I Introduction
The 6G network is envisioned to support ubiquitous intelligent services with challenging requirements on data rates, latency, and connectivity, for which an AI empowered network architecture, i.e., network intelligentization, subnetwork evolution and intelligent radio, is embraced [7]. Inspired by these trends, a novel communication paradigm of ”smart radio environment” enabled by intelligent reflecting surfaces (IRSs) has been proposed to enhance the spectrum and energy efficiency of the wireless networks by reconfiguring the signal propagation environments [11, 1, 5]. Specifically, IRS is a reconfigurable metasurface of electromagnetic (EM) material consisting of massive passive reflecting elements, each of which is capable of independently reflecting the incident signal by changing its amplitude and phase [9, 8]. By smartly adjusting the reflected signal propagation, we can constructively integrate reflected signals with nonreflected ones and destructively cancel the interference at receiver, thereby achieving the desired performance of wireless networks [16].
Recently, there have been significant progresses on beamforming design for IRSempowered wireless network [15, 3, 12, 6]. The transmit power minimization problem was exploited via jointly optimizing active beamforming at the base station (BS) and passive beamforming at the IRS in the proposed IRSempowered MISO wireless system [15] and nonorthogonal multiple access (NOMA) [3]. The weighted sumrate maximization problem was further considered in [12] by fractional programming. Moreover, IRS was deployed in multiple access networks to boost the received signal power for overtheair computation [6].
However, existing works mainly focused on the wireless networks with a single IRS, where the channel rank deficiency problem and computation problem have emerged [10]
. Specifically, the rankone lineofsight (LoS) channel is often considered to model the BStoIRS link, which yields the rank deficiency problem and limits the system capacity to serve multiusers. In addition, IRS is individually designed according to local channel information, thereby inducing channel estimation and computation tasks for optimization on local IRS controller. One promising solution is to deploy distributed IRSs in wireless networks, where BStoIRS channel can be generated as the sum of multiple rankone channels, thereby guaranteeing high rank channels
[10]. Moreover, in the distributed IRSempowered communication network architecture, distributed IRSs are coordinated via a central network controller, which reduces the computation at IRSs [1].With the benefits of centralized computation resources and coordinated passive beamforming for phase shift matrix design at IRSs, the proposed distributed IRSempowered wireless system can significantly improve network spectral efficiency and energy efficiency in dense wireless networks [13]. In this paper, we shall propose to deploy distributed IRSs in distributed wireless system with multiple sourcedestination pairs. Our goal is to maximize the sumrate via jointly optimizing the transmit power vector at the sources and the phase shift matrix with passive beamforming at all distributed IRSs. The formulated problem turns out to be nonconvex and highly intractable due to the multiuser interference and joint optimization. We thus design an iterative alternating algorithm by decoupling the optimization variables, which divides the original problem into two tractable subproblems, i.e., power control problem at sources and coordinated passive beamforming problem at IRSs. The resulting multipleratio fractional programming subproblem can be reformulated as a biconvex problem via quadratic transform [12], supported by an alternating convex search approach to solve it [4].
Ii System Model and problem formulation
Iia System Model
We consider a distributed IRSempowered communication system consisting of singleantenna sourcedestination pairs and cooperative intelligent reflecting surfaces (IRS)s controlled by a coordinated IRS controller. The th IRS has passive elements that are used for assisting the communication from source to destination via dynamically adjusting the phase shift according to the channel state information (CSI). In particular, IRS can operate in two coordinated modes, i.e., receiving mode for sensing environment and reflecting mode for scattering the incident signals from the sources [15, 14]. In previous single IRSempowered systems, IRS individually changes the phase shift based on the local channel information estimated by its controller, thereby yielding the channel estimation and computation tasks for optimization on single IRS. However, in our distributed IRSempowered system, distributed IRSs are coordinated by a central IRS controller according to the global channel information received from each single IRS, thus avoiding the computation on each single IRS. Due to the magnitude path loss, we ignore the power of signals reflected by IRS twice or more times [15]. Furthermore, we assume a quasistatic flatfading channel model with prefect CSI for all channels. Thus, the system consists of three components, i.e., sourceIRS link, IRSdestination link, sourcedestination link. To further simplify, we assume that the direct link does not exist, which represents it is either blocked or has negligible receive power. With this assumption, the signals transmitted from the sources to the destinations experience two phases, as shown in Fig. 1. In the first phase, the sources transmit signals to the IRS. The received signal at the th IRS is given by
(1) 
where is the Gaussian noise with distribution , denotes the transmitted symbol from th sources with , represents the transmit power of the th sources with power constraint , and is the vector containing the channel coefficients from the th source to the th IRS. In the second phase, the th IRS reflects the received signal based on the diagonal phase shifts matrix with and as the amplitude reflection coefficient on the incident signals. Without loss of generality, we assume . Therefore, the reflected signal at the th IRS can be written as
(2) 
Then, the signal received at the th destination is
(3) 
where is the additive noise at the th destination with distribution , denotes the channel coefficients between the th IRS and the th destination. Let , , and . We can rewrite (IIA) as
(4) 
Based on the singleuser detection, each destination treats the interferences as Gaussian noise. Therefore, the signaltointerferenceplusnoise ratio (SINR) at the th destination can be written as
(5) 
Then, the achievable rate of the th destination is given by
(6) 
where .
IiB Problem Formulation
In this paper, we aim to maximize the sumrate by jointly designing the power coefficient vector and the reflection coefficient vector under the following power transmit constraint and the IRS phase constraint :
(7) 
Let denote the number of passive elements. The sumrate maximization problem can be formulated as
(8) 
This problem turns out to be nonconvex due to the multiuser interference and ratio operation. Another challenge introduced by jointly optimizing and makes problem highly intractable. In next section, we propose to design an efficient iterative algorithm to find a suboptimal solution via alternatively optimizing and .
Iii Optimal Coordinated Passive Beamforming
In this section, we propose a lowcomplexity iterative suboptimal algorithm, which divides problem into several tractable subproblems via alternatively optimizing and .
Iiia Lagrangian Dual Reformulation
Basically, we can directly employ fractional programming to solve the original problem via applying the quadratic transform to each SINR term. In this approach, convex optimaizations problem need to be solved numerically in each iteration, which incurs large computation. Another more desirable and efficient method, based on a Lagangian dual reformulation of the original problem, performs in closed form at each iteration. Although the original problem is nonconvex, we can always derive its upper bound from Lagrangian dual form to find a suboptimal solution. To develop this efficient algorithm, we consider the following Lagrangian dual reformulation of the original problem proposed in [12]:
(9) 
where denotes an auxiliary variable vector, and the new objective is given by
(10) 
Based on the proposed iterative algorithm, and can be fixed firstly. Then, optimal can be obtained via setting to be zero, i.e.,
(11) 
After updating as , the first two terms of remain constant. Therefore, problem can be further reduced to
(12) 
where the objective function of is defined by
(13) 
Problem is still nonconvex due to the sum of multipleratio form, which can be solved by fractional programming framework proposed in [12]. Another challenge induced by joint optimization need to be tackled via alternatively optimizing and in the following subsections. Specifically, in each iteration, we first update , then optimize and respectively. Repeat this procedure until converges.
IiiB Power Control
In this subsection, we shall optimize based on the fixed and . Problem can be recast as the following power control subproblem with given :
(14) 
The classic power control problem, as a multipleratio fraction programming problem, has been well exploited by a novel quadratic transform in [12]. Based on its FP framework, above power control problem can be further equivalently reformulate as following biconvex optimization problem:
(15) 
where denotes the auxiliary variable vector and the objective function is defined by
(16) 
By utilizing the convex substructures of above problem, an alternating convex search approach can be further developed to obtain a local optimal solution [4]. In particular, only one variable is optimized at each step while others are fixed. Given fixed , the optimal is
(17) 
For fixing , the optimal is given by
(18) 
IiiC Optimizing Reflection Coefficients
In this subsection, we propose to solve another fractional programming subproblem via optimizing over the fixed . Based on the quadratic transform proposed in [12], we can obtain the following equivalent problem:
(19) 
where denotes the auxiliary variable vector and the new objective function is given by
(20) 
where denotes its conjugate. We propose to solve above problem via alternatively optimizing and . For fixed , the optimal can be obtained by setting to be zero, i.e.,
(21) 
Consider the phase constraint , it can be further rewritten as
(22) 
where denotes an elementary vector with a one at th position. Then, optimizing based on the fixed is a convex QCQP problem, which can be recast as the following equivalent dual problem by Lagrange dual decomposition:
(23) 
where represents the dual variable, and denotes the dual objective function, which is given by
(24) 
where is the fixed optimal . is a concave function with respect to . Since the Slater’s condition is satisfied, the duality gap is zero [2]. Therefore, the optimal can be obtained via setting to be zero, i.e.,
(25) 
where denotes the optimal dual variable vector, which can be obtained via ellipsoid method.
IiiD Proposed Algorithm
In this subsection, we present the proposed iterative alternating optimization algorithm in Algorithm 1. Specifically, the algorithm starts with two arbitrary initial vectors with and with . Then, we iteratively update based on the fixed , and based on the fractional programming until converges.
Lemma 1.
The proposed alternating algorithm is guaranteed to converge, with the sumrate monotonically nondecreasing after each iteration.
Proof.
Please refer to Appendix for details. ∎
Iv Simulation
In this section, we simulate the proposed alternating algorithm to evaluate its effectiveness and show the performance of distributed IRSaided communication systems.
Iva Simulation Setting
We consider a distributed IRSempowered communication system illustrated in Fig. 2, where all singleantenna sources and destinations are uniformly and randomly distributed in two circles centred at meters and meters with with radius 50 meters, respectively. Four distributed IRSs with a uniform rectangular array of passive reflecting elements are respectively located in meters and meters. The path loss model we consider is given by , where denotes the path loss at the reference distance meter, is the link distance and is the path loss exponent. In this simulation, we assume dB, and the path loss for the sourceIRS link and the IRSdestination link are respectively set to 2.2 and 2.8. We further assume all the considered channels suffer from Rayleigh fading. To be specific, the channel coefficients are given by
(26) 
where , , and respectively denote the distance between th source and th IRS, the distance between th IRS and destination. In addition, we set .
IvB Simulation Results
We simulate three different alternating algorithms denoted as Joint Optimization, Power Optimization and Phase Optimization, respectively.

Joint Optimization. Jointly optimizing both and .

Power Optimization. Only optimizing with random phase shift.

Phase Optimization. Only optimizing under the setting .
All the simulation results are obtained by averaging 100 channel realizations with fixed and .
Fig. 4 demonstrates that our proposed alternating algorithm converges under setting and dB. We further compare the sumrate versus different SNR in Fig. 4 with fixed . It can be observed that all the alternating algorithms perform better with the increasing SNR, since the transmit power of the signals reflected by IRS increases. However, joint optimization algorithm achieves higher sumrate than other two algorithms due to its jointly optimizing transmit power at sources and passive phase shifts at IRS.
Then, we investigate the impact of the number of distributed IRS on sumrate. Since there is no current work to exploit relevant problems, i.e., optimal distributed IRS positions, we randomly and uniformly deploy IRSs in the given region meters. Fig. 5 shows the sumrate increases with the increasing number of distributed IRSs, which demonstrates the admirable performance of proposed distributed IRSempowered system compared with existing single IRSempowered wireless networks.
V Conclusion
In this paper, we proposed a distributed IRSempowered wireless network to maximize the achievable sumrates by jointly optimizing the transmit power vector at the sources and the phase shift matrix with passive beamforming at all distributed IRSs. To solve this nonconvex and intractable problem, we presented an alternating algorithm by decoupling transmit power vector and passive beamforming optimization variables, yielding two multipleratio fraction programming subproblems. We further transformed the fractional programming problem into biconvex problem, for which an alternating convex search approach with closedform expressions was developed. Simulation results demonstrated the admirable performance of proposed distributed IRSempowered system.
Vi appendix
We now show that our proposed alternating algorithm converges. According to the Lagrangian dual reformulation proposed in [12], we have
(27) 
where . Based on the quadratic transform in [12], we have
(28) 
where and are updated by (17) and (18), respectively. Similarly, we can also obtain
(29) 
Therefore, we can proof the convergence behavior of proposed alternating algorithm by combining the above equations:
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