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Hybrid Beamforming for Reconfigurable Intelligent Surface based Multi-user Communications: Achievable Rates with Limited Discrete Phase Shifts

Boya Di, Hongliang Zhang, Lingyang Song, Yonghui Li, Zhu Han, H. Vincent Poor

arXiv:1910.14328v1cs.ITcs.ETeess.SP

TL;DR

The paper addresses hybrid beamforming for RIS-assisted multi-user communications with discrete phase shifts. It proposes continuous digital and RIS-based analog beamforming for sum-rate maximization, finding that rate improves with RIS size and quantization resolution while hardware costs can be reduced.

  • Problem

    Hybrid beamforming for reflection-dominated RIS communications must accommodate the practical constraint of limited discrete phase shifts.

  • Method

    The paper formulates RIS-assisted sum-rate maximization using continuous digital beamforming at the BS, discrete analog beamforming at the RIS, and an iterative solution.

  • Results

    The sum rate rises rapidly with quantization bits, approaches the continuous-phase case, and increases with RIS size before converging to a stable value.

  • Takeaways & Limitations

    RIS-based hybrid beamforming can achieve strong sum-rate performance while greatly reducing the cost of dedicated hardware.

Abstract

from arXiv · show

Reconfigurable intelligent surface (RIS) has drawn considerable attention from the research society recently, which creates favorable propagation conditions by controlling the phase shifts of the reflected waves at the surface, thereby enhancing wireless transmissions. In this paper, we study a downlink multi-user system where the transmission from a multi-antenna base station (BS) to various users is achieved by the RIS reflecting the incident signals of the BS towards the users. Unlike most existing works, we consider the practical case where only a limited number of discrete phase shifts can be realized by the finite-sized RIS. Based on the reflection-dominated one-hop propagation model between the BS and users via the RIS, a hybrid beamforming scheme is proposed and the sum-rate maximization problem is formulated. Specifically, the continuous digital beamforming and discrete RIS-based analog beamforming are performed at the BS and the RIS, respectively, and an iterative algorithm is designed to solve this problem. Both theoretical analysis and numerical validations show that the RIS-based system can achieve a good sum-rate performance by setting a reasonable size of RIS and a small number of discrete phase shifts.

I. INTRODUCTION

The paper addresses RIS-assisted downlink multi-user communication with limited discrete phase shifts by developing a hybrid beamforming scheme that jointly exploits BS digital and RIS analog beamforming. It formulates sum-rate maximization and analyzes how RIS size and phase-shift resolution affect performance.

  • Motivation and RIS background: RIS shapes the propagation environment by controlling multiple scatterers, directing reflected signals toward users without extra power sources or hardware.Limited discrete phase shifts arise from the RIS elements’ programmable PIN-diode operation.
  • System setting: The considered system uses a reflection-dominated one-hop path from a multi-antenna BS through a limited-resolution RIS to users whose direct links suffer deep shadowing.The resulting coupling between propagation paths and RIS phase shifts distinguishes this model from traditional relays and direct-link MIMO systems.
  • Proposed approach: The proposed hybrid beamforming scheme performs continuous digital beamforming at the BS and discrete RIS-based analog beamforming at the RIS.The design must address coupling between propagation and analog beamforming, mixed-integer optimization, and possible rate degradation from dense-element correlation.
  • Optimization algorithm: The mixed-integer sum-rate maximization is decomposed into digital and RIS analog subproblems solved iteratively using ZF beamforming with power allocation and outer approximation.The resulting solution is sub-optimal according to the paper’s organization of the algorithmic development.
  • Theoretical and numerical analysis: The analysis proves that the RIS-based scheme can save as much as half of the RF chains compared with traditional hybrid beamforming schemes.The paper also studies the effects of RIS size and the number of discrete phase shifts on sum rate theoretically and numerically, including the pure LoS case.

II. SYSTEM MODEL · A. Scenario Description

The paper models a RIS-based downlink multi-user system in which a multi-antenna BS serves single-antenna users through a configurable RIS. The RIS is introduced to shape the propagation environment and mitigate unstable or outage-prone direct BS–user links.

  • II. SYSTEM MODEL: The system comprises a BS with multiple antennas serving various single-antenna users via an RIS.
  • II. SYSTEM MODEL: The propagation environment can be pre-designed and configured to optimize system performance.
  • II. SYSTEM MODEL: The system model constructs discrete RIS phase shifts and channel models for the considered communication scenario.
  • A. Scenario Description: The BS has N_t antennas and transmits to K single-antenna users in a downlink multi-user communication system.
  • A. Scenario Description: Unexpected fading and potential obstacles can make the direct BS–user link unstable or cause outage.
  • A. Scenario Description: An RIS deployed between the BS and users reflects BS signals and directly projects them toward users by shaping the propagation environment.
  • A. Scenario Description: The RIS contains N_R × N_R electrically controlled sub-wavelength elements whose electromagnetic responses are manipulated by a controller using PIN diodes.
  • A. Scenario Description: The RIS requires no extra active power source or signal-processing capability and functions as a low-cost reconfigurable phased array.

B. Reconfigurable Intelligent Surface with Limited Discrete Phase Shifts · C. Reflection-dominated Channel Model

The RIS uses b-bit controllable elements to realize 2^b discrete phase configurations constrained by Lorentzian resonance and frequency-selective propagation. The channel model treats the passively reflected BS–RIS–user path as dominant, represents it with a Ricean model, and derives geometry-based propagation for planar RIS and linear BS arrays.

  • B. Reconfigurable Intelligent Surface with Limited Discrete Phase Shifts: The 2D RIS is a planar array of radiative elements whose internal propagation depends on element location and frequency-dependent wave number.Within the considered narrow band, the wave number is assumed identical across RIS elements.
  • B. Reconfigurable Intelligent Surface with Limited Discrete Phase Shifts: RIS element phase and amplitude are controlled through tunable polarizability, while practical implementation samples phase values from a finite feasible set.Multiple PIN diodes provide low-cost voltage-controlled manipulation of discrete phase values.
  • B. Reconfigurable Intelligent Surface with Limited Discrete Phase Shifts: Each RIS element is encoded to conduct 2^b possible phase shifts, with b denoting the number of quantization bits.The configurations are defined according to the Lorentzian resonance response.
  • C. Reflection-dominated Channel Model: The channel is modeled through a one-hop reflection ray because closely spaced RIS elements couple their scattered waves into a passively reflected path.This differs from relay-style two-hop propagation, where BS–RIS and RIS–user paths are treated independently.
  • C. Reflection-dominated Channel Model: Directional RIS reflections make the BS–RIS–user link stronger than other multipaths and the degraded direct link, motivating its treatment as the dominant component.The model therefore represents the BS–RIS–user link as LoS and all other paths as NLOS.
  • C. Reflection-dominated Channel Model: For geometry-based propagation, the RIS uses a uniform planar array and the BS uses a uniform linear array, with spherical coordinates defining element positions and distances.The formulation uses array orientations, inter-element separations, and user coordinates to derive BS–RIS and RIS–user distances.
  • C. Reflection-dominated Channel Model: Because array-element spacing is much smaller than the BS–user distance, the model assumes identical path loss for each BS–user link across antennas and RIS elements.The resulting propagation expressions use the derived distances while ignoring path-loss differences caused by individual antenna or RIS-element locations.

III. RIS-BASED HYBRID BEAMFORMING AND PROBLEM FORMULATION FOR MULTI-USER COMMUNICATIONS

This section presents a RIS-based hybrid beamforming scheme for multi-user communications with discrete phase shifts. Digital beamforming is performed at the BS, analog beamforming at the RIS, and the resulting sum-rate maximization is decomposed into digital beamforming and RIS configuration subproblems.

  • RIS-Based Hybrid Beamforming: RIS elements realize analog beamforming through RIS configuration but lack digital processing capacity.Signal processing is therefore carried out at the BS.
  • RIS-Based Hybrid Beamforming: The proposed HBF scheme uses RIS configuration to realize reflected waves toward preferable directions.The scheme is developed for RIS-based multi-user communications under the specified phase-shift and channel models.
  • RIS-Based Hybrid Beamforming: Digital beamforming is performed at the BS, while the RIS achieves analog beamforming using discrete phase shifts.This division assigns digital processing to the BS and discrete analog control to the RIS.
  • Problem Formulation: The HBF design formulates a sum-rate maximization problem and decomposes it into digital beamforming and RIS configuration-based analog beamforming subproblems.The decomposition separates optimization of the BS digital beamformer from RIS configuration.

A. Hybrid Beamforming Scheme … 3) Received Signal at the User:

The hybrid beamforming scheme combines digital beamforming at the BS with RIS-based analog beamforming, yielding a received-signal model for K users that includes accumulated RIS reflections and additive noise.

  • 1) Digital Beamforming at the BS:: The BS encodes K different data streams with a digital beamformer V_D of size N_t × K, requiring N_t ≥ K.The encoded signals are up-converted, power-allocated, and transmitted through N_t antennas.
  • 1) Digital Beamforming at the BS:: The BS represents the intended signals for K users by the vector s ∈ C^K×1 before transmission.The transmitted-signal expression follows from digital encoding, carrier-frequency up-conversion, and power allocation.
  • 2) RIS Configuration based Analog Beamforming:: After reflection-dominated propagation, user k receives a signal formed from RIS-element contributions and the digitally beamformed transmissions.The received expression includes the RIS reflection coefficient associated with element (l_1,l_2), antenna n, and user k.
  • 2) RIS Configuration based Analog Beamforming:: The RIS reflection coefficient depends in practice on the incidence and reflection angles, while the phase-shift treatment assumes φ^(k,n).The angle dependence is stated as a practical property of the RIS reflection coefficient.
  • 2) RIS Configuration based Analog Beamforming:: Ignoring coupling between RIS elements, each user’s received signal is modeled as the accumulated radiation from all RIS elements.The paper identifies this as a common assumption for metasurfaces and traditional antenna arrays.
  • 3) Received Signal at the User:: Each user down-converts the received signal z_k to baseband and recovers the final signal.This operation completes the per-user receiver processing in the transmission model.
  • 3) Received Signal at the User:: The K-user transmission model includes a noise vector w = [w_1, · · ·, w_K]^T and a transmission matrix F.The matrix F is defined through H_l1,l2 and Φ, both K × N_t matrices, combined by element-by-element multiplication.

B. Sum Rate Maximization Problem Formulation … A. Digital Beamforming Algorithm

The paper formulates sum-rate maximization over the BS digital beamformer and discrete RIS configuration, then decomposes the mixed-integer non-convex problem into digital beamforming and RIS analog-beamforming subproblems. The digital subproblem is addressed using zero-forcing beamforming with power allocation obtained by water-filling.

  • B. Sum Rate Maximization Problem Formulation: User-k achievable rates are evaluated from the matrix-form received signal to characterize sum-rate performance in the RIS-based system.The formulation rewrites each user’s received signal and derives its achievable rate.
  • B. Sum Rate Maximization Problem Formulation: The optimization jointly selects the digital beamformer VD and RIS configuration {ql1,l2} to maximize users’ achievable rates under the BS total transmit power PT.The problem optimizes both digital transmission and discrete RIS phase configuration.
  • C. Problem Decomposition: Problem (16) is a mixed integer non-convex optimization with many discrete RIS variables and coupling between propagation and RIS analog beamforming.These properties make direct optimization challenging.
  • C. Problem Decomposition: The formulation is decomposed into a digital beamforming subproblem with fixed RIS configuration and a RIS configuration subproblem with fixed digital beamformer VD.The two subproblems separately optimize the BS and RIS components.
  • IV. SUM RATE MAXIMIZATION ALGORITHM DESIGN: The SRM algorithm iteratively solves the digital and RIS subproblems, producing a suboptimal solution to problem (16).The overall method also includes convergence and complexity analysis.
  • A. Digital Beamforming Algorithm: For the digital subproblem, zero-forcing beamforming with power allocation is used to alleviate interference among users and obtain a near optimal solution.The ZF beamformer is constructed from the effective transmission matrix F.
  • A. Digital Beamforming Algorithm: The digital beamforming procedure solves the power allocation problem, obtains its optimal allocation, and derives the beamformer matrix from that allocation.The power allocation problem’s optimal solution is obtained by water-filling, with normalized factor µ satisfying the transmit-power condition.

B. RIS Configuration based Analog Beamforming Algorithm

The RIS analog beamforming subproblem is reformulated as a mix-integer semidefinite program and solved discretely using linearization and outer approximation. This approach avoids performance degradation from rounding discrete phase variables and is summarized in Algorithm 2.

  • Problem reformulation: Assuming ZF precoding, RIS configuration optimization is reformulated as a power-constrained problem because the data rate depends on RIS configuration through the power constraint.The reformulation follows from alternating digital beamforming and RIS configuration optimization.
  • Problem reformulation: The reformulated problem becomes a semidefinite program by exploiting the symmetric positive semidefinite structure of ˜F ˜F^H and applying the Schur complement.Here, ˜F = P −1 2F .
  • Discrete optimization: Because the formulation is a coupled mix-integer SDP and generally NP-hard, continuous relaxation followed by rounding can degrade performance when quantization uses only 2 or 3 bits.The discrete variables are coupled through constraint (25b), making the problem more complicated.
  • Discrete optimization: The algorithm instead solves the SDP discretely by linearizing nonlinear phase functions with binary vectors for phase shifts and phase differences.Proposition 1 provides the transformation, after which outer approximation is applied.
  • Outer approximation: Outer approximation enforces the SDP constraint through linear cuts, converts the problem into a mix-integer linear program, and solves it with branch-and-bound.When a candidate matrix is not positive semidefinite, an eigenvector associated with its smallest eigenvalue generates a valid cut.

C. Overall Algorithm Description … 2) Complexity:

The SRM algorithm alternates digital beamformer optimization with RIS-based analog beamforming until the objective difference falls below a threshold. Its objective is non-decreasing and bounded, guaranteeing convergence, while subproblem complexity is characterized separately.

  • C. Overall Algorithm Description: The SRM algorithm iteratively solves the original problem by alternating beamformer V_D optimization with fixed RIS configuration and RIS configuration optimization.Algorithm 1 solves the beamformer, while Algorithm 2 optimizes the RIS configuration; each iteration uses the obtained results to initialize subsequent iterations.
  • C. Overall Algorithm Description: The RIS configuration algorithm removes the semi-definite constraint, obtains an initial solution, and applies branch-and-bound to solve the resulting problem.It checks feasibility, adds a cut when necessary, repeats until feasibility, and then derives the phase shifts.
  • C. Overall Algorithm Description: The alternating iterations stop when the objective-function difference between adjacent iterations satisfies R(t+1) −R(t) ≤π.The two subproblems are solved alternately until the predefined threshold π is reached.
  • 1) Convergence:: In the digital beamforming subproblem, Algorithm 1 obtains a better result for the given RIS configuration at each iteration.This improvement supports the convergence analysis of the alternating procedure.
  • 1) Convergence:: The RIS optimization maximizes the sum rate for the updated beamforming result, making the original objective value non-decreasing after each SRM iteration.The convergence argument combines the improvements from both alternating subproblems.
  • 1) Convergence:: Because the objective value is upper bounded, the proposed SRM algorithm is guaranteed to converge.This conclusion follows from the objective being non-decreasing across iterations and upper bounded.
  • 2) Complexity:: O(K) is the computational complexity of the digital beamforming subproblem, which optimizes received power separately for each user.The per-user optimization follows equation (22).
  • 2) Complexity:: O(2bN2 is the stated complexity scale for each linear program in the RIS configuration subproblem solved by branch-and-bound.The bound follows because only one element in x_l1,l2 can equal 1, giving at most 2b possible solutions.

V. PERFORMANCE ANALYSIS OF RIS-BASED MULTI-USER COMMUNICATIONS · A. Comparison with Traditional Hybrid Beamforming · B. Special Case: Pure Line-of-Sight Transmissions

The RIS-based hybrid beamforming scheme is compared with traditional hybrid beamforming through RF-chain requirements, while the pure LoS case analyzes how RIS size, placement, and discrete phase shifts affect orthogonal links and achievable rate.

  • A. Comparison with Traditional Hybrid Beamforming: The RIS configuration is coupled with propagation, providing greater freedom to shape the propagation environment than traditional analog beamforming.This characteristic motivates a distinct condition for RIS-based HBF to achieve fully digital beamforming.
  • A. Comparison with Traditional Hybrid Beamforming: Nt ≥K is required, meaning the number of BS transmit antennas must be no smaller than the number of single-antenna users.This is the second condition identified for RIS-based HBF to achieve fully digital beamforming.
  • A. Comparison with Traditional Hybrid Beamforming: For RIS-based HBF to achieve fully digital beamforming, the RIS size must be no smaller than the product of the number of users and the BS antenna-array size.The result is stated as one of two required conditions for matching fully digital beamforming.
  • A. Comparison with Traditional Hybrid Beamforming: The RIS-based HBF requires half as many RF chains as the traditional scheme to achieve fully digital beamforming and can save phase shifters through the RIS.The RIS inherently realizes analog beamforming because of its flexible physical implementation.
  • B. Special Case: Pure Line-of-Sight Transmissions: The pure LoS data rate is treated as a lower bound of achievable rate, and RIS placement is analyzed to provide orthogonal communication links.The pure LoS case is used to study how RIS design influences achievable data rate.
  • B. Special Case: Pure Line-of-Sight Transmissions: Sub-wavelength RIS element spacing causes spatial correlation and a low-rank channel matrix, degrading achievable data rate.Unlike multipath decorrelation, the RIS-based one-hop LoS component is typically stronger than other multipaths and degraded direct links.
  • B. Special Case: Pure Line-of-Sight Transmissions: The pure LoS design targets a high-rank channel by selecting RIS size and BS antenna-array design while accounting for discrete phase shifts.Proposition 3 gives conditions for making different BS-to-user links through the RIS orthogonal.
  • B. Special Case: Pure Line-of-Sight Transmissions: Achievable data rate is highly related to RIS size and placement, with a threshold defined for RIS size when other parameters are fixed.The pure LoS sum-rate problem adds one constraint and can be solved using the proposed SRM algorithm after linearization.

VI. SIMULATION RESULTS

Simulations evaluate the RIS-based hybrid beamforming algorithm under varying SNR, user count, RIS size, and discrete phase-shift resolution. Results show near-continuous-phase performance with sufficient quantization, strong RIS-size effects, and performance close to simulated annealing while exceeding random phase shifts.

  • Antenna separation and pure LoS: In the pure LoS case, the optimal BS antenna separation is around 5–6.75 m and converges to 6.75 m when b = 8.The fluctuation is attributed to discrete RIS phase shifts, and b = 8 achieves the theoretical optimum.
  • RIS size: The sum rate grows rapidly with RIS size before flattening, with an inflection point around NR = 40.When NR exceeds 40, extra RIS elements can be turned off to maintain sum-rate performance.
  • RIS size: For small RIS sizes, small-scale fading outperforms pure LoS because multipath reduces channel-link correlation; the gap narrows as RIS size increases.Larger RISs orthogonalize the pure-LoS channel links, reducing the difference between the two cases.
  • Quantization resolution: As quantization bits increase, the proposed discrete-phase algorithm approaches continuous-phase performance, while larger RISs further shrink the gap.Implementation difficulty increases dramatically with quantization bits, creating a trade-off between sum rate and phase-shift resolution.
  • SNR and user count: The sum rate increases with SNR and the number of users, attributed respectively to greater BS power resources and higher diversity gain.These experiments use NR = 6, b = 2, and equal numbers of transmit antennas and users.
  • Algorithm comparison: Across SNR and user-count experiments, the proposed algorithm performs close to simulated annealing and much better than the random algorithm.This supports the proposed algorithm’s efficiency for solving the RIS-based HBF problem.

VII. CONCLUSIONS AND DISCUSSION

The paper develops an RIS-based hybrid beamforming scheme for downlink multi-user transmission without direct BS–user links, using continuous digital beamforming at the BS and discrete analog beamforming at the RIS. Analysis and simulations indicate that moderate RIS size and few quantization bits can provide satisfying sum rates at low cost.

  • System model: The RIS-based system uses a reflection-dominated one-hop propagation model with limited discrete phase shifts and no direct BS–user links.Signals are transmitted from the BS to users through the reflection-based RIS.
  • Proposed scheme: The proposed HBF scheme performs continuous digital beamforming at the BS and discrete analog beamforming at the RIS through configuration pattern selection.The sum-rate maximization problem is decomposed into two subproblems and solved iteratively.
  • Design insights: The sum rate increases rapidly with small quantization-bit counts and gradually approaches the continuous-phase result when b is sufficiently large.This establishes that only a small number of discrete phase shifts may be needed for strong performance.
  • Design insights: The sum rate increases with RIS size and converges to a stable value once the RIS reaches the threshold determined by Proposition 3.Increasing the RIS beyond that threshold does not continue producing the same rate growth.
  • Design insights: The minimum BS transmit-antenna count required for any fully digital beamforming scheme is half that of traditional HBF schemes, reducing dedicated-hardware cost.The conclusion recommends a moderate RIS size and very small quantization-bit count for satisfying sum rate at low cost.

APPENDIX A · APPENDIX B · APPENDIX C

The appendices establish linear transformations for discrete phase-shift variables, characterize when RIS-based hybrid beamforming can realize fully digital beamforming, and derive the condition for orthogonal links from one user’s BS antennas.

  • APPENDIX A: Discrete phase-shift properties transform the nonlinear functions involving θ_l1,l2 into linear ones.The construction uses variables x_l1,l2 and y_l1,l2,l1′,l2′, with the former satisfying ∥x_l1,l2∥1 = 1.
  • APPENDIX A: After these transformations, ˜F ˜F H is linear with respect to x_l1,l2 and y_l1,l2,l1′,l2′.The sine and cosine phase-difference terms are represented through y_l1,l2,l1′,l2′.
  • APPENDIX B: When N_t = K, any fully digital beamforming scheme can be achieved by the RIS-based HBF scheme if N_R^2 ≥ K N_t.The same sufficient construction extends to N_t > K by setting extra variables to zero.
  • APPENDIX B: When N_t < K, RIS-based HBF cannot implement fully digital beamforming, making N_t ≥ 2K a sufficient condition.The proof compares rank(H_FD V_FD) = K with rank(F V_D) = min {K, N_t}.
  • APPENDIX C: Appendix C orthogonalizes different BS-to-RIS links for the same user while accounting for user-dependent path losses and equal path loss across links to that user.The links are indexed by distinct transmit antennas n_a and n_b.
  • APPENDIX C: The orthogonality requirement is imposed for every pair of distinct transmit antennas, with n_b − n_a ∈ Z.Substitution into the channel expressions yields the final condition through the principle of geometric sums.
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