Energy efficiency maximization for full duplex MIMO cloud radio access networks

This paper studies the energy efficiency (EE) in full-duplex (FD) multiple-input multiple-output (MIMO) cloud radio access networks (CRANs). A cloud controlunit (CU) transfers information signals with multiple downlink (DL) and uplink (UL) users through FD radio units (RUs) by limited capacity FH links. In DL transmission, the signals intended to the DL users are centrally processed at the CU and, then, compressed to transfer to the RUs before they are forwarded to the DL users. On the other hand, the signals from the UL users to the RUs are compressed and forwarded to the CU for signal detection processing. Thus, the precoding designs and compression strategies for the UL and DL transmission are critical for the system performance. The conventional methods commonly focus on maximizing the spectral efficiency (SE) in the networks. Although the SE maximization based methods can offer the superior SE performance, they can result in the inefficient usage of the energy in the networks. Thus, this paper focuses on the joint design of the precoders and quantization covariance matrices to maximize the EE. The EE maximization problem is formulated as an optimization problem of the precoders and quantization covariance matrices subject to the transmit power constraints at each RU, each user and the limited capacity of FH links. To deal with the non-convexity of the formulated design problem, we exploit a successive convex approximation (SCA) method to derive the concave lower bound function of the sum rate and a convex upper bound of capacity functions of FH links. Then, the Dinkelbach approach is exploited to derive an efficient iterative algorithm (IA) guaranteeing convergence. Numerical results are given to demonstrate the EE performance of the proposed algorithm.


INTRODUCTION
Over the past years, the explosive surge in the amount of mobile data has resulted in the challenging requirements of advanced wireless communication possessing techniques to obtain not only higher achievable data rates but also lower power consumption 1 . Cloud radio access networks (CRANs) are a modern architecture for the next generation of mobile wireless systems. CRANs rely on the leverage of a baseband processing with central unit (CU) that leads to not only the effective joint transmission but also the replacement high-cost high-power base stations with the low-cost low-power radio units (RUs). As such, CRAN approaches can both enhance spectrum efficiency and reduce power consumption. CRANs have thus attracted significant investigations, see, e.g., 2 and references therein. On the other hand, multi-antenna techniques have also been recognized as the efficient transmission methods because they provide more degrees of freedom enable to steer signal's directions to the desired users which are called beamforming or precoding techniques. Hence, they are able to simul-taneously transfer more than one data streams over a specific resource, that can thus potentially achieve greater data rates. On this view, CRAN systems with multiple antennas have studied extensively as a potential approach to improve spectral efficiency (SE). There are studies on various implementation scenarios for CRAN systems including either only multiantenna at RUs and single-antenna user equipments (UEs) namely as multiple-input single-output CRAN (MISO CRAN) [3][4][5][6][7] , or multi-antenna at both RUs and UEs, referred as multiple-input multiple-output CRAN (MIMO CRAN) [8][9][10] . Reference 3 addressed the problem of concurrently optimizing both sum rate (SR) and total transmit power (TP) by using a multiobjective framework. They investigated on the algorithm for jointly beamforming, turning RUs on/off, and associating UEs to RU groups. Moreover, the authors in 4 first integrated nonorthogonal multiple access techniques into downlink MISO CRAN systems and, then tackled the problem of the beamforming design to maximize minimum data rate under TP and limited fronthaul (FH) rate constraints. Additionally, the work in 5 proposed a wireless powered MISO CRAN in which UEs support information decoding and energy harvesting in downlink (DL) periods and, then, they transmit their own signals by using harvested energy in uplink (UL) phases. The minimum UL rate maximization problem then was addressed under minimum DL rate, TP, and FH rate constraints. Alternatively, to further improve the SE, the hybrid digital-analog beamforming 6 and rate-splitting multiple access scheme 7 were considered in DL MISO CRAN systems. In addition, by considering a MIMO CRAN system, Park et. al. proposed SR maximization algorithms by jointly designing precoding and FH compressing for either DL or UL CRAN systems in 8 and 9 , respectively. Then, reference 10 investigated on a multi-cluster DL MIMO CRAN system in which the effective design is then studied by taking intercluster interference into account. Along with these two emerging approaches, i.e., CRANs and multi-antenna, full-duplex (FD) transmission is also a powerful technique to achieve the SE improvement. Thanks to the fact that the selfinterference (SI) caused by FD transmission mode can be significantly suppressed, FD communication can efficiently support both transmission and reception in the same frequency and time resources. Thus, FD schemes can potentially double channel capacity compared to traditional half-duplex (HD) counterparts [11][12][13][14][15] . Recently, the combination of FD transmission into CRAN systems has drawn considerable attention and conducted in the works [16][17][18][19][20][21] . Reference 16 studied the system capacity in a basic FD CRANs and demonstrated that the system SE of FD mode is superior to that of both HD mode and single cellular design approach. The authors in 17 first developed wireless power transfer techniques for FD MISO CRAN systems and, then, proposed an optimal approach for minimizing TP with the constraints of both UL, DL rate qualities and minimum amount of harvested energy. In 18 , the authors aimed at optimizing the SR of FD MIMO CRAN systems under the constraints of TP and maximum supported rates over FH links. Alternatively, the work in 19 exploited the relationship between the weighted minimum mean squared error (MMSE) problem and weighted SR problem in order to develop a low computational complexity iterative algorithm (IA) for maximizing the system SE. The computing time is then reduced as compared to the method in 18 . Additionally, the joint design of transmit beamformers, power allocation, FH compression ratio and RU selection was investigated to either minimize TP under quality-ofservice (QoS) constraints in 20 or maximize the DL capacity under UL QoS constraints in 21 .
However, the transmit strategies which are designed merely on the system data rate as mentioned above can result in the negative impacts on the system energy efficiency (EE) [22][23][24] . Hence, the transmission approaches with EE have been taken into consideration recently for CRAN systems [25][26][27][28][29][30][31][32] . Reference 25 studied the joint design of beamforming and RU selection to minimize TP under the constraints on quality of each user's links. The authors in 26 considered the power minimization problem along with minimum data rate requirements. They developed the optimal solution for both data sharing and data compressing MISO CRANs. Furthermore, the work in 27 investigated the EE problem in DL MIMO CRANs by jointly designing precoding matrices and turning on/off RUs. The results shown that the system EE is significantly enhanced as compared to some benchmark schemes. In addition, concerning a DL contentcentric MIMO CRANs, 28 addressed both SE and EE problems by successive convex quadratic programming for jointly designing user association, RU activation, data rate allocation, and precoding matrices. In 29 , the EE of DL MISO CRAN systems was maximized by optimizing both beamformers and RU-UE grouping. Similarly, the EE of downlink MISO CRANs was maximized by jointly designing the transmit beamforming, RU selection, and RU-UE association with difference of convex (D.C.) programming in 30 and sequence of second order cone programming in 31 . More recently, the EE optimization problem (OP) for FD MISO CRAN based data sharing system was tackled in 32 , however, the FH rate constraints was ignored. Motivated from above discussions, this paper is concerned with a FD MIMO CRAN system in which multiple-antenna DL and UL users communicate with a cloud CU through FD multiple-antenna RUs. We aim at designing the optimal precoder and the quantization covariance matrices to maximize the system EE under TP and FH link rate constraints. The design problem is mathematically expressed as a constrained OP which is highly non-convex in design variables. The challenges in solving the OP are due to not only a fraction form and non-concavity of the objective function but also non-convexity in FH rate constraints. Thus, the D.C. programming is used to obtain bounds of the achievable SR and FH rate. Then, the Dinkelbach method is applied to develop an efficient IA with a convex OP solved in each iteration. The numerical simulations are conducted to verify the convergence of our IA and to examine the EE performance of the proposed EE optimization.
The remainder of the paper is organized as follows. Sec. II describes the system model of FD MIMO CRANs and, then, formulates the design of precoding and quantization noise covariance matrices as an OP. Then, an IA to seek the optimal matrices is presented in Sec. III. Numerical simulation results are given in Sec. IV. Finally, the conclusions are presented in Sec. V. Notations: Boldface upper (lower) case letters denote matrices (vectors), An identity matrix with appropriate dimension is presented by I. X T , X H , ⟨X⟩ and |X| are the transpose, Hermitian transpose, trace and determinant of matrix X. X ± 0 denotes a positive semidefinite matrix. A complex Gaussian random

SYSTEM MODEL AND PROBLEM FORMULATION
Consider a FD MIMO CRAN model similar to that in 18,19 in which K R FD RUs are connected to a CU via the FH links, as shown in Figure 1. All the RUs simultaneously serve a set of K UL UL and K DL DL users via radio access links. Denote RU k , DLU k and ULU k as the kth FD RU, kth DLU and kth ULU, respectively. We also denote S UL , S DL and S RU as the set of all ULUs, DLUs, and RUs, respectively. RU k has M k transmit antennas (TAs) and N k receive antennas (RAs). Thus, M DL = ∑ K R k=1 M k and N DL = ∑ K R k=1 M k are total TAs and RAs at the RUs. Also, T k and R k are the number of antennas at ULU k and DLU k . To reliably detect the signals, it is assumed that the sum of data streams of all UL users should be less than or equal to the total number of the TAs at the UL users and the sum of RAs of all RUs. Assume that the ideal channel state information is available at the RUs and users 33 . In the DL transmission, denote the data symbol s DL denotes the precoding matrix of the k -th DLU in RU i . By defining all the signals generated at the CU as where From (1) and (2), we can ob- Before transferring the signals to the corresponding RU on the FH links, the baseband signal x DL i is first compressed by quantizing. Then, the received signal at RU i is written as where is the quantization noise associated with the signals to RU i in the DL channel and Ξ DL i ∈ C M k ×M k is the covariance matrix of downlink quantization noise. From (3), the DL FH rate at RU i is imposed by the following constraint 18,19 where C DL i is the maximum DL FH rate associated with RU i . It is worth noting that RU i will transmit signal x DL i in (3) to the DLUs. In light of (1), (2), (3), the TP constraint at RU i is given by C R k ×M i the channel matrix from all RUs to DLU k and H DU kl ∈ C R k ×R l the CCI channel from ULU k to DLU k . Before forwarding to the DLUs, the compressed signal from the CU is decompressed by the RUs. Hence, the received signal at DLU k is where we have defined (6), the achievable data rate R DL k at DLU k , is where △ k,DL is the interference-plus noise covariance matrix at DLU k and is defined as where Concerning the UL transmission, denote H UL ik ∈ C N i ×T k as the channel matrix from ULU k to RU i , and H UD i j ∈ C N i ×M j represents the residual SI channel matrix from RU j to RU i . Then, in the UL channel, the received signal y UL i at RU i is where F UL k ∈ C T k ×d UL k denotes the precoding matrix for the data symbol s UL is the number of data streams and the vector n UL After receiving the signal from (9), RU i then compresses it and sends to the CU. Thus, the received signal at the CU is given by where q UL i ∼ CN (0, Ξ UL i ) is the quantization noise at the RU i and Ξ UL i ∈ C N k ×N k represents the covariance matrix of uplink quantization noise. From (11), the CU can retrieve the signal from RU i if the UL FH rate is subjected to 18,19 where C UL i represents the UL FH rate constraint at the RU i , and where is the interference-plus-noise covariance matrix, and and . Different from the previous work 19 which maximizes the SE, this paper focuses on the system EE. The system EE is defined as the ratio of the total SR to the total power consumption and computed by where i ⟩+(K DL + K UL ) P UE + K R P RU is total system power consumed by all the RUs and UEs 35 . Here, P UE and P RU are the circuit power consumed by a user device and RU, respectively. We assume that circuit power consumed by UEs is same, i.e. P UE i = P UE , and also for the RUs. Our paper jointly optimizes the precoding and quantization noise covariance matrices to maximize the EE of the FD CRAN system subject to TP constraints in (5) and (10), and FH rate constraints in (4) and (12). Therefore, the design problem is mathematically formulated as s.t. (4), (5), (10), (12) (17b) The sets of optimization variables are . It can be shown that problem (17) is not a convex problem due to the non-convex fractional form of the objective function and non-convex constraints in (4), (12). Therefore, it is mathematically difficult to solve problem (17) directly. In next section, we will develop an IA based on sequential convex approximation and Dinkelbach approaches to make problem (17) amenable.

METHODOLOGY
In this section, we will recast problem (17) as a convex programming so that it can be efficiently solved by convex solvers. Firstly, to facilitate the nonlinearity of the OP, we define Q X k = F X k (F X k ) H , k ∈ S X , x ∈ {UL, DL}. The FH rate constraints in (4) and (12) can be respectively expressed as and Then the achievable downlink rate in (7) can also be written as where Similarly, the achievable uplink rate in (14) can be written as where On the other hand, the TP of RUs (5) and uplink users (10) also can be rewritten respectively as Then, the design of EE maximization is equivalently formulated as s.t. (18), (19), (25), (26) is the optimization variables. Next, we approximate the objective function (27a) and constraints (18), (19) using the D.C. approximation and then use the Dinkelbach method. The D.C. procedure in 15,24,36 is used to deal with the nonconvexity of the achievable rate function. Due to the fact that ∇ x log |I + X| = (I + X) −1 , by applying the first-order Taylor approximation at X 0 , we have an inequality log |I + X| ≤ log |I + X 0 | + ⟨(I + X 0 ) −1 (X − X 0 )⟩. Therefore, given feasible points Q (n) at iteration n, the rate functions R DL k and R UL k are respectively defined by Similarly, constraints (18), (19) are upper bounded by convex functions respectively defined as and where Then, problem (27)  Given λ (n) , Q (n) and Ξ (n) , solve (34) using CVX to obtain Q * and λ * Update Q (n) = Q * and Ξ (n) = Ξ * until convergence Update n=n+1 update Q (n) = Q (n) and Ξ (n) = Ξ (n) until convergence Output: Q opt , Ξ opt

RESULTS AND DISCUSSION
In this section, we simulate a FD CRAN system with the following parameters: K R = 4, M k = N k = 2, ∀k ∈ S RU , K DL = K UL = 4 and T k = R k = 2, ∀k. Moreover, we assume that the same TP constraints, i.e., P DL i = P DL , i ∈ S RU are imposed on RUs and UL users, and P UL i = P UL = 0.7 × P DL with k ∈ S UL . The noise powers in both UL and DL channels are set to σ 2 DL = σ 2 UL = −107 dBm. Both UL and DL channels are assumed to have the same FH rate constraints for all RUs, i.e., C X i = C FH (10 7 bps), ∀i ∈ S RU , X ∈ {UL, DL}, the bandwidth of the wireless system is 10 MHz, and the circuit power consumed by a RU and UE are P UE = 0.1 W, P RU = 10.65 W, respectively 35 . The users are randomly distributed in a square area of side length 100 m. The square is divided into four equal small squares with a side length 50 m. RUs are 493 (28) Figure 2 also show that the achievable EE tends to increase when the residual SI value decreases. Example 2: The achievable EE and SR over different TP budgets are evaluated. In this example, we set C FH = 10 8 bps. The achievable EE and SR are illustrated in Figure 3 and Figure 4, respectively. As expected, the higher EE performance is achieved at higher TP budgets. On the other hand, in these simulation settings, the EE obtained in FD transmission is higher than that in the HD counterpart. In addition, when a certain level of TP is reached, the resulting EE performance is saturated. Thus, these results can give useful information to set the appropriate TP at the transmitters. To provide more information about the achievable SR of the EE optimization Algorithm 1, the achievable SRs are plotted in Figure 4. It can be observed from Figure 4 that the achievable SRs in the considered settings are superior to that of the HD schemes. The smaller SI in FD transmission can lead to its better SR improvement.
Example 3: The EE and SR over different FH capacity limitations are investigated in this example. In this example, the TP budgets of the RUs are set 25 dBm. The average achievable EE is depicted in Figure 5 while the average achievable SR is shown in Figure 6. From the two figures, we can see that both achievable EE and SR are improved when the FH capacity increases. Noting that when the FH capacity is low, it is a bottleneck to restrict the system performance. In such cases, to reduce the FH traffic, the system will switch to a single RU association 19 . Therefore, when the FH capacity increases, the achievable EE and SR are increased. On the other hand, the achievable EE and SR increase as SI decreases.

CONCLUSION
In this paper, a joint design of FH compression and precoding has been studied for a FD MIMO CRAN. We have addressed the EE maximization problem for this considered system model by developing the IA using successive convex approximation and Dinkelbach approaches to deal with the non-convex problem. Finally, by numerical simulation results, we have shown that the considered FD CRAN system can bring significant EE gains over the HD scheme when SI is low. In addition, the results have unveiled the interesting influence of FH rate levels on the system EE performance. Particularly, the bottleneck situation caused by the limited FH capacity leads to the degradation in the EE system performance.