter requires high computational complexity in massive MIMO system. Throughout the whole paper, hence, we utilize the ZF beamformer which is the simplest form and has the optimal performance for high SNR [8]. Let Fj denote the ZF beamformer transmitted in the jth BS, i.e, Fj = [f1j , · · · , fKj ] ∈ CNt ×K . The ZF beamformer Fj is designed for removing the inter-user interference as follows: H −1 , F j = HH j (Hj Hj )
(5)
where Hj ∈ CK×Nt is the channel vector between the jth BS and all users. Since the transmit beamforming vector wkj is normalized, then, that is denoted as wkj =
fkj . fkj
(6)
Therefore, the received signal of the kth user is simplified by removing the inter-user interference as follows: J √ pkj yk = sk + nk , ∀k. (7) f kj j=1 Since the inter-user interference is removed by ZF beamformer, then, the SINR for the kth user can be simply expressed as: J √ 2 pkj . SINRk = (8) j=1 fkj III. T OTAL P OWER M INIMIZATION WITH S UM P OWER C ONSTRAINT In this section, we first consider the sum power constraint scenario which is to share the available powers of each BS. The goal is to minimize the total transmit power for cooperative BS system while meeting individual SINR constraints of each user in a large antenna arrays. As mentioned Section II, the ZF beamformer is an appropriate beamforming technology in a large antenna arrays. Therefore, the focus of this section is to design the power allocation pkj with ZF transmit beamforming vector for minimizing the transmit power while satisfying the SINR requirements of each user and sum power constraint. We find the optimum allocation of
439
3
power by a solution of the problem which satisfies the KKT conditions of the original problem. The original problem for power minimization can be formulated as min
w,p≥0
s.t
J K
pkj
k=1 j=1
SINRk ≥ γk , 1 ≤ k ≤ K J K pkj ≤ P ,
general. Therefore, we introduce the power allocation pkj to minimize the power while satisfying SINR requirements of each user and per BS power constraints. In this case, the transmit beamforming vector is also fixed for ZF beamformer. Let P j denote the available transmit power of the jth BS. Then, we can formulate a total transmit power minimization problem with the SINR and per BS power constraints as follows:
(9)
p≥0
k=1 j=1
where γk and P denote the target SINR for the kth user and the available total power of BSs, respectively. As shown in this problem, the objective function is equal to the left polynomial of the sum power constraint. If the constraints are feasible, due to this reason, we can consider only individual SINR constraints except for sum power constraint. Without loss of generality, it is assumed to hold that the set of target SINRs are feasible throughout this paper. Since the solution is intuitively extended according to the number of cooperative BS, we solved the optimal problem (9) in case of two cooperative BSs for simplicity. By fixing the transmit beamforming vector and the number of cooperative BS, therefore, we can simply reformulate above problem as follows: min p≥0
s.t
2 K
K J
min s.t
SINRk ≥ γk , 1 ≤ k ≤ K K pkj ≤ P j , 1 ≤ j ≤ J.
(12)
k=1
In scenario of per BS power constraint, we can also assume two cooperative BSs by the reasons in Section III. Therefore, this problem is also reformulated as follows: min p≥0
s.t
2 K
pkj
k=1 j=1
2 √ 2 pkj ≥ γk , 1 ≤ k ≤ K j=1 fkj K
pkj
pkj ≤ P j ,
1 ≤ j ≤ 2.
(13)
k=1
k=1 j=1
2 √ 2 pkj ≥ γk , 1 ≤ k ≤ K. j=1 fkj
pkj
k=1 j=1
(10)
Since all parameters in this problem are a constant, we can know easily that this power minimization problem is the convex problem. By the method of Lagrange multipliers which is generalized by the KKT conditions, therefore, it can be seen that the optimal power allocation with the given ZF beamformer is denoted as ⎛ ⎞2 2 2 f f kj kj ⎟ ⎜ ⎜ j =1,j =j ⎟ pkj = ⎜ (11) ⎟ γk , ∀k, j. 2 ⎠ ⎝ 2 fki i=1
If the number of cooperative BSs is more than 2, the solution easily produces more complex results by the method of Lagrange multipliers. Therefore, we do not present the optimal power allocation in above case for lack of space. IV. T OTAL P OWER M INIMIZATION WITH P ER BS P OWER C ONSTRAINTS The BSs in the cooperative BS system fully cooperate both the CSI and data, but it does not denote sharing the constraints of transmit power at each BS. Since each BS has a different available transmit power in practical, the cooperative BS system has different power constraints per BS in more
Since this problem (13) has per BS power constraints, it makes more complex problem in comparison with the problem (10). Thus, we consider two scenarios according to target SINR γk and available BS power P j . In other words, first scenario is that all available BS power constraints are satisfied. Next is that an available BS power is not satisfied and so the remaining power of the other BS is used to help the insufficient BS. Depending to two scenarios, we find the two different power allocation methods. A. Satisfied All Power Constraints Scenario This scenario is when all available per BS power constraints are enough in comparison with target SINR constraints of each user. In other words, it is the power minimization problem without consideration for the per BS power constraints. When the transmit powers by the solution (11) satisfy the per BS power constraints, this scenario is considered. The condition is analytically given by ⎛ ⎞2 2 2 fkj f kj K ⎜ ⎟ ⎜ j =1,j =j ⎟ (14) ⎜ ⎟ γk ≤ P j , ∀j. 2 ⎝ ⎠ 2 k=1 fki i=1
In this case, therefore, the solution of optimal power allocation is equal to that of power minimization with sum power constraint (11) in Section III.
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4
B. Unsatisfied Power Constraint Scenario
30
We consider this scenario that the available transmit power of any BS is insufficient for meeting the target SINR of each user. In other words, per BS power constraint of any BS is small compared with the power for satisfying the target SINR of users. Therefore, the other BS which has sufficient available power should use more transmit power for insufficient BS in this scenario. Since this scenario should consider the per BS power constraints unlike previous cases, also, the method to find the solution for minimizing the transmit power should be different to the previous method. The condition of this case is the opposite condition of the first case as follows: ⎛ ⎞2 2 2 fkj f kj K ⎟ ⎜ ⎜ j =1,j =j ⎟ ⎜ ⎟ γk ≥ P j , any j ∈ [1, 2] . 2 ⎝ ⎠ 2 k=1 fki
25
Non-cooperative BS in multicell (Jk =20dB)
i=1
(15) Without loss of generality, we assume that jth BS has insufficient available power. Since using the power of jth BS is more efficient than using the power of the other BS in terms of satisfying the target SINR of each user, it is optimal to use the available all power of jth BS for minimizing the transmit power. Nevertheless, it is complex to find the power allocation method for minimizing the total power due to many and complex constraints in this case. Thus, we consider the property of massive MIMO for obtaining the power allocation method. By Hj HH j ≈ KNt I obtained from the law of large numbers in the massive MIMO [7], the ZF beamformer (5) 1 HH becomes more simply as Fj ≈ KN j . Since the norm value t of each channel vector hkj is approximately same from the law of large numbers, the norm value of all beamformers is also approximately same, i.e, fkj ≈ m, ∀k, j, where m is any constant value. Then, the optimal problem (13) is the convex problem by the above property of massive MIMO. Then, the transmit power which is obtained by solving the sub-optimal problem is allocated as pkj
=
pkj
=
γk P j , ∀k, K γl l=1 ⎞ ⎛ √ pkj √ ⎠ fkj · ⎝ γk − , ∀k, j(j = j), (16) fkj
The above power allocation results does not include the approximate norm value of beamforming due to meeting the target SINR of each user, the approximate norm value of beamforming is used only in the process for finding the power allocation. As mentioned above Section III, the solution easily produces more complex results by the method of Lagrange multipliers when the number of cooperative BSs is more than 2. Therefore, this case have not been discussed here for lack of space.
Cooperative BS based ZF (Jk=20dB) Optimal beamforming in singlecell (Jk=20dB) Non-cooperative BS in multicell (Jk =10dB)
20
Total sum power (dBW)
Cooperative BS based ZF (Jk=10dB) 15
Optimal beamforming in singlecell (Jk=10dB)
10
5
0
-5
-10
-15 20
40
60
80
100 120 140 # of Tx Ant. at each BS
160
180
200
Fig. 2. Performance comparison of total transmit power versus the number of Tx antenna
V. P ERFORMANCE E VALUATION We consider the benefit of proposed algorithm introduced in the previous sections through numerical experiments. In all considered systems, we utilize the power allocation method for meeting the SINR constraints. If we do not consider the power allocation method like equal power allocation, it is infeasible because the SINR constraints of each user are unsatisfied. Throughout this section, also, we only consider the cooperative BS system with per BS power constraints for more generality. We assume that the perfect CSI is available at all BSs and users. In addition, all BSs in the cooperative BS system perfectly cooperate the transmit signal and CSI. Also, we assume that the set of target SINRs and available transmit power of each BS are feasible. In this section, we consider the total transmit power of optimal beamforming in singlecell [6] for the upper bound of the performance in cooperative BS system. If the number of antennas in single cell is equal to that of total antennas of all BSs in cooperative multi-cell and the path loss doesn’t exist, we intuitively know that the performance of single cell is the upper bound of that for cooperative multi-cell with per BS power constraints due to no power constraint in single cell. In Fig. 2, we compare the total transmit power of the proposed algorithm in cooperative BS system with that of the non-cooperative BS system and optimal beamforming in singlecell in case that target SINR is 10 and 20 dB and each available transmit power is 30dBW. The performance gap between proposed method and non-cooperative BS is smaller and smaller according to decreasing the target SINR of each user. The received SINR is proportional to the number of antennas. In case of many transmit antennas, therefore, the performance of proposed algorithm is convergent to the upper bound of that, since the ZF transmit beamformer is near optimal. Therefore, there is little difference of performances between proposed and optimal beamforming algorithm. Since
441
5
1
R EFERENCES
Non-cooperative BS system Cooperative BS system
0.9 0.8
Outage Probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1 10
0
10 Available power P 2 (W)
1
10
Fig. 3. Performance comparison of outage probability versus available power P2
the method of designing the optimal beamformer in single cell is very complex by the iterative algorithm until convergence, however, we can know the superior of proposed algorithm in Fig. 2. Fig. 3 shows the outage probability of proposed power allocation for cooperative BS and power allocation for noncooperative BS system with fixed available power of any BS versus available power of the other BS. Without loss of generality, we assume that 1st BS has fixed available transmit power. The outage probability means the infeasible probability to meet the SINR or power constraints. In Fig. 3, we consider a Nt = 100 transmit antennas, γk = 20dB target SINR of each user and P 1 = 10W fixed available power. As shown the Fig. 3, the outage probability of proposed allocation is convergent to zero at lower available power than that of noncooperative BS system. It shows that the cooperative BS is able to satisfy the target SINR and power constraints at lower available power than non-cooperative BS system.
[1] H. Dahrouj and W. Yu, “Coordinated beamforming for the multi-cell multi-antenna wireless system,” in Proc. Conference on Information Sciences and Systems 2008, CISS 2008, 2008. [2] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell, multi-antenna wireless system,” IEEE Trans. Wireless Commun., May 2010. [3] O. Somekh, O. Simeone, Y. Bar-Ness, A.M. Haimovich, and S. Shamai, “Cooperative multicell zero-forcing beamforming in cellular downlink channels,” IEEE Trans. Inf. Theory, July 2009. [4] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constriants,” in IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2646-2660, June 2007. [5] K. Karakayali et al., “On the maximum common rate achievable in a coordinated network,” in Proc. IEEE International Conference on Communications, ICC, 2006. [6] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 1828, 2004. [7] T. L. Marzetta, “How much training is required for multiuser MIMO?” in Proceedings of the 40th Asilomar Conference on Signals, Systems, and Computers (ACSSC 06), pp. 359363, Pacific Grove, Calif, USA, November 2006. [8] N. Jindal, “High SNR Analysis of MIMO Broadcast Channels”, in Proc. IEEE International Symposium on Information Theory (ISIT), pp. 23102314, Adelaide, SA, 2005. [9] S. A. Jafar, G.J. Foschini, and A. J. Goldsmith, “PhantomNet: exploring optimal multicellular multiple antenna systems,” in Proc. Vehicular Technology Conference 02 Fall, vol. 1, pp. 261-265, Vancouver, BC, Canada, Sep. 2002. [10] S. A. Jafar and A. J. Goldsmith, “Transmitter optimization for multiple antenna cellular systems,” in Proc. IEEE International Symposium on Information Theory (ISIT), p. 50, Palais de Beaulieu, Lausanne, Switzerland, 2002.
VI. C ONCLUSION This paper provides a solution to the optimal power allocation problem for cooperative BS system with massive MIMO to minimize total transmit power. Since ZF transmit beamformer is optimal performance and has low complexity in massive MIMO, we use the ZF transmit beamformer. We formulated the problem for both the sum power constraint and per BS power constraints. In case of sum power constraint, we provide the simple power allocation method while meeting the target SINR constraints. For the per BS power constraints case, we use the properties of a massive MIMO for finding the power allocation algorithm of closed form. The proposed algorithm is efficient, and it has near optimal performance while satisfying the target SINR constraints and per BS power constraints.
442
Minimizing Transmit Power for Cooperative Multicell System with Massive MIMO Jinkyu Kang and Joonhyuk Kang
Namjeong Lee, Byung Moo Lee and Jongho Bang
Department of Electrical Engineering Korea Advanced Institute of Science and Technology (KAIST) Daejeon, South Korea Email: [email protected] and [email protected]
Samsung Advanced Institute of Technology (SAIT) Samsung Electronics, Yoinin-si, Gyeoggi-do, 446-712, Korea Email: {namjeong.lee, bm37.lee and jh0278.bang}@samsung.com
Abstract—We consider the problem of designing transmit beamformer and power for downlink cooperative base-station (BS) system with a large antenna arrays. Since the design of the beamforming vector at the transmitter requires high computational complexity, in a large antenna arrays, we utilize the zero-forcing transmit beamformer, which is the simplest form and the optimal performance in a large antenna arrays. Therefore, this paper focuses on the design of power allocation with fixed transmit beamformer for minimizing the transmit power while meeting target signal-to-interference-and-noise-ratio (SINR) of each user and power constraints. We consider two scenarios according to the power constraints of cooperative BSs. One scenario is the sum power constraint on the cooperative basestations. In this case, the cooperative BSs share the total available transmit power. However, each BS exists a maximum available transmit power in practical implementations. Thus, we consider a more realistic per BS power constraints. We proposed the solution strategies for both scenarios: For the sum power constraint case, a simple intuitive solution, where the power is allocated without regard to the power constraint until the SINR constraints is satisfied, is presented. For the per BS power constraints case, we use the properties of a large antenna arrays to find the solution of closed form. We also demonstrate, via numerical simulation, the performance of proposed strategy is convergent to the optimal performance which is achieved by using the iterative algorithm.
I. I NTRODUCTION Conventional wireless networks are designed with a cellular architecture in which BSs of different cells communicate with their individual mobiles independently. In conventional cellular systems, signal processing is independently performed on each cell and inter-cell interference is treated as noise. As a result, conventional network can be considerably improved by designing joint signal processing for minimizing or removing inter-cell interference. Cooperation and coordination of BSs for multicell downlink system have been recently dealt in the literature [1]-[5]. Cooperative BSs fully cooperate both the channel state information (CSI) and data for being able to perfectly control the intercell interference from other cell [4]. In other words, they are available at all BSs, who share their antennas together to serve the users jointly. In multicell system, new algorithms are developed through many papers for meeting various proposed performance objectives (e.g. power minimization under given
978-1-4673-3133-3/13/$31.00 ©2013 IEEE
SINR constraints [1], [2], or minimum SINR maximization [5]). This paper considers the total transmit power minimization of various objectives. The power minimization problem for a single cell multi-user multiple input multiple output (MuMIMO) system with a single antenna at each user is researched in [6]. In [4], the optimization problem of cooperative BS system is to minimize the transmitted power subject to same SINR requirements of all users and same power constraints of all BSs. Although above explained cooperative BS system can lead to considerable decreasing the transmit power, it is unlikely that they will be able to carry the exponentially growing the performance gains. Due to this reason, the massive MIMO system which has the large number of antennas is introduced in [7]. In [7], it is particularly interesting that with an infinite number of antennas, the transmit power can be made extremely small and the received signal-to-noise-ratio (SNR) of each user becomes infinite. Due to the large number of antennas, however, the design of the beamforming vector at the transmitter requires high computational complexity. Hence, we utilize the zero-forcing (ZF) beamformer that is the simplest form and becomes optimal in the high SNR [8]. In this paper, we focus on the design of power allocation with fixed transmit beamformer for minimizing the transmit power while meeting target SINR of each user and power constraints in cooperative massive MIMO system. We consider two scenarios according to the power constraints of cooperative BS. In first scenario, the total transmit power of cooperative BSs should be smaller than or equal to the sum power constraint on the cooperative BSs [9], [10]. That is, the cooperative BSs share the transmit power constraint. Since each BS exists a maximum available transmit power in practical implementations, however, the cooperative BSs do not share the total transmit power constraint. Thus, we consider a more realistic per BS power constraints [3]. We proposed the solution strategies for both scenarios: For the sum power constraint case, a simple intuitive solution, where the power is allocated without regard to power constraint until the SINR constraints is satisfied, is presented. For the per BS power constraints case, we use the properties of a large antenna
438
2
Through the received signal yk , the SINR for kth user can be expressed as 2 √ 2 hkj wkj pkj j=1 , ∀k. (4) SINRk = 2 2 K √ 2 hkj wk j pk j + σn k =1,k =k j=1 Fig. 1.
B. Transmit Beamforming
Cooperative BS system with massive Mu-MIMO
arrays to find the solution of closed form. The remainder of the paper is organized as follows. Section II contains the system model and ZF transmit beamforming. In section III, we consider the power minimization problem with the sum power constraint in cooperative massive MIMO system. The total power minimization with the per BS power constraints is treated in Section IV. Section V provides numerical results. Concluding remarks are made in Section VI. II. S YSTEM M ODEL A. Channel Model Consider a cooperative BS system with Nt (note that Nt 1) transmit antennas at each of J BSs and a single receive antenna at each of K active users as illustrated in Figure 1. We assume that the channels between each BS and user are flat fading and independent. Let hkj ∈ C1×Nt denote the channel vector between the jth BS and the kth user whose element is complex Gaussian random variable with zero mean and unit variance. Then, the received signal of the kth user is given by yk =
J
hkj xj + nk ,
(1)
j=1
where nk is the additive white Gaussian noise with variance σn2 and xj ∈ CNt ×1 denotes transmit signal of jth BS, which consists of the linear beamformer and data symbol. In the cooperative BS system, the BSs fully cooperate: both CSI and data are available at all transmitters, who pool their antennas together to serve the users jointly. That is, the transmit beamforming for multiuser downlink is employed at each base-station. Let us suppose sk and wkj ∈ CNt ×1 as the data symbol for kth user with sk = 1 and the beamformer performed at the jth BS for kth user with wkj = 1, respectively. Then, the transmit signal xj is of the form xj =
K
√
wkj pkj sk ,
(2)
k=1
where pkj is the transmission power at the jth BS for kth user. The received signal at the kth user denotes a summation of the intended signal, intra-cell interference and noise: yk =
J j=1
√ hkj wkj pkj sk +
K
J
k =1,k =k j=1
√ hkj wk j pk j sk +nk (3)
We consider the beamforming vectors wkj and power allocation pkj for cooperative BS system in a large antenna arrays. In a large antenna arrays, the received SNR is proportional to the number of transmit antenna [7]. Although massive MIMO system has high received SNR, the design of the beamforming vector at the transmitter requires high computational complexity in massive MIMO system. Throughout the whole paper, hence, we utilize the ZF beamformer which is the simplest form and has the optimal performance for high SNR [8]. Let Fj denote the ZF beamformer transmitted in the jth BS, i.e, Fj = [f1j , · · · , fKj ] ∈ CNt ×K . The ZF beamformer Fj is designed for removing the inter-user interference as follows: H −1 , F j = HH j (Hj Hj )
(5)
where Hj ∈ CK×Nt is the channel vector between the jth BS and all users. Since the transmit beamforming vector wkj is normalized, then, that is denoted as wkj =
fkj . fkj
(6)
Therefore, the received signal of the kth user is simplified by removing the inter-user interference as follows: J √ pkj yk = sk + nk , ∀k. (7) f kj j=1 Since the inter-user interference is removed by ZF beamformer, then, the SINR for the kth user can be simply expressed as: J √ 2 pkj . SINRk = (8) j=1 fkj III. T OTAL P OWER M INIMIZATION WITH S UM P OWER C ONSTRAINT In this section, we first consider the sum power constraint scenario which is to share the available powers of each BS. The goal is to minimize the total transmit power for cooperative BS system while meeting individual SINR constraints of each user in a large antenna arrays. As mentioned Section II, the ZF beamformer is an appropriate beamforming technology in a large antenna arrays. Therefore, the focus of this section is to design the power allocation pkj with ZF transmit beamforming vector for minimizing the transmit power while satisfying the SINR requirements of each user and sum power constraint. We find the optimum allocation of
439
3
power by a solution of the problem which satisfies the KKT conditions of the original problem. The original problem for power minimization can be formulated as min
w,p≥0
s.t
J K
pkj
k=1 j=1
SINRk ≥ γk , 1 ≤ k ≤ K J K pkj ≤ P ,
general. Therefore, we introduce the power allocation pkj to minimize the power while satisfying SINR requirements of each user and per BS power constraints. In this case, the transmit beamforming vector is also fixed for ZF beamformer. Let P j denote the available transmit power of the jth BS. Then, we can formulate a total transmit power minimization problem with the SINR and per BS power constraints as follows:
(9)
p≥0
k=1 j=1
where γk and P denote the target SINR for the kth user and the available total power of BSs, respectively. As shown in this problem, the objective function is equal to the left polynomial of the sum power constraint. If the constraints are feasible, due to this reason, we can consider only individual SINR constraints except for sum power constraint. Without loss of generality, it is assumed to hold that the set of target SINRs are feasible throughout this paper. Since the solution is intuitively extended according to the number of cooperative BS, we solved the optimal problem (9) in case of two cooperative BSs for simplicity. By fixing the transmit beamforming vector and the number of cooperative BS, therefore, we can simply reformulate above problem as follows: min p≥0
s.t
2 K
K J
min s.t
SINRk ≥ γk , 1 ≤ k ≤ K K pkj ≤ P j , 1 ≤ j ≤ J.
(12)
k=1
In scenario of per BS power constraint, we can also assume two cooperative BSs by the reasons in Section III. Therefore, this problem is also reformulated as follows: min p≥0
s.t
2 K
pkj
k=1 j=1
2 √ 2 pkj ≥ γk , 1 ≤ k ≤ K j=1 fkj K
pkj
pkj ≤ P j ,
1 ≤ j ≤ 2.
(13)
k=1
k=1 j=1
2 √ 2 pkj ≥ γk , 1 ≤ k ≤ K. j=1 fkj
pkj
k=1 j=1
(10)
Since all parameters in this problem are a constant, we can know easily that this power minimization problem is the convex problem. By the method of Lagrange multipliers which is generalized by the KKT conditions, therefore, it can be seen that the optimal power allocation with the given ZF beamformer is denoted as ⎛ ⎞2 2 2 f f kj kj ⎟ ⎜ ⎜ j =1,j =j ⎟ pkj = ⎜ (11) ⎟ γk , ∀k, j. 2 ⎠ ⎝ 2 fki i=1
If the number of cooperative BSs is more than 2, the solution easily produces more complex results by the method of Lagrange multipliers. Therefore, we do not present the optimal power allocation in above case for lack of space. IV. T OTAL P OWER M INIMIZATION WITH P ER BS P OWER C ONSTRAINTS The BSs in the cooperative BS system fully cooperate both the CSI and data, but it does not denote sharing the constraints of transmit power at each BS. Since each BS has a different available transmit power in practical, the cooperative BS system has different power constraints per BS in more
Since this problem (13) has per BS power constraints, it makes more complex problem in comparison with the problem (10). Thus, we consider two scenarios according to target SINR γk and available BS power P j . In other words, first scenario is that all available BS power constraints are satisfied. Next is that an available BS power is not satisfied and so the remaining power of the other BS is used to help the insufficient BS. Depending to two scenarios, we find the two different power allocation methods. A. Satisfied All Power Constraints Scenario This scenario is when all available per BS power constraints are enough in comparison with target SINR constraints of each user. In other words, it is the power minimization problem without consideration for the per BS power constraints. When the transmit powers by the solution (11) satisfy the per BS power constraints, this scenario is considered. The condition is analytically given by ⎛ ⎞2 2 2 fkj f kj K ⎜ ⎟ ⎜ j =1,j =j ⎟ (14) ⎜ ⎟ γk ≤ P j , ∀j. 2 ⎝ ⎠ 2 k=1 fki i=1
In this case, therefore, the solution of optimal power allocation is equal to that of power minimization with sum power constraint (11) in Section III.
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4
B. Unsatisfied Power Constraint Scenario
30
We consider this scenario that the available transmit power of any BS is insufficient for meeting the target SINR of each user. In other words, per BS power constraint of any BS is small compared with the power for satisfying the target SINR of users. Therefore, the other BS which has sufficient available power should use more transmit power for insufficient BS in this scenario. Since this scenario should consider the per BS power constraints unlike previous cases, also, the method to find the solution for minimizing the transmit power should be different to the previous method. The condition of this case is the opposite condition of the first case as follows: ⎛ ⎞2 2 2 fkj f kj K ⎟ ⎜ ⎜ j =1,j =j ⎟ ⎜ ⎟ γk ≥ P j , any j ∈ [1, 2] . 2 ⎝ ⎠ 2 k=1 fki
25
Non-cooperative BS in multicell (Jk =20dB)
i=1
(15) Without loss of generality, we assume that jth BS has insufficient available power. Since using the power of jth BS is more efficient than using the power of the other BS in terms of satisfying the target SINR of each user, it is optimal to use the available all power of jth BS for minimizing the transmit power. Nevertheless, it is complex to find the power allocation method for minimizing the total power due to many and complex constraints in this case. Thus, we consider the property of massive MIMO for obtaining the power allocation method. By Hj HH j ≈ KNt I obtained from the law of large numbers in the massive MIMO [7], the ZF beamformer (5) 1 HH becomes more simply as Fj ≈ KN j . Since the norm value t of each channel vector hkj is approximately same from the law of large numbers, the norm value of all beamformers is also approximately same, i.e, fkj ≈ m, ∀k, j, where m is any constant value. Then, the optimal problem (13) is the convex problem by the above property of massive MIMO. Then, the transmit power which is obtained by solving the sub-optimal problem is allocated as pkj
=
pkj
=
γk P j , ∀k, K γl l=1 ⎞ ⎛ √ pkj √ ⎠ fkj · ⎝ γk − , ∀k, j(j = j), (16) fkj
The above power allocation results does not include the approximate norm value of beamforming due to meeting the target SINR of each user, the approximate norm value of beamforming is used only in the process for finding the power allocation. As mentioned above Section III, the solution easily produces more complex results by the method of Lagrange multipliers when the number of cooperative BSs is more than 2. Therefore, this case have not been discussed here for lack of space.
Cooperative BS based ZF (Jk=20dB) Optimal beamforming in singlecell (Jk=20dB) Non-cooperative BS in multicell (Jk =10dB)
20
Total sum power (dBW)
Cooperative BS based ZF (Jk=10dB) 15
Optimal beamforming in singlecell (Jk=10dB)
10
5
0
-5
-10
-15 20
40
60
80
100 120 140 # of Tx Ant. at each BS
160
180
200
Fig. 2. Performance comparison of total transmit power versus the number of Tx antenna
V. P ERFORMANCE E VALUATION We consider the benefit of proposed algorithm introduced in the previous sections through numerical experiments. In all considered systems, we utilize the power allocation method for meeting the SINR constraints. If we do not consider the power allocation method like equal power allocation, it is infeasible because the SINR constraints of each user are unsatisfied. Throughout this section, also, we only consider the cooperative BS system with per BS power constraints for more generality. We assume that the perfect CSI is available at all BSs and users. In addition, all BSs in the cooperative BS system perfectly cooperate the transmit signal and CSI. Also, we assume that the set of target SINRs and available transmit power of each BS are feasible. In this section, we consider the total transmit power of optimal beamforming in singlecell [6] for the upper bound of the performance in cooperative BS system. If the number of antennas in single cell is equal to that of total antennas of all BSs in cooperative multi-cell and the path loss doesn’t exist, we intuitively know that the performance of single cell is the upper bound of that for cooperative multi-cell with per BS power constraints due to no power constraint in single cell. In Fig. 2, we compare the total transmit power of the proposed algorithm in cooperative BS system with that of the non-cooperative BS system and optimal beamforming in singlecell in case that target SINR is 10 and 20 dB and each available transmit power is 30dBW. The performance gap between proposed method and non-cooperative BS is smaller and smaller according to decreasing the target SINR of each user. The received SINR is proportional to the number of antennas. In case of many transmit antennas, therefore, the performance of proposed algorithm is convergent to the upper bound of that, since the ZF transmit beamformer is near optimal. Therefore, there is little difference of performances between proposed and optimal beamforming algorithm. Since
441
5
1
R EFERENCES
Non-cooperative BS system Cooperative BS system
0.9 0.8
Outage Probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -1 10
0
10 Available power P 2 (W)
1
10
Fig. 3. Performance comparison of outage probability versus available power P2
the method of designing the optimal beamformer in single cell is very complex by the iterative algorithm until convergence, however, we can know the superior of proposed algorithm in Fig. 2. Fig. 3 shows the outage probability of proposed power allocation for cooperative BS and power allocation for noncooperative BS system with fixed available power of any BS versus available power of the other BS. Without loss of generality, we assume that 1st BS has fixed available transmit power. The outage probability means the infeasible probability to meet the SINR or power constraints. In Fig. 3, we consider a Nt = 100 transmit antennas, γk = 20dB target SINR of each user and P 1 = 10W fixed available power. As shown the Fig. 3, the outage probability of proposed allocation is convergent to zero at lower available power than that of noncooperative BS system. It shows that the cooperative BS is able to satisfy the target SINR and power constraints at lower available power than non-cooperative BS system.
[1] H. Dahrouj and W. Yu, “Coordinated beamforming for the multi-cell multi-antenna wireless system,” in Proc. Conference on Information Sciences and Systems 2008, CISS 2008, 2008. [2] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell, multi-antenna wireless system,” IEEE Trans. Wireless Commun., May 2010. [3] O. Somekh, O. Simeone, Y. Bar-Ness, A.M. Haimovich, and S. Shamai, “Cooperative multicell zero-forcing beamforming in cellular downlink channels,” IEEE Trans. Inf. Theory, July 2009. [4] W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constriants,” in IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2646-2660, June 2007. [5] K. Karakayali et al., “On the maximum common rate achievable in a coordinated network,” in Proc. IEEE International Conference on Communications, ICC, 2006. [6] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 1828, 2004. [7] T. L. Marzetta, “How much training is required for multiuser MIMO?” in Proceedings of the 40th Asilomar Conference on Signals, Systems, and Computers (ACSSC 06), pp. 359363, Pacific Grove, Calif, USA, November 2006. [8] N. Jindal, “High SNR Analysis of MIMO Broadcast Channels”, in Proc. IEEE International Symposium on Information Theory (ISIT), pp. 23102314, Adelaide, SA, 2005. [9] S. A. Jafar, G.J. Foschini, and A. J. Goldsmith, “PhantomNet: exploring optimal multicellular multiple antenna systems,” in Proc. Vehicular Technology Conference 02 Fall, vol. 1, pp. 261-265, Vancouver, BC, Canada, Sep. 2002. [10] S. A. Jafar and A. J. Goldsmith, “Transmitter optimization for multiple antenna cellular systems,” in Proc. IEEE International Symposium on Information Theory (ISIT), p. 50, Palais de Beaulieu, Lausanne, Switzerland, 2002.
VI. C ONCLUSION This paper provides a solution to the optimal power allocation problem for cooperative BS system with massive MIMO to minimize total transmit power. Since ZF transmit beamformer is optimal performance and has low complexity in massive MIMO, we use the ZF transmit beamformer. We formulated the problem for both the sum power constraint and per BS power constraints. In case of sum power constraint, we provide the simple power allocation method while meeting the target SINR constraints. For the per BS power constraints case, we use the properties of a massive MIMO for finding the power allocation algorithm of closed form. The proposed algorithm is efficient, and it has near optimal performance while satisfying the target SINR constraints and per BS power constraints.
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