# Power Optimization for Aerial Intelligent Reflecting Surface-Aided Cell-Free Massive MIMO-Based Wireless Sensor Network

Authors: Tao Zhou; Kui Xu; Chunguo Li; Zhexian Shen
Journal: Security and Communication Networks
 (2021)

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## Abstract

Intelligent reflecting surfaces (IRSs) have significant advantages in enhancing the coverage and reducing the deployment cost of wireless networks. This paper studies an aerial IRS- (AIRS-) enhanced cell-free massive multiple-input multiple-output- (MIMO-) based wireless sensor network (WSN) in which multiple access points (APs) serve several sensor users (SUs). Direct links between the APs and SUs are blocked due to occlusion by tall buildings. Hence, we deploy an AIRS to improve the communication quality of the SUs. Our goal is to minimize the total transmit power of all APs under a given minimum signal-to-interference-plus-noise ratio (SINR) requirement. We propose a joint iterative optimization algorithm by designing an active beamforming mechanism at each AP and a passive beamforming mechanism at the AIRS to solve this problem. Simulation results illustrate the good performance of the proposed method.


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## Body

## 1. Introduction

Compared with fifth-generation (5G) communication, sixth-generation (6G) communication will require higher speeds, lower latency, and Internet of Everything (IoE) capabilities [1–3]. In addition, 6G will achieve integrated air-space-ground coverage, providing safe and reliable service [4–6]. As one of the core technologies for 6G, intelligent reflective surfaces (IRSs) can effectively enhance the regional coverage and reduce the energy consumption of wireless networks [7]. Therefore, IRSs have received widespread attention in recent years.An IRS consists of many programmable subwavelength metamaterial array elements. Each element can control the amplitude and phase of electromagnetic waves to realize smart and reconfigurable transmission based on the wireless environment [8–10]. In contrast to traditional transmitters, an IRS does not perform any signal processing and merely modifies the phase and amplitude of incident electromagnetic waves. An IRS does not need to be equipped with a radio frequency link composed of filters, mixers, and power amplifiers. Therefore, IRSs can effectively reduce the costs and energy consumption of wireless networks [11, 12]. The signals reflected by an IRS can be made to be coherently superimposed at the receiver by suitably designing the reflection coefficients of the IRS. We call this approach passive beamforming [13, 14].Cell-free (CF) massive multiple-input multiple-output (MIMO) technology breaks away from the fixed architecture of traditional cellular massive MIMO. In a CF system, multiple distributed access points (APs) provide communication services for several users in the coverage area simultaneously. All the APs are connected through backhaul links and are controlled by a smart central processing unit (CPU). The CPU can control the resource allocation in real time, thereby eliminating the cell boundaries in traditional cellular networks. Thus, the interference from different cells can be expunged [15–18].To enhance the coverage of a wireless network, a large number of distributed APs need to be deployed. However, limited by their deployment cost and energy consumption, high-density deployment of APs is impractical. Based on the above analysis, an IRS can be deployed in a CF massive MIMO network to enhance the communication quality. The deployed IRS can replace some APs in providing services for users. Due to its low cost and flexible deployment characteristics, the IRS can enhance the coverage of the CF massive MIMO system while reducing its deployment cost. However, due to the complex distribution of urban buildings, the deployment of an IRS (which is usually located on a wall or roof) depends on the buildings that are present in the area. Some of the best IRS deployment locations may not be suitable in practice due to topographical or architectural constraints. Therefore, to overcome the terrain limitations of traditional ground IRS deployment and make full use of air-ground resources, an aerial IRS (AIRS) was proposed as a new IRS architecture in [19, 20]. An AIRS is usually installed on a launch platform (a launch balloon or unmanned aerial vehicle, UAV) to establish a strong line-of-sight (LoS) connection. The AIRS location can then be adjusted in accordance with current service needs, in line with the user-centred principle.In 6G communication, a large number of intelligent devices will not only provide convenience for our life but also constitute a huge wireless sensor network (WSN). The WSN requires higher transmission rate. To solve this problem, we can use the IRS to increase coverage, improve spectrum efficiency, and reduce the deployment cost of wireless networks.Motivations and Contributions. The current research on IRSs has mainly focused on traditional cellular networks, including IRS-assisted multiple-input single-output (MISO) communication [13, 21, 22], MIMO communication [23], and multicell interference cancellation scenarios [24]. There have been only a few studies on the application of IRSs in CF massive MIMO networks, mainly focusing on the rate optimization problem [25–28], while the problem of power minimization has not been addressed. In [13, 22], the problem of power optimization for single-IRS-assisted single-cell communication was studied. However, power optimization for an IRS-aided CF massive MIMO network has not yet been investigated. In addition, the existing research mainly focuses on the single-IRS case. There are a few studies that consider the multi-IRS case. For the multi-IRS case, the IRS deployment strategy is important in enhancing the communication quality. However, few studies consider the IRS deployment strategy. Moreover, the power optimization problem in the IRS-aided WSN system has not been considered yet. Therefore, in this study, we consider the AIRS-enhanced CF massive MIMO WSN system. The main contributions are summarized as follows:(1)
In this paper, we first consider the power optimization problem for the AIRS-aided CF massive MIMO-based WSN system. Several APs serve several sensor users (SUs) with the assistance of an AIRS. This paper aims to minimize the total transmit power of all APs under a signal-to-interference-plus-noise ratio (SINR) requirement.(2)
We propose a joint optimization algorithm to optimize the total transmission power for the single-AIRS case. Then, we extend to power optimization for the multi-AIRS case. The simulation results show the good performance of our proposed method in power savings.(3)
Moreover, we discuss the AIRS deployment strategy. Compared with centralized AIRS deployment, distributed AIRS deployment shows better performance, providing some guidelines for AIRS deployment.


### 1.1. Methods

Based on the idea of alternating iterations commonly used in IRS-assisted transmission optimization, we jointly optimize the beamforming vectors and AIRS reflection coefficients to solve the transmission power optimization problem. Specifically, the optimization problem is divided into two subproblems: fixing the AIRS reflection coefficients to optimize the beamforming vectors and fixing the beamforming vectors to optimize the AIRS reflection coefficients. We can obtain an optimal solution by iteratively optimizing the above two subproblems. The transmission power can be effectively reduced by using the proposed method.


### 1.2. Organization

In Section2, we introduce and analyze the system model and formulate the power optimization problem. We use the Ricean channel model to characterize the air-to-ground channels. In Section 3, we introduce our proposed optimization strategy and analyze it in detail. In Section 4, we present a numerical simulation performed to verify the performance of the proposed method. Finally, in Section 5, we summarize this paper, analyze the deficiencies of this work, and then identify future research directions.


### 1.3. Notations

We use italic letters for constants (B) and bold lowercase letters to denote vectors (p). Matrices are represented by bold capital letters. The argument of a complex number B is denoted by argB. For a vector p, we use argp to express its corresponding argument vector, and diagp denotes a diagonal matrix whose diagonal elements are the elements in p. AH denotes the conjugate transpose of a matrix A, and AT is the transpose of a matrix A. ⊗ is the Kronecker product.


## 1.1. Methods

Based on the idea of alternating iterations commonly used in IRS-assisted transmission optimization, we jointly optimize the beamforming vectors and AIRS reflection coefficients to solve the transmission power optimization problem. Specifically, the optimization problem is divided into two subproblems: fixing the AIRS reflection coefficients to optimize the beamforming vectors and fixing the beamforming vectors to optimize the AIRS reflection coefficients. We can obtain an optimal solution by iteratively optimizing the above two subproblems. The transmission power can be effectively reduced by using the proposed method.


## 1.2. Organization

In Section2, we introduce and analyze the system model and formulate the power optimization problem. We use the Ricean channel model to characterize the air-to-ground channels. In Section 3, we introduce our proposed optimization strategy and analyze it in detail. In Section 4, we present a numerical simulation performed to verify the performance of the proposed method. Finally, in Section 5, we summarize this paper, analyze the deficiencies of this work, and then identify future research directions.


## 1.3. Notations

We use italic letters for constants (B) and bold lowercase letters to denote vectors (p). Matrices are represented by bold capital letters. The argument of a complex number B is denoted by argB. For a vector p, we use argp to express its corresponding argument vector, and diagp denotes a diagonal matrix whose diagonal elements are the elements in p. AH denotes the conjugate transpose of a matrix A, and AT is the transpose of a matrix A. ⊗ is the Kronecker product.


## 2. System Model and Problem Formulation

As shown in Figure1, in the considered CF massive MIMO-based WSN system, L multiantenna APs serve K single-antenna SUs. Tall buildings block direct links between the APs and SUs. An AIRS is deployed to enhance the quality of service (QoS) of the SUs. Suppose that each AP has the form of a uniform linear array (ULA), while the AIRS has the form of an Nx×Ny uniform plane array. Each AP has M antennas. The AIRS consists of N=Nx×Ny elements, with half-wavelength spacing of the antennas/elements. Similar to [13, 22, 29–33], we characterize the channel from the l-th AP to the AIRS Gl∈ℂN×M as follows:(1)Gl=βGlκκ+1ψNθi,φiςMHθi+1κ+1G˜l,where G˜l denotes the non-line-of-sight (NLoS) component, whose elements are chosen from CN0,1. βGl is the path loss. Here, we define βGl as(2)βGl=a0dGl2,where a0 is the path loss at a reference distance of 1 m and dGl is the distance between the l-th AP and the AIRS. ψNθi,φi and ςMθi are string vectors that can be expressed as(3)ψNθi,φi=1,e−jπτsinθisinφ,…,e−jπτNx−1sinθisinφiT⊗1,e−jπτsinθicosφi,…,e−jπτNy−1sinθicosφiT.And(4)ςMθi=1,ejπτcosθi,…,ejπτM−1cosθiT.Figure 1
System model.Whenκ=0, the above model reduces to the Rayleigh fading channel model. An AIRS is usually deployed at a high altitude. From an elevated position, the AIRS can more easily establish LoS links with the ground nodes, which leads to LoS-dominated channel conditions [19, 20, 34, 35]. Similarly, the channel from the AIRS to the k-th SU can be modeled as(5)hk=βhkκκ+1ψNαk,ηk+1κ+1h˜k.hk∈ℂ1×N. The corresponding symbols are defined similarly to the case of Gl.All APs serve SUs simultaneously. The received signal at thek-th SU can be expressed as(6)yk=∑l=1Lclkxl+ϖk=∑l=1Lclk∑i=1Kwlisi+ϖk=∑l=1Lclkwlksk+∑l=1Lclk∑i≠kKwlisi+ϖk,where xl denotes the transmit signal of the l-th AP, and xl=∑i=1Kwlisi; wlk is the beamforming vector at the l-th AP to the k-th SU; sk is the information symbol sent to the k-th SU; ϖk is the additive white Gaussian noise (AWGN) at the k-th SU, and ϖk∼CN0,σ2; and clk∈ℂ1×M is the AIRS-reflected channel between the l-th AP and the k-th SU, which can be expressed as(7)cli=hiΦGl=vHdiaghiGl=vHUli,where Φ=diagejω1,ejω2,…,ejωN∈ℂN×N is the reflection coefficient matrix of the AIRS, ωn∈0,2π is the reflection coefficient of the n-th element, and v=ejω1,ejω2,…,ejωNH. The SINR of the k-th SU can be expressed as(8)SINRk=∑l=1Lclkwlk2∑i≠kK∑l=1Lcliwli2+σ2.In this paper, we aim to minimize the transmission powerP of all APs, P=∑l=1L∑i=1Kwli2. Thus, the problem to be optimized is constructed as follows:(9)P1:minv,wli∑l=1L∑i=1Kwli2,s.t.SINRk≥γk,k=1,2,…,K,ωn∈0,2π,n=1,2,…,N.


## 3. Joint Beamforming and Reflection Design

To solveP1, we need to design the beamforming mechanism at each AP and the AIRS reflection coefficients. As we can see from (7) and (9), all channels clk (l=1,…,L, k=1,…,K) vary with Φ. If we modify the AIRS reflection coefficients, the corresponding beamforming design for the APs will also change. Therefore, it is difficult to solve P1. Inspired by the traditional idea of alternating iterations in IRS-aided communication [22, 33], we propose a joint optimization strategy to solve this problem. That is, first, for fixed Φ, we optimize the beamforming at the APs, and then, for fixed beamforming design at the APs, that is, wli, we optimize the passive beamforming at the AIRS. The above steps are alternatively performed until convergence.


### 3.1. Beamforming Design for the APs

Note that when the AIRS reflection coefficientsΦ are fixed, we need only to optimize wli to minimize P. Then, P1 can be simplified as follows:(10)P2:minwli∑l=1L∑i=1Kwli2,s.t.SINRk≥γk,k=1,2,…,K.In accordance with (8), we define ck=c1k,c2k,…,cLk and wk=w1kw2k…wLkT. ck∈ℂ1×ML is the reflected channel vector of the k-th SU, and wk∈ℂML×1 is the beamforming vector for the k-th SU. Thus, we have ∑l=1Lcliwlj=ciwj. Therefore, P2 can be converted into(11)P3:minwi∑i=1Kwi2,s.t.ckwk2∑i≠kKckwi2+σ2≥γ0,k=1,2,…,K.From (11), we can find that P3 is a convex problem when Φ is fixed. We can use a semidefinite programming- (SDP-) based method to solve this problem. Note that(12)ckwk2=ckwkckwkH=TrckckHwkwkH=TrCkWk,where Ck=ckckH and Wk=wkwkH. Then, P3 can be considered to be equivalent to(13)P4:minwli∑i=1KtrWi,s.t.trCkWk≥γ0∑i≠kKtrCkWi+σ2,k=1,2,…,K,Wk⋟0,k=1,2,…,K.P4 is an SDP problem, and Wk can be solved by using a convex optimizer such as CVX [36]. Similar to [37], Wk satisfies the rank-one property, and the optimal beamforming design wkopt can be obtained via the rank-one decomposition (eigenvalue decomposition) of Wk.


### 3.2. Reflection Design of the AIRS

When the beamforming design is determined, we can optimizeΦ to further reduce the total transmission power. When the beamforming vectors wk (k=1,…,K) are fixed, the objective function can be rewritten as follows:(14)P5:limx⟶∞maxΦ∑i=1Kwli2,s.t.∑l=1Lclkwlk2∑i≠kK∑l=1Lclkwli2+σ2⩾γ0,k=1,2,…,K,ωn∈0,2π,n=1,…,N.According to (7), cli=vHUli, where Uli=diaghiGl. Then, the SINR constraint in (14) can be written as(15)∑l=1LvHUlkwlk2∑i≠kK∑l=1LvHUlkwli2+σ2⩾γ0.Here, we defineUk=U1k,U2k,…,ULk∈ℂN×ML. Then, (15) can be written as(16)∑l=1LvHUkwk2∑i≠kK∑l=1LvHUkwi2+σ2⩾γ0.P5 can be expressed as(17)P6:maxv∑k=1Kwk2,s.t.vHUkwk2∑i≠kKvHUkwi2+σ2⩾γ0,k=1,2,…,K,ωn∈0,2π,n=1,…,N.Note thatP6 is not a convex problem with a unit-modulus constraint ωn∈0,2π, n=1,…,N. Moreover, the objective function does not contain v. To solve P6, we can only find a vector v to satisfy the constraints in (17). Therefore, it is difficult to solve P6. Instead, we will rewrite P6 as an equivalent optimization problem and relax P6 into an SDP problem to obtain an optimal solution.When the beamforming vectors are obtained, the current setting ofv already satisfies the SINR constraint condition. In other words, the SINR of each SU satisfies SINRk⩾γ0. To reduce the total transmission power, we need to find a vector v∗ such that the new SINR constraints calculated with v∗ satisfy the following conditions:(18)SINRk∗⩾γ0∗,k=1,2,…,K,γ0∗⩾γ0.v∗ leads to a higher SINR threshold value of γ0∗. Thus, the transmission power can be reduced. From (17), we can find that, for the k-th SU, vHUkwk2 is the useful component, while ∑i≠kKvHUkwi+σ2 is the interference-plus-noise component. Define the SINR margin of the k-th SU as χk, which can be expressed as(19)χk=vHUkwk2−γ0∑i≠kKvHUkwi2+σ2,k=1,…,K.As discussed earlier, when the beamforming vectors are obtained, the current setting already satisfies the SINR constraint condition in (17). Therefore, χk satisfies the following constraint condition:(20)χk≥χ0,k=1,…,K,where χ0 is a constant that is greater than zero. χ0 denotes the base lower bound of χk,k=1,…,K. If the base lower bound χ0 is increased, it is easy to know that higher γ0 can be obtained and the SINR constraints in (20) are satisfied. Therefore, to satisfy the conditions in (20), we need to maximize χ0.Based on the above analysis, we rewriteP6 as(21)P7.1:maxvminχk,s.t.χk=vHUkwk2−γ0∑i≠kKvHUkwi2+σ2;χk≥0,k=1,…,K,ωn∈0,2π,n=1,…,N.Then, we use the SDP-based method to relaxP7.1 into an SDP problem. We have the following equations:(22)vHUkwk2=vHUkwk×vHUkwkH=vHUkwkwkHUkHv=vHQk,kv=trQk,kV,where Qk,k=UkwkwkHUkH is a determined matrix and V=vvH. Then, we can rewrite P7.1 as(23)P7.2:maxVminχk,s.t.χk=trQk,kV−γ0∑i=1,i≠kKtrQk,kV+σ2;χk≥0,k=1,…,K;V⋟0;diagV=1,1,1,…,1.P7.2 is an SDP problem that can be solved using a convex optimization tool such as CVX.By solvingP7.2, we can obtain an optimal solution for V. However, V cannot be guaranteed to be of rank one. If V is of rank one, we can use eigenvalue decomposition to obtain optimal v. When V is not of rank one, an approximate optimal solution can be obtained using the Gauss randomization method presented in [38].


### 3.3. Algorithm Implementation

The main steps of the optimization process are as follows:(1)
First, we randomly set the AIRS reflection coefficient vectorv0 and use v0 to optimize the beamforming vectors.(2)
Then, we use the optimized beamforming vectors to update the AIRS reflection coefficient vectorv.(3)
We use the new AIRS reflection coefficient vectorv to update the beamforming vectors at each AP. This iterative process is repeated until the total transmission power converges.The specific steps are given in Algorithm1.Algorithm 1: Joint optimization algorithm.
(1)
Initialize the AIRS reflection coefficient vectorv0.(2)
Update the beamforming vectorswk by solving P4 and calculate the total transmission power.(3)
Use the updated beamforming vectorswk to obtain V by solving P7.2.(4)
IfV is of rank one Obtain the optimal v via rank-one decomposition. Else Obtain the optimal v via Gauss randomization. End(5)
Repeat steps 2–4. If the absolute value of the difference of transmission power between then-th and the n−1-th iterations is less than 0.0001, the convergence is achieved. Then output the current wk and v.


### 3.4. Extension to the Multi-AIRS Case

Now, we will extend the optimization process to the multi-AIRS case. Let there beT AIRSs assisting the communication between L APs and K SUs. The channel from the l-th AP to the t-th AIRS is represented by Glt. The channel from the t-th AIRS to the k-th SU is represented by htk. Glt and htk are defined according to (4) and (5), respectively. The reflection coefficient matrix of the t-th AIRS is Φt (t=1,2,…,T). The channel from the l-th AP to the k-th SU reflected by the t-th AIRS can be expressed as(24)cl,t,k=htkΦtGlt=vtHdiaghtkGlt.The joint channel matrix from thel-th AP to the k-th SU can be expressed as(25)yk=∑l=1L∑t=1Tcl,t,k∑i=1kwlisi=∑l=1L∑t=1TvtHdiaghtkGlt∑i=1kwlisi=∑l=1Lv˜Hdiagh˜kG˜l∑i=1kwlisi=∑l=1Lc˜lk∑i=1kwlisi,where c˜lk=v˜Hdiagh˜kG˜l, v˜H=v1H,v2H,…,vTH∈ℂ1×NT, h˜k=h1k,h2k,…,hTk∈ℂNT×1, and G˜l=Gl1,Gl2,…,GlTT∈ℂNT×LM.The SINR expression is similar to (8), and the objective function is similar to (9). By replacing clk with c˜lk, we can use the proposed Algorithm 1 to solve this problem as well.


## 3.1. Beamforming Design for the APs

Note that when the AIRS reflection coefficientsΦ are fixed, we need only to optimize wli to minimize P. Then, P1 can be simplified as follows:(10)P2:minwli∑l=1L∑i=1Kwli2,s.t.SINRk≥γk,k=1,2,…,K.In accordance with (8), we define ck=c1k,c2k,…,cLk and wk=w1kw2k…wLkT. ck∈ℂ1×ML is the reflected channel vector of the k-th SU, and wk∈ℂML×1 is the beamforming vector for the k-th SU. Thus, we have ∑l=1Lcliwlj=ciwj. Therefore, P2 can be converted into(11)P3:minwi∑i=1Kwi2,s.t.ckwk2∑i≠kKckwi2+σ2≥γ0,k=1,2,…,K.From (11), we can find that P3 is a convex problem when Φ is fixed. We can use a semidefinite programming- (SDP-) based method to solve this problem. Note that(12)ckwk2=ckwkckwkH=TrckckHwkwkH=TrCkWk,where Ck=ckckH and Wk=wkwkH. Then, P3 can be considered to be equivalent to(13)P4:minwli∑i=1KtrWi,s.t.trCkWk≥γ0∑i≠kKtrCkWi+σ2,k=1,2,…,K,Wk⋟0,k=1,2,…,K.P4 is an SDP problem, and Wk can be solved by using a convex optimizer such as CVX [36]. Similar to [37], Wk satisfies the rank-one property, and the optimal beamforming design wkopt can be obtained via the rank-one decomposition (eigenvalue decomposition) of Wk.


## 3.2. Reflection Design of the AIRS

When the beamforming design is determined, we can optimizeΦ to further reduce the total transmission power. When the beamforming vectors wk (k=1,…,K) are fixed, the objective function can be rewritten as follows:(14)P5:limx⟶∞maxΦ∑i=1Kwli2,s.t.∑l=1Lclkwlk2∑i≠kK∑l=1Lclkwli2+σ2⩾γ0,k=1,2,…,K,ωn∈0,2π,n=1,…,N.According to (7), cli=vHUli, where Uli=diaghiGl. Then, the SINR constraint in (14) can be written as(15)∑l=1LvHUlkwlk2∑i≠kK∑l=1LvHUlkwli2+σ2⩾γ0.Here, we defineUk=U1k,U2k,…,ULk∈ℂN×ML. Then, (15) can be written as(16)∑l=1LvHUkwk2∑i≠kK∑l=1LvHUkwi2+σ2⩾γ0.P5 can be expressed as(17)P6:maxv∑k=1Kwk2,s.t.vHUkwk2∑i≠kKvHUkwi2+σ2⩾γ0,k=1,2,…,K,ωn∈0,2π,n=1,…,N.Note thatP6 is not a convex problem with a unit-modulus constraint ωn∈0,2π, n=1,…,N. Moreover, the objective function does not contain v. To solve P6, we can only find a vector v to satisfy the constraints in (17). Therefore, it is difficult to solve P6. Instead, we will rewrite P6 as an equivalent optimization problem and relax P6 into an SDP problem to obtain an optimal solution.When the beamforming vectors are obtained, the current setting ofv already satisfies the SINR constraint condition. In other words, the SINR of each SU satisfies SINRk⩾γ0. To reduce the total transmission power, we need to find a vector v∗ such that the new SINR constraints calculated with v∗ satisfy the following conditions:(18)SINRk∗⩾γ0∗,k=1,2,…,K,γ0∗⩾γ0.v∗ leads to a higher SINR threshold value of γ0∗. Thus, the transmission power can be reduced. From (17), we can find that, for the k-th SU, vHUkwk2 is the useful component, while ∑i≠kKvHUkwi+σ2 is the interference-plus-noise component. Define the SINR margin of the k-th SU as χk, which can be expressed as(19)χk=vHUkwk2−γ0∑i≠kKvHUkwi2+σ2,k=1,…,K.As discussed earlier, when the beamforming vectors are obtained, the current setting already satisfies the SINR constraint condition in (17). Therefore, χk satisfies the following constraint condition:(20)χk≥χ0,k=1,…,K,where χ0 is a constant that is greater than zero. χ0 denotes the base lower bound of χk,k=1,…,K. If the base lower bound χ0 is increased, it is easy to know that higher γ0 can be obtained and the SINR constraints in (20) are satisfied. Therefore, to satisfy the conditions in (20), we need to maximize χ0.Based on the above analysis, we rewriteP6 as(21)P7.1:maxvminχk,s.t.χk=vHUkwk2−γ0∑i≠kKvHUkwi2+σ2;χk≥0,k=1,…,K,ωn∈0,2π,n=1,…,N.Then, we use the SDP-based method to relaxP7.1 into an SDP problem. We have the following equations:(22)vHUkwk2=vHUkwk×vHUkwkH=vHUkwkwkHUkHv=vHQk,kv=trQk,kV,where Qk,k=UkwkwkHUkH is a determined matrix and V=vvH. Then, we can rewrite P7.1 as(23)P7.2:maxVminχk,s.t.χk=trQk,kV−γ0∑i=1,i≠kKtrQk,kV+σ2;χk≥0,k=1,…,K;V⋟0;diagV=1,1,1,…,1.P7.2 is an SDP problem that can be solved using a convex optimization tool such as CVX.By solvingP7.2, we can obtain an optimal solution for V. However, V cannot be guaranteed to be of rank one. If V is of rank one, we can use eigenvalue decomposition to obtain optimal v. When V is not of rank one, an approximate optimal solution can be obtained using the Gauss randomization method presented in [38].


## 3.3. Algorithm Implementation

The main steps of the optimization process are as follows:(1)
First, we randomly set the AIRS reflection coefficient vectorv0 and use v0 to optimize the beamforming vectors.(2)
Then, we use the optimized beamforming vectors to update the AIRS reflection coefficient vectorv.(3)
We use the new AIRS reflection coefficient vectorv to update the beamforming vectors at each AP. This iterative process is repeated until the total transmission power converges.The specific steps are given in Algorithm1.Algorithm 1: Joint optimization algorithm.
(1)
Initialize the AIRS reflection coefficient vectorv0.(2)
Update the beamforming vectorswk by solving P4 and calculate the total transmission power.(3)
Use the updated beamforming vectorswk to obtain V by solving P7.2.(4)
IfV is of rank one Obtain the optimal v via rank-one decomposition. Else Obtain the optimal v via Gauss randomization. End(5)
Repeat steps 2–4. If the absolute value of the difference of transmission power between then-th and the n−1-th iterations is less than 0.0001, the convergence is achieved. Then output the current wk and v.


## 3.4. Extension to the Multi-AIRS Case

Now, we will extend the optimization process to the multi-AIRS case. Let there beT AIRSs assisting the communication between L APs and K SUs. The channel from the l-th AP to the t-th AIRS is represented by Glt. The channel from the t-th AIRS to the k-th SU is represented by htk. Glt and htk are defined according to (4) and (5), respectively. The reflection coefficient matrix of the t-th AIRS is Φt (t=1,2,…,T). The channel from the l-th AP to the k-th SU reflected by the t-th AIRS can be expressed as(24)cl,t,k=htkΦtGlt=vtHdiaghtkGlt.The joint channel matrix from thel-th AP to the k-th SU can be expressed as(25)yk=∑l=1L∑t=1Tcl,t,k∑i=1kwlisi=∑l=1L∑t=1TvtHdiaghtkGlt∑i=1kwlisi=∑l=1Lv˜Hdiagh˜kG˜l∑i=1kwlisi=∑l=1Lc˜lk∑i=1kwlisi,where c˜lk=v˜Hdiagh˜kG˜l, v˜H=v1H,v2H,…,vTH∈ℂ1×NT, h˜k=h1k,h2k,…,hTk∈ℂNT×1, and G˜l=Gl1,Gl2,…,GlTT∈ℂNT×LM.The SINR expression is similar to (8), and the objective function is similar to (9). By replacing clk with c˜lk, we can use the proposed Algorithm 1 to solve this problem as well.


## 4. Simulation Results

In this section, we present a numerical simulation of an actual communication scenario conducted to study the performance of the proposed algorithm. First, we analyse the performance of the single-AIRS case. Then, we extend to the Multi-AIRS case, and we mainly focus on the performance comparison for AIRS deployment.


### 4.1. Single-AIRS Case

As shown in Figure2, three APs serve four SUs with the assistance of an AIRS. The AIRS is deployed at a fixed point (0, 75, 40). All the APs and SUs are in the x-o-y plane. The parameter settings are detailed in Table 1.Figure 2
Simulation scenario of the single-AIRS case.Table 1
Simulation settings.
ParameterValueBandwidth1MHzNumber of antennas at each AP, M20Number of AIRS elements, N30Noise power−110dBmRice factor,κ10Coordinates ofAP1 (m)(0, 0, 0)Coordinates ofAP2 (m)(0, 50, 0)Coordinates ofAP3 (m)(50, 50, 0)Coordinates of AIRS (m)(0, 90, 40)Coordinates ofSU1 (m)(0, 100, 0)Coordinates ofSU2 (m)(10, 95, 0)Coordinates ofSU3 (m)(20, 100, 0)Coordinates ofSU4 (m)(−10, 100, 0)Antenna or element spacing,d0.5λIn Figure3, we compare the performance under three different strategies:(1)
Beamforming optimization with random reflection design: In this setting, we randomly set the AIRS reflection coefficients and optimize the beamforming vectorswk (k=1,2,…,K).(2)
Zero forcing (ZF) beamforming with random reflection design: In this setting, we use the ZF beamforming strategy to set the beamforming vectorswk (k=1,2,…,K). In this system, we randomly set the AIRS reflection coefficients and then use the global channel to design the ZF beamforming vectors. The global channel is denoted by C∈ℂK×ML and is expressed as follows:(26)C=c11c21…cL1c12c22…cL2⋮⋮⋮⋮c1Kc2K…cLK=c1c2⋮cK.The pseudoinverse ofC is(27)C¯=CHCCH−1=c¯1,c¯2,…,c¯K.We set the beamforming vectors as follows:(28)wk=pkc¯kc¯k2.Then, we calculate the minimum transmit powerP that satisfies the SINR constraint. P is defined as(29)P=p1+p2,…,+pk.(3)
ZF beamforming with exhaustion-based AIRS reflection design: In this setting, we randomly setv and use the ZF beamforming method 1000 times. We choose the best performance of 1000 experiments as the output.(4)
ZF beamforming with the channel gain maximization-based AIRS reflection design: In this setting, we use ZF beamforming and a channel energy maximization-based AIRS phase optimization method to design the AIRS reflection coefficient. The main consideration is that after ZF beamforming, the interference between SUs has been eliminated. One method is to maximize the total channel gain so that the beamforming gain is optimal.First, we optimize the AIRS reflection vectorv by the channel gain maximization-based method. Then we use v to design the ZF beamforming. The total channel gain is given as:(30)C=∑k=1Kck2=∑k=1KckckH=∑k=1K∑l=1LvHUlkUlkHv=∑k=1K∑l=1LtrU˜lkV,whereU˜lk=UlkUlkH. The objective function for optimizing v is formulated as follows:(31)maxV∑k=1K∑l=1LtrU˜lkV,s.t.V⋟0;diagV=1,1,1,…,1.Here, we use a similar approach to [22, 39] to solve this problem. Then the optimal v can be obtained. For brevity, the specific details have been omitted.(5)
Proposed method: In this setting, we use the proposed method to optimize both the beamforming vectors and the AIRS reflection coefficients.Figure 3
Performance comparison.As shown in Figure3, By optimizing the AIRS reflection coefficients, the total transmission power can be effectively reduced. Overall, the proposed method shows better performance in saving the total transmission power.In Figure4, we analyze the convergence of the proposed algorithm. We set the target SINR to 10 dB, while the other parameters are set as before. If the absolute value of the difference of transmission power between the n-th and the n−1-th iterations is less than 0.0001, convergence is achieved. As seen from Figure 4, through iteration, the total transmission power is significantly reduced and finally converges to a value. The proposed method has strict convergence.Figure 4
Convergence analysis.In Figure5, we study the impact of the AIRS scale on the algorithm performance. We set the number of AIRS elements N to 10, 20, 30, 40, 50, and 60, and the target SINR is set to 10 dB. Figure 5 shows that as the number of AIRS elements increases, the total transmission power decreases if both the beamforming and AIRS reflection are optimized, indicating that more AIRS elements can be deployed to further reduce the transmission power with the proposed method.Figure 5
Impact of the number of AIRS elements.


### 4.2. Multi-AIRS Case

Then, we study the multi-AIRS case. We mainly analyze the impact of the AIRS deployment strategy on performance. Assume that there are six SUs. As shown in Figure6, the SUs are distributed in three communities. We aim to compare the performance of centralized and distributed AIRS deployment.Figure 6
Simulation scene of multi-AIRS case.For centralized AIRS deployment, we use one AIRS to assist communication. The number of AIRS elements is set as 30. The AIRS is deployed at0,90,40. For distributed AIRS deployment, we use three AIRSs to assist communication. To be fair, the number of elements of each AIRS is set as 10. The AIRSs are deployed at 0,90,40, 60,90,40, and −60,90,40. The coordinates are detailed in Table 2. The other settings are the same as the single-AIRS case.Table 2
Coordinate settings.
ParameterValueCoordinates ofAP1 (m)(0, 0, 0)Coordinates ofAP2 (m)(50, 50, 0)Coordinates ofAP3 (m)(0, 50, 0)Coordinates ofAIRS1 (m)(0, 90, 40)Coordinates ofAIRS2 (m)(−60, 90, 40)Coordinates ofAIRS3 (m)(60, 90, 40)Coordinates ofSU1 (m)(60, 100, 0)Coordinates ofSU2 (m)(60, 95, 0)Coordinates ofSU3 (m)(0, 105, 0)Coordinates ofSU4 (m)(5, 100, 0)Coordinates ofSU5 (m)(−60, 100, 0)Coordinates ofSU6 (m)(−60, 105, 0)As we can see from Figure7, the distributed AIRS deployment shows a better performance in reducing the total transmission power. One reasonable explanation is that distributed AIRS deployment can provide more flexible deployment according to the location of the SUs so that the AIRS can be deployed closer to the SUs to better assist communication. Another reason is that distributed AIRS deployment can provide a relatively rich scattering environment because the reflection is increased. The rich scattering environment can reduce the correlation among the channels of different SUs; thus, the interference between different SUs can be suppressed.Figure 7
Impact of the AIRS deployment strategy.


## 4.1. Single-AIRS Case

As shown in Figure2, three APs serve four SUs with the assistance of an AIRS. The AIRS is deployed at a fixed point (0, 75, 40). All the APs and SUs are in the x-o-y plane. The parameter settings are detailed in Table 1.Figure 2
Simulation scenario of the single-AIRS case.Table 1
Simulation settings.
ParameterValueBandwidth1MHzNumber of antennas at each AP, M20Number of AIRS elements, N30Noise power−110dBmRice factor,κ10Coordinates ofAP1 (m)(0, 0, 0)Coordinates ofAP2 (m)(0, 50, 0)Coordinates ofAP3 (m)(50, 50, 0)Coordinates of AIRS (m)(0, 90, 40)Coordinates ofSU1 (m)(0, 100, 0)Coordinates ofSU2 (m)(10, 95, 0)Coordinates ofSU3 (m)(20, 100, 0)Coordinates ofSU4 (m)(−10, 100, 0)Antenna or element spacing,d0.5λIn Figure3, we compare the performance under three different strategies:(1)
Beamforming optimization with random reflection design: In this setting, we randomly set the AIRS reflection coefficients and optimize the beamforming vectorswk (k=1,2,…,K).(2)
Zero forcing (ZF) beamforming with random reflection design: In this setting, we use the ZF beamforming strategy to set the beamforming vectorswk (k=1,2,…,K). In this system, we randomly set the AIRS reflection coefficients and then use the global channel to design the ZF beamforming vectors. The global channel is denoted by C∈ℂK×ML and is expressed as follows:(26)C=c11c21…cL1c12c22…cL2⋮⋮⋮⋮c1Kc2K…cLK=c1c2⋮cK.The pseudoinverse ofC is(27)C¯=CHCCH−1=c¯1,c¯2,…,c¯K.We set the beamforming vectors as follows:(28)wk=pkc¯kc¯k2.Then, we calculate the minimum transmit powerP that satisfies the SINR constraint. P is defined as(29)P=p1+p2,…,+pk.(3)
ZF beamforming with exhaustion-based AIRS reflection design: In this setting, we randomly setv and use the ZF beamforming method 1000 times. We choose the best performance of 1000 experiments as the output.(4)
ZF beamforming with the channel gain maximization-based AIRS reflection design: In this setting, we use ZF beamforming and a channel energy maximization-based AIRS phase optimization method to design the AIRS reflection coefficient. The main consideration is that after ZF beamforming, the interference between SUs has been eliminated. One method is to maximize the total channel gain so that the beamforming gain is optimal.First, we optimize the AIRS reflection vectorv by the channel gain maximization-based method. Then we use v to design the ZF beamforming. The total channel gain is given as:(30)C=∑k=1Kck2=∑k=1KckckH=∑k=1K∑l=1LvHUlkUlkHv=∑k=1K∑l=1LtrU˜lkV,whereU˜lk=UlkUlkH. The objective function for optimizing v is formulated as follows:(31)maxV∑k=1K∑l=1LtrU˜lkV,s.t.V⋟0;diagV=1,1,1,…,1.Here, we use a similar approach to [22, 39] to solve this problem. Then the optimal v can be obtained. For brevity, the specific details have been omitted.(5)
Proposed method: In this setting, we use the proposed method to optimize both the beamforming vectors and the AIRS reflection coefficients.Figure 3
Performance comparison.As shown in Figure3, By optimizing the AIRS reflection coefficients, the total transmission power can be effectively reduced. Overall, the proposed method shows better performance in saving the total transmission power.In Figure4, we analyze the convergence of the proposed algorithm. We set the target SINR to 10 dB, while the other parameters are set as before. If the absolute value of the difference of transmission power between the n-th and the n−1-th iterations is less than 0.0001, convergence is achieved. As seen from Figure 4, through iteration, the total transmission power is significantly reduced and finally converges to a value. The proposed method has strict convergence.Figure 4
Convergence analysis.In Figure5, we study the impact of the AIRS scale on the algorithm performance. We set the number of AIRS elements N to 10, 20, 30, 40, 50, and 60, and the target SINR is set to 10 dB. Figure 5 shows that as the number of AIRS elements increases, the total transmission power decreases if both the beamforming and AIRS reflection are optimized, indicating that more AIRS elements can be deployed to further reduce the transmission power with the proposed method.Figure 5
Impact of the number of AIRS elements.


## 4.2. Multi-AIRS Case

Then, we study the multi-AIRS case. We mainly analyze the impact of the AIRS deployment strategy on performance. Assume that there are six SUs. As shown in Figure6, the SUs are distributed in three communities. We aim to compare the performance of centralized and distributed AIRS deployment.Figure 6
Simulation scene of multi-AIRS case.For centralized AIRS deployment, we use one AIRS to assist communication. The number of AIRS elements is set as 30. The AIRS is deployed at0,90,40. For distributed AIRS deployment, we use three AIRSs to assist communication. To be fair, the number of elements of each AIRS is set as 10. The AIRSs are deployed at 0,90,40, 60,90,40, and −60,90,40. The coordinates are detailed in Table 2. The other settings are the same as the single-AIRS case.Table 2
Coordinate settings.
ParameterValueCoordinates ofAP1 (m)(0, 0, 0)Coordinates ofAP2 (m)(50, 50, 0)Coordinates ofAP3 (m)(0, 50, 0)Coordinates ofAIRS1 (m)(0, 90, 40)Coordinates ofAIRS2 (m)(−60, 90, 40)Coordinates ofAIRS3 (m)(60, 90, 40)Coordinates ofSU1 (m)(60, 100, 0)Coordinates ofSU2 (m)(60, 95, 0)Coordinates ofSU3 (m)(0, 105, 0)Coordinates ofSU4 (m)(5, 100, 0)Coordinates ofSU5 (m)(−60, 100, 0)Coordinates ofSU6 (m)(−60, 105, 0)As we can see from Figure7, the distributed AIRS deployment shows a better performance in reducing the total transmission power. One reasonable explanation is that distributed AIRS deployment can provide more flexible deployment according to the location of the SUs so that the AIRS can be deployed closer to the SUs to better assist communication. Another reason is that distributed AIRS deployment can provide a relatively rich scattering environment because the reflection is increased. The rich scattering environment can reduce the correlation among the channels of different SUs; thus, the interference between different SUs can be suppressed.Figure 7
Impact of the AIRS deployment strategy.


## 5. Conclusion

In this paper, we study a power minimization problem in an AIRS-aided CF massive MIMO WSN system. We propose a joint optimization method to design the beamforming vectors and the AIRS reflection coefficients. Simulation results illustrate the good performance of the proposed method. This work deserves further study in the following aspects. In this paper, we simply assume that perfect channel state information (CSI) is obtained. However, channel estimation is challenging in this system. Therefore, in our future work, we will consider a suitable channel estimation method for this system. Moreover, AIRS deployment requires the location of SUs. AIRS-aided user positioning also deserves further study.



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*Source: 9989098-2021-09-25.xml*