Low-complexity joint active and passive beamforming for RIS-aided MIMO systems

In this paper, we consider the reconﬁgurable intelligent surface (RIS)- aided MIMO systems to realize a high quality link between the base station (BS) and the users via a RIS. To achieve this goal, the ac- tive beamforming at the BS and the passive beamforming at the RIS should be jointly optimized. However, most of the existing joint opti- mization schemes that maximize the sum-rate have high computational complexity due to the complex derivative calculations and matrix in- version. To this end, a biologically inspired particle swarm optimization (PSO) algorithm is exploited to solve the non-convex sum-rate opti- mization problem with low-complexity. Simulation results show that the proposed scheme provides a better trade-off between performance and complexity than existing solutions for RIS-aided MIMO systems.

✉ Email: siddiqimz10@mails.tsinghua.edu.cn In this paper, we consider the reconfigurable intelligent surface (RIS)aided MIMO systems to realize a high quality link between the base station (BS) and the users via a RIS. To achieve this goal, the active beamforming at the BS and the passive beamforming at the RIS should be jointly optimized. However, most of the existing joint optimization schemes that maximize the sum-rate have high computational complexity due to the complex derivative calculations and matrix inversion. To this end, a biologically inspired particle swarm optimization (PSO) algorithm is exploited to solve the non-convex sum-rate optimization problem with low-complexity. Simulation results show that the proposed scheme provides a better trade-off between performance and complexity than existing solutions for RIS-aided MIMO systems.
Introduction: In the last decade, the network capacity has been significantly improved by using various wireless communication techniques such as multiple-input and multiple-output (MIMO), millimetre wave (mmWave) communications, and ultra-dense networks. However, these techniques require high hardware cost and power consumption [1]. Reconfigurable intelligent surface (RIS) is an emerging cost-effective technique to boost the network capacity [2]. In particular, RIS is a metasurface with a large number of small passive elements. These RIS elements can be artificially controlled by changing their phase shifts (PSs), that the incident signals can be redirected to some other directions. One exciting benefit of RIS is that, it does not add thermal noise while redirecting the received signals with high array gain [3].
To achieve the aforementioned benefit of RIS, the phase shifts of RIS elements should be properly optimized to realize passive beamforming at the RIS along with active beamforming at the BS. Some recent works on the sum-rate maximization problem are available by jointly optimizing the active and passive beamforming [4][5][6]. For instance, the authors in [4] presented a semidefinite relaxation (SDR) method to maximize the sum-rate of the entire system under the consideration of hardware constraints. Furthermore, a sum-rate maximization problem for the RISaided transmission was investigated in [5], while the multiple RIS-aided communication was studied in [6]. However, these solutions suffer from high computational complexity due to the complex derivative calculations and matrix inversion, which hampers their deployment in future 6G communication systems.
Motivated by the above discussion, in this letter, we consider an RISaided MIMO system, where the active beamforming at the BS and passive beamforming at the RIS are jointly optimized to improve the system spectral efficiency. To this end, we formulate the non-convex sum-rate maximization problem for the RIS-aided MIMO system at first. Then, to solve the non-convex sum-rate maximization problem, we present a low-complexity biologically inspired particle swarm optimization (PSO) algorithm. Specifically, the PSO algorithm firstly randomly generates a number of swarm particles (passive beamformers) with different positions based on the initial velocities. Then, with the sum-rates achieved by those passive beamformers, the PSO algorithm refines the velocities of the passive beamformers by minimizing the distance between the candidate beamformers. The velocity is subject to the update of the candidate's position of all beamformers. Repeating this procedure for several times, we can generate an optimal beamformer with the best position in the end. Simulation results indicate that our scheme achieves Notations: a is a vector and A is a matrix. The transpose and Hermitian (conjugate transpose) of a matrix A is denoted by A T and A H . E(·) represents the expectation operator, and CN (0, σ 2 ) denotes the Gaussian distribution with zero mean and variance σ 2 . A −1 , A + , and A F indicate the inverse, pseudo-inverse, and Frobenius norm of A, respectively. Finally, | · |, arg(·), and Tr(·) represent the modulus, argument of a complex number, and trace of a matrix.
System model and problem formulation: In this section, the system model of RIS-aided MIMO system will be introduced at first, and then we will formulate the capacity maximization problem in the considered system.
System model: As illustrated in Figure 1, in this paper we consider a typical downlink RIS-aided MIMO system. The system consists of a BS with M antennas and an RIS with N elements, which simultaneously serve K single-antenna users. Each RIS element is capable of changing its phase shift independently, so that the incident signals can be reflected in some other directions with high array gain [3].
Assuming that the data symbol x k for the kth user has the normalized power, where k = 1, 2, . . . , K, the transmit signal x ∈ C M×1 from M antennas at the BS is where g k ∈ C M×1 is the active beamforming vector at the BS. In practise, the power of the transmitted signal at the BS should be constrained as where G [g 1 , g 2 , . . . , g K ] ∈ C M×K is the active beamforming matrix at the BS, P T is the maximum transmission power at the BS. Then, the received signal y k for the kth user is given by where h d,k ∈ C 1×M denotes the direct channel link between BS and the kth user, β k is the channel attenuation coefficient for the kth user, and h r,k ∈ C 1×N denotes the reflecting channel between RIS with N elements and the kth single-antenna user; Furthermore, RIS beamforming matrix is denoted by the where |˜ n | 2 = 1 represents the unit modulus hardware constraint of the nth RIS element [4], ∀n = 1, 2, . . . , N; H ∈ C N×M stands for the BS-RIS channel matrix; The last term n k indicates the additive white Gaussian noise (AWGN) that follows CN (0, σ 2 ).
Based on the signal model (3), the signal-to-interference-plus-noise ratio (SINR) for the kth user is written by Based on the SINR in (4), the achievable sum-rate R of all K users in the RIS-aided MIMO system can be obtained as Problem formulation: For the considered RIS-aided MIMO system, our objective is to jointly optimize the active beamforming matrix G at the BS and the passive beamforming matrix at the RIS to maximize the achievable sum-rate R. Mathematically, the sum-rate optimization problem can be formulated as where F denotes all possible passive beamformers that satisfy the unit modulus hardware constraint. In this paper, we assume that the channel state information (CSI) is perfectly known, e.g. the channels information can be retrieved by a technique discussed in [7]. With the known channel information, BS exploits the classical zero-forcing (ZF) as the active beamforming scheme for the signal transmissions. To this end, for a given passive beamforming matrix , we obtain the equivalent chan- Then, the pseudo inverse of the equivalent channelH gives the ZF beamforming matrix G, i.e. G =H † . Substituting G =H † into (6), we can rewrite the sum-rate maximization problem (6) as s.t. c1 : ∈ F, where γ = G 2 F /P T is the optimal scaling factor, so the optimal active beamforming matrix at the BS is given byĜ = 1 γ G. However, the reformulated problem (7) is still non-convex due to the unit modulus constraint of in c1. In the following, we present a computationally efficient algorithm to solve this non-convex problem (7).

RIS beamforming design:
In this section, we use the particle swarm optimization (PSO) algorithm to solve the non-convex sum-rate maximization problem (7) with low-complexity.
PSO for RIS design: PSO is a stochastic iterative optimization technique, which is inspired by the social behavior of birds flocking. The population of individual candidates, like birds, are called particles. These particles are randomly initialized within the search space. The coordinate of a particle that represents a solution to a problem is called the position of a particle. Moreover, in each iteration, the velocity of each particle is adjusted towards the best position, and the best suboptimal particle can be finally obtained after several iterations.
Specifically, the PSO algorithm (summarized in Algorithm 1) works in the following three steps. First, the PSO algorithm generates O particle swarms (e.g., possible passive beamformers in our problem (7)) with random positions as˜ 1 (0),˜ 2 (0), . . . ,˜ O (0). All positions are normalized to ensure the unit power. Subsequently, the velocity of all O passive beamformers is randomly initialized, where the velocity of the oth passive beamformer is represented by v o . Based on each passive beamformer, we can easily obtain the active beamforming matrix G from the equivalent channel matrix. Then, the value of the objective function (e.g. the achievable sum-rate R in our problem (7)) can be computed. For all O, compare the objective function values. The o * th beamformer˜ o * that maximizes the objective function in (7) is denoted byŵ best , and is regarded as the best particle in one iteration. After that, we update the velocity and corresponding position of the oth particle for the next iteration as

Algorithm 1 PSO algorithm
where u 1 and u 2 are the uniformly distributed random vectors,˜ obest is the best and˜ ocurrent is the current position of the oth particle. The c 1 and c 2 are positive acceleration coefficients, and the element-wise multiplication is denoted by . In the first iteration when i = 0, we set˜ obest (0) = ocurrent (0) for any o. In the following iterations, each particle keeps track of its own best position that can achieve the maximum objective function value. Whenever the oth particle position is refined, its new objective function value will be updated accordingly. If the updated objective function value is larger than the previous best value of the candidate, then we set˜ obest (i) =˜ ocurrent (i). Finally, for all O new positions, we again compare the objective function values, and the particle that maximizes R in (7) will also be compared with the previous bestŵ best and the one that maximizes (7) will become the best particleŵ best . These steps will be repeated, until the number of iterations I is satisfied. In the end, w best =˜ obest .
Computational analysis: In this subsection, we provide the computational complexity comparison between the PSO algorithm and an existing scheme in [6]. It is observed that the computational complexity of the PSO algorithm mainly involves the computation of fitness values of O passive beamformers for K users. Thus, the computational complexity is O (OK log(O)). On the other hand, the complexity of the scheme in [6] is O(N 3 + N 2 + N ). It is clear that the complexity of the existing scheme [6] grows fast with N RIS elements, while the complexity of the PSO algorithm is only linearly increasing with the particles number O.
Simulation results: In this section, simulation results are provided to validate the performance of the RIS-aided MIMO communication system. We compare the spectral efficiency performance of the PSO algorithm with the existing work [6].
The configuration of the simulated scenario is shown in Figure 2, where BS and RIS are located at (0, −20 m) and (30 m, 3 m) respectively. Four users are randomly distributed in a uniform region with radius 1m. The distance between the BS and the uniform region is denoted as the horizontal distance D. Simulation setup: We consider a large-scale channel model as discussed in [3]. Let d BS,k , d BS,RIS , d RIS,k denote the distance between BS-kth user, BS-RIS and RIS-kth user, respectively. Thus, the distance-dependent path loss (PL) for the BS-user link can be written as and the PL for the BS-RIS-user link can be written as where C d and C r are the channel fading variables depending on the wavelength, channel status, and angle of departure (AoD); Moreover, C r is also related to the angle of arrival (AoA), the material and size of RIS element [3]; G BS and G k denote the antenna gain of the BS and the kth user; Here, we assume C d G BS G k = −30 dB and C r G BS G k = −40 dB [6]; The PL exponents for the BS-RIS link and RIS-user link are set as ν BS,RIS = ν RIS,k = 2, while BS-user link is set as ν BS,k = 3 [6].
To exploit the small-scale fading, we adopt a Rician fading channel model for the BS-user channel h d,k as where α BS,k is the Rician factor, and h LoS d,k and h NLoS d,k denote the LoS and Rayleigh fading components.
In the simulation, we consider M = 8, N = 64, K = 4 and O = 40. The number of iterations is I = 30 and the noise power is set as σ 2 =−120 dBm. The distance between BS and RIS is defined as 30m. Finally, the channel attenuation coefficient β k in (3) is set as 1.
Achievable spectral efficiency comparison: The spectral efficiency performance of the PSO algorithm is shown in Figure 3, [6] in the whole transmit power range. Moreover, the PSO algorithm achieves about 93% of the spectral efficiency achieved by the existing scheme [6], but with much lower complexity as we have discussed before. Figure 4 shows the spectral efficiency performances against the horizontal distance D between BS and users. From this figure, it is observed Fig. 4 Achievable spectral efficiency performance against the distance D that as the horizontal distance D between the BS and users increases, the achievable spectral efficiency performance decreases rapidly due to the serious attenuation of the signals. However, a peak at D = 30 m indicates that the spectral efficiency of both PSO and existing scheme [6] increases when the users approach the RIS, since the users receive strong signals reflected from the RIS. On the other hand, no peak is observed for the conventional without RIS scheme.

Conclusions:
In this letter, we focused on the joint active and passive beamforming problem for the RIS-aided MIMO system. We formulated the sum-rate optimization problem with practical hardware constraints. To solve this non-convex problem, the biologically inspired lowcomplexity PSO algorithm was adopted. Complexity analysis showed the advantage of the presented scheme over the recently proposed solution for real-time implementation. Simulation results demonstrated that, PSO algorithm achieves about 93% of the spectral efficiency achieved by the existing scheme but with low-complexity.