Ziyi Yang,Yu Zhang

School of Information and Electronics,Beijing Institute of Technology,Beijing 100086,China.

Abstract: Reconfigurable intelligent surface(RIS)is a promising technology that utilizes massive low-cost elements to reflect incident signals to desired direction.In this paper,we consider a RIS-aided cell free multiple-input multiple-output (MIMO) network that composed of multiple access points (APs),multiple RISs,multiple information receivers(IRs),and multiple energy receivers(ERs).Under the maximum transmit power constraints of the APs and the minimum total energy harvesting power constraints of the ERs,we aim to maximize the weighted sum-rate(WSR)by jointly optimizing the beamforming matrices and the passive RIS reflection matrices.However,the formulated problem is nonconvex and very challenging to solve.Therefore,we propose an alternating optimization algorithm to obtain a suboptimal solution.Explicitly,the proposed algorithm decomposes the original optimization problem into three subproblems that can be efficiently solved.To overcome the nonconvexity of the subproblems,a successive convex approximation (SCA) based algorithm is proposed to tackle the energy harvesting constraints.Moreover,an alternating direction method of multipliers(ADMM)based algorithm is proposed to optimize the constant modulus constrained RIS reflecting elements.Finally,simulations are conducted to demonstrate the performance advantages of the proposed algorithms.

Keywords: cell-free networks; SWIPT; Reconfigurable intelligent surface;SCA;ADMM

I.INTRODUCTION

Reconfigurable intelligent surface (RIS) is a promising technology that can improve the network capacity for next generation wireless networks[1-3].A typical RIS consists of a large number of passive low-cost reflecting elements,each element can dynamically and independently adjust the phase shift and amplitude of the incident signals with the aid of a smart controller.Different from traditional relay technologies,RIS reflecting process does not introduce extra noise.As a result,RIS is capable of constructing favorable wireless channels and can be applied to various kinds of communication scenarios[2,3].

Recently,the research on RIS-aided wireless communications have gained increasing prominence from academia[4-7].Specifically,the research on RISaided wireless communications can be mainly categorized into two classes,i.e.,channel estimation and transceiver optimization.In[8,9],the authors investigated the channel estimation for the ISR-aided systems.Since the RIS operates without radio frequency (RF) chains and thus lacks the baseband processing capability.Therefore,the traditional trainingbased channel estimation schemes cannot be implemented.In[10],different cascaded transmitter-RISreceiver channel estimation schemes using the reflection grouping strategy are proposed for both narrowband and wideband communications.

The second category aims to study RIS-aided transceiver designs for different communication systems,e.g.,point-to-point multiple-input multipleoutput (MIMO) systems[11]and multiuser MIMO systems[12,13].Generally,the optimization of transceivers is jointly investigated with the passive RIS reflection matrix design.In terms of the transmit beamforming optimization of access point(AP),it is widely investigated for different performance metrics,e.g.,weighted sum-rate(WSR),mean square error (MSE),and total transmit power.Furthermore,deep learning (DL) based algorithms are also integrated in RIS-aided networks that utilizes the probabilistic sampling theory to select the optimal locations of these active antennas[14-16].Moreover,some other works have also investigated the integration of the RIS with other advanced techniques,such as nonorthogonal multiple-access[17,18]and hybrid beamforming[4,19].With respect to signal processing complexity and energy consumption,it is widely conceived that RIS has competitive advantages over the existing MIMO relay and massive MIMO technologies.

Although the implement of RIS can improve the efficiency of wireless communications,the energy scarcity issue is still an important and challenging problem,especially in ultra-dense cell-free networks that composed of multiple APs and devices.To address this issue,simultaneous wireless information and power transfer (SWIPT) has been proposed to exploit RF signals to enable the simultaneous transfer of information and energy.A multiuser multipleinput single-output (MISO) SWIPT system is investigated in[20],in which a multi-antenna transmitter sends energy-carrying information signals to multiple single-antenna information receivers(IRs)and an energy receivers(ERs).In[21],the authors investigated the performance of multi-relay cooperative networks with the aid of SWIPT.Overall,SWIPT technology is one of the key technologies that enabling efficient transmission in energy-limited ultra-dense networks.

Recently,a wide range of SWIPT-based applications have been studied for wireless communications.The performance tradeoffs between data rate and energy harvesting was investigated in[22].Moreover,the authors in[23]investigated the performance for both frequency-selective fading channel and flat fading channels.In[24],the authors studied the MIMO SWIPT system and proposed a practical transceiver architecture for IRs and ERs.Motivated by the importance of SWIPT,there are many works focus on the performance of SWIPT for various MIMO systems,e.g.,multiuser MIMO systems[25,26],MIMO relay systems[27,28],and partially wireless powered communication networks[29,30].For instance,[31]utilized the semi-definite relaxation (SDR) technique to study the joint transmit beamforming,power splitting receiver,and time switching receiver optimization to minimize the transmit power under SINR and EH constraints.In[32],the authors proposed a second-order cone programming (SOCP) based algorithm to solve the joint optimization problem,which has significantly reduced computational complexity compared with the SDR based algorithm.

In order to reap the performance advantages of SWIPT and RIS,a variety of combinations of these two techniques have been investigated recently.It can be used to enhance many different communication paradigms,e.g.,secure communications[33],heterogeneous networks[34],mobile edge computing[35]and Internet of Things (IoT)[36].For instance,the minimum transmit power at APs and the minimum power received at all ERs are optimized by adjusting their transmit beamforming matrices and the RIS reflecting phase shifts,while guaranteeing the qualityof-service (QoS) constraints at all receivers,namely,the individual signal-to-interference-plus-noise ratio(SINR) constraints at IRs and the energy harvesting(EH) constraints at ERs[37-39].The security of the RIS-aided SWIPT system was studied in[33,40].Specifically,the objective is to maximize the secrecy rate by jointly optimizing the transmit beamforming and artificial noise covariance matrix at the AP and RIS reflecting phase shifts,while guaranteeing the predefined transmit power at the BS and EH constraints at the ERs.In[41],the weighted harvested energy at ERs was maximized for RIS-aided SWIPT system,while subjecting to a given set of minimum SINR constraints at RIS.Simulation results demonstrated the effectiveness of RIS for enhancing the performance of SWIPT systems as well as its superior performance gains.

In this paper,we investigate the joint beamforming and reflecting phase shifts optimization for RISaided cell-free networks MIMO networks,in which the APs transmit two types of signals to multiple RISs,ERs and IRs.To the best of authors’knowledge,this work is the first attempt that focus on the beamforming optimization of RIS-aided cell-free MIMO network with SWIPT[42].Due to the intricately coupled optimization variables in both the objective function and constraints,it is difficult to derive a globally optimal closed-form solution to the formulated non-convex WSR maximization problem.Therefore,we propose an alternating optimization algorithm to overcome the inherent nonconvexity of the optimization problem.The main contributions of our work are listed as follows.

·First,we consider a RIS-aided SWIPT MIMO cell free network,in which the beamforming and RIS reflecting phase shifts are jointly optimized.There are three kinds of nonconvex constraints in the considered optimization problem.For cell free networks,multiple cooperative APs transmit both information signals and energy-carrying signals to multiple ERs and IRs.The considered SINR,energy harvesting constraints and the constant modulus constraints are nonconvex with respect to the optimization variables.

·Second,we propose an alternating algorithm to address the aforementioned nonconvexity.Specifically,both SDR and successive convex approximation (SCA) techniques are exploited to solve the nonconvex beamforming optimization problem.In particular,the SCA-based algorithm has less computational complexity compared to the SDR-based algorithm.

·Third,we propose an ADMM-based algorithm to optimize the RIS reflecting phase shifts.Explicitly,the optimization problem is transferred into vector form and the constant modulus constraints can be iteratively satisfied by applying the auxiliary variables and the penalty terms.Simulation results demonstrate the superior performance of proposed algorithm compared to the benchmarks.

The rest of this paper is organized as follows.Section II introduces the system model and the problem formulation of the joint optimization in RIS-aided cell free networks.In Section III,an alternating optimization algorithm is proposed to solve the WSR maximization problem.Numerical results and discussions are presented in Section IV.Finally,Section V concludes this paper.

Notation:Throughout this paper,the following notations are used.Scalar,vector,and matrix are denoted by non-bold,bold-lower-case,and bold-uppercase letters,respectively.The symbolsA∗,AH,and Tr(A) represent the conjugate,Hermitian,and trace of matrixA,respectively.N+denotes the set of positive integers.The symbol Diag{a}denotes a square diagonal matrix whose diagonal elements are the same as that ina.The real and imaginary parts of a complex variable are represented byℜ{·}andℑ{·},respectively.The statistical expectation is denoted by E[·].“independent and identically distributed”is abbreviated as“i.i.d.”.

II.SYSTEM MODEL AND PROBLEM FORMULATION

In this paper,we investigate a RIS-aided SWIPT network which is shown in Figure.1.There areIAPs and each AP is equipped withNSantennas.The information signals and energy signals are simultaneously transmitted from each AP toMRISs,JIRs andKERs.In cell-free MIMO networks,all APs cooperate to transmit signals to all RISs and receivers.Specifically,the transmitted signal from APiequals

Figure 1. System structure.

Figure 2. Convergence behavior of the proposed SDRbased alternating algorithm.

wheresjdenotes the information signal for IRjwith zero mean and unit variance.The vectorpi,jis the corresponding beamforming vector forsjat APi.The vectordiis the energy symbol for ERkand its covariance matrix is defined as follows

The channel matrix between theith AP and thejth IR is denoted as

where|θm,n|= 1,where∀m ∈{1,2,...,M}and∀n ∈{1,2,...,NR}.In other words,each diagonal element of the diagonal matrix Φmis only phase adjustable.

Based on the previous definitions,the received signalypat thepth IR equals

At thepth IR,a linear receiverrpis performed to recover the desired signalsp.Then the corresponding SINR at thepth IR can be derived to be in(6)

whererpis the receiver at thepth information user.In order to make the SINR more compact for the following analysis,the following auxiliary variables are introduced

the SINR in (6) can be written in (8)

based on which,the weighted sum rate of the whole network equals

whereωpis the corresponding weighting factor for thepth information receiver.

Meanwhile,the channel matrix from APito ERjis denoted as follows

the energy received at ERlequals

Considering SWIPT,in our work,we investigate the optimization problem aiming at maximizing the weighted sum rate of all RIS under both transmit power constraints at each AP and the energy harvesting constraints at different ERs.In other words,the beamforming algorithm makes sure the minimum energy harvesting requirements are satisfied.This is important for practical internet of things(IoT)networks.In a nutshell,under both the transmit power constraints and the energy harvesting constraints the beamforming optimization problem is formulated as (13)

whereEi,Sis the maximum transmit power at APiandEl,conis the minimum threshold at thelth ER.It is obvious that this optimization is nonconvex with respective to the variables.The main difficulty is three folds.First,the mathematical formula of SINRpis very complicated.Multiple APs cooperate in communicating with numerous IR simultaneously for cell free networks,and there is severe mutual interference in the spatial domain.Second,the second kind of constraint is nonconvex.It is well-known that energy harvesting constraints are nonconvex with respect to the beamforming vectors at the source nodes or the covariance matrices of the energy signalsdi’s.Third,there is a series of constant modulus variables to optimize,which are the phases of antenna elements of RISs.The optimization for constant modulus variables is much more challenging as constant modulus constraints are nonconvex.Motivated by the logic of the famous WMMSE algorithms,a series of auxiliary variablesμp’s have been introduced and the following

equality holds

It is because based on the first order condition the minimum value of the function on the righthand side of the above equality corresponds to the following solution

Substituting (15) into the righthand side,the equality holds.Therefore,after removing the irrelative constant parameters,the optimization problem (13) is equivalent to the following optimization problem(16).

The main idea behind this transformation is how to exploit an iterative optimization procedure to overcome the nonconvexity of the original optimization problems.It is worth highlighting that the optimal solutions of P2 corresponds to a lower bound of P1.Generally speaking,the optimization algorithm is iteratively implemented among the beamforming vectorspj’s at the different APs,the covariance matricesQdi’s of the energy signals,the phase shift matrices Φm’s at the different RISs.This is the focus of the following section.

III.BEAMFORMING OPTIMIZATION

When the weighting factorsμp’s in P2 are fixed,It is obvious that 1/(1 + SINRp) is not convex with respect to the involved optimization problems.It means that it is still very challenging to solve the considered optimization problem directly.Common logic is how to transfer the original nonconvex function into convex ones.To the best of our knowledge,the most widely used convex functions in wireless communications are only four kinds of functions,i.e.,linear functions,quadratic functions,fractional functions and logarithmic functions.The last two kinds of functions can be removed directly as they are the original formulas in the optimization problems (13).Quadratic functions usually correspond clear physical meanings e.g.,mean square errors,signal energy,etc.Based on this fact,in the objective function the fractional equation in the objective function can be further rewritten as (17),

where

At this time,the original weighted sum rate maximization has been transferred into the weighted MSE minimization under transmit power constraints and energy harvesting constraints.From a physical meaning perspective,it is straightforward that minimizing MSE is equivalent to maximizing SINR.An alternating optimization algorithm is proposed for the optimization problem (19).Generally speaking,using the corresponding equivalence,the proposed algorithm optimizes the considered optimization problem P3 iteratively among the involved optimization variables.

3.1 The Optimization of Receivers

When all the other optimization variables are fixed,because there is no constraint on the receiver,the optimal solution of the linear receiverrpat thepth IR can be derived based on the complex vector derivative based on which the optimal solution ofrpis derived to be

It is worth pointing out that the optimalrpfor the considered optimization problem is in nature the LMMSE receiver at IRp.

3.2 The Optimization of Energy Beamforming

The energy signaldiis only characterized by its covariance matrixIn order to realize effective energy transmission,the covariance matricesat different APs should be jointly optimized.For the optimization of the covariance matrices of energy signals transmitted from each AP,after removing the unrelated variables the optimization problem is equivalent to the following one

It is evident that the optimization problem(22)concerningis a standard semidefinite programming(SDP)problem,and it can be efficiently solved by using interpoint algorithms.

3.3 The Optimization of Precoders

For the optimization of the beamforming vectors at each AP for the corresponding information receivers,the optimization problem equals to(23).

It is worth noting that in the optimization problem(23) the second kind of constraints,i.e.,the energy harvesting constraints is nonconvex because of the inequality direction.Therefore,the optimization problem (23) is not a standard QCQP problem.In order to overcome the nonconvexity,here we use two kinds of techniques to solve it.The first is the SDP-based algorithm whose idea is to transfer the original nonconvex optimization problem into a convex SDP problem.The second is the SCA based algorithm.Specifically,successive convex approximation(SCA)is used for the nonconvex energy harvesting constraints..

3.3.1 SDP-Based Algorithm

Introducing the following positive semidefinite matrices

with theith submatrix satisfying

the optimization problem P5 can be transferred into(26).

It is obvious that the optimization problem P7 is a standard SDP problem which can be efficiently solved via interior point algorithm.

3.3.2 SCA-Based Algorithm

To overcome the convex nature of P5,SCA can be used.For the non-convex energy harvesting constraint,based on the first order Taylor expansion the first term satisfies the following inequality

Based on the lower bound,the optimization problem(23)can be transferred into(28).

The optimization problem (28) is a standard quadratic programming (QP) problem.It can be concluded that the optimization problem (28) can be solved efficiently using interior point algorithms.In the SCA based optimization algorithm,the values ofare updated iteratively and the superscripttdenotes the iteration index.The gap between the optimization problems(23)and(28)will shrink asymptotically.

3.4 The Optimization of RIS Reflecting Matrices

In this subsection,we will focus on how to optimize the phase shift matrices Φm’s at the RISs.With respect to the diagonal matrices Φm’s,the corresponding optimization is(29).

We would like to highlight that the diagonal matrices Φm’s are contained inHp,WlandWl,i.In order to make the following derivations clear,the diagonal matrices Φm’s will be rewritten in an explicit manner.

At the beginning,the following equality is given

and the diag[Φm]denotes the row vector constructed by the diagonal elements of the diagonal matrix Φm,i.e.,

based on which and together with(30)and the definition ofHpwe have where the vector

and the matrixBpare defined as follows

with

Then based on the equation given in(33),in the objective function of(29)the following equation holds

Therefore,the objective function of P8 can be rewritten as(37).

Meanwhile,in the constraint of (29),similarly we have

where the vectorcj(l)and the matrixCj(l)are defined in the following

with

As a result,we have the following inequality for the quadratic term

BecauseQdis a positive semidefinite matrix,based on matrix decomposition we have

based on whichQdequals

whereqn’s are column vectors ofThen the second term in the constraint can be rewritten as

Similar to the previous discussions,we have

where the vectoren(l) and the matrixEn(l) are defined in the following

As a result,we have the following equality for thequadratic term

Based on the above discussions and derivations,after introducing the auxiliary matrix variables(48).

The optimization problem P7 for the phase shit matrices at different RISs is transferred into the following compact formula

It can be seen that in(49)both the objective function and the constraint functions are quadratic with respect to the vectorf.Moreover,each element offhas a constant modulus,i.e.,unit modulus constraint.

3.4.1 SDP-Based Algorithm

By introducing an auxiliary matrix variable

the optimization problem P9 can be translated into the following optimization

Based on the widely used SDR technique and relaxing the elementsfto be unconstrained in modulus,P10 becomes a standard SDP problem.It is obvious that there are two many relaxations in the SDP-based algorithm.However,the solutions of P10 can be used as an initial value for more advanced algorithms.In the following,an ADMM based algorithm is proposed for the optimization P9.

3.4.2 ADMM Based Algorithm

In our work,the alternating direction method of multipliers(ADMM)based framework is exploited to solve the above optimization problem with constant modulus constraints.In order to implement ADMM algorithm,a series of auxiliary variables{zl}forl=0,...,Lare introduced first and then the optimization problem(49)is reformulated as

where{zl}are auxiliary variables.The function 1F(z0) is an indicator function of the unit modulus setF,which is defined as

The corresponding Augmented Lagrange of P11L(f,{zl},{αl},{βl})can be written in the following form

where{αl},{βl}are the Lagrange multipliers for the corresponding constraints.In addition,the parameters{μl}are scalar penalty parameters (step size).Denote the superscripttas the iteration index,and then the iterative procedure of the proposed ADMM based algorithm is summarized as (55)

where{βl}can be obtained by utilizing the following complementarity slackness condition

Remark:In this paper,the channel matrices are assumed to be known perfectly.In this remark,we want to highlight that the proposed algorithms carry over to the cases in which the CSI is imperfect.With imperfect CSI,the channel matrix can be written in the formwhereis the estimated channel matrix and ΔHis the estimation error.When statistical channel error model is adopted,the channel error can be written asand ΨTare the row and column correlation matrices,respectively.The elements ofHware i.i.d.random variables.Based on this error model,the Bayesian robust beamforming algorithm can be designed.

IV.NUMERICAL RESULTS AND DISCUSSIONS

In this section,several numerical results will be presented to demonstrate the performance advantages of the proposed beamforming optimization algorithm.Unless specified otherwisely,the number of APs,RISs,ERs and IRs areI= 2,M= 2,L= 2,J= 3,respectively.Moreover,we assume that the APs are located at(0,0,2)m and(0,80,2)m with radiusrI= 1 m,rE= 1 m,respectively.The RISs are located at (0,5,2)m and (10,50,2)m,respectively.The centers of ER cluster and IR cluster are (0,5,0)m and(20,50,0)m,respectively.The radius isrI= 10m andrE= 5m.The maximum number of iterations of proposed algorithm is set as 50.The transmit power at the APs are set asEi,S ∈[10,30]dBm.

For comparison,we adopt three benchmarks to evaluate the performance of proposed algorithm,which are:1)Without RIS case:the RIS reflecting matrices are set as Φm= 0,∀m,the proposed alternating algorithm is performed without the optimization of RIS reflecting matrices.2)Random RIS:the RIS reflecting matrices are randomly generated and remain constant through the alternating optimization.3) Equal power transmission(EPT):At each AP,the transmit power is equally allocated to all RIS,i.e.,Moreover,the noise power at each receiver is set to beσ2=−70dBm.The free-space path loss models of RIS-assisted wireless communications in[43]are considered.In detail,in the considered large-scale fading model,the BS-RIS-user link suffers the double-fading effect,which gives us access to an ideal system model.

In Figure.2,we present the convergence behavior of the proposed SDR-based alternating algorithm.Specifically,the simulation setups areEi= 20dBm andEL=−30dBm.It is clear that the weighted sum rate of the proposed alternating algorithm increases monotonically and converges to its maximum value within 20 iterations.

In Figure.3,we present the weighted sum rate versus the energy threshold at the ERs,where the without RIS case is also considered as a benchmark.It is clear that the deployment of RIS can significantly improve both the weighted sum rate and the harvested energy.For both the RIS-aided and the without RIS schemes,the trade-off between the weighted sum rate and the harvested energy can be observed,which also indicates that the RIS reflecting matrices should be carefully designed.

Next,the weighted sum rate as a function of the maximum transmit power at the APs is illustrated in Figure.4.Note that the maximum achievable energy threshold varies according to the maximum transmit power.Therefore,the energy thresholds are set asEl,con=0.8El,max,∀l.It can be observed that the proposed alternating algorithm outperforms the existing benchmarks.Moreover,it is obvious that the weighted sum rates of all the schemes increase with the maximum transmit power at the APs.Furthermore,the EPT scheme performs worse with the increased maximum transmit power due to limited design freedom of power allocation among the RIS.

Figure 3. Weighted sum rate with respect to the energy threshold at the ERs.

Figure 4. Weighted sum rate with respect to the maximum transmit power at the APs.

Finally,the weighted sum rate for the number of AP antennas is depicted in Figure.5.The maximum transmit power at the APs and the energy threshold are set asEi,S= 25dBm,El,con=−25dBm,∀i,l,respectively.Clearly,all the schemes’ weighted sum rates increase with the number of AP antennas due to increased multiplexing gains.

Figure 5. Weighted sum rate with respect to the AP antennas.

V.CONCLUSIONS

In this paper,we investigate beamforming optimization for RIS-aided SWIPT in cell free networks.The beamformers at each AP for different RIS,the covariance matrices of the energy signals,the phase shift matrices for different RISs are jointly optimized.Exploiting the equivalence between information rate and weighted MSE,the considered sum rate maximization problem for RIS-aided SWIPT is transferred into a weighted MSE minimization problem under energy harvesting constraints and power constraints.Based on an SCA algorithm,the nonconvex energy harvesting constraints have been replaced by their first-order Taylor expansion.Furthermore,an ADMM algorithm is taken advantage of to optimize the phases of the RIS elements.At the end of the paper,the performance advantages of the proposed optimization algorithm are demonstrated by several numerical results.

ACKNOWLEDGEMENT

This work was supported by National Natural Science Foundation of China(Fund number 61620106001).