Joint Antenna Selection and Precoding Optimization for Small-Cell Network with Minimum Power Consumption

We focus on the power consumption problem for a downlink multiuser small-cell network (SCN) considering both the quality of service (QoS) and power constraints. First based on a practical power consumption model taking into account both the dynamic transmit power and static circuit power, we formulate and then transform the power consumption optimization problem into a convex problem by using semidefinite relaxation (SDR) technique and obtain the optimal solution by the CVX tool.We further note that the SDR-based solution becomes infeasible for realistic implementation due to its heavy backhaul burden and computational complexity. To this end, we propose an alternative suboptimal algorithm which has low implementation overhead and complexity, based on minimum mean square error (MMSE) precoding. Furthermore, we propose a distributed correlation-based antenna selection (DCAS) algorithm combining with our optimization algorithms to reduce the static circuit power consumption for the SCN. Finally, simulation results demonstrate that our proposed suboptimal algorithm is very effective on power consumption minimization, with significantly reduced backhaul burden and computational complexity.Moreover, we show that our optimization algorithms with DCAS have less power consumption than the other benchmark algorithms.


Introduction
Power consumption has increased considerably with the explosive rise of mobile data traffic demand over the past decade.Base stations have to consume much power to transmit a large amount of data traffic to meet the higher quality of service (QoS) required by the users.As pointed out in [1], base stations consume much more than sixty percent of the total power in the cellular networks.However, this is not in conformity with the lower power consumption and higher energy efficiency (EE) advocated by the green communications.
Spurred by growing environmental and economic concerns about how to sustain the exponential traffic growth, it is important to design energy saving wireless networks.The small-cell network (SCN) and massive MIMO are recognized as the key technologies for the decrease of power consumption because they have a great potential to enhance the EE [2,3].However, massive MIMO improves the EE, but at the cost of deploying more hardware infrastructure which means high power consumption.Besides, SCN achieves higher EE than massive MIMO in whatever crowded or sparse areas [4].Unlike massive MIMO, SCN consists of a number of smallcell access points (SAPs), where each SAP is connected to a central processing unit (CPU) through a limited-capacity backhaul.SCN has less propagation losses and higher spatial reuse due to the short access distance provided [5], thereby resulting in higher EE.
So far, methods of improving the SCN's EE have been focused on in many related investigations such as communication mode [6], hardware improvement [7], and network deployment [8,9].All the above literature provides comprehensive insights into how the EE can be improved, while the optimization for power consumption minimization of the SCN has also been well investigated in [10][11][12].However, the optimization methods in [10][11][12] are all 2 International Journal of Antennas and Propagation based on coherent joint transmission, which requires full channel state information (CSI) of all jointly processed SAPs, a strict synchronization across the SAPs, and large backhaul capacity for information exchange [13].Since the overhead of exchanging all information and computational complexity of joint processing is usually prohibitive for practical implementations, it is not suitable for coherent joint transmission in the SCN.Facing these challenges, we aim at designing efficient transmission schemes on the coordination and loosening the backhaul requirements for the SCN.
In addition, it is well known that antenna selection can significantly reduce the complexity and power consumption to improve the spectral efficiency (SE) [14] and the EE [15,16], which has been extensively studied for MIMO systems.However, most of these previous works for enhancing the SE and the EE ignored the fact that increasing the number of antennas is not always the best choice because of more power consumed in the circuits part [17].Therefore, we believe that antenna selection is also an efficient approach to further reduce power consumption for the SCN.
To the best knowledge of the authors, the power consumption optimization, which combines the antenna selection and transmission scheme design for the SCN, is not presented before.Therefore, it is attractive to analyze and optimize this issue.In this work, we aim to minimize the power consumption under the quality of service (QoS) constraint per user and the power constraint per antenna of each SAP.First, we transform the initial power consumption minimization problem into a convex problem by using semidefinite relaxation (SDR) to obtain the system power consumption minimization and find the relationship among power consumption, the number of users, and the number of antennas per SAP.However, we know that it is difficult for this optimal algorithm to implement precoding in real-time due to heavy backhaul burden and high computational complexity when the number of SAPs is large.Then we propose a suboptimal algorithm which is on the basis of noncoherent joint transmission, for example, multi-SAP minimum mean square error (MMSE) precoding, to reduce the backhaul overhead and computational complexity.In addition, SAPs equipped with redundant antennas would consume a large amount of power consumption.In this case, we propose joint antenna selection and precoding optimization algorithms to further reduce power consumption.The simulation results show that there exists a small performance gap between the optimal algorithm and our proposed suboptimal algorithm.Moreover, our optimization algorithms with distributed correlation-based antenna selection (DCAS) are effective ways to minimize power consumption when the number of antennas is larger than the number of users per SAP.
The rest of this paper is organized as follows.In Sections 2 and 3, we present the system model and formulate the power consumption minimization problem, respectively.In Section 4, we propose a low overhead and complexity suboptimal algorithm.In Section 5, we propose the DCAS algorithm to further reduce the power consumption.The simulation results are presented in Section 6.Finally, the paper is concluded in Section 7.
Notation.Capital and small bold letters represent matrices and vectors, respectively; A  and A  stand for conjugate transpose and transpose of matrix A, respectively.C × is the set of complex matrix with  rows and  columns, tr(A) is the trace of matrix A, I is an identity matrix, and ⟨a, b⟩ denotes correlation of a and b.CN(⋅, ⋅) stands for a multivariate circularly symmetric complex Gaussian distribution and we use ‖ ⋅ ‖ and | ⋅ | to denote the Euclidean norm and absolute value, respectively.The basic notations are given in the Notations.

System Model
As shown in Figure 1, we consider a downlink multiuser SCN consisting of  small cells, where each small cell deploys a SAP with   antennas and all the SAPs are connected to the CPU through the high-speed backhaul, via which some critical information, such as the CSI, is shared among all SAPs.There are  single-antenna users simultaneously being served on the same frequency band.We assume the following aspects for the SCN: users and the number of users being served stays constant in each small cell.
(ii) Uniform Network Topology.SAPs are deployed uniformly and randomly in the SCN and the users are distributed uniformly and randomly in the coverage area of each SAP.
(iii) Perfect CSI.Perfect CSI is globally available at all SAPs and users.
Conventionally, each user is associated with only one SAP.Named as single-SAP transmission, it means that each SAP only serves the users in its own coverage area.However, in this paper, we assume a user can be associated with multiple SAPs and consider noncoherent multi-SAP transmission, which is much less complicated to be implemented in practice as compared to the coherent joint transmission which requires strict phase-synchronization [18].Under noncoherent multi-SAP transmission, each user can be associated by multiple SAPs but the information flow is encoded and transmitted independently at every SAP.In this way, it is able to strike a compromise between low implementation complexity (e.g., the single-SAP transmission) and good performance (e.g., the coherent joint transmission); thereby the noncoherent multi-SAP transmission can be attractive for practical system setups.
The channel vector between SAP  and user  is denoted by h  ∈ C   ×1 , modeled as where g  is the small scale fading component whose elements are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and unit variance and   represents the large scale fading including shadow fading, path loss, and penetration loss [3].
International Journal of Antennas and Propagation The received signal at user  is given by where   ∼CN(0,  2  ) is the circularly symmetric complex Gaussian noise.At SAP , the transmitted signal x  is given by where   ∈ CN(0, 1) is the transmitted signal from SAP  to user  and w  ∈ C   ×1 is the precoding vector, which is to be optimized to minimize the power consumption in the next section.

Power Consumption Optimization
The goal of our work is to minimize the power consumption in the considered SCN and show how the result is affected by the system parameters.We first formulate the power consumption minimization problem as follows.Unlike most ideal power consumption models where the circuits consumption is ignored, in this work, we consider a practical power consumption model which consists of two terms, respectively, corresponding to the transmitted power and the cost of circuits [19,20] and denoted by  1 and  2 as follows: where   ≥ 1 and   ≥ 0 are parameters describing the inefficiency of the power amplifier at SAP  and the power dissipation in the circuits of each antenna, respectively.The objective is to minimize  1 +  2 , under two practical constraints stated in the following.
(1) The power constraint per antenna of each SAP: where D  is a positive semidefinite weighting matrix whose th diagonal elements are one and zero elsewhere.The weighting matrix can describe any combination of perantenna constraints.  is fixed power constraint for each antenna at SAP , defined in mW.
(2) The QoS constraint per user (defined as the information rate in bits/s/Hz): where   is a fixed QoS target and the signal-to-interferenceand-noise ratio of user  is given by The first and second part of the denominator in ( 8) are interference and noise, respectively.To minimize the power consumption of the system, the optimization problem is formulated as min w  , ∀, Due to the fact that  2 is not related to w  in expression (5),  2 is considered as a static part which is mainly determined by the number of antennas.While  1 is dynamic, problem (9) can be solved if we make  1 minimum while satisfying the QoS and power constraints.However, the constraints are not convex so that the CVX cannot work on problem (9).
To solve this problem, we reformulate (9) as a convex optimization problem by using SDR method.According to the definition in (7), we have Plugging ( 8) into (10) yields which is alternatively expressed as Then, we define the positive semidefinite matrix with rank(V  ) ≤ 1.
Using the above results, we obtain the inequality In addition, we can further relax the rank constraint rank(V  ) ≤ 1.The constraint can be removed with definitely satisfying the optimality as before [21].Now (9) can be rewritten as min Thus we can transform the initial power consumption minimization problem (9) into a convex problem (15), which can be solved by using CVX [22].

Proposed Suboptimal Algorithm
Solving the optimization problem in (15) requires centralized processing, for which the CSI of all SAP-user links should be known at the CPU.This induces large backhaul overhead and high computational complexity, which cannot be ignored especially when  becomes large.In this section, we propose a suboptimal algorithm, which first alleviates the backhaul overhead by designing the multi-SAP MMSE precoding.Then, we transform (9) into a convex power allocation problem, reducing the computational complexity.The details of our proposed algorithm are shown as in the following steps.

Proposed Suboptimal Algorithm
Step 1.Each SAP computes its MMSE precoding vector; for example, SAP  designs the precoding vector for user  given by where Step 3. The result of power allocation optimized in ( 18) by the CPU is sent to each SAP; for example,  *  is informed to SAP .
Step 4. Each SAP transmits the data to the served users with the suboptimal precoding vector We observe that the precoding vector is obtained by expression (16) rather than optimization (15), which means the distributed precoding design has been done by each SAP before the power allocation problem (18) is optimized.As a result, the CPU needs little backhaul overhead to complete the power allocation.Compared with the optimal algorithm which has to share complete CSI of each user, the proposed suboptimal algorithm exchanges only two scalars, that is,   and   , between each SAP and the CPU.Moreover, the optimization variable is the vector w in the optimal algorithm, while it is the scalar  in our proposed suboptimal algorithm.All of these point out that the backhaul overhead and computational complexity of the proposed suboptimal algorithm can be significantly reduced as compared to the original problem in (15).

Antenna Selection Algorithm
In this section, we show that the system power consumption can be further reduced by combining the proposed precoding optimization with properly designed antenna selection algorithm.Note that when the circuit power consumption is not considered in the analysis, having more antennas will always lead to lower power consumption for the reason that it provides more degree of freedom.However, as we are considering the practical power consumption model where the circuit power consumption cannot be ignored, increasing the number of antennas (i.e., degree of freedom) comes at a cost of consuming more circuit power.Therefore, it may be not optimal to use all   antennas at every SAP for transmission, and there always exists an optimal number of antennas to minimize the total power consumption.
In general, antenna selection algorithm can be formulated as a combinatorial optimization problem, which can be solved by exhaustive search (ES) method, but leading to a prohibitive computational burden even for a small number of antennas and users.The simplest antenna selection algorithm is known as the random antenna selection (RAS) which has very low complexity but poor performance.Alternatively, in this work, we propose the DCAS algorithm.With this algorithm, each SAP individually selects the best  out of   antennas of its own before doing optimization algorithms.The channel matrix between SAP  and its corresponding with   rows and   columns (it is shown by simulations that the optimal  should be set equal to the number of served users   in the corresponding small cell).The th and th row of the channel matrix H  are denoted by [H  ]  and [H  ]  , respectively.With these notations, the DCAS algorithm is described as in the following steps.
Step 4. Repeat Steps 2 and 3 until   −  rows are all deleted; then each SAP obtains a selected channel matrix Ĥ = [ ĥ1 ĥ2 ⋅ ⋅ ⋅ ĥ   ] with  rows and   columns, where the channel vector between SAP  and user  is denoted by ĥ ∈ C ×1 .This algorithm is explained as follows: we select two rows in H  with the highest correlation and eliminate the one with lower power and then repeat this process until   −  rows are all deleted.After this antenna selection procedure, the dimension of the channel matrix is reduced from   ×   to  ×   .The resultant channel matrix will have rows with minimal correlation and maximal power; therefore a large capacity is expected for this dimension-reduced channel matrix.

Simulation Results
In this section, we give numerical simulations to evaluate the power consumption performance of our proposed precoding optimization and antenna selection algorithms.The simulation is implemented on the Matlab platform with the toolbox CVX.We set the number of antennas and users per SAP to be   ∈ {1, 2, 3, 4, 5} and   ∈ {1, 2, 3, 4, 5}, respectively.The SAPs are deployed arbitrarily in the cell and the users are distributed uniformly and randomly in the area of each SAP as shown in Figure 1 (we find that the different SAP density scenarios will not affect the conclusion in Figure 2 through extensive simulations).Some of the parameters follow LTE standard; others are based on [23,24], as Table 1 shows.
In Figure 2, we show the total power consumption of the precoding optimization for different numbers of users per SAP when the QoS target is 2 bits/s/Hz.In ideal power consumption models, we generally believe that the power consumption decreases with the increasing number of antennas, while, in our practical model, the power consumption rises as the number of antennas increases.We find that no feasible solution can be obtained when the number of users is larger than the number of antennas per SAP due to QoS constraint per user.Figure 3 describes the distribution of solutions of the precoding optimization.Combining with Figure 2, we see that it is optimal to serve the user by only one SAP which is the nearest instead of all of the SAPs.It is interesting to observe that in our practical power consumption model, under the condition that the ratio of the number of antennas and the number of users is 1 : 1, the total power consumption of the SCN is globally minimal.This means that the SCN has the best performance when the number of selected antennas equals the number of served users per SAP.Motivated by the above conclusion, we will consider antenna selection to further reduce circuit power consumption for the SCN.
In the simulation of Figure 4, we set   = 2 and   = 2 per SAP.Three schemes are compared: the optimal optimization algorithm, the suboptimal optimization algorithm, and the original scheme without any optimization.It is shown that without any antenna selection, the suboptimal optimization algorithm shows satisfactory performance as the gap between the suboptimal algorithm and the optimal algorithm is narrow and acceptable.It should be emphasized that although there is slight performance loss as compared with the optimal algorithm, the proposed suboptimal algorithm is able to greatly reduce the execution time consumption, as will be shown later.Table 2 compares the execution times of simulation work between the optimal optimization algorithm and the suboptimal optimization algorithm (the execution time is measured under the following conditions: Matlab: R2014a version; CPU: Intel Core i7-4720HQ; RAM: 8 GB DDR3L).We set  = 3 and   =   in the simulation.The execution time shown in Table 2 describes the time used by the optimal and suboptimal algorithms for the different number of users.We find that the time used for the suboptimal algorithm is always in the scale of millisecond, which is much less than the time consumption of the optimal algorithm.With increasing number of users, the gap of the execution time between these two algorithms becomes larger.The comparison results show that the optimal optimization algorithm is time-consuming to be implemented in practice, especially for dense networks, where the number of users is always large.On the other hand, the proposed suboptimal algorithm provides a more efficient solution with negligible performance loss.
Figure 5 demonstrates the effectiveness of the antenna selection algorithm.We consider   = 5 and   = 2 per SAP.Assume that five antennas are all available and we use DCAS algorithm to select two, three, four, and five of them to serve the two users, respectively.We can conclude that the total power consumption increases with the increasing number of selected antennas when the QoS target is below 3.5 bits/s/Hz.The total power consumption is minimal if we select the same number of antennas as the number of users before our optimization algorithm.The reason is that when the QoS constraint is in the range of IMT-Advanced requirement,  2 is the major part of the total power consumption (the downlink SE of the IMT-Advanced requirement is up to 3 bits/s/Hz and 2.25 bits/s/Hz for indoor usage.In this paper, we mainly consider the QoS target is below 3.5 bits/s/Hz, which meets the IMT-Advanced requirement [18]).In this case, the total power consumption depends mostly on the number of selected antennas.When the QoS target per user is larger than 3.5 bits/s/Hz,  1 becomes to dominate the total power consumption to satisfy all the users' requirements, resulting in a better performance by selecting as many antennas as possible.
Figure 6 focuses on the total power consumption versus QoS target per user when the number of antennas is larger than that of users.We consider   = 3 and   = 2 per SAP and compare the following seven schemes: (i) S1: The power consumption with optimal algorithm and DCAS algorithm.
(ii) S2: The power consumption with proposed suboptimal algorithm and DCAS algorithm.(iii) S3: The power consumption with optimal algorithm but without any antenna selection.
(iv) S4: The power consumption with proposed suboptimal algorithm but without any antenna selection.
(v) S5: The power consumption with optimal algorithm and RAS algorithm.
(vi) S6: The power consumption with proposed suboptimal algorithm and RAS algorithm.
(vii) S7: The original power consumption without any optimization or antenna selection.
The comparison result demonstrates that the performance of our proposed suboptimal algorithm S4 is superior to the original scheme S7 but a little worse than the optimal algorithm S3.Table 3 shows the comparison of overhead and complexity between the optimal algorithm S3 and our proposed suboptimal algorithm S4.We see that the optimal algorithm has to exchange 2  backhaul data in total, while only 2 parameters need to be exchanged in the proposed suboptimal algorithm, resulting in less backhaul overhead.The CPU has to compute the trace of matrices in the optimal algorithm, while the computation of our proposed suboptimal algorithm just involves scalars, yielding lower computational complexity.
In addition, Figure 6 shows the gap between the optimal algorithm and the suboptimal algorithm is rather small,

Figure 1 :
Figure 1: The system model of SCN.

Figure 2 :
Figure 2: Total power consumption versus the number of antennas per SAP.

Figure 4 :
Figure 4: Total power consumption versus QoS target per user.Results are shown for different optimization algorithms.

Figure 5 :
Figure 5: Total power consumption versus QoS target per user.Results are shown for DCAS algorithm when the number of antennas is no less than that of users.
user (bits/s/Hz) Total power per subcarrier (mW) Opt with DCAS Sub with DCAS Opt without any AS Sub without any AS Opt with RAS Sub with RAS Original scheme

Figure 6 :
Figure 6: Total power consumption versus QoS target per user.Results are shown for different schemes.
2 in this paper; then each SAP can easily obtain the following parameters.

Table 2 :
Comparison of execution times used by the optimal and suboptimal algorithms.