EnergyEfficient Coordinated MultiCell Multigroup Multicast Beamforming with Antenna Selection
Abstract
This paper studies energyefficient coordinated beamforming in multicell multiuser multigroup multicast multipleinput singleoutput systems. We aim at maximizing the network energy efficiency by taking into account the fact that some of the radio frequency chains can be switched off in order to save power. We consider the antenna specific maximum power constraints to avoid nonlinear distortion in power amplifiers and userspecific quality of service (QoS) constraints to guarantee a certain QoS levels. We first introduce binary antenna selection variables and use the perspective formulation to model the relation between them and the beamformers. Subsequently, we propose a new formulation which reduces the feasible set of the continuous relaxation, resulting in better performance compared to the original perspective formulation based problem. However, the resulting optimization problem is a mixedBoolean nonconvex fractional program, which is difficult to solve. We follow the standard continuous relaxation of the binary antenna selection variables, and then reformulate the problem such that it is amendable to successive convex approximation. Thereby, solving the continuous relaxation mostly results in nearbinary solution. To recover the binary variables from the continuous relaxation, we switch off all the antennas for which the continuous values are smaller than a small threshold. Numerical results illustrate the superior convergence result and significant achievable gains in terms of energy efficiency with the proposed algorithm.
I Introduction
The energy efficiency (EE) of the multiantenna technologies is a performance indicator growing its importance for wireless communications and future 5G network design [1, 2, 3, 4, 5, 6]. The total power consumption consists of various elements, such as base stations (BSs) direct current (DC) power, baseband processing, radio frequency (RF) processing, coding, decoding, and backhaul operations. In fact, the data transmit power takes only a small share of the total power consumption when moving towards 5G networks [7]. Each active antenna element requires an RF chain to preprocess the signal which has a significant impact on the power consumption. An efficient method to improve the energy efficiency is to use only some of the antennas for transmission by switching off a portion of the RF chains, giving rise to the problem of joint beamforming and antenna selection [8, 5].
Coordinated beamforming [9] has been one of the most efficient methods to deal with the intercell interference in the research on the multicell networks. The technique, adopted in the current LTEA standards, uses the coordination of channel state information or intercell interference conditions to improve the system performance. Energyefficient coordinated beamforming has been extensively studied recently, e.g., in [6, 10, 11, 12].
The research presented before focuses mostly on unicast beamforming where each user is assigned with an independent data stream. However, the increasing demand for high data rate applications such as video broadcasting services creates new challenges. This gives rise to the concept of multicast beamforming, where multiple users desire to receive the same information. Multicasting is a particularly powerful technique in the context of cacheenabled cloud radio access networks proposed for 5G systems, where it can be used to transmit the same popular contents to multiple users to improve both spectral and energy efficiency [13]. The physical layer multicasting has been also included in the LTE standards and it has applications in satellite communications [14, 15] as well. Multicast beamforming problems have been studied in singlecell systems for different optimization targets, e.g., transmit power minimization [16, 17], maxmin fairness [17, 18], and sum rate maximization [19]. Joint beamforming and antenna selection for transmit power minimization was studied in [20]. Coordinated multicast beamforming for transmit power minimization and maxmin fairness has been studied in [21]. He et al. [22] proposed a centralized algorithm for the energy efficiency maximization problem in multicell system with singlegroup per cell. However, [22] only considered the beamforming problem without taking into account the fact that significant energy efficiency savings can be achieved by switching off some of the RF chains, i.e., antenna selection.
In this paper, we study energyefficient coordinated beamforming in multicell multigroup multiuser multicast multipleinput singleoutput (MISO) systems with joint beamforming and antenna selection. We first introduce a novel way to reduce the feasible set of the continuous relaxation, resulting in better performance compared to the original formulation. Then, by proper equivalent transformations, the continuous relaxation of binary variables, and the CharnesCooper transformation, we propose a successive convex approximation (SCA) based algorithm to solve the problem. Thanks to the proposed formulation, the continuous relaxation yields some continuous values being close to binary ones. To this end, we switch off antennas which have continuous values close to zero.
Contrary to [22], where an energyefficient coordinated beamforming method for fixed antenna sets was derived with single group per cell, we focus on energyefficient joint beamforming and antenna selection problem with multiple groups per cell which has not been considered in the previous multicasting research. In addition, we set antenna specific maximum power constraints whereas [22] only considered the BSspecific sum power constraints which can result in large dynamic range of the PA output powers, causing large phase deviations due to phase nonlinearities [23].
The rest of the paper is organized as follows. Section II presents the system model, power consumption model and the considered optimization problem. The proposed algorithms are provided in Section III while numerical evaluation and conclusions are presented in Section IV and V, respectively.
The following notations are used in this paper. denotes the cardinality of if is a set, and absolute value of , otherwise. is the Euclidean norm of and boldcase letters are vectors. and mean transpose, Hermitian transpose, real part and imaginary part of , respectively.
Ii System Model and Problem Formulation
Iia System Model
We consider a multicell multigroup multicasting system with BSs, where each BS , transmits independent information to groups of users in its cell. The set of groups served by BS is denoted by . The total number of users and groups in the network is denoted by and , respectively. BS is equipped with transmit antennas, whereas each user has only one receive antenna. The serving BS of user group is denoted as . The set of users in group is denoted by . Since each user belongs to only one group, the sets of users belonging to different groups are disjoint, i.e., , . The received signal at user in group is given by
(1)  
where is the channel (row) vector from BS to user , is the transmit beamforming vector of group , is the corresponding normalized data symbol and is the complex white Gaussian noise sample with zero mean and variance . The SINR of user can be written as
(2) 
and the data rate towards user is given as
(3) 
IiB Power Consumption Model
The total power consumption at the transmitter side is [5]
(4) 
where is the antenna selection variable for the th transmit antenna of BS , i.e., if the th antenna is selected and otherwise, and is the power amplifier efficiency. Note that the first term in (4) is the power consumption of the PA’s to get the desired output powers, is the power consumption of all circuit blocks in each active RF chain, is the static power spent by cooling systems, power supplies, etc.
IiC Problem Formulation
The optimization target is to maximize network energy efficiency
(5a)  
(5d)  
where is the SINR threshold for user , , includes the beamforming coefficients related to antenna of BS and index refers to a group index served by BS . In the numerator of the objective (5a), the rate for user group is defined as a minimum of the user rates in the group, because common information is transmitted to the group. The constraint in (5d) guarantees that if , then the beamformers associated with antenna are set to zero. It also sets the maximum antenna specific power constraint. The above problem is a nonconvex mixedBoolean fractional program which is hard to tackle as such. One of the main challenges is that the problem is nonconvex even when the binary variables are relaxed to continuous. More specifically, in that case, (5d) and the numerator of the objective function are nonconvex.
Iii Proposed Solution
Iiia Equivalent Transformation
To find a more tractable reformulation, we first equivalently transform (5) as
(6a)  
(6g)  
where are new variables representing the SINR of each user , the power transmitted from antenna of BS [5], and the rate of the weakest link in user group , respectively. It is not difficult to see that (5) and (6) are equivalent, because the constraints in (6g) and (6g) are active and in (6g) is satisfied at the optimum. We have equivalently replaced with in (6g), because if the minimum user rate satisfies the constraint, then all the user rates in that group have to satisfy it. In constraint (6g), we have introduced a new formulation which is equivalent for any value of because each is binary. The exponent in (6g) can be interpreted as a penalty parameter which penalizes the values of so that they are encouraged towards binary solution in case of continuous relaxation. The new formulation (6g) reduces the feasible set of the problem when continuous relaxation is performed, so that the larger , the smaller feasible set. In other words, we have the following inequalities
(7) 
where and are the optimal objective of Boolean formulation (5), continuous relaxation of (6) with , continuous relaxation of (6) with and continuous relaxation of (5). In fact, the proposed formulation is a generalization of the approach used in [5] for singlecell multiuser MISO unicasting, where and were used. However, the numerical results show that the value of has a significant effect on the algorithm performance, and, thus, the energy efficiency.
IiiB Solving the Equivalent Transformation
By taking a look at (6), we observe that if the binary variables are relaxed as , the objective function is a concaveconvex fractional function and the difficulty in solving (6) is due to the constraints (6g) and (6g). By using the continuous relaxation and introducing new variables to represent the total interferenceplusnoise of user , the problem becomes
(8a)  
(8f)  
Now the left side of (8f) is linear and the right side is convex quadraticoverlinear function. To deal with the right side of (8f), we can write its linear lower bound approximation at point as
(9) 
To find a more tractable formulation for (8f), we first equivalently write it as
(10) 
where the left side is a convex quadraticoverlinear function and the right side is convex. We can write the linear lower bound approximation of the right side at point as
(11) 
With the approximations (IIIB) and (11), the problem becomes
(12a)  
(12e)  
At this point, we note that the problem is a concaveconvex fractional program which can be transformed to a convex one with the CharnesCooper transformation [24]. Thus, solving (12) is equivalent to the following convex problem
(13a)  
(13i)  
After solving (13), the optimal solutions for the original fractional program (12) can be found as , where are the optimal variables of (13). As a result, we use the idea of successive convex approximation [25], where the convex problem (13) is solved at iteration . After solving the problem at iteration , the optimal solutions are then used to update and for the next iteration. The monotonic convergence of the objective function (13a) is not difficult to see, and a more detailed convergence analysis for the problem with similar structure can be found, e.g., in [26, Appendix A].
Remark 1.
If at least one of the SINR targets in some user group of BS is nonzero, we can further reduce the feasible set of (13) by adding the constraints
(14) 
where is the number of groups served by BS which have at least one user having nonzero SINR target. This can be done because it is known that at least antennas have to be active to be able to transmit independent data streams.
Recovering the Binary Solution from Continuous Relaxation
Generally, solving the continuous relaxation usually results in a solution where many of the antenna selection variables are nonbinary. However, due to the new formulation in (8f), many of the continuous antenna selection variables converge to binary. Thus, we propose to switch off all the antennas for which , where is a small threshold. After performing the antenna selection, the algorithm needs to be run again with the selected antenna set to find the beamformers with lower dimensions. However, in the numerical results, we also illustrate that the beamformers returned by the relaxed problem already yields a good energy efficiency with the correct choice of . This method is called ‘Alg. 1 ‘simple’’ in the numerical results. The proposed joint beamforming and antenna selection method is summarized in Algorithm 1.
Initial Points
One of the challenges is to find feasible initial points especially due to the QoS constraints. The initial can be set to allones. To find , we can apply a similar approach as that in [27]. We first initialize any and then consider a penalized formulation of (13), which can be written in a general form as
(15a)  
(15c)  
where is a concave objective function, and are convex constraints, includes all the variables in (13), is a positive penalty parameter and are new slack variables, one for each constraint in (13). The above problem is iteratively solved by updating the constraints (i.e., in (13)) after each iteration, until . The feasible point is found very efficiently because the elements of are encouraged to zero due to the penalty function in the objective.
Iv Numerical Results
We evaluate the performance for a quasistatic frequency flat Rayleigh fading channel model with base stations. Each BS serves two groups of users with users per group, i.e., the total number of users in the network is 8. The worstcase interference scenario is assumed so that the average path loss from all the BSs to all the users is 0 dB. We assume a unit bandwidth and noise power is normalized to dBW, . We set for all , i.e., is the number of antennas at each BS, and the algorithms have been stopped when the change of the objective value has been smaller than between two last iterations. We set dBW, W, , and the other simulation parameters are given in the figures.
Fig. 1 illustrates the convergence of the relaxed problem and the achieved energy efficiency for two different values of , and two random channel realizations (denoted as ‘CH’ in the figure). The same initial points have been used for both values of . We can see that the convergence speed is fast in the considered settings for both cases. However, we observe that they converge to different solutions in both examples. Specifically, the objective value of the relaxed problem after convergence is higher for , but the achieved energy efficiency (after recovering the binary solution) is worse than with . This is because with , more antenna selection variables are nonbinary after convergence, which results in worse antenna selection result. Another observation is that with (which achieves better energy efficiency), the objective value returned by the relaxed problem and the achieved energy efficiency are very close to each other. This means that the solution of the relaxed problem is already very close to binary. The examples demonstrate the effectiveness of the proposed formulation. The impact of on the average energy efficiency is studied in the next experiment.
Fig. 2 illustrates the effect of on the average energy efficiency for 0 dB and 6 dB. We also illustrate the performance of the simplified algorithm (Alg. 1 ’simple’), where the beamformers achieved from the relaxed problem (i.e., step 7 is ignored in Alg. 1) are used for transmission. We can see that the choice of affects the achieved energy efficiency, and the choice of the best depends on the system parameters. In this case, gives the best performance when 0 dB, and for larger , it is better to use larger . The reason for this is that with larger , the transmit (TX) power has to be higher to satisfy the SINR constraints. As a result, in the relaxed problem, also the relaxed variables have to be larger, implying that the larger ’penalty’ (i.e. ) is required to force them to zero. We can also see that the larger the , the better results are achieved with the simple method, implying that the relaxed problem yields nearbinary solutions. This again demonstrates the benefit of the proposed formulation.
Fig. 3 illustrates the average energy efficiency versus . Specifically, the proposed algorithm is run with and , and compared with the method where only beamformers are optimized (Alg. 1, no AS). First, we can observe that Alg. 1 with achieves the best results, and the gap between and increases with . It is interesting to observe that with , even the beamformers obtained from the relaxed problem result in very good performance, which again verifies that the relaxed problem yields nearbinary solutions. However, the simple method with is even worse than the method without antenna selection for larger . Finally, we can see that the proposed algorithm provides significant energy efficiency gains over the method without antenna selection (i.e., roughly 550% with the considered system parameters).
Fig. 4 plots the average EE versus the number of antennas per BS. First, we can see that EE starts to decrease without antenna selection when . As can be observed, significant gains are achieved with the proposed algorithm and the gains increase with the number of antennas. Once again, significant gains are achieved with compared to the method with . In addition, the simple method with gives even better performance than the original algorithm with and the gain increases with . The simple method with , on the other hand, yields poor performance.
Fig. 5 illustrates the transmit powers versus with the proposed algorithm and a method without antenna selection. We can see that it is energyefficient to increase sum power when the number of antennas increases. However, the average transmit power per active antenna decreases with increasing also with antenna selection. This means that the more available antennas, the more antennas are chosen to maximize the energy efficiency.
V Conclusions
This paper has studied energyefficient multicell multigroup coordinated joint beamforming and antenna selection with antennaspecific maximum power constraints and userspecific QoS constraints. The resulting mixedBoolean nonconvex optimization problem was tackled by continuous relaxation and successive convex approximation, where the antennas for which continuous antenna selection variables converge to zero are switched off. The numerical results have illustrated the achieved energy efficiency gains of the proposed methods over the method without antenna selection.
Vi Acknowledgements
This work was supported in part by Infotech Oulu Doctoral Program and the Academy of Finland under projects MESIC belonging to the WiFIUS program with NSF, and WiConIE. It was also supported by a research grant from Science Foundation Ireland and is cofunded by the European Regional Development Fund under Grant 13/RC/2077, projects FNR SEMIGOD, SATSENT, INWIPNET and H2020 SANSA. The first author has been supported by KAUTE Foundation, HPY Research Foundation, Walter Ahlström Foundation, Tauno Tönning Foundation, and Nokia Foundation.
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