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Power Scaling Law Analysis and Phase Shift Optimization of RIS-aided Massive MIMO Systems with Statistical CSI

Kangda Zhi, Cunhua Pan, Hong Ren, Kezhi Wang

arXiv:2010.13525v5eess.SP

TL;DR

The paper studies uplink RIS-aided massive MIMO with Rician channels when RIS phase shifts rely on statistical CSI, addressing an underinvestigated setting relevant to coverage extension. It derives finite-antenna rate expressions and scaling laws, then optimizes phase shifts for sum rate and minimum user rate. The analysis and simulations support RIS integration, including large-size low-resolution RIS.

  • Problem

    Statistical-CSI design for RIS-aided massive MIMO systems under the Rician channel model had not been investigated, despite RIS-related channels potentially containing line-of-sight components.

  • Method

    The paper derives closed-form uplink achievable-rate expressions and optimizes RIS phase shifts using statistical CSI for RIS-aided massive MIMO with Rician channels.

  • Results

    The analysis reveals power-scaling laws and a trade-off between achievable spatial multiplexing gain and unwanted path loss, while simulations validate RIS integration into massive MIMO systems.

  • Takeaways & Limitations

    The results support using RIS to enhance coverage in massive MIMO systems and indicate the feasibility of large-size RIS with low-resolution phase control.

Abstract

from arXiv · show

This paper considers an uplink reconfigurable intelligent surface (RIS)-aided massive multiple-input multiple-output (MIMO) system with statistical channel state information (CSI). The RIS is deployed to help conventional massive MIMO networks serve the users in the dead zone. We consider the Rician channel model and exploit the long-time statistical CSI to design the phase shifts of the RIS, while the maximum ratio combination (MRC) technique is applied for the active beamforming at the base station (BS) relying on the instantaneous CSI. Firstly, we reveal the power scaling laws and derive the closed-form expressions for the uplink achievable rate which holds for arbitrary numbers of base station (BS) antennas. Based on the theoretical expressions, we discuss the rate performance under some special cases and provide the average asymptotic rate when using random phase shifts. Then, we consider the sum-rate maximization and the minimum user rate maximization problems by optimizing the phase shifts at the RIS. However, these two optimization problems are challenging to solve due to the complicated data rate expression. To solve these problems, we propose a novel genetic algorithm (GA) with low complexity but can achieve considerable performance. Finally, extensive simulations are provided to validate the benefits by integrating RIS into conventional massive MIMO systems. Besides, our simulations demonstrate the feasibility of deploying large-size but low-resolution RIS in massive MIMO systems.

I. INTRODUCTION

RIS-aided massive MIMO is positioned as a way to extend coverage to users that cannot directly communicate with the BS, while avoiding the cost and energy burden of equipping BSs with hundreds of antennas. This paper addresses statistical-CSI phase-shift design for Rician RIS-aided massive MIMO and derives rate-scaling results while evaluating optimization strategies.

  • Motivation: Conventional massive MIMO can provide high throughput, but may require hundreds of BS antennas, increasing cost and energy consumption.Dense deployment of small BSs is also described as costly.
  • Motivation: RIS can assist users that cannot directly communicate with the BS and help provide coverage in blocked environments.The RIS is described as an efficient and cost-effective solution for conventional massive MIMO blockage problems.
  • Motivation: Prior RIS studies predominantly designed phase shifts using instantaneous CSI, which suits fixed or low-mobility scenarios with channel estimation and coherence-time updates.The paper contrasts this with high-mobility settings having short channel coherence times.
  • Motivation: Statistical CSI is more practical for high mobility and can reduce RIS feedback, controller power, and BS computational complexity because updates occur on a longer timescale.The phase-shift matrix is updated only when statistical CSI changes, rather than with instantaneous CSI.
  • Research gap: Statistical-CSI RIS-aided massive MIMO under the Rician channel model had not been investigated, although RIS-related channels may contain line-of-sight components.A cited prior work considered correlated Rayleigh channels and set RIS phase shifts as an identity matrix.
  • Contributions: The paper derives a closed-form uplink achievable-rate expression for finite BS antenna numbers, reveals scaling laws, evaluates random phase shifts, and studies phase-shift optimization.The reported results include a trade-off between spatial multiplexing gain and path loss and support large-size, low-resolution RIS deployment.

II. SYSTEM MODEL

The system models an uplink RIS-aided massive MIMO network in which blocked direct links are assisted by an RIS, with Rician cascaded channels and MRC at the BS. Long-term geometric information determines the line-of-sight components and RIS phase shifts, while users transmit through the cascaded RIS-BS channel.

  • System setting: The network contains a BS with M antennas, an RIS with N reflecting elements, and K single-antenna users.
  • System setting: Blocked direct BS-user links motivate deploying an RIS on a building to assist communication in dead zones.The RIS is positioned to create propagation paths when buildings, humans, or trees obstruct ground links.
  • Channel model: The user-RIS and RIS-BS channels follow Rician models combining LoS components with i.i.d. complex Gaussian NLoS components.The model includes path-loss parameters and Rician factors for both links.
  • Channel model: The LoS channel components depend on azimuth and elevation angles that remain invariant over the considered period and can be obtained from known locations.Uniform square planar arrays are assumed at the BS and RIS.
  • Signal processing: The RIS applies a diagonal reflection matrix with phase shifts θ_n, and MRC uses instantaneous CSI for BS-side active combining.The cascaded channel column for user k is represented by H2Φh_k.

III. UPLINK ACHIEVABLE RATE ANALYSIS

The analysis derives closed-form uplink achievable rates and power-scaling laws for arbitrary BS antenna counts, then examines how RIS phase alignment, propagation conditions, interference, and phase resolution affect scaling.

  • General rate analysis: Closed-form achievable-rate expressions capture the effects of BS antennas, RIS elements, transmit power, and Rician factors.The expressions apply to the uplink multi-user system with arbitrary antenna counts.
  • Scaling with RIS phases: Phase alignment to user k makes |f_k(Φ)| grow without bound as N increases, while other users’ corresponding terms remain bounded except for nearly identical angles.The analysis excludes the rare case of nearly matching azimuth and elevation AoAs.
  • Scaling with RIS phases: When RIS phases align to user k, the relevant signal term scales as O(N^3), whereas other principal terms can scale as O(N^4).
  • Multi-user interference: RIS-aided MRC retains interference in LoS settings because users share the RIS-BS channel, unlike conventional massive MIMO where the interference term can vanish.Proper phase-shift design can compensate for the resulting rate degradation by increasing desired signal power and mitigating interference.
  • Power scaling: With pk = Eu/M, users can reduce transmit power by 1/M while their rates converge to a non-zero value as M →∞.This massive-MIMO power scaling parallels the conventional system in the BS-antenna dimension.
  • Power scaling: With aligned RIS phases and large M and N, user k can reduce transmit power to Eu/(MN^2) while retaining a non-zero rate.The rate improves when the propagation environment has fewer scatters and larger Rician factors.
  • Phase-shift effects: Random RIS phases leave the sum achievable rate bounded for large M and N, whereas discrete b-bit phase shifts still provide O(log2(N)) sum-rate gain.These results support optimizing RIS phases and deploying large low-resolution RISs.
  • Special propagation cases: In uncorrelated Rayleigh environments, RIS phases can be set arbitrarily, while RIS deployment with many elements still yields significant performance gain.

IV. PHASE SHIFTS DESIGN

The paper formulates RIS phase-shift optimization using long-term statistical CSI for both sum-rate and minimum-user-rate objectives. Both continuous and b-bit discrete phase shifts are considered to reduce training and update overhead.

  • Design framework: Long-term statistical CSI is used to design RIS phase shifts, reducing training overhead and phase-shift update frequency.
  • Phase constraints: The formulations include continuous phase shifts and discrete phase shifts with b-bit precision.The discrete feasible values are quantized over the interval [0, 2π).
  • Optimization objectives: Two optimization problems target sum user rate and minimum user rate, respectively representing system capacity and fairness objectives.

A. Special Cases

The paper identifies special cases where RIS phase-shift design is optimal or unnecessary, then uses a genetic algorithm to address the general coupled optimization problems.

  • Special cases: When N = 1, any phase shift satisfying the optimization constraints is optimal for both problems.The resulting phase-dependent terms have equal magnitude for every θ1.
  • Special cases: When δ = 0 or εk = 0 for all users, any feasible RIS phase shifts are optimal for both problems.All terms related to Φ become zero in this case.
  • Special cases: Rich scattering between the BS, RIS, or users removes the need to design RIS phase shifts based on statistical CSI.This follows from the phase-dependent terms vanishing when the corresponding Rician factors are zero.
  • Special cases: If only user k transmits, aligning the RIS phase shifts to user k is optimal for sum-rate maximization.A user located very close to the RIS can make this alignment nearly optimal even in the multi-user setting.
  • General optimization: In the general case, complicated rates and coupled active-passive beamforming make global optimization difficult, so the paper applies a GA-based method.The GA evolves populations whose chromosomes represent RIS phase shifts for continuous or discrete configurations.
  • GA procedure: The GA evaluates objective-function fitness, preserves elite individuals, selects parents by stochastic universal sampling, and creates new generations through crossover and mutation.Scaled fitness determines selection probability; the remaining individuals undergo mutation, and the process stops at a generation limit or small fitness change.

4) Crossover:

The paper specifies crossover and mutation operations for the GA, then evaluates phase-shift optimization and system behavior across channel and RIS conditions. Simulations verify the analytical expressions and expose trade-offs involving scattering, path loss, fairness, and multiplexing.

  • 4) Crossover:: For N > 2, the GA uses two-point crossover; otherwise, it uses single-point crossover to recombine parent chromosomes.Crossover generates Nc offspring from selected parents.
  • Mutation: Mutation applies with probability pm and uses uniform random phase values for continuous RIS shifts or randomly selected discrete values.Mutation increases population diversity and can generate offspring with better fitness.
  • Simulation setup: The simulations average results over 10000 random channel generations using a GA population of Nt = 200, Ne = 10, Nc = 152, and Nm = 38.The baseline setup uses N = 16 RIS elements and M = 64 BS antennas unless otherwise stated.
  • Numerical validation: The derived desired-signal and interference expressions perfectly match Monte Carlo simulations under random RIS phase shifts.This verifies the accuracy of the analytical results for the evaluated setting.
  • A. Trade-off between path-loss and spatial multiplexing: As the RIS-BS Rician factor δ increases, rate performance degrades toward random-phase performance because stronger LoS components increase inter-user interference and reduce spatial multiplexing.When δ → ∞, the cascaded channel rank approaches one, preventing support for multiple-user communication.
  • A. Trade-off between path-loss and spatial multiplexing: As βRB increases, optimized sum-rate and minimum-rate performance decreases toward random-phase performance, revealing a trade-off between spatial multiplexing gain and path loss.At small βRB, max-sum and max-min optimization achieve similarly good performance; high throughput and fairness are harder to balance as δ increases.
  • A. Trade-off between path-loss and spatial multiplexing: The cascaded-channel condition number decreases quickly as the number of RIS elements N increases.The figure reports the average condition number of G versus N.

B. The interplay between RIS and massive MIMO

RIS passive beamforming improves rate, fairness, power efficiency, and channel conditioning in massive MIMO, while increasing RIS size can offset hardware and antenna requirements. Low-resolution phase shifts cause marginal degradation that does not grow with system size.

  • Channel conditioning: The cascaded-channel condition number decreases quickly as the number of RIS elements N increases.Lower condition numbers are associated with better high-SNR performance.
  • Channel conditioning: RIS optimization makes the channel nearly well-conditioned, reducing disparity among singular values and increasing high-SNR capacity.The result indicates that RIS can reshape the massive-MIMO channel.
  • Rate and hardware trade-offs: Increasing N significantly improves data rate, while MRC sum rate and minimum user rate approach saturation as M →∞ because of inter-user interference.Conventional massive MIMO may require extremely large antenna arrays to serve excessive users.
  • Rate and hardware trade-offs: 100 antennas with 64 RIS elements can outperform 400 antennas with 16 RIS elements.The comparison illustrates the passive beamforming gain of RIS and supports moderate BS-array sizes.
  • Multiuser support: The minimum user rate decreases as K increases, whereas sum rate increases; more RIS elements and optimized phase shifts can substantially improve the minimum rate.The users are placed on the same circle, with randomly generated angles for six users.
  • Power scaling: Carefully designed RIS phase shifts can further reduce user transmit power, and larger RIS size positively affects power consumption.The power-scaling experiment uses p_k = 100/M for every user.
  • Phase-shift resolution: Low-resolution reflecting elements cause marginal rate degradation that can be compensated by increasing N and does not enlarge with BS antenna count.Random continuous and discrete phase shifts have the same rate performance in the reported comparison.
  • Contributions and conclusion: The study derives finite-antenna rate expressions, analyzes power scaling and random phases, and uses a GA for sum-rate and minimum-user-rate maximization.The design uses statistical CSI for RIS phase shifts, reducing implementation complexity and signaling overhead.

APPENDIX A

Appendix A derives intermediate expectations for the Rician cascaded channel by expanding moments, removing zero-expectation terms, and exploiting independence and Gaussian-moment properties.

  • Moment derivation: The derivation rewrites cascaded-channel entries under the Rician model and evaluates required expectations term by term.Zero terms are removed during binomial expansions and simplification.
  • Gaussian moments: Independent Gaussian components and their raw moments provide the identities needed for fourth-order expectation calculations.The appendix uses E{s^4} = E{t^4} = 3/4 and E{s^2} = E{t^2} = 1/2.
  • Proof assembly: The appendix decomposes higher-order matrix and vector expectations into indexed terms, applies independence properties, and substitutes the resulting identities into Lemma 1.The derivation includes deterministic-matrix identities and entrywise notation.
  • Cross-user dependence: Because users share the RIS-BS channel H2, cascaded channels gk and gi are no longer independent as in systems without RIS.This shared-channel dependence is incorporated into the cross-user expectation calculations.

APPENDIX B

Appendix B examines asymptotic rate behavior when RIS and BS dimensions grow, identifying power scalings that preserve one user’s rate while potentially sacrificing others.

  • Asymptotic alignment: Aligning RIS phase shifts to user k gives f_k(Φ) = N, while |f_i(Φ)| remains bounded as N →∞ for other users.The resulting asymptotic orders distinguish the aligned user from non-aligned users.
  • Power scaling: User k can maintain a non-zero rate as M and N grow when its power scales as p_k = E_u/(MN^2).Other users use p_i = E_u/(MN), but their rates become zero under the stated asymptotic conditions.
  • Power scaling: The dominant terms in the aligned-user rate expression have order MN^2.The appendix identifies the corresponding coefficient involving the Rician and channel parameters.
  • Parameter dependence: The achievable-rate expression increases with α_k, β, and δ under the stated special case.These monotonicity properties characterize parameter effects in the asymptotic expression.

APPENDIX C

Appendix C derives asymptotic average rates for random continuous and finite-resolution discrete RIS phase shifts using trigonometric identities, independence, and bounded cross terms.

  • Random phase statistics: Independent uniformly distributed phase shifts make the relevant first-order complex expectation zero.This property is used when replacing phase-dependent rate terms by their expectations.
  • Random phase statistics: For b > 1, continuous and discrete random phase shifts have identical expectations for the relevant first- and second-order cosine terms.The derivation therefore treats both phase-shift cases together.
  • Asymptotic rate: The phase-dependent cross term is bounded as N →∞, while the dominant asymptotic terms have order O(M^2N^2).The appendix substitutes these orders into the achievable-rate expression to obtain the random-phase asymptotics.
  • Asymptotic rate: The asymptotic expression includes a coefficient depending on β, α_i, α_k, δ, ε_i, and ε_k.The reported coefficient contains the factor β^2α_iα_k(δ + 1)^2(ε_i + 1)(ε_k + 1).
  • Parameter dependence: The resulting asymptotic rate expression is a decreasing function of the Rician parameter δ.This monotonicity follows from the derived expression under the random-phase setting.

APPENDIX D

The appendix derives rate expressions for Rician-channel cases, including a conventional uplink non-RIS massive MIMO system with a ULA and MRC detection.

  • The derivation selects non-zero terms as all Rician factors grow to infinity and completes the rate expression in (29).
  • The conventional comparison system contains one BS and K users with deterministic line-of-sight channels from users to the BS.
  • A uniform linear array represents the BS channel, with ϑ_k denoting the angle of arrival from user k.
  • Using maximum ratio combining, the appendix gives the rate of user k and incorporates the ULA structure in the analysis.
  • The resulting analysis includes inter-user interference terms in the rate derivation.

APPENDIX E

The appendix examines discrete RIS phase shifts and phase alignment, showing that low-resolution control can preserve logarithmic rate scaling while phase noise affects aligned-user performance.

  • With b-bit precision, RIS phase shifts are restricted to a discrete set of adjustable values.
  • Quantization error is defined relative to the optimal phase shifts designed under the continuous-phase assumption.
  • Aligning RIS phase shifts to an arbitrary user k provides a simple sub-optimal solution for sum-rate maximization.
  • For even N, the appendix quantifies the worst influence of phase noise, while f_i(Φ) remains bounded for users i ≠ k as N approaches infinity.
  • User k’s rate retains the stated orders of magnitude under the discrete-phase analysis.
  • O(log2(N)) rate scaling remains achievable with low-resolution RIS phase shifts.
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