Reconfigurable Intelligent Surface-Aided AirComp for Large-Scale IoT Networks

Abstract

Traditional wireless data aggregation (WDA) technology based on the principle of separated communication and computation is difficult to achieve large-scale access under the limited spectrum resources, especially in scenarios with strict constraints on time latency. As an outstanding fast WDA technology, over-the-air computation (AirComp) can reduce transmit time while improving spectrum efficiency. Most edge devices in wireless networks are battery-powered. Therefore, optimizing the transmit power of devices could prolong the life cycle of nodes and save the system power consumption. In this research, we aim to minimize the device transmit power subject to aggregation error constraint. Additionally, to improve the harsh wireless transmission environment, we use reconfigurable intelligent surface (RIS) to assist AirComp. To solve the presented nonconvex problem, we present a two-step solution method. Specifically, we introduce matrix lifting technology to transform the original problems into semidefinite programming problems (SDP) in the first step and then propose an alternate difference-of-convex (DC) framework to solve the SDP subproblems. The numerical results show that RIS-assisted communication can greatly save system power and reduce aggregation error. And the proposed alternate DC method is superior to the alternate semidefinite relaxation (SDR) method.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Supporting access to massive nodes is one of the main visions of future wireless communication networks represented by the fifth-generation (5G) mobile communications, and the number of nodes will continue to explode. The implementation of large-scale Internet of Things (IoT) scenarios will depend on deploying a large number of intelligent edge nodes. In 5G mobile communication networks, the number of IoT nodes will reach 100 million and the deployment density will reach one million per square kilometer. Unlike traditional wireless networks, whose primary purpose is to achieve end-to-end transmission, IoT systems pay more attention to the data function rather than the data of a single node. Future IoT scenarios need to accommodate large numbers of edge nodes to monitor the environment and gather large amounts of node data for analysis. For example, IoT-based monitoring systems focus not only on the large number of individual observations but also on their sum or average [1]. In the process of big data calculation, the authors of [2] extracted meaningful data from large-scale big data sets and deleted a lot of meaningless data before communication. Future wireless networks will shift from data-centric to computation-centric, making traditional wireless data aggregation (WDA) technologies based on the separation of communication and computation inefficient. To cope with communication limitations and high transmission delays caused by massive access to nodes, AirComp provides a solution for large-scale IoT deployment. AirComp is regarded as a promising technology in the IoT network to solve communication limitations and transmission delay caused by the large number of nodes connected, which can quickly aggregate data from large numbers of nodes. AirComp technology makes use of the superposition characteristic of the wireless channel to realize the WDA of multiple nodes in concurrent transmission [3, 4]. The authors of [5] analyze the potential application value of AirComp technology in massive IoT.

There is extensive research on the AirComp networks; e.g., the authors of [6] carried out relevant researches from the point of information theory, the authors of [7] carried out relevant researches from the aspect of signal processing, and the authors of [8] studied the design of transceiver beamforming. The authors of [9] derived achievable aggregation rates for sensitive functions and threshold functions of data. The authors of [8] proposed a transmitter design using zero-force transmission to compensate for fading between antennas in IoT networks. The authors of [10] presented a fast global model aggregation using AirComp-assisted federated learning. The authors of [11] jointly optimized transmit digital beamforming at wireless nodes and receive hybrid beamforming at the access point (AP) to minimize aggregation errors. In order to improve the accuracy of the calculation and reduce the aggregation errors, authors of [12] proposed an appropriate coding method. The authors of [13] presented a multicell AirComp network and studied the optimal strategy to distribute the transmit power of edge devices to minimize the error of the aggregated signal. The MIMO-AirComp equalization and wireless channel feedback technologies were designed for spatial multifunction computing in [14]. A closed-form approximate optimal equalizer was derived using differential geometry to minimize the error of data aggregation. In [15], an analog gradient aggregation solution to overcome the communication bottleneck of wireless federated learning applications is studied by using the idea of analog AirComp.

None of the researches mentioned above considers the bad characteristics of wireless links. In many cases, the wireless propagation environment is closely related to the performance of the communication system. Particularly, high-frequency signals are susceptible to obstacles [16]. In recent years, reconfigurable intelligent surfaces (RIS) can significantly improve the wireless communication environment, which has attracted a great deal of attention from researchers. RIS is considered an effective method to enhance the energy efficiency and spectral efficiency of wireless networks. A RIS usually does not require any dedicated power sources and can be easily integrated into the walls of buildings. A RIS consists of a number of passive components, each individually adjusts the phase shift of the incident signal [16, 17]. By adjusting the phase shifts of all components jointly, we are able to effectively combine reflected and direct signals to significantly increase the power of the received signal, thereby enhancing the performance of AirComp networks.

In order to overcome the unfavorable wireless channel environment of AirComp, the authors of [18] proposed to deploy a RIS in the AirComp system to increase the power of the received signal and thus reduce the aggregation error. The authors of [19] innovatively proposed sum power constraints in the RIS-aided AirComp to save system energy consumption. To minimize the aggregation error, the authors of [20] investigated the advantages of RIS-assisted AirComp in a large-scale cloud wireless access network. In [14], the authors proposed to deploy RIS to help AirComp achieve wireless data aggregation in IoT networks with imperfect CSI.

As far as we know, most of the researches on AirComp are aimed at optimizing the wireless data aggregation error of the system. Due to the popularization of IoT technology, the dramatic increase in the number of smart edge devices has led to a huge amount of energy consumption. Therefore, energy savings have become an urgent problem to solve. In IoT networks, compared to data sense and processing, the wireless transmission of data expends most of the power of sensor devices. While most edge devices are powered by batteries, optimizing the transmit power of edge devices is of great significance for reducing system energy consumption and prolonging the life cycle of devices.

Inspired by the above observation, we explore the application of RIS-assisted AirComp in IoT networks. In our study, the CSI is assumed to be perfect. However, due to the passivity of RIS, there are some challenges in correlation channel estimation. Therefore, the research works on RIS-related channel estimation are also very meaningful. The authors of [21] studied the uplink cascade channel estimation problem of RIS-assisted MISO system and proposed a low complexity alternate optimization algorithm with efficient initialization to achieve the local optimal solution. Our goal is to minimize system transmit power under the maximum tolerable aggregation error constraint, while ensuring that each device rate meets the user minimum rate constraint. With the RIS phase shift unit module, device rate, and aggregation error constraints, the problem presented is nonconvex. To solve the proposed thorny problem, we present a two-step solution method. Specifically, we introduce the difference-of-convex (DC) framework for optimization problems with rank-1 constraint in the first step and then present an alternate optimization (AO) method in the second step to optimize the transmit power of node devices.

The main contributions of this study are summarized as follows: (1) to minimize the device transmit power, we jointly optimized the RIS phase shift and emission vector. However, due to the coupling between variables, the proposed problem is a thorny nonconvex quadratic constrained quadratic programming (QCQP) problem. We introduce matrix lifting technology to convert the original problem to a semidefinite programming problem (SDP). (2) We introduce a DC description framework of SDPs and reexpress SDPS as DC forms. (3) Finally, we propose an alternate DC algorithm based on convex approximation, which can solve the QCQP problem effectively. (4) The simulation results confirm the advantages of the proposed DC algorithm and the advantages of RIS in energy savings in IoT systems.

The rest of the work is arranged as follows: we give the IoT network model and form the optimization problems in Section 2. An AO framework is designed in Section 3. We develop an alternate DC method to handle the original optimization problem in Section 4. The numerical results and discussions are organized in Section 5, and in Section 6, we present the conclusion. In addition, acronyms are listed in Table 1.

Table 1

Comparison table of abbreviations.

Acronyms	Corresponding phrase
RIS	Reconfigurable intelligent surface
AirComp	Over-the-air computation
QCQP	Quadratic constrained quadratic programming
DC	Difference-of-convex
AO	Alternate optimization
WDA	Wireless data aggregation
MLT	Matrix lifting technology
IoT	Internet of things
SDP	Semidefinite programming problem

2. System Model

Here, we first give the IoT network model and then propose the optimization problem of system performance.

2.1. Model of IoT System

We consider an RIS-aided IoT system, which consists of $K$ single-antenna edge nodes and an AP with $N$ antennas, as shown in Figure 1. In the AirComp-based scenario, the objective of the AP is to obtain the data characteristics of the aggregated data for all edge nodes. The observational data of edge node $k$ is expressed as $d_k\in ℂ$, where $k=1,2,\cdots ,K$.

[figure(s) omitted; refer to PDF]

The objective function to aggregate node data in AirComp-based IoT is written as $$\begin{array}{}\text{(1)}& f=\phi \sum _{k=1}^K{\varphi }_k{d}_k,\end{array}$$where ${\varphi }_k\bullet$ denote the precoding scalar of IoT node $k$ and $\phi \bullet$ is the decoding function at AP. The symbols for each local node after precoded ${\text{s}}_k≔{\varphi }_k{\text{d}}_{\text{k}}$ are considered to have the property of unit variance, i.e., $E{\text{s}}_k{\text{s}}_k^H=1$. The objective function to be obtained at AP is defined as $$\begin{array}{}\text{(2)}& \text{s}≔\sum _{k=1}^K{\text{s}}_k.\end{array}$$

To overcome the disadvantages of the wireless environment, we propose to deploy a RIS with $M$ reflection components to assist the IoT network. Specifically, we define the RIS phase shift matrix as $\mathbf{\Theta }=\mathrm{diag}\beta e^{j{\theta }_1},\cdots ,\beta e^{j{\theta }_M}\in {ℂ}^{M\times M}$, where${\theta }_m\in 0,2\pi$ is the reflection angle of the component $m$ and $\beta \in 0,1$ represents the reflect coefficient of the incident signal. Without loss of generality, we define $\beta =1$. In addition, signals reflected more than twice could be ignored due to transmission loss [22]. Therefore, as shown in Figure 1, the uplink between edge nodes and AP consists of the AP-node link, the RIS-AP link, and RIS-node link, represented by ${\mathit{h}}_k^d\in {ℂ}^N$, $\mathit{G}\in {ℂ}^{N\times M}$, and ${\mathit{h}}_k^r\in {ℂ}^M$, respectively.

The AP can adopt traditional multiple access (e.g., TDMA) for data transmission. However, in this way, the AP needs to gather the data first and then calculate the objective function, which leads to high latency.

If we adopt AirComp to aggregate the data of edge nodes, the objective function can be calculated in one time slot as shown in Figure 1. The estimated signal at AP can be described as $$\begin{array}{}\text{(3)}& \widehat{f}={\mathit{m}}^{\mathit{H}}\sum _{k=1}^K\mathit{G}\mathbf{\Theta }{\mathit{h}}_k^r+{\mathit{h}}_k^d{p}_k{s}_k+\mathit{n},\mathit{n}\in {ℂ}^N~\mathcal{C}\mathcal{N}0,{\mathit{I}\sigma }^2,\end{array}$$where $p_k\in ℂ$ is the transmit beamforming of node $k$, $\mathit{n}$ is the transmit noise, and $\mathit{m}\in {ℂ}^N$ denotes the receiver beamforming vector at the AP.

Subtracting the estimated function in (3) from the objective function in (1), the corresponding data aggregate error can be expressed as $e=f-\widehat{f}$. We define MSFE to measure the error performance of the AirComp-based IoT system, which is as follows: $$\begin{array}{}\text{(4)}& \text{MSFE}=\frac{E{e}^2}{E{f}^2}=\frac{{\sum }_{k=1}^{K}E{{\mathit{m}}^{\mathit{H}}{\mathit{h}}_k{p}_k-{\varphi }_k}^2{s_k}^2+{\mathit{m}}^2{\sigma }^2}{{\sum }_{k=1}^{K}E{{\varphi }_k{s}_k}^2},\end{array}$$where ${\mathit{h}}_k=\mathit{G}\mathbf{\Theta }{\mathit{h}}_k^r+{\mathit{h}}_k^d$ denotes the combined channel for AP user. By substituting $E{s}_k{s}_k^H=1$ into (4), equation (4) can be simplified as $$\begin{array}{}\text{(5)}& \text{MSFE}=\frac{{\sum }_{k=1}^{K}E{{\mathit{m}}^{\mathit{H}}{\mathit{h}}_k{p}_k-{\varphi }_k}^2+{\mathit{m}}^2{\sigma }^2}{{\sum }_{k=1}^{K}E{{\varphi }_k}^2}.\end{array}$$

2.2. Problem Presentation

In this study, we explore the transmit power minimization subject to the aggregation error and the node transmit rate constraint. We describe the system transmit power as $P_{\text{sum}}={\sum }_{k=1}^K{p_k}^2$.

Specifically, the optimization problem of minimizing the system transmit power is described as follows: $$\begin{array}{c}\text{(6)}& \underset{{\theta }_m,\mathit{m},p_k}{\min }\hspace{1em}\sum _{k=1}^K{p_k}^2,\hfill \text{s}.\text{t}.\hspace{1em}\text{MSFE}=\frac{{\sum }_{k=1}^{K}E{{\mathit{m}}^{\mathit{H}}{\mathit{h}}_k{p}_k-{\varphi }_k}^2+{\mathit{m}}^2{\sigma }^2}{{\sum }_{k=1}^{K}E{{\varphi }_k}^2}<ϵ,\forall \text{k},\hfill \\ 0\le {\theta }_m\le 2\pi ,\forall m.\end{array}$$

When given the receiver beamforming vector $\mathit{m}$ and the RIS phase shift matrix $\mathbf{\Theta }$, the optimal transmit scalar $p_k$ can be given as follows [9, 10]: $$\begin{array}{}\text{(7)}& p_k={\varphi }_k\frac{{{\mathit{m}}^H{\mathit{h}}_k}^H}{{{\mathit{m}}^H{\mathit{h}}_k}^2}.\end{array}$$

For the sake of calculation, we assume ${\sigma }^2=1$. According to (7), the transmit power optimization problem is rewritten as $$\begin{array}{c}\text{(8)}& {\mathcal{P}}_A:\underset{{\theta }_m,\mathit{m}}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{{{\mathit{m}}^{\mathit{H}}{\mathit{h}}_k}^2},\hfill \text{s}.\text{t}.\hspace{1em}{\mathit{m}}^2<ϵ,\hfill \\ 0\le {\theta }_m\le 2\pi ,\forall m.\end{array}$$

However, ${\mathcal{P}}_A$ is a QCQP problem which is highly intractable to solve. In the next section, we will deal with the problem by using an alternate DC method.

3. Matrix Lifting

Since receiver beamforming vector $\mathit{m}$ and RIS phase shift matrix $\mathbf{\Theta }$ in problem ${\mathcal{P}}_A$ are coupled, we cannot solve both variables simultaneously. Therefore, we will alternately optimize the variables $\mathit{m}$ and $\mathit{\Theta }$ to solve problem ${\mathcal{P}}_A$.

For a given RIS phase shift matrix $\Theta$, problem ${\mathcal{P}}_A$ is reduced to the following expression: $$\begin{array}{c}\text{(9)}& \underset{\mathit{m}}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{{{\mathit{m}}^{\mathit{H}}{\mathit{h}}_k}^2},\hfill \text{s}.\text{t}.\hspace{1em}{\mathit{m}}^2<ϵ.\hfill \end{array}$$

On the other hand, for a given $\mathit{m}$, let ${\mathit{c}}_k={\mathbf{m}}^H{\mathit{h}}_k^d$,$z_m=e^{j{\theta }_m}$, $m=1,\cdots ,M$, and ${\mathit{m}}^H\mathit{G}\mathbf{\Theta }{\mathit{h}}_k^r=a_k^H\mathit{z}$, where $\mathit{z}={{\text{e}}^{j{\theta }_1},\cdots ,{\text{e}}^{\text{j}{\theta }_{\text{M}}}}^{\text{T}}$ and $a_k^H={\mathit{m}}^H\mathit{G}\mathrm{diag}{\mathit{h}}_k^r$. Then, problem ${\mathcal{P}}_A$ can be reduced to the follows: $$\begin{array}{c}\text{(10)}& \underset{\mathbf{\Theta }}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{{a_k^H\mathit{z}+c_k}^2},\hfill \text{s}.\text{t}.\hspace{1em}{z_m}^2=1,\forall m.\hfill \end{array}$$

However, due to it being nonconvex and nonhomogeneous, problem (10) is difficult to solve. Fortunately, problem (10) can be converted to a homogeneous nonconvex QCQP by introducing auxiliary variables [22]. By introducing the auxiliary variable $t$, problem (10) can be equivalent transformed as follows: $$\begin{array}{c}\text{(11)}& \underset{\mathbf{\Theta }}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{{\stackrel{⃛}{\mathit{z}}}^{\mathit{H}}{\mathit{E}}_{\mathit{k}}\stackrel{⃛}{\mathit{z}}+c_k^2},\hfill \text{s}.\text{t}.\hspace{1em}{z_m}^2=1,\forall m,\hfill \end{array}$$where ${\mathit{E}}_{\mathbf{k}}=\begin{array}{cc}{a}_k{a}_k^H,& a_k^H{c}_k\\ c_k{a}_k^H,& 0\end{array}$ and $\stackrel{⃛}{\mathit{z}}=\begin{array}{c}\mathit{z}\\ t\end{array}$.

If ${\stackrel{⃛}{\mathit{z}}}^{\ast }={\stackrel{\wedge }{\mathit{z}},\stackrel{\wedge }{t}}^H$ is an optimal solution to (11), then it is easy to get the optimal solution of (11) as ${\mathit{z}}^{\ast }=\widehat{\mathit{z}}/\widehat{t}$ . At the same time, the optimal RIS phase shift matrix $\mathbf{\Theta }$ is recovered as $\mathbf{\Theta }=\mathrm{diag}{\mathit{z}}^{\ast }$.

Summarily, we can effectively solve ${\mathcal{P}}_A$ by solving (9) and (11) alternately. Due to their nonconvexity, problems (9) and (11) are tricky to handle. In the following subsection, we shall use matrix lifting technology (MLT) to transform these two nonconvex problems into SDPs.

3.1. Matrix Lifting Technology

To change the nonconvexity of (9) and (11), an effective method is to convert them into SDPs using MLT [23]. By denoting $\mathit{M}=\mathit{m}{\mathit{m}}^H$, we can lift variable $\mathit{m}$ into a symmetric positive semidefinite (PSD) matrix with rank-1. Then, (9) can be converted to the following form: $$\begin{array}{c}\text{(12)}& {\mathcal{P}}_{\text{A}1}:\underset{\mathbf{m}}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{Tr\mathbf{M}{\mathbf{H}}_{\mathbf{k}}},\hfill \text{s}.\text{t}.\hspace{1em}Tr\mathbf{M}<ϵ,\hfill \\ \mathbf{M}\succcurlyeq 0,\mathrm{rank}\mathbf{M}=1,\end{array}$$where ${\mathbf{H}}_{\mathbf{k}}={\mathbf{h}}_{\mathbf{k}}{\mathbf{h}}_{\mathbf{k}}^{\mathbf{H}}$.

In the same way, we convert (11) to an SDP problem by using LMT technology. By denoting $\mathit{Z}=\stackrel{⃛}{\mathit{z}}{\stackrel{⃛}{\mathit{z}}}^H$, problem (11) can be rephrased as follows: $$\begin{array}{c}\text{(13)}& {\mathcal{P}}_{A2}:\underset{\mathbf{\Theta }}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{Tr{\mathbf{R}}_k\mathbf{Z}+p_k^2},\hfill \text{s}.\text{t}.\hspace{1em}{\mathbf{Z}}_{m,m}=1,\forall m,\hfill \\ \mathbf{Z}\succcurlyeq 0,\mathrm{rank}\mathbf{Z}=1.\end{array}$$

Next, we can use SDR technology to solve SDPs ${\mathcal{P}}_{A1}$ and ${\mathcal{P}}_{A2}$ [23]. The SDPs, after simply dropping the rank-1 constraint by SDR technology, can be effectively solved through the CVX tool [24]. If the gained solution satisfies the rank-1 condition, the optimal solution of original problems can be obtained by rank-1 decomposition. Otherwise, if the gained solutions do not satisfy the rank-1 condition, the Gaussian randomization technique will be adopted to get the suboptimal solution of original problems [23]. However, when the variable to be optimized is high-dimensional, the solution gained by the SDR method usually does not satisfy rank-1, and the complexity of the algorithms is very high. Considering the shortcomings of the SDR method, we present a DC method in the next section to handle the SDPs ${\mathcal{P}}_{\text{A}1}$ and ${\mathcal{P}}_{\text{A}2}$.

4. Alternate DC

Here, we first carry out the equivalent transformation of rank-1 constraints of problems ${\mathcal{P}}_{\text{A}1}$ and ${\mathcal{P}}_{\text{A}2}$, then give the DC algorithm for the SDP, and finally solve original problem ${\mathcal{P}}_{\text{A}}$by using the AO method.

4.1. DC Transformation

The most common low-rank matrix optimization problem with the rank-1 constraint can be described in the following form: $$\begin{array}{c}\text{(14)}& \underset{X\in \mathcal{H}}{\min }\hspace{1em}\text{Tr}{T}_0\mathbf{X},\hfill \text{s}.\text{t}.\hspace{1em}\text{Tr}{T}_i\mathbf{X}=b_i,\forall i,\hfill \\ \mathbf{X}\succcurlyeq 0,\mathrm{rank}\mathbf{X}=1,\end{array}$$where $X$ is the Hermitian matrix.

For a PSD matrix, if $\text{Tr}\mathbf{X}\ge 1$, then we can conclude as follows [25]: $$\begin{array}{}\text{(15)}& \mathrm{rank}\mathbf{X}=1\iff \text{Tr}\mathbf{X}-{\mathbf{X}}_2=0.\end{array}$$

According to (15), we can substitute $\text{Tr}\mathbf{X}-{\mathbf{X}}_2$ for rank-1 constraints in ${\mathcal{P}}_{A1}$ and ${\mathcal{P}}_{A2}$ equivalently. We propose to add $\text{Tr}\mathbf{X}-{\mathbf{X}}_2$ to the objective function in place of the rank-1 constraint, resulting in the equivalent optimization problem as follows: $$\begin{array}{c}\text{(16)}& \underset{Xϵ\mathcal{H}}{\min }\hspace{1em}\text{Tr}{\text{T}}_0\mathbf{X}+\delta Tr\mathbf{X}-{\mathbf{X}}_2,\hfill \text{s}.\text{t}.\hspace{1em}\text{Tr}{T}_i\mathbf{X}=b_i,\forall i,\hfill \\ \mathbf{X}\succcurlyeq 0,\end{array}$$where $\delta >0$ is the penalty factor. We can get rank-1 solutions when the penalty component is zero.

4.2. DC Framework

Problem (16) is still nonconvex, and we will solve it by adopting the maximization-minimization technology [26]. Specifically, we transform (16) into a series of subproblems and thus convert the nonconvex term $-\rho X_2$ to convex. Then, we only need to handle the subproblem described below in iteration $t$. $$\begin{array}{c}\text{(17)}& \underset{X\in \mathcal{H}}{\min }\hspace{1em}\text{Tr}{\text{T}}_0\mathbf{X}+\delta \bullet \mathit{X},I-\partial {{\mathit{X}}^{t-1}}_2,\hfill \text{s}.\text{t}.\hspace{1em}\text{Tr}{T}_i\mathbf{X}=b_i,\forall i,\hfill \\ \mathbf{X}\succcurlyeq 0,\end{array}$$where ${\mathit{X}}^{t-1}$ is the optimal solution of this series of subproblems at the $t-1‐\text{th}$ iteration. It is easy to see that (18) is convex, and we can solve it by the CVX toolkit. Furthermore, $\partial {{\mathit{X}}^t}_2$ can be gained by $\partial {{\mathbf{X}}^t}_2={\mathit{\nu }}_{\mathbf{1}}{\mathit{\nu }}_{\mathbf{1}}^{\mathit{H}}$, where ${\nu }_1\in {ℂ}^N$ is the eigenvector corresponding to the maximum eigenvalue of $\mathbf{X}$.

The DC framework presented above can converge to the optimal critical solution of (17) from any initial values [26]. The presented DC method is described in detail in Algorithm 1.

Algorithm 1: DC Algorithm for (17).

1: Initialize X⁰ and threshold ${ϵ}_{dc}$.

2: t$⟵$1;

3: repeat

4:Compute the leading eigenvalue of ${\mathbf{X}}^{t-1}$, and obtain $\partial {{\mathbf{X}}^{t-1}}_2$;

   Solve problem (17) to obtain ${\mathbf{X}}^t$;

5: $t=t+1$;

6: until${\mathbf{X}}^t-{\mathbf{X}}^{t-1}<{ϵ}_{dc}$;

7: Then${\mathbf{X}}^{\ast }={\mathbf{X}}^t$;

8: Cholesky decomposition ${\mathbf{X}}^{\ast }={\mathbf{x}}^{\ast }{{\mathbf{x}}^{\ast }}^H$to get ${\mathbf{x}}^{\ast }$.

4.3. Alternate DC

According to the DC framework proposed above, we can convert problems ${\mathcal{P}}_{\text{A}1}$and ${\mathcal{P}}_{\text{A}2}$to SDP. Specifically, we use the DC framework to translate problem ${\mathcal{P}}_{\text{A}1}$ into the DC problem as follows: $$\begin{array}{c}\text{(18)}& \underset{\mathbf{m}}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{Tr\mathbf{M}{\mathbf{H}}_{\mathbf{k}}}+\delta \text{Tr}\mathbf{M}-{\mathbf{M}}_2,\hfill \text{s}.\text{t}.\hspace{1em}Tr\mathbf{M}<ϵ,\hfill \\ \mathbf{M}\succcurlyeq 0.\end{array}$$

We can get the exact rank-1 solution ${\mathit{M}}^{\ast }$ of (18) when the penalty component is zero. And we could get the optimal solution ${\mathit{m}}^{\ast }$ of (9) via Cholesky decomposition ${\mathit{M}}^{\ast }={\mathit{m}}^{\ast }{{\mathit{m}}^{\ast }}^H$.

By the same method, we can obtain the DC transformation form of problem ${\mathcal{P}}_{\text{A}2}$ as follows: $$\begin{array}{c}\text{(19)}& \underset{\mathbf{\Theta }}{\min }\hspace{1em}\sum _{k=1}^K\frac{1}{Tr{\mathbf{R}}_k\mathbf{Z}+p_k^2}+\delta \text{Tr}\mathbf{Z}-{\mathbf{Z}}_2,\hfill \text{s}.\text{t}.\hspace{1em}{\mathbf{Z}}_{m,m}=1,\forall m,\hfill \\ \mathbf{Z}\succcurlyeq 0,\mathrm{rank}\mathbf{Z}=1.\end{array}$$

When $\text{Tr}\mathbf{Z}-{\mathbf{Z}}_2$ in (19) is forced to zero, the solution ${\mathit{Z}}^{\ast }$with rank-1 can be obtained. And we can get the optimal solution of (11) by Cholesky decomposition of ${\mathit{Z}}^{\ast }$.

Summarily, the presented alternate DC framework for solving ${\mathcal{P}}_A$ is described in detail in Algorithm 2.

Algorithm 2: : Alternate DC algorithm for ${\mathcal{P}}_A$.

1: Initialize$\mathbf{\Theta }$⁰ with a random value and set threshold $>0$.

2: t $⟵$1

3: repeat

4: For a given ${\mathbf{\Theta }}^t$, solve (18) via Algorithm 1 to

   get ${\mathit{M}}^t$.

5: For a given ${\mathit{M}}^t$, solve (19) via Algorithm 1 to

   get $\mathbf{\Theta }$^t+1

6: $t=t+1$;

7: until$P_{sum}^t-P_{sum}^{t-1}<ϵ$

8: then$P_{sum}^{\ast }=P_{sum}^t$.

5. Numerical Result

Here, we analyze the numerical results of transmit power optimization of node devices with the MSFE constraint for a RIS-aided IoT system. We compare the alternate SDR technology with the alternate DC technology for solving problems $P_{A1}$ and $P_{A2}$. For the SDR method, we simply drop the rank-1 constraint and then use the CVX toolkit to solve $P_{A1}$ and $P_{A2}$ alternately. When the dimension of the optimization variable increases, the probability of the SDR method returning the rank-1 solution becomes very low, which will lead to the degradation of system performance. We compare two ways to solve SDP problems and analyze the impact of RIS deployment on system performance.

The simulation settings for this paper are given below, unless otherwise specified. We adopt a two-dimensional coordinate system. The AP and the RIS are located at (0,0) meters and (20,20) meters, respectively. Additionally, the IoT edge nodes are evenly distributed at region ([30,40], [-10,10]) meters. Suppose that the path loss function is defined as follows: $$\begin{array}{}\text{(20)}& Ld=C_0{\frac{d}{d_0}}^{-\beta },\end{array}$$where $\beta$ is the link loss exponent and $C_0$ is the link loss when $d_0=1m$. We set $\beta =3.5,2.2,2.3$, which represent path loss exponent of node-AP link, RIS-AP link, and node-RIS link, respectively. The corresponding link coefficient is defined as ${\mathit{h}}_k^d=\sqrt{L{d}_k^d{\gamma }^d}$, ${\mathit{h}}_k^r=\sqrt{L{d}_k^r{\gamma }^r}$, and $\mathit{G}=\sqrt{Ld\mathcal{T}}$, where ${\gamma }^d$, ${\gamma }^r$, and $\mathcal{T}~\mathcal{C}\mathcal{N}0,\mathit{I}$. Here, $d_k^d$, $d_k^r$, and $d$ are the link distances between node $k$ and the AP, node $k$ and RIS, and the RIS and the AP, respectively. Furthermore, we set the aggregation threshold ${ϵ}_{DC}={10}^{-8},$$\text{MSFE}\text{tolerance}={10}^{-3}$, $\rho =5$, $C_0=30\text{dB}$, and ${\sigma }^2=1$.

Figure 2 shows system transmit power under different MSFE constraints for AirComp-based IoT systems with and without a RIS when node number $K=8$, RIS reflection component $M=20$, and numbers of AP antennas $N=10$. As the data aggregation error threshold is relaxed, the transmit power decreases, which indicates that high precision data transmission is at the cost of the system power. It can also be seen from Figure 2 that the alternate DC technology and the alternate SDR technology can bring better system performance than the scenario without RIS. In addition, the proposed alternate DC technology is superior to the alternate SDR.

[figure(s) omitted; refer to PDF]

Figure 3 illustrates the transmit power versus AP antenna $N$ for AirComp-based IoT systems with and without a RIS when node number $K=8$ and number of RIS reflection component $M=20$. We can find that the deployment of a RIS in IoT networks can significantly save transmit power compared to the absence of RIS. As can be seen from Figure 3, as the number of antennas increases, the transmit power decreases. The results show that the system power can be saved by increasing the number of AP antennas. Furthermore, compared to the alternate SDR technology, the alternate DC algorithm proposed in this paper jointly optimizes active and passive beamforming, thus reducing transmit power.

[figure(s) omitted; refer to PDF]

Figure 4 shows the influence of the number of RIS reflection components on the system transmit power when $N=20$ and $K=8$. As the number of RIS components increases, the transmit power of all methods decreases significantly. This indicates that RIS has more reflection components, leading to better system performance. This is because RIS with more components can produce a more accurate reflection beam to the incident signal. Furthermore, the proposed alternate DC method is superior to the alternate SDR method for a different number of RIS reflection components.

[figure(s) omitted; refer to PDF]

Figure 5 explains the relationship between the transmit power and the number of nodes in IoT systems with and without RIS when $N=20$ and $M=20$. As can be seen from Figure 5, when the number of nodes increases, the transmit power increases sharply. Therefore, with the increase of nodes in the IoT networks, power optimization becomes a problem that cannot be ignored.

[figure(s) omitted; refer to PDF]

We further illustrate the decoding power versus the data aggregation error MSFE constraint of the IoT system in Figure 6 with $K=8$, $N=20$, and $M=20$. As the data aggregation error is relaxed, the power spent on decoding increases dramatically; i.e., the Frobenius norm of the received beamforming increases. This indicates that high precision data transmission can reduce the decoding complexity of the AP. We can also see from Figure 6 that deploying the RIS increases the decoding complexity of the AP, which indicates that there exists a trade-off between decoding complexity and transmit power of IoT systems.

[figure(s) omitted; refer to PDF]

6. Conclusion

In this work, we propose to deploy a RIS to reduce the transmit power of WDA through AirComp in IoT system. We propose joint optimization of the node’s transmit beamforming and RIS phase shift matrix to minimize the transmit power of the system nodes. We propose an alternate DC technology to deal with the proposed nonconvex QCQP problem. Specifically, we first use MLT to deal with nonconvexity of the optimization problems. Then, we propose an alternate DC technology to get the optimal rank-1 solution by alternately solving the DC function problem. The numerical results verify the effectiveness of RIS deployment in saving system power, and the proposed alternate DC technology is superior to the traditional alternate SDR technology. In our future works, we will expand this work. Due to the passivity of RIS, it is difficult to obtain the perfect channel information associated with RIS, so we will carry out research on channel estimation.

Disclosure

A preprint of this paper has previously been published [27].

Acknowledgments

This work is funded by the Education Reform Project Fund of Guizhou Province (2022SJG005); the Guizhou Provincial Department of Education Project (No. KY[2020]208); the Fund for High-Level Talents Project of Qiannan Normal University for Nationalities (QNSY2019RC12); and the Natural Science Foundation of Guizhou Education Department (Characteristic Project) (No. KY[2019]074).

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