Intelligent Reflecting Surface Aided Multigroup Multicast MISO Communication Systems

Intelligent reflecting surface (IRS) has recently been envisioned to offer unprecedented massive multiple-input multiple-output (MIMO)-like gains by deploying large-scale and low-cost passive reflection elements. By adjusting the reflection coefficients, the IRS can change the phase shifts on the impinging electromagnetic waves so that it can smartly reconfigure the signal propagation environment and enhance the power of the desired received signal or suppress the interference signal. In this paper, we consider downlink multigroup multicast communication systems assisted by an IRS. We aim for maximizing the sum rate of all the multicasting groups by the joint optimization of the precoding matrix at the base station (BS) and the reflection coefficients at the IRS under both the power and unit-modulus constraint. To tackle this non-convex problem, we propose two efficient algorithms. Specifically, a concave lower bound surrogate objective function has been derived firstly, based on which two sets of variables can be updated alternately by solving two corresponding second-order cone programming (SOCP) problems. Then, in order to reduce the computational complexity, we further adopt the majorization--minimization (MM) method for each set of variables at every iteration, and obtain the closed-form solutions under loose surrogate objective functions. Finally, the simulation results demonstrate the benefits of the introduced IRS and the effectiveness of our proposed algorithms.

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I Introduction

In the era of 5G and Internet of Things by 2020, it is predicted that the network capacity will increase by 1000 folds to serve at least 50 billions devices through wireless communications [1] and the capacity is expected to be achieved with lower energy consumption. To meet those Quality of Service (QoS) requirements, intelligent reflecting surface (IRS), as a promising new technology, has been proposed recently to achieve high spectral and energy efficiency. It is an artifical passive radio array structure where the phase of each passive element on the surface can be adjusted continuously or discretely with low power consumption [2, 3], and then change the directions of the reflected signal into the specific receivers to enhance the received signal power [4, 5, 6] or suppress interference as well as enhance security/privacy [7, 8].

The IRS, as a new concept beyond conventional massive multiple-input and multiple-output (MIMO) systems, mantains all the advantages of massive MIMO systems, such as being capable of focusing large amounts of energy in three-dimensional space which paves the way for wireless charging, remote sensing and data transmissions. However, the differences between IRS and massive MIMO are also obvious. Firstly, the IRS can be densely deployed in indoor spaces, making it possible to provide high data rates for indoor devices in the way of near-field communications [9]. Secondly, in constrast to conventional active antenna array equipped with energy-consuming radio frequency chains and power amplifiers, the IRS with passive reflection elements is cost-effective and energy-efficient [4], which enables IRS to be a prospective energy-efficient technology in green communications. Thirdly, as the IRS just reflects the signal in a passive way, there is no thermal noise or self-interference imposed on the received signal as in conventional full-dulplex relays.

Due to these significant advantages, IRS has been investigated in various wireless communication systems. Specifically, the authors in [4] first formulated the joint active and passive beamforming design problem both in downlink single-user and multiple-users multiple-input single-output (MISO) systems assisted by the IRS, while the total transmit power of the base station (BS) is minimized based on the semidefinite relaxation (SDR) [10] and alternating optimization (AO) techniques. In order to reduce the high computational complexity incurred by SDR, Yu et al. proposed low complexity algorithms based on MM (Majorization–Minimization or Minorization–Maximization) algorithm in [7] and manifold optimization in [11] to design reflection coefficients with the targets of maximizing the security capacity and spectral efficiency communications, respectively. Pan et al. considered the weighted sum rate maximization problems in multicell MIMO communications [5] and simultaneous wireless information and power transfer (SWIPT) aided systems [6], both demonstrating the significant performance gains achieved by deploying an IRS in the networks. By adopting the alternating algorithm combined with the popular bisection search method, Shen et al. in [8] derive closed-form transmit covariance matrix of the source and semi-closed-form phase shift matrix of the IRS for the security rate maximization problem.

However, all the above-mentioned contributions only investigated the performance benefits of deploying an IRS in unicast transmissions, where the BS sends an independent data stream to each user. However, unicast transmissions will cause severe interference and high system complexity when the number of users is large. To address this issue, the multicast transmission based on content reuse [12] (e.g., identical content may be requested by a group of users simultaneously) has attracted wide attention, especially for the application scenarios such as popular TV programme or video conference. From the perspective of operators, it can be envisioned that multicast transmission is capable of effectively alleviating the pressure of tremendous wireless data traffic and play a vital role in the next generation wireless networks. Therefore, it is necessary to explore the potential performance benefits brought by an IRS during the multigroup multicast transmission.

A common performance metric in multicast transmissions is the max-min fairness (MMF), where the minimum signal-to-interference-plus-noise-ratio (SINR) or spectral efficiency of users in each multicasting group or among all multicasting groups is maximized [13, 14, 15, 16, 17]. Prior seminal treatments of multicast transmission in single-group and multigroup are presented in [13, 14], where the MMF problems are formulated as a fractional second-order cone programming (SOCP) and are NP-hard in general. The SDR technique [10] was adopted to approximately solve the SOCP problem with some mathematical manipulations. In order to reduce the high computaional complexity of SDR, several low-complexity algorithms, such as asymptotic approach [15], successive convex approximation approach [16]

, and heuristic algorithm

[17], have been proposed by exploiting the special feature of near-orthogonal massive MIMO channels.

In this paper, we consider an IRS-assisted multigroup multicast transmission system in which a multiple-antenna BS transmits independent information data streams to multiple groups, and the single-antenna users in the same group share the same information and suffer from interference from those signals sent to other groups. Unfortunately, the popular SDR-based method incurs a high computational complexity which hinders its practical implementation when the number of design parameters (e.g., precoding matrix and reflection coefficient vector) becomes large. Furthermore, the aforementioned low-complexity techniques designed for the IRS-aided unicast communication schemes cannot be directly applied in the multigroup multicast communication systems since the MMF metric is a discrete and complex objective function.

Against the above background, the main contributions of our work are summarized as follows:

  • To the best of our knowledge, this is the first work exploring the performance benefits of deploying an IRS in multigroup multicast communication systems. Specifically, we jointly optimize the precoding matrix and the reflection coefficient vector to maximize the sum rate of all the multicasting groups, where the rate of each multicasting group is limited by the minimum rate of users in the group. This formulated problem is much more challenging than previous problems considered in unicast systems since our considered objective function is discrete and complex due to the nature of the multicast transmission mechanism. In addition, the highly coupled variables and complex sum rate expression aggravates the difficulty to solve this problem.

  • The formulated problem is solved efficiently in an iterative manner based on the alternating optimization method. Specifically, we firstly minorize the original non-concave objective function by a surrogete function which is biconcave of precoding matrix and reflection coefficient vector, and then apply the alternating optimization method to decouple those variables. At each iteration of the alternating optimization method, the subproblem corresponding to each set of variables is reformulated as an SOCP problem by introducing auxiliary variables, which can help to transform the discrete concave objective function into a series of convex constraints.

  • To further reduce the computational complexity, we use the MM method to derive closed-form solutions of each subproblem, instead of solving the complex SOCP problems with a high complexity at each iteration. Specifically, we firstly apply the log-sum-exp lower bound to approximate the discrete concave objective function, yielding a differentiable continuous concave function. Then, we use the MM method to derive a tractable surrogate objective function of the log-sum-exp function, based on which we derive the closed-form solutions of each subproblem. Finally, we prove that the proposed algorithm is guaranteed to converge and the solution sequences generated by the algorithm converge to stationary points.

  • Finally, the simulation results demonstrate the superiority of the IRS-assisted multigroup multicast system over conventional massive MIMO systems in terms of spectral efficiency and energy efficiency.

The remainder of this paper is organized as follows. Section II introduces the system model and formulates the optimization problem. An SOCP-based method is developed to solve the problem in Section III. Section IV further provides a low-complexity algorithm. Finally, Section V and Section VI show the simulation results and conclusions, respectively.

Notations: The following mathematical notations and symbols are used throughout this paper. Vectors and matrices are denoted by boldface lowercase letters and boldface uppercase letters, respectively. The symbols , , , and denote the conjugate, transpose, Hermitian (conjugate transpose), Frobenius norm of matrix , respectively. The symbol denotes 2-norm of vector . The symbols , , , and denote the trace, real part, modulus and angle of a complex number, respectively. is a diagonal matrix with the entries of on its main diagonal. means the element of the vector . The Kronecker product between two matrices and is denoted by . means that is positive semidefinite. Additionally, the symbol denotes complex field, represents real field, and is the imaginary unit.

Ii System Model

Ii-a Signal Transmission Model

Fig. 1: An IRS-aided multigroup multicast communication system.

As shown in Fig. 1, we consider an IRS-aided multigroup multicast MISO communication system. There is a BS with transmit antennas serving multicasting groups. Users in the same group share the same information data and the information data destined for different groups are independent and different, which means there exists inter-group interference. Let us define the set of all multicast groups by . Assuming that there are users in total, the user set belonging to group is denoted as and each user can only belong to one group, i.e., =. The transmit signal at the BS is

(1)

where is the desired information of group and is the correspongding precoding vector. Let us denote the collection of all precoding vectors as satisfying the power constraint , where is the maximum available transmit power at the BS.

In the multigroup multicast system, we propose to employ an IRS with the goal of enhancing the received signal strength of users by reflecting signals from the BS to the users. It is assumed that that the IRS has reflection elements, and the reflection coefficient matrix of the IRS is modelled by a diagonal matrix , where . The channels spanning from the BS to user , from the BS to the IRS, and from the IRS to user are denoted by , , and , respectively.

It is assumed that the channel state information (CSI) is perfectly known at the BS. The BS is responsible for designing the reflection coefficients of the IRS and sends them back to the IRS controller as shown in Fig. 1. As a result, the received signal of user belonging to group is

(2)

where is the received noise at user

, which is an additive white Gaussian noise (AWGN) following circularly symmetric complex Gaussian (CSCG) distribution with zero mean and variance

. Then, its achievable data rate (bps/Hz) is given by

(3)

Denoting by the equivalent channel spanning from the BS to user and by the equivalent reflection coefficient vector, we have

(4)
(5)

Note that belongs to the set . Then, the data rate expression in (3) can be rewritten in a compact form as

(6)

Due to the nature of the multicast mechanism, the achievable data rate of group is limited by the minimum user rate in this group and is defined as follows

(7)

Ii-B Problem Formulation

In this paper, we aim to jointly optimize the precoding matrix and reflection coefficient vector to maximize the sum rate of the whole system, which is defined as the sum rate achieved by all groups. Mathematically, the optimization problem is formulated as

(8)

Problem (8) is a non-convex problem and difficult to solve since the objective function is discrete and non-concave, while the unit-modulus constraint set is also non-convex. In the following, we propose two efficient algorithms to solve Problem (8).

Iii SOCP-based Alternating Optimization method

In this section, we propose an SOCP-based alternating optimization method to solve Problem (8). Specifically, we first handle the non-convex objective function by introducing its concave surrogate function. Then, we adopt the alternating optimization method to solve the subproblems corresponding to different sets of variables iterately.

Note that is a composite function which is the linear combinations of some pointwise minimum with non-concave subfunction . We first tackle the non-concave property of . To this end, We introduce the following theorem.

Theorem 1

Let be the solutions obtained at iteraion , then is minorized by a concave surrogate function defined by

(9)

where

at fixed point .

Proof: Please refer to Appendix A.                                                                                      

Based on the above theorem, Problem (8) can be transformed into the following surrogate problem:

(10)

We note that is biconcave of and [18], since with given is concave of and with given is concave of . This biconvex problem enables us to use the alternating optimization (AO) method to alternately update and .

Iii-a Optimizing the Precoding Matrix

In this subsection, we aim to optimize the precoding matrix with given . With some manipulations, in (9) can be shown to be a quadratic function of :

(11)

where

(12)
(13)

and is a selection vector in which the element is equal to one and all the other elements are equal to zero.

By using (11), the subproblem of Problem (10) for the optimization of is

(14)

We then tackle the pointwise minimum expressions in the objective function of Probelm (14) by introducing auxiliary variales for , as follows

(15)

Problem (15) is an SOCP problem and can be solved by the CVX [19] solver, such as MOSEK [20].

Iii-B Optimizing the reflection coefficient vector

In this subsection, we focus on optimizing the reflection coefficient vector with given , then can be rewritten as

(16)

where

(17)
(18)

Upon replacing the objective funcion of Problem (10) by (16), the subproblem for the optimization of is given by

(19)

Also introducing auxiliary variables for , Problem (19) is equivalent to

(20)

The above problem is still non-convex due to the non-convex unit-modulus set . To address this issue, we replace it with a relaxed convex one as

where is a selection vector where the element is equal to one and all the other elements are equal to zero. Let us denote by the optimal solution of the following relaxed version of the SOCP problem in (21)

(21)

Then, the optimal in the iteration is

(22)

where

(23)

and symbol denotes the element of the vector . Here the and the are both element-wise operations.

Iii-C Algorithm development

Based on the above analysis, Algorithm 1 summarizes the alternating update process between precoding matrix and reflection coefficient vector to maximize the sum rate of the whole system.

0:  Initialize and , and .
1:  repeat
2:     Set and calculate by solving Problem (15);
3:     Set and calculate by solving Problem (21);
4:     ;
5:  until The value of function in (8) converges.
Algorithm 1 SOCP-based alternating optimization algorithm

Iii-C1 Complexity analysis

Now we analyze the computational complexity of Algorithm 1, which mainly comes from optimizing in the SOCP problem in (15) and optimizing in the SOCP problem in (21).

According to [21], the complexity of solving an SOCP problem, with second order cone constraints where the dimension of each is , is , where is the convergence accuracy. Problem (15) contains one power constraint with dimension and rate constraints with dimension . Therefore, the complexity of solving Problem (15) per iteration is . Problem (21) has constant modulus constraints with dimension one for sparse vector and rate constraints with dimension . Therefore, the complexity of solving Problem (21) per iteration is . Therefore, the approximate complexity of Algorithm 1 per iteration is .

Iii-C2 Convergence analysis

In the following, we analyze the convergence of Algorithm 1. Specifically, we have

where the first equality holds when and in (9); the first inequality follows from the globally optimal solution of Problem (15); the second one follows from the locally optimal solution of Problem (21); and the last inequality comes from (9). Then, we have a nondecreasing sequence . In addition, the sequence generated at each iteration of Algorithm 1 converges to a stable point as because and are bounded in their feasible sets and , respectively [22]. Hence, sequence can guarantee to converge to a local optimum.

Iv Log-Sum-Exp-based Majorization-Minimization method

As seen in Algorithm 1, we need to solve two SOCP problems in each iteration, which incurs a high computational complexity. In this section, we aim to derive a low-complexity algorithm with good performance.

Since in Problem (10) is non-differentiable, we approximate it as a smooth function by using the following smooth log–sum–exp lower-bound [23]

(24)

where is a smoothing parameter which satisfies

Theorem 2

is biconcave of and .

Proof: According to [24], if the Hessian matrix of a function is semi-negative definite, that function is concave. In particular, we derive the Hessian matrix of the exp-sum-log function as

(25)

where . Then for all , we have

(26)

where the components of vectors and are and , respectively. The inequality follows from the Cauchy-Schwarz inequality. Then , and the log-sum-exp function is concave. Therefore, is an increasing and concave function w.r.t. . Recall that is biconcave of and . Finally, according to the composition principle [24], is biconcave of and . The proof is complete.      

Large leads to high accuracy of the approximation, but it also causes the problem to be nearly ill-conditioned. When is chosen appropriately, Problem (10) is approximated as

(27)

This problem is still a biconvex problem of and , which enables us to alternately update and by adopting the alternating optimization method.

Iv-a Optimizing the Precoding Matrix

Given , the subproblem of Problem (27) for the optimization of is

(28)

Even is a concave and continuous function of precoding matrix , it is still very complex and difficult to be optimized directly. Since the aim of the MM algorithm [25, 26] is to find an easy-to-solve surrogate objective function and optimize it instead of the original complex one, we use this algorithm to find a locally optimal solution of Problem (28).

Let denote a real-valued function of variable with given . The function is said to minorize at given point if they satisfy the following conditions [26]:

where , defined as the direction derivative of in the direction , is

In this subsection, the minorizing function of is given in the following theorem.

Theorem 3

Since is twice differentiable and concave, we minorize at any fixed with a quadratic function satisfying conditions (A1)-(A4), as follows

(29)

where

(30)
(31)
(32)
(33)
(34)

Proof: Please refer to Appendix B.                                                                                  

Upon replacing the objective function of Problem (28) with (29), we obtain the following surrogate problem

(35)

The optimal could be obtained by introducing a Lagrange multiplier associated with the power constraint, yielding the Lagrange function

By setting the first-order derivative of w.r.t. to zero, we have

Then the optimal solution of in iteration can be derived as

(36)

By substituting (36) into the power constraint, one has

(37)

It is obvious that the left hand side of (37) is a decreasing function of .

  • If the power constraint inequality (37) holds when , then

    (38)
  • Otherwise, there must exist a that (37) holds with equality, then

(39)

Iv-B Optimizing the Reflection Coefficient Vector

Given , the subproblem of Problem (27) for the optimization of is

(40)

Upon adopting the MM algorithm framework, we first need to find a minorizing function of and denote it as . Since is a non-convex set, we should modify (A3) so as to claim stationarity convergence [27, 28]:

where is the Boulingand tangent cone of at . Therefore should satisfy the following conditions:

and is given in the following theorem.

Theorem 4

Since is twice differentiable and concave, we minorize at any fixed with a function satisfying conditions (B1)-(B4), as follows

(41)

where

(42)
(43)
(44)
(45)
(46)

Proof: Please refer to Appendix C.                                                                                      

Upon replacing the objective function of Problem (40) by (41), we obtain the following surrogate problem as

(47)

Then, the optimal at the iteration is

(48)

where is an element-wise operation.

Iv-C Convergence Analysis

In this section, we adopt alternating optimization algorithm to alternately optimize precoding matrix and reflection coefficient vector . In each iteration, we adopt the MM algorithm to update each set of variables. The monotonicity of the MM algorithm has been proved in [26] and [29]. In the following, we claim the monotonicity of Algorithm 2 which will be shown later. At the iteration, with given , we have

where the first equality follows from (A1), the first inequality follows from (35), and the second one follows from (A2). Subsequently, with given , it is straightforward to have