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Two-Timescale Design for Reconfigurable Intelligent Surface-Aided Massive MIMO Systems with Imperfect CSI

Kangda Zhi, Cunhua Pan, Hong Ren, Kezhi Wang, Maged Elkashlan, Marco Di Renzo, Robert Schober, H. Vincent Poor, Jiangzhou Wang, Lajos Hanzo

arXiv:2108.07622v4cs.ITeess.SP

TL;DR

The paper addresses two-timescale RIS-aided massive MIMO transmission under imperfect aggregated CSI. It combines LMMSE estimation, MRC detection, closed-form achievable-rate analysis, and statistical-CSI RIS optimization, deriving power-scaling results and a limitation under EMI.

  • Problem

    The paper studies RIS-aided massive MIMO under imperfect aggregated CSI, including achievable-rate scaling and the previously unexamined effects of spatial correlation and EMI.

  • Method

    The paper derives LMMSE aggregated-channel estimates, applies MRC detection, obtains UatF achievable-rate bounds, and optimizes RIS phase shifts using gradient ascent.

  • Results

    The analysis establishes transmit-power scaling laws while maintaining a non-zero rate for Rician and Rayleigh RIS-BS channels.

  • Takeaways & Limitations

    Two-timescale RIS operation supports reduced CSI adaptation overhead while enabling statistical-CSI passive-beamforming optimization in massive MIMO systems.

  • Takeaways & Limitations

    The scaling laws obtained without EMI may not be preserved in the severe-EMI region.

Abstract

from arXiv · show

This paper investigates the two-timescale transmission design for reconfigurable intelligent surface (RIS)-aided massive multiple-input multiple-output (MIMO) systems, where the beamforming at the base station (BS) is adapted to the rapidly-changing instantaneous channel state information (CSI), while the passive beamforming at the RIS is adapted to the slowly-changing statistical CSI. Specifically, we first propose a linear minimum mean square error (LMMSE) estimator to obtain the aggregated channel from the users to the BS in each channel coherence interval. Based on the estimated channel, we apply the low-complexity maximal ratio combining (MRC) beamforming at the BS, and then derive the ergodic achievable rate in a closed form expression. To draw design insights, we perform a detailed theoretical analysis departing from the derived ergodic achievable rate. If the BS-RIS channel is Rician distributed, we prove that the transmit power can be scaled proportionally to $1/M$, as the number of BS antennas, $M$, grows to infinity while maintaining a non-zero rate. If the BS-RIS channel is Rayleigh distributed, the transmit power can be scaled either proportionally to $1/\sqrt{M}$ as $M$ grows large, or proportionally to $1/N$ as the number of reflecting elements, $N$, grows large, while still maintaining a non-zero rate. By capitalizing on the derived expression of the data rate under the statistical knowledge of the CSI, we maximize the minimum user rate by designing the passive beamforming at the RIS. Numerical results confirm that, even in the presence of imperfect CSI, the integration of an RIS in massive MIMO systems results in promising performance gains. In addition, the obtained results reveal that it is favorable to place the RIS close to the users rather than close to the BS.

I. INTRODUCTION

RIS-aided massive MIMO is studied as a two-timescale system that reduces CSI-related overhead by adapting BS beamforming to instantaneous aggregated CSI and RIS phase shifts to slower statistical CSI. The paper analyzes imperfect CSI, channel effects, achievable-rate scaling, and statistical-CSI RIS optimization.

  • Motivation: Instantaneous-CSI RIS designs incur pilot, computation, feedback, and energy overhead because RIS beamforming must be recalculated in every coherence interval.Pilot overhead grows with the number of RIS elements, while frequent phase-shift updates require repeated feedback to the RIS controller.
  • Two-timescale design: Two-timescale designs use instantaneous aggregated CSI for BS beamforming, whose dimension is independent of the number of RIS elements.This reduces the required pilot signals to a number larger than the number of users.
  • Two-timescale design: RIS phase shifts are optimized from long-term statistical CSI and updated only when large-scale channel information changes.Examples of statistical CSI include user locations and angles of arrival and departure.
  • Research scope: The paper analyzes uplink RIS-aided massive MIMO under imperfect aggregated CSI and examines achievable-rate scaling, power scaling, spatial correlation, and EMI.The introduction identifies spatial correlation and EMI under the two-timescale scheme with imperfect CSI as open research problems.
  • Contributions: The proposed analysis combines LMMSE channel estimation, MRC detection, closed-form achievable-rate bounds, and gradient-ascent RIS optimization based on statistical CSI.The RIS optimization targets minimum user-rate maximization for spatially independent and spatially correlated channel models.
  • Findings: Numerical results indicate benefits from spatial correlation for large RISs, while severe EMI may eliminate the advantage over conventional massive MIMO.The study also reports particular benefits when RISs are deployed near cell-edge users.

II. SYSTEM MODEL

The system models uplink transmission in an RIS-aided massive MIMO system, with an RIS near the users and MRC at the BS based on an LMMSE estimate of the aggregated channel. The analysis considers Rician and Rayleigh fading, imperfect channel estimation, and pilot-based estimation overhead.

  • The BS has M active antennas, the RIS has N nearly-passive reflecting elements, and each of K users has one transmit antenna.
  • The cascaded channel for user k is g_k = H2Φh_k, and the users’ cascaded channels form G = H2ΦH1 ∈ C^M×K.
  • The aggregated instantaneous channel is Q = G + D, combining the RIS-assisted cascaded channel with the direct user-BS channel.
  • The BS estimates Q using a standard LMMSE estimator and applies low-complexity MRC based on the estimated channel.
  • The channel model adopts Rician fading for the user-RIS and RIS-BS links, while the direct user-BS link is modeled as Rayleigh fading.
  • Only the aggregated channel Q is estimated, giving the same dimension as a conventional user-BS channel and thereby reducing estimation overhead and computational complexity.

IV. ANALYSIS OF THE ACHIEVABLE RATE

The section derives a tractable lower bound for the ergodic achievable rate under imperfect CSI and uses it to analyze RIS-aided massive MIMO design. The expression captures CSI uncertainty through auxiliary parameters and supports statistical-CSI phase-shift optimization.

  • The UatF bound is tractable and incorporates performance degradation caused by imperfect CSI through auxiliary parameters e_k1, e_k2, and e_k3.
  • Theorem 2 derives a closed-form lower bound for each user’s ergodic achievable rate under imperfect CSI.
  • The closed-form rate avoids inverse matrices, numerical integrals, and costly Monte Carlo evaluation even when M and N are large.
  • Because the rate expression relies only on statistical CSI, it can serve as an objective for optimizing RIS phase shifts using long-term channel information.
  • The interference term scales as O(M^2), indicating stronger multi-user interference than in conventional massive MIMO systems.
  • The analysis permits phase-shift choices that control |f_k(Φ)| from 0 to N when N > 1, while aligning one user can disadvantage others.

B. Multi-user Case

The multi-user analysis characterizes how RIS-BS fading, user-RIS fading, antenna and element counts, and imperfect CSI determine rate behavior and power scaling. It also identifies spatial diversity and fairness considerations for RIS deployment and phase design.

  • Fairness requirements are necessary because RIS phase alignment can increase one user’s rate while driving another user’s rate toward zero.
  • With Rician RIS-BS and user-RIS channels, transmit power scaled as 1/M maintains a non-zero rate as M grows.
  • For a Rayleigh RIS-BS channel, the rate is independent of RIS phase shifts, so phase-shift optimization is unnecessary when the user-RIS links are fully NLoS.
  • With a Rayleigh RIS-BS channel, transmit power scaled as 1/N maintains a non-zero rate as N grows, regardless of user-RIS Rician factors.
  • In the Rayleigh RIS-BS case, desired signal power scales as O(M) while interference scales as O(1), producing logarithmic rate growth with M.
  • A small RIS-BS Rician factor is beneficial because it corresponds to a high-rank channel that provides spatial diversity for multi-user communication.
  • Placing the RIS near users can reduce the RIS-BS Rician factor and favor a high-rank RIS-BS channel, whereas placement near the BS can yield rank deficiency.

C. Single-user Case

The single-user analysis gives power-scaling laws and pilot-design insights for large antenna and RIS configurations. It shows how channel fading, pilot overhead, and RIS phase alignment shape achievable rates.

  • For single-user systems with |f_k(Φ)| = N, the rate is maximized by phase shifts satisfying the aligned condition |f_k(Φ)| = N.
  • With |f_k(Φ)| = N, power scaled as 1/(MN^2) maintains a lower-bounded rate when both M and N grow.
  • Under the joint scaling p = E_u/(MN^2), the desired signal scales as O(M^2N^4) and signal leakage as O(M^2N^3).
  • Power scaled as 1/N^2 supports a lower-bounded rate as N grows under the single-user aligned-phase configuration.
  • The single-user asymptotic SNR does not depend on pilot length except for a pre-log factor, making τ = K = 1 optimal in the analyzed cases.
  • Rates increase with Rician factors, while the 1/N^2 scaling cases collapse when either the RIS-BS or user-RIS link lacks a LoS component.
  • If δ = 0 or ε_k = 0, the 1/N^2 law no longer holds, and power can be reduced only as 1/N to maintain a non-zero rate.

B. Channel Estimation

The channel-estimation analysis develops an LMMSE estimator and corresponding rate expressions for aggregated channels under imperfect CSI. It further shows how spatial correlation and EMI alter estimation, rate, and asymptotic power-scaling behavior.

  • The LMMSE estimator uses pilot observations to estimate each user’s aggregated channel q_c,k under spatial correlation and EMI.
  • Spatial correlation allows RIS phase shifts to influence channel-estimation accuracy, unlike the negligible-correlation case where the relevant correlation matrices are identity matrices.
  • Theorem 4 provides a UatF lower bound for the achievable rate that explicitly includes signal, noise, EMI, and leakage terms.
  • Spatial correlation can enhance RIS channel-shaping ability because RIS-adjustable terms are fixed under negligible correlation but become shapeable when correlation is present.
  • In the presence of EMI, the previously summarized 1/M and 1/N power-scaling laws are not guaranteed to hold.
  • This failure occurs because intended signal power weakens under scaling while EMI power at the RIS remains unaffected and can dominate the received signal.

VI. DESIGN OF THE RIS PHASE SHIFTS

The paper designs RIS phase shifts using statistical CSI to maximize achievable or minimum user rates, with special-case criteria and an accelerated gradient method for multi-user optimization.

  • Design principle: Statistical CSI permits less frequent RIS phase-shift updates, reducing channel acquisition overhead and computational complexity.The ergodic rate depends only on statistical CSI, so RIS updates follow long-term CSI variations.
  • Single-user design: In the single-user case, phase-shift optimization reduces to selecting x = |f_k(Φ)|^2 over 0 ≤ x ≤ N^2.The rate is rewritten using constants s1, s2, t1, and t2, enabling derivative-based optimization.
  • Single-user design: The optimal single-user configuration chooses |f_k(Φ)| = N, |f_k(Φ)| = 0, or an endpoint based on x_R^0 and the endpoint SNR comparison.For intermediate x_R^0, the maximum is attained at x = 0 or x = N^2 and is selected by comparing SNR_k(0) and SNR_k(N^2).
  • Single-user design: As N →∞, |f_k(Φ)| = N is optimal because SNR_k(0) remains bounded while SNR_k(N^2) →∞.This large-N result supports the corresponding asymptotic analysis.
  • Multi-user design: For multiple users, the paper maximizes the minimum user rate with gradient ascent over real phase variables θ under the RIS unit-modulus constraint.The objective’s minimum is smoothly approximated, and Nesterov acceleration with backtracking line search improves the optimization procedure.
  • Multi-user design: The proposed real-variable gradient method avoids projection operations because complex exponentials preserve the unit-modulus constraint for every phase vector θ.This differs from projected gradient ascent over complex variables and avoids projection-induced suboptimality.

VII. NUMERICAL RESULTS

Numerical results validate the analytical channel-estimation and power-scaling behavior, and show that RIS-aided massive MIMO can improve rates with fewer BS antennas. They also favor placing the RIS near users under the studied conditions.

  • Channel estimation: In general Rician channels, MSE increases with N while NMSE decreases because communication paths grow without a corresponding pilot-length increase.Increasing channel-gain intensity can nevertheless reduce normalized estimation error as N grows.
  • Channel estimation: In purely LoS RIS-assisted channels, MSE and NMSE are independent of N, while NMSE tends to zero as N →∞.The independence is attributed to deterministic LoS channels.
  • Two-timescale comparison: The two-timescale scheme outperforms instantaneous CSI-based design at large N because instantaneous CSI requires pilot length proportional to N.The resulting estimation overhead leaves fewer data symbols and causes a severe rate decrease for instantaneous CSI-based transmission.
  • Power scaling: In Rician fading, scaling transmit power as 1/N^2 preserves a limiting rate, whereas in NLoS-only channels 1/N^2 drives rate to zero and 1/N is required.The rate limit under 1/N^2 is maximized in LoS-only channels.
  • Rician factors: The achievable rate decreases with the BS-RIS Rician factor δ and increases with the RIS-user factor ε_k.A rank-deficient cascaded channel can result when the BS-RIS channel approaches a rank-1 condition, limiting multi-user transmission.
  • Deployment: Placing the RIS near users is preferred when accounting for spatial diversity, while placing it near the BS can create rank deficiency under the studied Rician model.Increasing the RIS-user LoS component, including by installing the RIS at a certain height, is beneficial.

B. Spatial-correlated Channels in the Presence of EMI

The section evaluates spatial correlation and EMI in RIS-aided systems, showing that their effects depend on RIS spacing, element count, and EMI strength. It also assesses the accelerated optimization method and deployment placement.

  • Spatial correlation: When d_ris = λ/2 and ρ = 30 dB, spatial correlation can be safely ignored because EMI is light.This supports using spatially independent analytical insights under these conditions.
  • Spatial correlation: For smaller RIS spacings, spatial correlation reduces the achievable rate at small N because the channel rank decreases.The analyzed spacings include d_ris = λ/4 and λ/8.
  • Spatial correlation: At large N, RIS beamforming gains outweigh the negative impact of spatial correlation, producing a better achievable rate.The RIS retains greater ability to customize wireless channels despite reduced channel rank.
  • EMI: When ρ < 60 dB, EMI has negligible impact on achievable rate, but strong EMI can become dominant and make RIS-aided systems worse than conventional systems.The power-scaling law p = 10/N is validated for mild EMI but no longer holds under strong EMI.
  • Deployment: RIS deployment still provides performance gains, but gains are reduced when the RIS is not optimized near cell-edge users.The placement study compares RIS-aided systems with conventional massive MIMO under the stated deployment assumptions.

APPENDIX A SOME USEFUL RESULTS

This appendix develops auxiliary expectation results used in the paper’s channel and rate analysis. It establishes matrix and estimator identities under independence, Gaussianity, and deterministic-matrix assumptions.

  • Matrix expectations: For independent zero-mean random matrices, the expectation of XWX^H is diagonal with diagonal entries equal to v_x Tr{W}.The result applies to deterministic W and extends to random-vector cases by setting m = 1 or n = 1.
  • Matrix expectations: The appendix uses independence and zero-mean properties to eliminate cross terms and derive expectations involving random channel matrices.Several lemmas specialize these identities to the matrices and vectors used in the system model.
  • LMMSE estimation: The LMMSE channel estimate is formed from the observation vector, with its mean and covariance obtained through the preceding lemmas.The corresponding estimation error and normalized mean-square error are then derived.

APPENDIX D

Appendix D derives the signal, noise, and interference expectations needed for the achievable-rate expression. The derivation accounts for imperfect CSI and dependence among users sharing the RIS-BS channel.

  • Signal and noise: The appendix expands the signal and noise terms into expectations over the estimated channels, direct channels, RIS channels, and noise.Independence and zero-mean properties remove terms whose expectations vanish.
  • Interference: Because different users share the RIS-BS channel, their aggregated channels are not independent, requiring explicit treatment of interference cross terms.The LMMSE error is uncorrelated with, but dependent on, the channel estimate because the cascaded channel is non-Gaussian.
  • Interference: Only 8 cross terms have non-zero expectations among the interference expansion’s cross terms.The remaining terms vanish through independence, zero-mean properties, and the relevant lemmas.
  • Interference: The interference term I_ki(Φ) is obtained by combining the derived modulus-square and cross-term expectations with direct simplifications.This completes the analytical interference component used in the rate expression.

C. Signal Leakage

This section derives the signal leakage term by expanding its expectation and evaluating modulus-square and cross terms. The calculation uses independence and zero-mean properties to remove vanishing contributions.

  • Signal leakage: The signal leakage term is expanded into 16 modulus-square terms followed by the remaining cross terms.The terms are evaluated sequentially within the derivation of the signal leakage expectation.
  • Signal leakage: Cross terms involving independent zero-mean direct channels, cascaded channels, and noise are removed when their expectations vanish.The derivation explicitly invokes independence and zero-mean properties for these simplifications.
  • Signal leakage: The resulting signal leakage expression combines the evaluated terms after applying the relevant Wishart-distribution identity.The Wishart property is used for one of the expectation terms in the expansion.

2 H2Φ˜hk

The appendix reduces the expectation calculation to a limited set of nonzero cross-terms and analyzes how RIS phase alignment controls the magnitude of user-dependent terms.

  • The derivation uses conditional expectations, independence, and previously established lemmas to simplify products involving the channel variables.The conditional expectation of ˜H2 given ˜h_k is identified with its unconditional expectation when independence applies.
  • Only 20 of 16 × 15 cross-terms are non-zero, and these are combined into 10 terms using independence and zero-mean properties.The derivation then evaluates the 10 terms sequentially, omitting the real operator because the results have only real parts.
  • For N > 1, |f_k(Φ)| is bounded between zero and N, with the upper equality attained when all RIS phase shifts are aligned.The minimum value zero is shown achievable for both even and odd N.
  • When RIS phases maximize |f_k(Φ)|, the corresponding term |f_i(Φ)| for a user with a different angle of arrival remains bounded as N →∞.The one-dimensional ULA result is used to support the corresponding deduction for the two-dimensional USPA model.

APPENDIX G

This appendix derives channel statistics and LMMSE estimates, then describes instantaneous-CSI benchmark processing and its pilot-overhead limitation.

  • The LMMSE estimator for the aggregated channel is obtained from the derived first- and second-order statistics of the channel and observation vector.The derivation uses covariance relations and independence among channels, noise, and electromagnetic interference.
  • Several interference and signal-leakage derivation details are omitted, with the appendix referring to analogous calculations instead.The omitted portions concern the interference, signal-leakage, and related proof calculations.
  • The instantaneous-CSI scheme estimates the direct channel and cascaded channel using two pilot phases requiring 1 and N pilot symbols, respectively.Its total pilot requirement is N + 1 symbols per coherence interval.
  • Based on estimated channels, the BS applies MRC beamforming, while RIS phases are optimized to maximize desired-signal power through a low-complexity suboptimal procedure.Alternating optimization updates the beamforming vector and RIS phase shifts, aligning cascaded-channel phases with the direct-channel phase.
  • The achievable rate includes a pilot-overhead loss factor, and if N + 1 > τ_c, the rate becomes zero because no symbols remain for data transmission.The rate is also negatively affected by channel-estimation overhead under imperfect CSI.

APPENDIX K EXPRESSIONS FOR GRADIENT VECTORS

The appendix presents gradient-vector expressions for the objective components used in RIS phase optimization, including signal, leakage, and EMI terms.

  • Theorem 6 gives the gradient of f(θ) with respect to the RIS phase vector θ.The surrounding expressions include gradients associated with signal and signal-leakage components.
  • The displayed formulas combine channel-dependent trace and coefficient terms to form the gradient entries used by the RIS optimization.The expressions explicitly include terms associated with signal leakage and electromagnetic interference.
  • Theorem 7 gives the gradient of f_c(θ) with respect to θ for the corresponding optimization component.The listed expressions include separate signal and EMI gradient terms.
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