Yu ZHANG, Xuelu WU, Hong PENG, Caijun ZHONG, Xiaoming CHEN

1College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China

2National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

3College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China

Abstract: Owing to the inherent central information processing and resource management ability, the cloud radio access network (C-RAN) is a promising network structure for an intelligent and simplified sixth-generation (6G)wireless network. Nevertheless, to further enhance the capacity and coverage, more radio remote heads (RRHs) as well as high-fidelity and low-latency fronthaul links are required, which may lead to high implementation cost. To address this issue, we propose to exploit the intelligent reflecting surface (IRS) as an alternative way to enhance the C-RAN, which is a low-cost and energy-efficient option. Specifically, we consider the uplink transmission where multi-antenna users communicate with the baseband unit (BBU) pool through multi-antenna RRHs and multiple IRSs are deployed between the users and RRHs. RRHs can conduct either point-to-point (P2P) compression or Wyner-Ziv coding to compress the received signals, which are then forwarded to the BBU pool through fronthaul links. We investigate the joint design and optimization of user transmit beamformers, IRS passive beamformers,and fronthaul compression noise covariance matrices to maximize the uplink sum rate subject to fronthaul capacity constraints under P2P compression and Wyner-Ziv coding. By exploiting the Arimoto-Blahut algorithm and semidefinite relaxation (SDR), we propose a successive convex approximation approach to solve non-convex problems,and two iterative algorithms corresponding to P2P compression and Wyner-Ziv coding are provided. Numerical results verify the performance gain brought about by deploying IRS in C-RAN and the superiority of the proposed joint design.

Key words: Cloud radio access network (C-RAN); Intelligent reflecting surface (IRS); Transmit beamforming;Fronthaul compression

1 Introduction

With the development of artificial intelligence(AI), Internet of Things (IoT), mobile Internet,and industrial Internet, to provide diversified wireless network services to the whole society, it is necessary to build a cross-scenario, cross-business,reconfigurable, powerful, extremely intelligent,and extremely simplified wireless network, which integrates communication, computing, caching, and control (the 4Cs) with AI. With this aim, the concept of“intellicise” wireless network(Zhang P et al.,2022)has been proposed,among which the cloud radio access network (C-RAN) is one of the potential candidate network structures(Peng et al.,2020). In a C-RAN, the baseband processing function of conventional base stations is backward migrated into a baseband unit (BBU) pool, and radio remote heads(RRHs)are deployed close to users. Therefore,joint signal processing and flexible resource management can be inherently achieved in a C-RAN. Nevertheless, high-speed fronthaul links are required to connect RRHs and the BBU pool, which leads to high implementation cost and complexity for dense deployment of RRHs(Pizzinat et al.,2015). Therefore,a more cost-effective way to improve the capacity and coverage of C-RAN is required in the face of increase of access equipment and data traffic.

To tackle this issue, we propose to exploit the recently emerging intelligent reflecting surface(IRS)technology to enhance the C-RAN. An IRS consists of a large number of reflection elements, with which controllable phase shifts can be imposed on the impinging waves (Wu and Zhang, 2020). Aided by this, the IRS can generate the desired reflection beams and create favorable propagation conditions. Since the IRS is basically a passive device and solely requires a low-rate control link, it provides an energy-efficient and cost-effective way to enhance the C-RAN. Recently, a lot of works have studied the IRS-assisted wireless communication systems, e.g., multiple-input multiple-output (MIMO)systems (Wu and Zhang, 2018;Guo et al., 2019),simultaneous wireless information and power transfer(SWIPT) (Pan et al., 2020), two-way communication systems (Zhang Y et al., 2020),non-orthogonal multiple access (NOMA) (Zeng et al., 2021), and multi-cell systems (Hua et al., 2021). For IRSassisted millimeter-wave(mmWave)communication,Zhu et al. (2020) verified the system performance improvement based on stochastic geometry analysis. A low-complexity channel information acquisition method for IRS-assisted mmWave communication systems was proposed in Cui and Yin(2019).

1.1 Related works

One of the main concerns in C-RAN is the capacity-limited fronthaul link between the BBU pool and RRHs, and there are a lot of works considering fronthaul compression optimization to improve the system performance. Del Coso and Simoens(2009) investigated distributed fronthaul compression based on Wyner-Ziv coding at RRHs and optimized the compression noise covariance matrix in a gradient approach. In Park et al. (2013b), optimization of the per-RRH compression noise covariance matrix was proposed. Zhou YH and Yu(2014) considered both point-to-point (P2P) compression and Wyner-Ziv coding at RRHs, and then they optimized the corresponding quantization noise covariance matrices to maximize the uplink sum rate.For both uplink and downlink transmission,the distributed fronthaul compression techniques were surveyed by Park et al. (2014), and the design frameworks were proposed. In the C-RAN scheme in which users and RRHs are equipped with multiple antennas,the user/RRH beamforming and fronthaul compression can be jointly designed to further improve the system performance. Zhou YH and Yu(2016) jointly optimized the user transmit beamforming and fronthaul compression for a C-RAN uplink using the weight minimum-mean-square-error(WMMSE) approach and the successive convex approximation (SCA) approach, and thereafter proposed a low-complexity design under a high signalto-quantization-noise ratio(SQNR).For the C-RAN downlink, Park et al. (2013a) considered multivariate compression across RRHs and proposed to jointly optimize RRH beamforming and fronthaul compression,wherein a robust design with respect to imperfect channel state information (CSI) was also considered. For the C-RAN with wireless fronthauling,the beamforming corresponding to the fronthaul link(i.e., RRH beamforming for uplink and BBU pool beamforming for downlink)should also be taken into consideration. The joint design for uplink transmission was considered in Park et al.(2017a)and Najafi et al. (2019), and the design for downlink has been assessed by Park et al. (2016,2017b).

Up to now, there have been limited works devoted to IRS-assisted C-RAN or cell-free networks.Recently,Huang et al.(2021)proposed a distributed design framework to jointly optimize the transmit beamformer and IRS passive beamformer to maximize the weighted sum rate. An IRS-aided wideband cell-free network was considered by Zhang ZJ and Dai (2021), where a joint design was investigated to maximize the system capacity. Nevertheless,limited fronthaul capacity and fronthaul compression optimization were not addressed in the above works. For the C-RAN with wired fronthaul,Yu et al.(2020)exploited the IRS to improve the accuracy of over-theair computation. Similarly,Weinberger et al.(2021)investigated methods to obtain the full benefit from the cooperation of rate splitting and IRS techniques to improve the energy efficiency of C-RAN. In addition, Zhang Y et al. (2021) considered the deployment of a single IRS on the wireless fronthaul in a C-RAN and jointly optimized transmit beamforming, IRS passive beamforming, and fronthaul compression using a WMMSE-SCA approach.

1.2 Contributions

The main contributions of this paper are summarized as follows:

1. We consider IRS-assisted C-RAN uplink transmission, wherein multiple IRSs are deployed between multi-antenna users and RRHs. RRHs conduct P2P compression or Wyner-Ziv coding to compress the received uplink signals. We investigate the joint design of user transmit beamforming,IRS passive beamforming, and fronthaul compression under the two compression schemes, with the goal of maximizing the uplink sum rate. To the best of our knowledge, the above joint design has not been investigated in the literature.

2. The design problem is non-convex, wherein the transmit beamformers, passive beamformers,and fronthaul compression noise covariance matrices are coupled. By exploiting the Arimoto-Blahut algorithm (Blahut, 1972) and semi-definite relaxation(SDR),we propose two iterative algorithms based on SCA for the cases of P2P compression and Wyner-Ziv coding, respectively. Moreover, inspired by the results in the conventional C-RAN (Zhou YH and Yu, 2016), we modify the proposed algorithms for high SQNR regime, to reduce the computational complexity.

Note that the design of IRS passive beamforming is coupled with that of fronthaul compression through the fronthaul constraints. Therefore, the joint design investigated in this paper is not a trivial extension of the aforementioned works in conventional C-RAN. Besides, we consider Wyner-Ziv coding with joint decompression instead of sequential decompression as in previous works(Park et al.,2014;Zhou YH and Yu, 2016;Zhang Y et al.,2021),which is more general and leads to different fronthaul constraints. In particular, we propose a novel approach based on the Arimoto-Blahut algorithm to treat the non-convex problem.

2 System model

We consider the uplink transmission of a C-RAN, whereKmulti-antenna users communicate with the BBU pool throughLRRHs, as depicted in Fig. 1. Each user and each RRH are equipped withNUandNRantennas, respectively.MIRSs are deployed to aid the communication between the users and RRHs,each of which hasNIreflection elements.For simplicity, we assume global CSI at the BBU pool. Note that the CSI acquisition for the IRS link has been widely discussed(Cui and Yin,2019;Wang et al.,2020).

Fig. 1 System model

On the access link,userk(k ∈K)transmits the signal to the RRHs, which is given by the following

whereHk,l ∈CNR×NU,Gl,m ∈CNR×NI, andHk,R,m ∈CNI×NUrepresent the channel matrices between userkand RRHl, between IRSmand RRHl, and between userkand IRSm, respectively.HK,l= [H1,l,H2,l,...,HK,l] represents the channel matrix between each user and RRHl,Gl,M= [Gl,1,Gl,2...,Gl,M] represents the channel matrix between each IRS and RRHl,Hk,R,M=[HTk,R,1,HTk,R,2,...,HTk,R,M]Tdenotes the channel matrix between userkand each IRS, andHK,R,M= [H1,R,M,H2,R,M,...,HK,R,M] denotes the channel matrix between each user and each IRS.Θm= diag(θm,1,θm,2,...,θm,NI) represents the passive beamformer of IRSm(we assume that the IRS can adjust only the phase shift, i.e.,|θm,n|= 1),Θ= diag({Θm}m∈M),FK= diag({Fk}k∈K), andsK= [sT1,sT2,...,sTK]T. Finally,nl～CN(0,σ2I) is the additive white Gaussian noise. In the system,each IRS is deployed near an RRH,and the distance between two arbitrary IRSs is quite large. Therefore,the double reflection among the IRSs is ignored due to the large path-loss.

RRHlcompresses its received signals and then transmits the quantization bits to the BBU pool through a wired fronthaul link with limited capacity. By adopting the Gaussian test channel model,the compressed signal recovered by the BBU pool can be expressed as follows(Park et al.,2014):

whereql～CN(0,Ωl) represents the quantization noise for RRHlandΩldenotes its covariance matrix which is determined by the corresponding quantization codebook.

3 Joint design of beamforming and fronthaul compression under P2P compression

In this section we investigate the joint design of user transmit beamformers, IRS passive beamformers,and fronthaul compression when each RRH conducts P2P compression, aiming to maximize the system uplink sum rate.

3.1 Problem formulation

From Eqs. (1) and (3), the achievable uplink sum rate of the considered C-RAN is given by

whereClrepresents the fronthaul capacity from RRHlto the BBU pool. According to Eqs. (2)and (3), the random variables in the mutual information term on the left jointly follow the Gaussian distribution. LetVl=HK,l+Gl,MΘHK,R,M. Constraint(5)can be expressed as

With the goal of maximizing the uplink sum rate in Eq. (4) under the fronthaul constraint (6) under P2P compression, we jointly optimize the transmit beamformers of users,the passive beamformers,and the quantization noise covariance matrices for fronthaul compression. The problem is formulated as

It is non-trivial to find the optimal solution to the above problem, due to the fact that the objective function,the fronthaul constraints,and the constraint|θm,n|= 1 are not convex. In the following subsection,we adopt the SCA approach(Scutari et al., 2014)to make the problem tractable.

3.2 SCA approach

First,for the objective in problem(7),we exploit the Arimoto-Blahut algorithm,which is given by the following lemma:

Lemma 1(Cover and Thomas,2006) For a channel with inputs,outputy,and transition probabilityp(y|s), the mutual informationI(s;y) with an arbitrary input probability distributionp(s) is given by the following expression:

Using the Arimoto-Blahut algorithm, the formula of the achievable sum rate is transformed into an expectation term,with which this non-convex objective function can be transformed into a convex form, as will be discussed later. Then we tackle constraint (6) in problem (7). First, we state the following lemma:

Lemma 2 (Zhou YH and Yu, 2016) For positive definite Hermitian matricesΓ,E ∈CL×L, we have

Remark 1According to inequality (16), any feasible solution to problem (19) satisfying constraint (18) also satisfies constraint (6) for problem (7). Therefore,it is also feasible for the original problem.

With the above problem reformulation, we can tackle the original problem (7) using SCA as follows: In each iteration,we first update the auxiliary variablesEl,W, andΣ, under fixedFk,Θ, andΩl. Obviously, the optimalWandΣare given by Eqs. (11) and (12), respectively.Elis updated according to Eq. (17). Then, we optimizeFk,Θ,andΩlby solving problem(19). The convergence is discussed in Section 4.

3.3 Alternative optimization

In this subsection, we solve problem (19) by an alternative approach. The expectation term in the objective can be evaluated as follows:

Similarly,we can rewrite constraint(18)in problem(19)as follows:

We apply SDR by relaxing the rank-one constraint on ˉΘ,and the resulting problem now becomes convex. Thus, it can be effectively solved by standard convex optimization tools such as CVX. Note that the obtained ˉΘmay not be exactly of rankone in general. We apply the efficient randomization techniques given in Sidiropoulos et al.(2006)to generate suboptimal candidates and choose the one achieving the minimum objective function.

To this end, we summarize the proposed SCA algorithm for P2P compression (SCA-P2P) (Algorithm 1).

4 Joint design of beamforming and fronthaul compression under Wyner-Ziv coding

Algorithm 1 Successive convex approximation for point-to-point compression(SCA-P2P)1: Initialize Fk, Θ, and Ωl feasible for problem (7).2: Update W, Σ, and El using Eqs. (11), (12), and (17),respectively.3: Update Fk by solving the convex problem (21).4: Solve problem (25). Update (Θ,Ωl) if the objective decreases.5: Repeat steps 2-4 until convergence.

In this section we focus on the joint design of beamforming and fronthaul compression under Wyner-Ziv compression. Note that under the P2P compression discussed in Section 3, the BBU pool processes the compressed signals from each RRH individually. Since the received signals at different RRHs are statistically dependent, RRHs can perform more efficient compression (Park et al., 2014)by exploiting this dependency via Wyner-Ziv coding. The BBU pool jointly recovers the compressed signals from all RRHs,whereby the signal dependencies are used as side information. The corresponding fronthaul constraints are given by the following expression(Zhou YH and Yu, 2016):

where ˉSis the complement set ofS. Similar to the case of P2P compression, one can evaluate the mutual information term on the left and rewrite constraint(26)as follows:

With the goal of maximizing the uplink sum rate in Eq. (4) under the fronthaul constraint (27)for Wyner-Ziv coding, the problem is formulated as follows:

Again,problem(28)is non-convex and we adopt the SCA to tackle it. Nevertheless, note that the fronthaul constraint (27) for Wyner-Ziv coding cannot be treated in the same way as that for P2P compression,i.e., using solely Lemma 2.

4.1 SCA method

The objective function of problem (28) can be handled similarly as in Section 3. In the following,we focus on the fronthaul constraint (27). First, we rewrite inequality (27)as follows:

Similar to the P2P compression case,according to inequalities(30)and(32),any feasible solution to problem (36) is also feasible for the original problem (28). Based on this, we still use SCA to solve problem (28) as follows: In each iteration, we first update the auxiliary variablesEL,W,Σ,WˉS, andΣˉS, under fixedFk,Θ, andΩL. Obviously, the optimalWandΣare given by Eqs. (11) and (12),respectively. On the other hand,EL,WˉS, andΣˉSare updated according to Eqs. (31), (33), and (34),respectively. After updating and fixing the auxiliary variables, we optimizeFk,Θ, andΩLby solving problem(36).

4.2 Alternative optimization

We solve problem (36) in an alternative manner. First, the expectation term on the LHS of constraint (35) of problem (36) can be evaluated as follows:

Problem (41) has a similar structure to problem(25). We also exploit SDR to solve the problem.We summarize the proposed SCA-based iterative optimization algorithm under Wyner-Ziv coding(SCAWZ)(Algorithm 2).

Algorithm 2 Successive convex approximation for Wyner-Ziv coding (SCA-WZ)1: Initialize Fk, Θ, and ΩL feasible for problem (28).2: Update W, Σ, EL, WˉS, and ΣˉS using Eqs. (11), (12),(31), (33), and (34), respectively.3: Update Fk by solving the convex problem (38).4: Solve problem (41). Update (Θ,ΩL) if the objective decreases.5: Repeat steps 2-4 until convergence.

Remark 2In this study, we consider the joint design of user transmit beamformers, IRS passive beamformers, and fronthaul compression assuming perfect CSI knowledge. With the CSI error, one approach is to add a “safety region” on the feasible region as shown in Najafi et al. (2019). Explicitly,we can multiply a constantb, which is smaller than“1” on the right-hand side of the first constraint in problems(7)and(28)(i.e.,inequalities(6)and(27)).Then,the proposed algorithms can still be used without modification. Another approach is to calculate the expectation of the system’s sum rate and fronthaul compression rate over all possible CSI errors(Zhou G et al., 2020). Then, the resulting problem is different from that considered in this study.

4.3 Simplified design under high SQNR

In Zhou YH and Yu (2016), it was shown that for a C-RAN uplink in high SQNR,i.e.,in the case of a large fronthaul capacity,the quantization noise covariance matrices at RRHs are nearly optimal when proportional to the background noise covariance matrices, i.e.,Ω?l ≈βlIfor RRHl, whereβl> 0 is chosen to satisfy the fronthaul constraint. Although it was proved for P2P compression and Wyner-Ziv coding with sequential decompression,it is straightforward to extend to the case of Wyner-Ziv coding with joint decompression considered in this work.Therefore,we propose an efficient scheme under high SQNR for both Wyner-Ziv coding and P2P compression. Explicitly, we setΩl=βlIand then jointly optimizeFk,Θ, andβl, wherein the non-convex objective function and constraints can be treated in the same way as before. Since we optimize the scalarβlinstead of the whole covariance matrixΩl, the computational complexity can be reduced. Moreover,suchΩlcan be achieved by low complexity perantenna signal quantization at each RRH in practice.

4.4 Convergence and complexity analysis

In this subsection,we discuss the convergence of the proposed SCA-P2P algorithm and SCA-WZ algorithm,as well as their computational complexities.In the following paragraphs,we take SCA-WZ as an example for detailed discussion, and the computational complexity for SCA-P2P is similar to that for SCA-WZ.

The computational complexity of the SCA-WZ algorithm is dominated by steps 3 and 4, which involve solving problems (38) and (41). In each iteration, by using the interior point method (Potra and Wright, 2000), problem (38) can be efficiently solved by the polynomial in the problem size given byKNUd, and the size of problem (41) is given by(MNI)2+LN2R(Zhou YH and Yu,2016). The overall complexity of the SCA-WZ algorithm is given as the product of the number of iterations and the above complexity. The complexity of SCA-P2P can be similarly analyzed;it is of the same order as that of SCA-WZ.

5 Numerical results

5.1 Simulation setup

In the simulation, we consider a C-RAN with four users, two IRSs, and two RRHs. The positions of the two RRHs are(?30, 100,15)m and (30,100,15) m, and the two IRSs are located at (?40, 80,15) m and (40, 80, 15) m. The users are randomly distributed within a circle centered at the origin with a radius of 30 m,and the height is 1 m. For the channels between the users and RRHs,between the users and IRSs, and between the IRSs and RRHs, we apply the Rician fading model with limited numbers of propagation paths (Kim et al., 2021). For example, the uplink channel between userkand RRHlis modeled as

whereu0is the path-loss at distanced0, anddU,RandαU,Rdenote the distance and path-loss exponent between the user and RRH,respectively. The RicianK-factor is denoted byKU,R, andGU,Ris the total number of non-line-of-sight (NLoS) paths. For thegthpath,ηU,R,g～CN(0,1)is the complex path gain.The array response vector at the RRH,i,e.,aRRH(·),is given by

In the simulation, we set the path-lossu0to?30 dB when the reference distanced0=1 m. The path-loss exponentsαU,R,αU,I, andαI,Rare set to 3.6, 2.2, and 2.2, respectively. For small-scale fading, we set the Rician factors asKU,I=10 dB,KI,R=10 dB,andKU,R=0 dB(we assume that there is no LoS path for the direct links between users and RRHs). The numbers of NLoS paths are set asGU,I=10,GI,R=4, andGU,R=10, where the vertical and horizontal AoAs (AoDs) of each NLoS path are assumed to be uniformly distributed within[?π/4,π/4]and[?π/2,π/2],respectively. Each user or RRH is equipped with a uniform linear array,and each IRS is a uniform planar array. The transmit power of each user is set asPk=10 dBm. Finally,the Gaussian noise variance is set to?87 dBm, and the stop condition threshold is set asΔ=10-2.

5.2 Simulation results

First, we verify the convergence of the proposed algorithms under different numbers of reflection elements per IRS (Fig. 2), wherein “WZ-simp”and “P2P-simp” denote the low-complexity schemes designed for the high SQNR regime where we setΩl=βlI. It can be observed that the uplink sum rate gradually increases as the number of iterations rises, and that all the proposed algorithms converge in all cases and have good convergence performance.Comparing the convergence speeds for the schemes with the number of reflection elements as 30 and 70,it can be found that more iterations are required with a larger number of reflection elements. Moreover, it is shown that the iteration numbers of the simplified algorithms for high SQNR, i.e., “WZ-simp” and“P2P-simp,” are smaller than those of “SCA-WZ”and“SCA-P2P,” respectively.

Fig. 2 Average sum rate versus the number of iterations (Pk =10 dBm, Cl =5 bits/(s·Hz), ?l, NR = 4,NU =2)

Next,we investigate the performance of the proposed algorithms under different numbers of RRH antennas but fixed numbers of user antennas and IRS elements. It can be seen from Fig. 4 that the uplink sum rate increases along with the increase of the number of RRH antennas. This verifies the receiving array gain brought about by the two multiantenna RRHs. Similar to Fig.3,the proposed algorithms SCA-WZ and SCA-P2P still outperform all the benchmark schemes under all numbers of RRH antennas.

Fig. 3 Average sum rate versus the number of reflection elements per IRS (Pk =10 dBm, Cl =5 bits/(s·Hz), ?l, NR =4, NU =2)

Fig. 4Average sum rate versus the number of RRH antennas (Pk =10 dBm, NI =50, NU = 2,Cl =5 bits/(s·Hz), ?l)

Fig.5 plots the average uplink sum rate achieved by the proposed algorithms for Wyner-Ziv coding and P2P compression as well as the low-complexity schemes for high SQNR, wherein the user transmit power is set asPk= 10 dBm and each IRS has 50 reflection elements. As the sum fronthaul capacity increases,the rate loss of“WZ-simp”and“P2P-simp”becomes smaller. This validates the premise that setting the quantization covariance matrix proportional to the identity matrix is nearly optimal under high SQNR.It can also be observed that“SCA-WZ”and “WZ-simp” always outperform “SCA-P2P” and“P2P-simp,” respectively, as the fronthaul capacity increases.

Finally, we simulate the system’s sum rate achieved by the proposed algorithms and the benchmarks under different numbers of IRSs (the number of reflection units for each IRS reflection unit is set to 30). In the simulation,we consider a C-RAN with four users and two RRHs. The positions of the two RRHs are(?30,100,15)m and(30,100,15)m,and the users are randomly distributed within a circle centered at the origin with a radius of 30 m,and the height is 1 m. The locations of IRSs are:A(?40,80, 15) m,B(40, 80, 15) m,C(?40, 80, 20) m, andD(40, 80,20)m.

Fig. 5 Average sum rate versus the sum fronthaul capacity (Pk =10 dBm, NI =50, NR =4, NU =2)

Table 1 Average sum rate versus the number of IRSs

The simulation results are shown in Table 1. It can be observed that the uplink sum rate increases with the increase of the number of IRSs, and that the proposed algorithms still outperform the corresponding benchmarks. Therefore, the uplink sum rate can be improved by increasing the number of IRSs or the number of reflection elements per IRS.This also verifies that deploying IRSs can enhance the performance of the system.

6 Conclusions

In this paper, we have studied a joint design of user transmit beamforming, IRS passive beamforming, and fronthual compression for multi-IRS-aided C-RAN uplink under P2P compression and Wyner-Ziv coding compression, aiming at maximizing the uplink sum rate. We have used the Arimoto-Blahut algorithm and an SDR to handle the non-convex objective function and constraints. Then a successive approximation and optimization approach has been proposed. Based on the approach two iterative optimization algorithms for P2P and WZ have been given. Numerical results have verified that deploying IRSs can considerably improve the C-RAN uplink sum rate with the proposed optimization algorithms.

As for the network energy consumption for uplink transmission, in this work, we have considered solely the transmit power. In practice, the energy consumed by RRHs, IRSs, and fronthauls for data processing and system operation should also be considered. Therefore, the joint design of beamforming and fronthaul compression to improve the network energy efficiency can be a possible extension of the current work.

Contributors

Yu ZHANG and Xuelu WU designed the research.Xuelu WU processed the data and drafted the paper. Yu ZHANG helped organize the paper. Yu ZHANG, Hong PENG, Caijun ZHONG, and Xiaoming CHEN revised and finalized the paper.

Compliance with ethics guidelines

Yu ZHANG, Xuelu WU, Hong PENG, Caijun ZHONG,and Xiaoming CHEN declare that they have no conflict of interest.

Appendix A: Derivation of inequality(24)

Then, we can obtain inequality(24).

Appendix B: Derivation of inequality(40)