Sum Rate Optimization for MIMO Multicasting Network with Active IRS

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Introduction
Recently, intelligent reflecting surface (IRS) has been proposed as an effective approach for wireless communication.An IRS is a metasurface consisting of several programmable elements that can be smartly controlled to manipulate wireless propagation environment [1].Since passive IRS only reflects incident signals, the power consumption is relatively much smaller than that of traditional active transmitter [2].For example, IRS-assisted scheme has been investigated in the millimeter-wave (mmWave) networks [3], the antijamming communication networks [4], the simultaneous wireless information and power transfer (SWIPT) networks [5], the unmanned aerial vehicle (UAV) networks [6], the hybrid satellite-terrestrial networks [7], etc. Besides, some novel IRS structures have been proposed, such as the multilayer IRS [8] and the simultaneous transmission and reflection intelligent surface [9].More and more research suggests the promising application prospect of IRS.However, to overcome the severe double path loss in the reflecting channel, the literature recently introduced a novel active IRS architecture [10], in which each element is integrated with an amplifier [11].By converting direct current bias power into radio fre-quency power, the active element can directly amplify the incident signal [12].Recently, the energy efficiency of active IRS has been studied [13], and the results suggested that active IRS can obtain higher energy efficiency than passive IRS.Then, the literature investigated the active IRS-assisted secure transmission [14] and extends to the spectral and energy efficiency tradeoff of active IRS-assisted network [15].While the literature indicates that active IRS can obtain higher energy efficiency than passive IRS in multiuser multiple-input single-output (MISO) network [16], recently, the authors studied the beamformer design in active IRSassisted multiuser multiple-input multiple-output (MIMO) radio networks [17], where a sum mean squared error (MSE) objective was optimized.And the authors investigated the secrecy beamformer in active IRS-assisted wiretap channel with energy harvesting [18], where a max-min fairness problem was addressed by the successive convex approximation algorithm.
However, when considering other objective such as the sum information rate, how to use active IRS in MIMO network deserves more research.Motivated by the above observations, we investigate an active IRS-aided MIMO system, where a multiple-antenna transmitter (Tx) transmits the signal to several multiple-antenna users.Different from the commonly used passive IRS, active IRS is used to assist the signal transmission and to improve the system performance.The main contribution of the work is summarized as follows: (i) We consider a multiuser MIMO multicasting network, where both the direct link and the cascaded link are existing.Then, to overcome the main system bottleneck in the passive IRS-aided network incurred by the double fading effect, we propose to use active IRS to amplify and alter the phase of the incident signal, thus improving the spectral efficiency of this network (ii) Specifically, we aim to maximize the sum information rate of these users, by jointly designing the precoding matrix at the Tx and the reflecting coefficient (RC) at the IRS.The formulated problem is nonconvex, mainly due to the coupled variables and the complicated logarithmic determinant functions.To handle this obstacle, we reformulate the sum rate maximization problem as an equivalent problem by using some important facts about matrices, where the objective is linearized.Then, a block coordinate descent (BCD) algorithm is developed to handle the reformulated problem, where the subproblems are solved alternatingly until convergence.In fact, several subproblems have closed-form solutions, and the others can be solved by the quadratically constrained quadratic program (QCQP).Thus, the proposed algorithm has polynomial time computational complexity (iii) Finally, simulation results verify the effectiveness of the proposed algorithm as well as the superiority of active IRS when compared with other benchmarks.
Besides, some meaningful insights can be attained: (

System Model and Problem Formulation
2.1.System Model.As shown in Figure 1, the MIMO system consists of one Tx, one IRS, and K users.It is assumed that the Tx and the k-th user have N t and N k antennas, while the IRS has M elements, respectively.We denote , and H k ∈ ℂ N k ×M as the channel coefficient from the Tx to the k-th user, the Tx to the IRS, and the IRS to the k-th user, respectively.Since active IRS only utilizes power amplifiers and phase-shift circuits that control and reflect the signals, no dedicated digital-to-analog converters (DACs), analog-todigital converters (ADCs), and mixers are required.In contrast, relays are usually equipped with these mentioned electronic components for transmission and low-noise amplifiers for reception, which leads to higher hardware cost and power consumption than active IRS [19].International Journal of Distributed Sensor Networks In this paper, all CSI is assumed perfectly obtained at the Tx, which is due to the fact that several channel estimation techniques have been investigated for obtaining the CSI such as the progressive refinement method [20] and the parallel factor decomposition method [21], which makes the estimation and acquirement of the CSI become practical.In addition, there exists an IRS controller which coordinates the CSI exchange between the Tx and the IRS [22], thus achieving the joint design.
The intended signal s k ∈ ℂ d×1 ðd ≤ N t Þ for the k-th user is precoded by the precoder W k ∈ ℂ N t ×d .Thus, the transmit signal x ∈ ℂ d×1 can be expressed as Without loss of generality (W.l.o.g), we assume that E½s k s H k = I, ∀k and s k , s i are independent for ∀i ≠ k.Thus, the received signal at the k-th user is where Θ = Diagðθ 1 , ⋯, θ M Þ ∈ ℂ M×M denotes the RC matrix of the IRS, with θ m = α m e jϕ m being the RC of the m-th element.Here, α m ∈ ½0, α m,max and ϕ m ∈ ½0, 2πÞ represent the amplitude and the phase, respectively.For an active IRS, α m,max is not necessary 1.Here, we assume that α m and ϕ m are independent for any m.In fact, α m and ϕ m are coupled with each other for practical IRS.Thus, the authors have studied the function relationship between the α m and ϕ m for passive IRS [23][24][25].However, the coupled relationship between α m and ϕ m for active IRS has not been studied yet.
We will try to improve this in our future work.
In addition, n r ~CN ð0, σ 2 r IÞ and n k ~CN ð0, σ 2 k IÞ denote the additive noise at the IRS and the k-th user, respectively.
Accordingly, the information rate at the k-th user is [26] where H k denotes the covariance matrix for the k-th user.

Problem Statement.
We aim to maximize the sum rate among these users, via jointly designing the precoder and the RC, subject to the transmit power constraints.The problem is given as follows: where P s and P r are the maximum achievable power for the Tx and IRS, respectively.The main difficulties in solving ((4a)-(4d)) are twofold: firstly, (4a) is nonconvex; secondly, the variables are highly coupled.

The Joint Precoder and RC Design
To tackle the difficulty of (4a), we extend the key idea of the popular weighted mean square error minimization (WMMSE) algorithm [27], to reformulate ((4a)-(4d)), and then use the BCD method [28].We first introduce the following lemmas.
Lemma 1 (see [28]).Define matrix function where D is any positive definite matrix.Then, we have the following equations: Equation 1: for any positive definite matrix P ∈ ℂ m×m , Equation 2: for any positive definite matrix Q, we have Equation 3: we have For more detailed derivation about these equations, reader can refer to [29,30].
3.1.Reformulation of the Objective (4a).By using the above equations, we rewrite R k ðW k , ΘÞ as where 3 International Journal of Distributed Sensor Networks Furthermore, the matrix functions Ξ k ðU k , W k , QÞ are given as Combining these relationships, we obtain the following problem: ) is still hard to solve due to the coupled variables.However, when several variables are fixed, ((12a)-(12d)) can be turned into convex problem with respect to (w.r.t) the other variables, thus motivating us to propose the BCD algorithm.
(A) Subproblem 1: optimizing fU k , Ψ k g with given fW k , Θg By the above lemmas, we obtain the following optimal fU k , Ψ k g: Specifically, the subproblem is given as Then, by utilizing the identity TrðP H RSTÞ = vecðPÞ H ðT T ⊗ RÞvecðSÞ and defining w k = vecðW k Þ, ((14a)-(14c)) can be further recast as where [31] and can be solved by the toolbox CVX [32].

Summarization and Discussion
. The whole BCD algorithm is summarized in Algorithm 1.
Here, we analyze the computational complexity of Algorithm 1.In fact, the main complexities of Algorithm 1 lie in the subproblems ((15a)-(15c)) and ((23b)-(23c)).Besides, the complexity for solving a QCQP problem is given by Oð ffiffiffiffi m p ðmn 2 + n 3 Þ ln ð2m/ϵÞÞ, where m denotes the number of variables, n denoted the number of constraints, and ϵ is the solution accuracy [34].Thus, the complexities of ( Algorithm 1: The BCD algorithm.
5 International Journal of Distributed Sensor Networks Therefore, the total computational complexity of Algorithm 1 is given by In addition, the convergence of Algorithm 1 has been analyzed [29]; thus, we omit the details for brevity.

Simulation Results
The deployment is given in Figure 2, where there exists one Tx, one IRS, and 3 users.The coordinates of Tx and IRS are ð0m, 0m, 10mÞ and ð10m, 50m, 10mÞ, while the users are randomly located in a circle with radius 5m and centered at ð0m, 60m, 1:5mÞ, respectively [5].
Firstly, we studied the convergence of the BCD method.Figure 3 shows R s versus the number of iterations with different N t and M. From this figure, we can see that for different N t and M, R s always increase with the iteration numbers and gradually converge within 20 iteration, which confirms the convergence of the BCD method.6 International Journal of Distributed Sensor Networks Then, we show the sum rate versus P s .From Figure 4, we can see that for all these schemes, R s increase with P s , and all IRS-aided schemes obtain better performance than the no IRS-aided designs.Besides, the active IRS scheme significantly outperforms the passive IRS design, mainly owing to the power amplification effect which alleviates the impact of double fading attenuation.Thus, the received signal power at the users is enhanced and R s is improved.
Next, in Figure 5, we show R s versus M. From this figure, we can see that for all these IRS-aided methods, R s tends to increase with the increase of M, since more signal can incident the IRS with larger M. In addition, the reflected signal at the IRS will increase with M when the RC is properly designed.However, when using random IRS, only array gain can be obtained, thus suffering from certain performance loss.This result further verifies that more R s improvement can be achieved by using a larger IRS and optimizing the RC properly.
Last, Figure 6 plots R s versus the Tx-IRS distances, where we assume that the IRS moves along the y-axis from the Tx to the user's area.From this figure, we can see that the active IRS scheme always outperforms the passive IRS scheme in the considered region.Moreover, for active IRS, the R s increases when IRS moves from the Tx to the user's area, while for passive IRS, the R s first decreases to a low point and then increases.Thus, deploying the active IRS near the users is beneficial to improve the R s .

Conclusion
This work studied the active IRS-aided MIMO multicasting network, where the sum rate was maximized by jointly optimizing the precoder and RC, subject to the transmit power constraint at the Tx and IRS.We decouple the original problem into several subproblems and then proposed a BCD algorithm to optimize the variables.Simulation result verified the performance of the proposed design as well as the superiority of active IRS when compared with other benchmarks.

Figure 4 :
Figure 4: R s versus P s .

Figure 6 :
Figure 6: R s versus IRS location.

0 and covariance A.
:Initialization: Set a feasible point fW k , Θg and i = 0.