I Introduction
The past decade has witnessed an enormous increase in the number of mobile devices [1], triggering urgent needs for highspeed and seamless data services in future wireless systems. To meet such demands, one fundamental issue is how to improve the link quality in the complicated timevarying wireless environments involving unpredictable fading and strong shadowing effects. Various technologies have been developed such as relay [2] and massive multiple input and multiple output (MIMO) [3], aiming to actively strengthen the target signals by forwarding and taking advantage of multipath effects, respectively. However, these techniques require extra hardware implementation with inevitable power consumption and high complexity for signal processing, and the quality of services is also not always guaranteed in harsh propagation environments.
Recently, the development of metasurfaces [4] has given rise to a new transmission technique named reconfigurable intelligent surface (RIS), which shapes the propagation environment into a desirable form by controlling the electromagnetic response of multiple scatters [5]. Specifically, the RIS is an ultrathin surface inlaid with multiple subwavelength scatters, i.e., RIS elements, whose electromagnetic response (such as phase shifts) can be controlled by simple programmable PIN diodes [6]. Based on the ON/OFF functions of PIN diodes, only a limited number of discrete phases shifts can be achieved by the RIS [7]. Instead of scattered waves emanated from traditional antennas, the subwavelength separation between adjacent RIS elements enables the refracted and reflected waves to be generated via superposition of incident waves at the surface [8]. Benefited from such a programmable characteristic of molding the wavefronts into desired shapes, the RIS serves as a part of reconfigurable propagation environment such that the received signals are directly reflected towards the receivers without any extra cost of power sources or hardwares [9], thereby improving the link quality and coverage.
To exploit the potential of RIS techniques, many existing works have considered the RIS as a reflectiontype surface deployed between sources and destinations in either pointtopoint communications [5, 10, 11, 12] or multiuser systems [13, 14, 15, 16] for higher data rates or energy saving. In [11], a pointtopoint RISassisted multiinput singleout system has been investigated where the beamformer at the transmitter and continuous phase shifts of the RIS are jointly optimized to maximize the sum rate. In [15]
, a channel estimation protocol has been proposed for a multiuser RISassisted system and the continuous phase shifts have been designed to maximize the minimum user date. In
[16], the authors have minimized the transmit power of the access point by optimizing the continuous digital beamforming and discrete phase shifts. An algorithm has been designed for the singleuser case and extended to the multiuser case.However, two major issues still remain to be further discussed in the open literature.

For the multiuser case, how to determine the limited discrete phase shifts directly such that the interuser interference can be eliminated? How does the quantization level influence the sum rate of the system?

Considering the strengthened coupling between propagation and discrete phase shifts brought by the dominant reflection ray via the RIS, how do we design the size of RIS and perform beamforming in a multiantenna system to achieve the maximum sum rate?
To address the above issues, in this paper, we consider a downlink multiuser multiantenna system where the direct links between the multiantenna base station (BS) and users suffer from deep shadowing. To provide highquality data services, a RIS with limited discrete phase shifts is deployed between the BS and users such that the signals sent by the BS are reflected by the RIS towards the users. Since the incident waves are reflected rather than scattered at the RIS, it is the reflectionbased onehop ray [17] via the RIS that dominates the propagation between the BS and users. Therefore, the propagation model in the RISbased system differs from those for traditional twohop relays and onehop direct links in MIMO systems, revealing an inner connection between the phase shifts and the propagation paths.
To achieve better directional reflection rays towards the desired users, it is vitally important to determine the phase shifts of all RIS elements, the process of which is also known as the RIS configuration. Such builtin programmable configuration [5] is actually equivalent to analog beamforming, realized by the RIS inherently. Since the RIS elements do not have any digital processing capability, we consider the hybrid beamforming (HBF) [18] consisting of the digital beamforming at the BS and RIS configuration based analog beamforming. A novel HBF scheme for RISbased communications with discrete phase shifts is thus required for better shaping the propagation environment and sum rate maximization.
Designing an HBF scheme presents several major challenges. First, the reflectiondominated onehop propagation and the RIS configuration based analog beamforming are coupled with each other, rendering the optimal scheme very hard to be obtained. The traditional beamforming schemes with separate channel matrix and analog beamformer do not work any more. Second, discrete phase shifts required by the RIS renders the sum rate maximization to be a mixed integer programming problem, which is nontrivial to be solved especially in the complex domain. Third, given the dense placing of the RIS elements, the correlation between elements may degrade the data rate performance. Thus, it is necessary to explore how the achievable rate is influenced by the size of RIS, which is challenging due to the complicated propagation environments especially in a multiuser case.
Through solving the above challenges, we aim to design an HBF scheme for the RISbased multiuser system with limited discrete phase shifts to maximize the sum rate. Our main contributions can be summarized below.

We consider a downlink RISbased multiuser system where a RIS with limited discrete phase shifts reflects signals from the BS towards various users. Given a reflectionbased onehop propagation model, we design an HBF scheme where the digital beamforming is performed at the BS, and the RISbased analog beamforming is conducted at the RIS.

A mixedinteger sum rate maximization problem for RISbased HBF is formulated and decomposed into two subproblems. We propose an iterative algorithm in which the digital beamforming subproblem is solved by zeroforcing (ZF) beamforming with power allocation and the RISbased analog beamforming is solved by the outer approximation.

We prove that the proposed RISbased HBF scheme can save as much as half of the radio frequency (RF) chains compared to traditional HBF schemes. Extending from our theoretical analysis on the pure LineofSight (LoS) case, we reveal the influence of the size of RIS and the number of discrete phase shifts on the sum rate both theoretically and numerically.
The rest of this paper is organized as follows. In Section II, we introduce the system model of the downlink RISbased MU multiantenna system. The frequencyresponse model of the RIS and the channel model are derived. In Seciton III, the HBF scheme for the RISbased system is proposed. A sum rate maximization problem is formulated and decomposed into two subproblems: digital beamforming and RIS configuration based analog beamforming. An iterative algorithm is developed in Section IV to solve the above two subproblems and a suboptimal solution is obtained. In Section V, we compare the RISbased HBF scheme with the traditional one theoretically, and discuss how to achieve the maximum sum rate in the pure LoS case. The complexity and convergence of the proposed algorithm are also analyzed. Numerical results in Section VI evaluate the performance of the proposed algorithm and validate our analysis. Finally, conclusions are drawn in Section VII.
Notations:
Scalars are denoted by italic letters, vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively. For a complexvalued vector
, denotes its Euclidean norm, and denotes a diagonal matrix whose diagonal element is the corresponding element in . For a square matrix , denotes its trace. For any general matrix , and denote its conjugate transpose and transpose, respectively. , , anddenote an identity matrix, allzero and allone vectors, respectively.
Ii System Model
In this section, we first introduce the RISbased downlink multiuser multiantenna system in which the BS with multiple antennas serves various singleantenna users via the RIS such that the propagation environment can be predesigned and configured to optimize the system performance. The discrete phase shift model of RIS and the channel model are then constructed, respectively.
Iia Scenario Description
Consider a downlink multiuser communication system as shown in Fig. 1 where a BS equipped with antennas transmits to singleantenna users. Due to the complicated and dynamic wireless environment involving unexpected fading and potential obstacles, the BS  user link may not be stable enough or even be in outage. To alleviate this issue, we consider to deploy an RIS between the BS and users, which reflects the signals from the BS and directly projects to the users by actively shaping the propagation environment into a desirable form.
An RIS consists of electrically controlled RIS elements as shown in Fig. 1, each of which is a subwavelength metamaterial particle with very small features. An RIS controller, equipped with several PIN diodes, can control the ON/OFF state of the connection between adjacent metal plates where the RIS elements are laid, thereby manipulating the electromagnetic response of RIS elements towards incident waves. Due to these builtin programmable elements, the RIS requires no extra active power sources nor do not have any signal processing capability such as decoding [5]. In other words, it serves as a lowcost reconfigurable phased array that only relies on the combination of multiple programmable radiating elements to realize a desired transformation on the transmitted, received, or reflected waves [9].
IiB Reconfigurable Intelligent Surface with Limited Discrete Phase Shifts
As shown in Fig. 2, the RIS is achieved by the bit reprogrammable metamaterial, which has been implemented as a set of radiative elements layered on a guiding structure following the wave guide techniques, forming a 2dimensional (2D) planar antenna array [7]. Being a miniature radiative element, the field radiated from the RIS element has a phase and amplitude determined by this element’s polarizability, which can be tuned by the RIS controller via multiple PIN diodes (ON/OFF) [5]. However, the phase and amplitude introduced by an RIS element are not generated randomly; instead they are constrained by the Lorentzian resonance response [4], which greatly limits the range of phase values. Based on such constraints, one common manner of implementation is to constrain the amplitude and sample the phase values from the finite feasible set [4] such that the voltagecontrolled diodes can easily manipulate a discrete set of phase values at a very low cost.
Specifically, we assume that each RIS element is encoded by the controller (e.g., via PIN diodes) to conduct possible phase shifts to reflect the radio wave. Due to the frequencyselective nature of the metamaterials, these elements only vibrate in resonance with the incoming waves over a narrow band centering at the resonance frequency. Without loss of generality, we denote the frequency response of each element at the th row and th column of the 2D RIS within the considered frequency range as , . Since the RIS is bit controllable, possible configuration modes (i.e., phases) of each can be defined according to the Lorentzian resonance response [4].
(1) 
where denotes the phase shift of RIS element . For convenience, we refer to as the number of quantization bits.
Since the 2D RIS is constructed based on the ultrathin wave guides, the propagation inside the 2D RIS is influenced by both the location and the wavenumber of each metasurface element. The latter one is also a function of the frequency, which reflects the frequencyselective nature of RIS. Denote the position of the element as . The propagation introduced by this element can then be given by , where is the wavenumber of RIS element , and is the corresponding wave length. For simplicity, we assume that the wavenumber remains the same for all RIS element within the considered narrow band^{1}^{1}1This model can be easily extended to a frequencyselective case where varies with the working frequency. The propagation can then be modelled by a filter with finite impulse response [9]..
IiC Reflectiondominated Channel Model
In this subsection, we model the channel between antenna of the BS and user . Specifically, instead of the traditional twohop channel for relays [10], we use the onehop reflection ray to model the dominant channel between the BS and users via the RIS which only passively reflects the received signals. The key reason can be detailed below. Due to small spacing between adjacent RIS elements (usually much less than the wavelength), the signals projected onto the surface are no longer just scattered randomly into the open space like those signals spread by the traditional antennas. Instead, the superposition of spherical waves facilitated by a number of miniature scatters enables refracted and reflected waves [8] without any extra decoding or signal forwarding procedures. Therefore, unlike the traditional scatteringbased propagation model where signals travel independently along the BS  RIS and RIS  user paths, in the considered scenario signals are only passively reflected by the RIS along the reflectionbased path due to the coupling effect of RIS elements.
Moreover, benefited from the directional reflections of the RIS, the BS  RIS  user link is usually stronger than other multipaths as well as the degraded direct link between the BS and the user [17]. Therefore, we model the channel between the BS and each user as a Ricean model such that the BS  RIS  user link acts as the dominant “LoS” component and all the other paths together form the “nonLoS (NLOS)” component.
Specifically, let and denote the distance between antenna and RIS element , and that between user and RIS element , respectively. The “LoS” channel between the signal transmitted by the BS at antenna to user via RIS element can be given by
(2) 
where is the path loss parameter. Therefore, the channel model between each antenna of the BS and user via RIS element can be written by
(3) 
where is the Rician factor, is the path loss model for NLOS transmissions, and is the smallscale NLOS component. Here we assume that the perfect channel state information is known to the BS via communicating with the RIS controller over a dedicated wireless link. A number of channel estimation methods can be found in [15, 16], which is out of the scope of this paper. For the case where channel information is partially known to the BS, we will consider the pure LoS transmission and discuss it in detail in Section V.
Geometric Model of the Propagation: We now derive the above distances and by using the geometry information [19]. Without loss of generality, we assume that the uniform planar arrays (UPA) and uniform linear arrays (ULA) are deployed at the RIS and the BS, respectively. The UPA is placed in a way that the surface is perpendicular to the ground. To describe the geometry of the above mentioned UPA and ULA, we employ the spherical coordinates as shown in Fig. 3. For the RIS, we define the local origin as the higher corner of the surface; while for the BS, its local origin is set as one end of the linear array. The axis is set along the direction of BS antenna 0  element link, and the axis is perpendicular to the ground. For convenience, the directions of the ULA and RIS on the  plane can be depicted by the angles and , respectively, as shown in Fig. 3. We can then define the principle directions of the linear antenna arrays at the BS and the RIS, denoted by and , as
(4a)  
(4b)  
(4c) 
where , , and are directions of the , , and axis, respectively.
We denote the uniform separation between any two adjacent elements in the above defined ULA and UPA as and , respectively. The position of any antenna at the BS can be represented by
(5) 
and the position of any RIS element can be given by
(6) 
Based on  , the distance between antenna and RIS element , i.e., , can be calculated as
(7a)  
(7b)  
(7c) 
where is obtained by adopting when . Denote the position of each user as . The distance between the RIS element and any user can be expressed by
(8) 
Since the distance between any two antennas or RIS elements is much smaller than the distance between the BS and the user, i.e., , we assume that the path loss of each BSuser link is the same, ignoring the influence brought by different antennas or RIS elements. Therefore, the channel propagation can be rewritten as
(9) 
where and are given above in and .
Based on the propagation characteristics introduced above, we will investigate how RIS can be utilized to assist multiuser transmissions in the following section.
Iii RISbased Hybrid Beamforming and Problem Formulation for Multiuser Communications
Note that the RIS usually consists of a large number of RIS elements, which can be viewed as antenna elements fay away from the BS, inherently capable of realizing analog beamforming via RIS configuration. However, these RIS elements do not have any digital processing capacity, requiring signal processing to be carried out at the BS.
In this section, to realize reflected waves towards preferable directions, we present an HBF scheme for RISbased multiuser communications given the phase shift model and the channel model of RISbased transmissions in Section II. As shown in Fig. 4, the digital beamforming is performed at the BS while the analog beamforming is achieved by the RIS with discrete phase shifts. Based on the considered HBF scheme, we formulate a sum rate maximization problem, and then decompose it into the digital beamforming subproblem and the RIS configuration based analog beamforming subproblem.
Iiia Hybrid Beamforming Scheme
IiiA1 Digital Beamforming at the BS
The BS first encodes different data steams via a digital beamformer, , of size , satisfying . After upconverting the encoded signal over the carrier frequency and allocating the transmit powers, the BS sends users’ signals directly through antennas. Denote the intended signal vector for users as . The transmitted signals of the BS can be given by
(10) 
IiiA2 RIS Configuration based Analog Beamforming
After travelling through the reflectiondominated channel introduced in Section IIC, the received signal at the antenna of user can then be expressed as
(11) 
where is the additive white Gaussian noise and denotes the th element in row of matrix . In , denotes the reflection coefficient of the RIS element with respect to the transmitting antenna and user . In practice, it is a function of the incidence and reflection angels, but here without loss of generality we assume that , . We ignore the coupling between any two RIS elements here for simplicity, and thus the received signal of each user comes from the accumulated radiations of all RIS elements, as shown in . This is a common assumption widely used in the literature on both metasurfaces [4] and traditional antenna arrays [20].
IiiA3 Received Signal at the User
For each user , after it receives the signal , it down converts the signal to the baseband and then recovers the final signal. The whole transmission model of users can be formulated by
(12) 
where is the noise vector. The transmission matrix F in is defined as
(13) 
where and are both matrices consisting of elements and , respectively. The notion implies the elementbyelement multiplication of two matrices.
IiiB Sum Rate Maximization Problem Formulation
To explore how the HBF design influences the sumrate performance, we evaluate the achievable data rates of all users in the RISbased system. Based on and , we first rewrite the received signal of user in matrix form as
(14) 
where and denote the th columns of matrices F and , respectively. The achievable rate of user can then be given by
(15) 
We aim to maximize the achievable rates of all users by optimizing the digital beamformer and the RIS configuration , as formulated below:
(16a)  
(16b)  
(16c)  
(16d) 
where is the total transmit power of the BS.
IiiC Problem Decomposition
Note that problem is a mixed integer nonconvex optimization problem which is very challenging due to the large number of discrete variables as well as the coupling between propagation and RIS configuration based analog beamforming. Traditional analog beamforming design methods with finite resolution phase shifters [18] may not fit well since it is nontrivial to decouple the transmission matrix F into the product of a channel matrix and a beamformer matrix in our case. To solve this problem efficiently, we decouple it into two subproblems as shown below.
IiiC1 Digital Beamforming
Given RIS configuration , the digital beamforming subproblem can be written by
(17a)  
(17b) 
where is fixed.
IiiC2 RIS Configuration based Analog Beamforming
Based on constraint , the RIS configuration subproblem with fixed beamformer is equivalent to
(18a)  
(18b) 
In the next section, we will design two algorithms to solve these subproblems, respectively.
Iv Sum Rate Maximization Algorithm Design
In this section, we will develop a sum rate maximization (SRM) algorithm to obtain a suboptimal solution of problem (16) in Section III. Specially, we iteratively solve subproblem (17) given RIS configuration , and solve subproblem (18) given beamformer . Finally, we will summarize the overall algorithm and provide convergence and complexity analysis.
Iva Digital Beamforming Algorithm
Subproblem (17) is a wellknown digital beamforming problem. According to the results in [21], the ZF digital beamformer can obtain a near optimal solution. Therefore, we consider ZF beamforming together with power allocation as the beamformer at the BS to alleviate the interference among users. Based on the results in [22], the beamformer can be given by
(19) 
where and is a diagonal matrix whose th diagonal element is the received power at the th user, i.e., .
In the ZF beamforming, we have the following constraints:
(20) 
With these constraints, subproblem (17) can be reduced to the following power allocation problem:
(21a)  
(21b) 
The optimal solution of this problem can be obtained by waterfilling [23] as
(22) 
where is the th diagonal element of and is a normalized factor which is selected such that . The algorithm can be summarized in Algorithm 1.
IvB RIS Configuration based Analog Beamforming Algorithm
Since we iterate between the digital beamforming and RIS configuration based analog beamforming, the latter can be optimized assuming ZF precoding as shown in (19). Since the data rate with ZF precoding in (21) only depends on the RIS configuration through the power constraint (21b), the RIS configuration based analog beamforming problem can be reformulated as a power minimization problem:
(23a)  
(23b) 
where
(24) 
Here, .
Since is a symmetric, positive semidefinite matrix, we can transform this problem into a semidefinite programming (SDP) problem. Let . According to Schur complement [25], the problem can be rewritten by
(25a)  
(25b)  
(25c) 
where means that matrix is a symmetric and positive semidefinite matrix.
Remark on the tractability of the formulated problem: This problem is a mixinteger SDP, which is generally NPhard. Moreover, any two discrete variables are coupled with each other via constraint , which makes the problem even more complicated. One commonly used solution is to first relax the discrete variables into continuous ones and then round the obtained solution to satisfy the discrete constraints. However, for the RISbased systems, the typical value of the number of quantization bits is usually very small (e.g., 2 or 3) such that the roundoff methods will lead to inevitable performance degrade.
To avoid the above issue, we consider to solve the SDP discretely. In the following, we first present the following Proposition 1 to transform the nonlinear functions in with respect to into linear ones, as proved in Appendix A. We then use the outer approximation method [26] to solve this problem.
Proposition 1.
Let
We introduce a binary vector , where indicates whether , and a binary vector for phase difference . Therefore, problem (25) can be rewritten by
(26a)  
(26b)  
(26c)  
(26d) 
Here, is a constant vector whose first elements are 1 and others are 0.
Problem (26
) is a mixinteger SDP with linear constraints, which can be solved by the outer approximation method. The basic idea of the outer approximation method is to enforce the SDP constraint via linear cuts and transform the original problem into a mixinteger linear programming one, which can be solved by the branchandbound algorithm
[27]. In the following, we will elaborate on how to enforce the SDP constraints via linear cuts.Assume that a solution is
. In most mixinteger programming problems, it is very common to use the gradient cuts to approach the feasible set. However, the function of smallest eigenvalues is not always differentiable. Therefore, we use the characterization instead
[26]. Note that is equivalent to for arbitrary . If withis not positive semidefinite, we compute eigenvector
associated with the smallest eigenvalue. Then(27) 
is a valid cut that cuts off . The RIS configuration based analog beamforming algorithm can be summarized in Algorithm 2.
IvC Overall Algorithm Description
Based on the results presented in the previous two subsections, we propose an overall iterative algorithm, i.e., the SRM algorithm, for solving the original problem in an iterative manner. Specially, the beamformer is solved by Algorithm 1 while keeping the RIS configuration fixed. After obtaining the results, we will optimize the RIS configuration by Algorithm 2. Those obtained results are set as the initial solution for subsequent iterations. Define as the value of the objective function. The two subproblems will be solved alternatively until in iteration the value difference of the objective functions between two adjacent iterations is less than a predefined threshold , i.e., .
IvD Convergence and Complexity Analysis
We now analyze the convergence and complexity of our proposed SRM algorithm.
IvD1 Convergence
First, according to Algorithm 1, in the digital beaforming subproblem, we can obtain a better result given RIS configuration in the th iteration. Therefore, we have
(28) 
Second, given the beamforming result , we maximize sum rate of all users, and thus, the following inequality holds:
(29) 
Based the above inequalities, we can obtain
(30) 
which implies that the objective value of the original problem is nondecreasing after each iteration of the SRM algorithm. Since the objective value is upper bounded, the proposed SRM algorithm is guaranteed to converge.
IvD2 Complexity
We consider the complexity of the proposed algorithms for two subproblems separately.

In the digital beamforming subproblem, we need to optimize the received power for each user according to (22). Therefore, its computational complexity is .

In the RIS configuration based analog beamforming subproblem, we solve a series of linear programs by the branchandbound method. Since only one element in can be 1, can have at most possible solutions. Thus, the scale of the computational complexity of each linear program is .
V Performance Analysis of RISbased Multiuser Communications
In this section, we compare the RISbased HBF scheme with the traditional ones in terms of the minimum number of required RF chains. A special case, i.e., the pure LoS transmission, is also considered to explore theoretically how the size of RIS and its placement influence the achievable rates.
Va Comparison with Traditional Hybrid Beamforming
We adopt the fully digital beamforming scheme as a benchmark to compare the traditional and RISbased HBF schemes. In the traditional HBF, it has already been proved that when the number of RF chains is not smaller than twice the number of target data streams, any fully digital beamforming matrix can be realized [18]. However, in the RISbased HBF, the inherent analog beamforming (i.e., the RIS configuration) is closely coupled with propagation, which offers more freedom for shaping the propagation environment than the traditional scheme. To capture this characteristic, we explore a new condition for the RISbased system to achieve fully digital beamforming.
We start by describing the fully digital beamforming scheme in an RISbased system. Consider an ideal case where each RIS element directly connects with an RF chain and ADC as if it is part of the BS^{2}^{2}2In practice, the RIS does not connect to the RF chain directly unless it is installed at the BS, which is not the truth in our case where RIS actually only reflects signals. Therefore, we only consider such an ideal scheme as a benchmark to evaluate the effectiveness of RISbased HBF. The fully digital beamformer can then be denoted by , based on which we present the following proposition, which will be proved in Appendix B.
Proposition 2.
For the RISbased HBF scheme with , to achieve any fully digital beamforming scheme, the number of transmit antennas at the BS should not be smaller than the number of singleantenna users, i.e., .
This implies two conditions for the proposed RISbased scheme to achieve the fully digial beamforming scheme. First, the size of RIS should be no smaller than the product of the number of users and the size of the antenna array at the BS. Second, the number of transmit antennas at the BS should be no smaller than the number of singleantenna users.
Remark on dedicated hardware reduction for analog beamforming: In the traditional HBF, to offload part of the digital baseband processing to the analog domain, a number of RF chains are required at the BS to feed the analog beamformer, equipped with necessary hardwares such as mixers, filters, and phase shifters [28]. In contrast, as shown in Proposition 2, the number of minimum RF chains required by the RISbased HBF to achieve the fully digial beamforming has already been reduced by half compared to the traditional scheme. Moreover, the phase shifters can be saved since the RIS inherently realizes analog beamforming owning to its flexible physical structure [29].
VB Special Case: Pure LineofSight Transmissions
We consider the data rate obtained by the pure LoS case as a lower bound of the achievable rate and analyze optimal RIS placement to provide orthogonal communication links. The LoS case also reveals insights on how the achievable data rate is influenced by RIS design and placement.
Since a multitude of RIS elements are placed in a subwavelength order, spacial correlation between these elements are inevitable. In this case, the channel matrix F approaches a lowrank matrix, leading to a degraded performance in terms of the achievable data rate. Traditionally, we can utilize the multipath effect to decorrelate different channel links between the transceiver antennas. However, it is also important to understand how the system works in the pure LoS case, especially when it comes to the RISbased systems where the reflectionbased onehop link between the BS and users acts as the dominated “LoS” component and is usually much stronger than other multipaths as well as the degraded direct links.
In the pure LoS case, we aim to achieve a highrank LoS channel matrix by designing the size of RIS and the antenna array at the BS. Different from those existing works on antenna array design for LoS MIMO systems, the discrete phase shifts depicted by needs to be considered in the RISbased propagation. We thus present the results in Proposition 3 below, which is proved in Appendix C.
Proposition 3.
In the RISbased system, to make different links between the BS and user via the RIS orthogonal to each other, the RIS design should satisfy the following conditions:
(31a)  
(31b) 
This proposition shows that the achievable data rate is highly related to the size of the RIS and its placement. For convenience, when all other parameters are fixed, we refer to the threshold of the RIS size as
(32) 
For the pure LoS case, the sum rate maximization problem can still be formulated as with one extra constraint . Our proposed SRM algorithm can be utilized to solve this problem after we convert the extra constraint into a linear one shown below by following the transformations in Proposition 1,
(33) 
Vi Simulation Results
In this section, we evaluate the performance of our proposed algorithm for RISbased HBF in terms of the sum rate. We show how the system performance is influenced by the SNR, number of users, the size of RIS, and the number of quantization bits for discrete phase shifts. For comparison, the following algorithms are performed as well.

Simulated annealing: We utilize the simulated annealing method [30] to approach the global optimal solution of the sum rate maximization problem with discrete phase shifts. The maximum number of iterations is set as .

Pure LoS case: We consider the pure LoS case as a lower bound to evaluate the performance of the HBF scheme. The proposed iterative SRM algorithm can still be utilized to solve the corresponding optimization problem .

Random phase shift: Algorithm 1 is first performed, followed by a random algorithm to solve the RISbased analog beamforming subproblem .

RISbased HBF with continuous phase shifts: This scheme only serves as a benchmark when we investigate how the discreteness level (i.e., the number of quantization bits, ) influences the sum rate. The discrete constraint in the original subproblem is relaxed to . The HBF solution is obtained by iteratively performing Algorithm 1 and the gradient descent method.
In our simulation, we set the distance between the BS and the RIS, , as 20m, and users are randomly deployed within a half circle of radius 60m centering at the RIS. The antenna array at the BS and the RIS are placed at angles of and to the x axis, respectively. The transmit power of the BS is 20 W, the carrier frequency is 5.9 GHz, the antenna separation at the BS is 1 m, the RIS element separation is 0.03 m, and the Rician fading parameter is 4 [31]. We set the size of the RIS ranging between , the number of antennas at the BS and the number of users between 5 15, the discreteness level of RIS between 1 5, and the SNR (defined as between 2 dB 10 dB. Specifically, for the LoS case, given the above parameters the designing rule in is only satisfied when the size of RIS is set as 40 or the antenna separation at the BS is set as 6.75 m.
We present Fig. 5 to verify Proposition 3 in the pure LoS case. The figure shows the sum rate of all users versus the antenna separation at the BS, , with different numbers of quantization bits in the pure LoS case. Given the parameters set above, according to Proposition 3, the optimal value of should be 6.75 m so as to orthogonalize the channel links between each antenna of the BS and a user via any RIS element. We observe that the optimal sum rate can be achieved when is around m. The numerically optimal value of approaches the theoretical result as the number of quantization bits, , grows, implying that such fluctuation around the optimal value comes from the discrete phase shifts of RIS. When is large enough, i.e., in Fig. 5, the optimal equals 6.75 m, which justifies Proposition 3.
Fig. 6 shows the sum rate of all users versus the size of RIS^{3}^{3}3For convenience, here we adopt to represent the size of RIS to better display the curves. with , . We observe that the sum rate grows rapidly with a small size of RIS and gradually flattens as the size of RIS continues to increase^{4}^{4}4We do not show the simulated annealing algorithm in this figure due to its high complexity with a large size of RIS.. The inflection point of each curve shows up around , which verifies the threshold given by Proposition 3 very well. When the RIS size exceeds 40, though Proposition 3 does not hold, the sum rate does not drop since the RIS can always turn off those extra RIS elements to maintain the sumrate performance. This figure also implies that though Proposition 3 is obtained in the pure LoS case, it also sheds insight into the RIS placement and array design in a more general case with smallscale fading.
Moreover, Fig. 6 also shows that the performance of RISbased beamforming with smallscale fading is much better than that of the pure LoS case when the size of RIS is small. Such a gain comes from the reduced correlation between different channel links owning to multipath effects. As the size of RIS grows, Proposition 3 is satisfied such that the channel links in the pure LoS case are orthogonalized, making the gap between these two cases smaller.
Fig. 7 depicts the sum rate of all users versus the number of quantization bits for discrete phase shifts in RIS configuration with SNR = 2 dB, , and . As the number of quantization bits increases, the sum rate obtained by our proposed algorithm with discrete phase shifts approaches that in the continuous case. When the size of RIS grows, the gap between the discrete and continuous cases shrinks since a larger RIS usually provides more freedom of generating directional beams. Note that the implementation difficulty increases dramatically in practice with the number of quantization bits. A tradeoff can then be achieved between the sum rate and number of quantization bits.
Fig. 8 and Fig. 8 show the sum rate of all users versus SNR and the number of users, respectively, obtained by different algorithms with a RIS of size (i.e., ), quantization bits for phase shifts, equal number of transmit antennas at the BS and the downlink users. In Fig. 8, the sum rate increases with SNR since more power resources are allocated by the BS. In Fig. 8, the sum rate grows with the number of users since a higher diversity gain is achieved. From both figures, we observe that the performance of our proposed algorithm is close to that of the simulated annealing method and much better than the random algorithm. This indicates the efficiency of our proposed algorithm to solve the RISbased HBF problem.
Vii Conclusions and Discussion
In this paper, we have studied a RISbased downlink multiuser multiantenna system in the absence of direct links between the BS and users. The BS transmits signals to users via the reflectionbased RIS with limited discrete phase shifts. To better depict the close coupling between channel propagation and the RIS configuration pattern selection, we haved considered a reflectiondominated onehop propagation model between the BS and users. Based on this model, we have carried out an HBF scheme for sum rate maximization where the continuous digital beamforming has been performed at the BS and the discrete analog beamforming has been achieved inherently at the RIS via configuration pattern selection. The sum rate maximization problem has been decomposed into two subproblems and solved iteratively by our proposed SRM algorithm.
Three remarks can be drawn from the theoretical analysis and numerical results, providing insights for RISbased system design.

The sum rate of the RISbased system with discrete phase shifts increases rapidly when the number of quantization bits is small, and gradually approaches the sum rate achieved in the continuous case if is large enough.

The sum rate increases with the size of RIS and converges to a stable value as the size of RIS grows to reach the threshold determined by Proposition 3.

The minimum number of transmit antennas at the BS required to achieve any fully digital beamforming scheme is only half of that in traditional HBF schemes, implying that the RISbased HBF scheme can greatly reduce the cost of dedicated hardware.
The above remarks have indicated that when designing the RISbased systems, a moderate size of RIS and a very small number of quantization bits are enough to achieve the satisfying sum rate at low cost.
Appendix A Proof of Proposition 1
Note that
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