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Reconfigurable Intelligent Surface-Assisted Cell-Free Massive MIMO Systems Over Spatially-Correlated Channels

Trinh Van Chien, Hien Quoc Ngo, Symeon Chatzinotas, Marco Di Renzo, Björn Ottersten

arXiv:2104.08648v3cs.IT

TL;DR

The paper studies RIS-assisted Cell-Free Massive MIMO under spatially correlated channels, where unreliable direct links and channel-estimation overhead are central concerns. It proposes aggregated channel estimation, statistical RIS phase-shift optimization, asymptotic analysis, and closed-form throughput expressions. Numerical results confirm improved net throughput, particularly when direct links are frequently blocked.

  • Problem

    The paper addresses the absence of performance analysis for RIS-assisted Cell-Free Massive MIMO with spatially correlated channels and potentially blocked direct links.

  • Method

    The paper combines aggregated channel estimation, statistical RIS phase-shift optimization, asymptotic analysis, and closed-form uplink and downlink net-throughput expressions.

  • Results

    RISs significantly enhance per-user net throughput, especially when direct AP-user links are blocked with high probability.

  • Takeaways & Limitations

    RIS assistance can improve channel-estimation quality and system performance when direct links are unreliable.

Abstract

from arXiv · show

Cell-Free Massive multiple-input multiple-output (MIMO) and reconfigurable intelligent surface (RIS) are two promising technologies for application to beyond-5G networks. This paper considers Cell-Free Massive MIMO systems with the assistance of an RIS for enhancing the system performance under the presence of spatial correlation among the engineered scattering elements of the RIS. Distributed maximum-ratio processing is considered at the access points (APs). We introduce an aggregated channel estimation approach that provides sufficient information for data processing with the main benefit of reducing the overhead required for channel estimation. The considered system is studied by using asymptotic analysis which lets the number of APs and/or the number of RIS elements grow large. A lower bound for the channel capacity is obtained for a finite number of APs and engineered scattering elements of the RIS, and closed-form expressions for the uplink and downlink ergodic net throughput are formulated in terms of only the channel statistics. Based on the obtained analytical frameworks, we unveil the impact of channel correlation, the number of RIS elements, and the pilot contamination on the net throughput of each user. In addition, a simple control scheme for optimizing the configuration of the engineered scattering elements of the RIS is proposed, which is shown to increase the channel estimation quality, and, hence, the system performance. Numerical results demonstrate the effectiveness of the proposed system design and performance analysis. In particular, the performance benefits of using RISs in Cell-Free Massive MIMO systems are confirmed, especially if the direct links between the APs and the users are of insufficient quality with high probability.

I. INTRODUCTION

The paper develops an RIS-assisted Cell-Free Massive MIMO framework for spatially correlated channels, addressing channel-estimation overhead and unreliable direct links. It derives analytical throughput results and evaluates how RIS configuration, correlation, pilot contamination, and system scaling affect performance.

  • Cell-Free Massive MIMO reduces intercell interference through distributed AP collaboration but may provide poor service under poor scattering or severe blockage.
  • The paper addresses the lack of prior analysis for RIS-assisted Cell-Free Massive MIMO with spatially correlated channels and blocked direct links.
  • Aggregated channel estimation jointly represents direct and RIS-assisted links, requiring only the existing pilot-training overhead when RIS phase shifts are fixed.The approach estimates the aggregated channel rather than every individual RIS-related coefficient.
  • As AP and RIS-element counts increase, MR processing averages out non-coherent interference, fading, and noise, while coherent interference remains and indirect links become dominant.
  • The analysis uses closed-form uplink and downlink net-throughput expressions to expose the effects of array gain, coherent transmission, estimation errors, pilot contamination, spatial correlation, and RIS phase shifts.
  • Numerical results show that RISs significantly enhance per-user net throughput, especially when direct AP-user links are blocked with high probability.

C. RIS Phase Shift Control and Optimization

The paper optimizes RIS phase shifts by minimizing aggregate channel-estimation NMSE, with a simple equal-phase solution when direct links are negligible. This design supports efficient estimation and improves net throughput under the stated conditions.

  • Objective: The RIS phase-shift scheme minimizes the total NMSE across users and APs to improve channel-estimation quality.The NMSE measures relative channel-estimation error per AP and is affected by pilot reuse.
  • Optimization: The optimal phase shifts depend only on statistical CSI, including large-scale fading coefficients and channel covariance matrices.They are independent of instantaneous CSI, although globally optimizing many independently tunable elements is difficult.
  • Optimization: When direct links are negligible, equal phase shifts θ_1 = ... = θ_N optimally minimize the sum-NMSE criterion.This result assumes the spatially correlated RIS-assisted channel model used in the paper.
  • Implications: Under completely blocked direct links, equal phase shifts provide a simple channel-estimation design for RISs with many engineered scattering elements.The design also offers good net-throughput gains when direct links cannot be completely ignored.
  • Limitation: If direct links are not weak enough, equal phase shifts are no longer optimal, so numerical optimization may be required.The paper mentions gradient descent as one possible locally optimal approach and postpones throughput-based optimization.

III. UPLINK DATA TRANSMISSION AND PERFORMANCE ANALYSIS WITH MR COMBINING

The uplink analysis combines simultaneous user transmissions with MR processing at the CPU and derives asymptotic ergodic net-throughput expressions. It considers regimes with many APs, with or without a growing RIS.

  • Uplink model: All users transmit simultaneously to the APs, with each symbol weighted by a power-control factor √η_k.Power control can compensate near-far effects and mitigate mutual interference.
  • Detection: The CPU uses distributed maximum-ratio combining based on the estimated channels to detect each user’s uplink data.The resulting decision statistic is formed from the AP processing outputs.
  • Throughput analysis: The uplink ergodic net throughput is analyzed from the combined observation for each user.The analysis follows the construction of the uplink decision statistic.
  • Asymptotic regimes: The asymptotic study covers fixed N with large M and jointly large N and M, conditioned on the channel statistics and pilot/data powers.The setup includes large-scale fading coefficients and covariance matrices.
  • Signal decomposition: The weighted uplink signal separates pilot-sharing-user signals, interference from users with orthogonal pilots, and post-combining additive noise.These components are represented by T_k1, T_k2, and T_k3, respectively.

1) Case I:

In Case I, the number of APs grows while the RIS size remains fixed, causing noncoherent interference, small-scale fading, and noise to vanish asymptotically. Pilot contamination remains the residual impairment and limits gains from adding APs with MR combining.

  • Case I: Fixed N, large M: As M→∞ with fixed N, the noncoherent interference and additive-noise terms converge to zero under favorable propagation and channel-noise independence.The post-processed decision statistic converges in probability to a deterministic value.
  • Case I: Fixed N, large M: For fixed N and large M, channels become asymptotically orthogonal, eliminating small-scale fading and noncoherent interference.The direct and RIS-assisted contributions remain present in the limiting expression.
  • Case I: Fixed N, large M: Pilot contamination is the only residual impairment and prevents further performance improvement from adding APs under MR combining.The limitation arises from users sharing the same pilot sequence.
  • Case I: Fixed N, large M: The limiting signal depends on both direct-link coefficients and RIS-assisted indirect-link covariance terms.These contributions appear through β_mk′ and tr(Θ_mk′), respectively.
  • Case II: Large N and M: The large-N, large-M regime requires assumptions ensuring bounded covariance-matrix spectral quantities.The stated assumptions constrain the largest singular value and eigenvalue sum of RIS-element covariance matrices.

C. Uplink Ergodic Net Throughput with a Finite Number of APs and RIS Elements

For finite AP and RIS-element counts, the paper derives an achievable lower-bound uplink net throughput with MR combining and a closed-form SINR. The analysis separates desired signal, beamforming uncertainty, mutual interference, RIS-induced interference, pilot contamination, and noise.

  • SINR structure: The effective SINR consists of desired-signal strength, beamforming uncertainty, mutual interference, RIS-induced interference, pilot contamination, and additive noise.These terms quantify the main signal, interference, estimation, and noise contributions in the uplink.
  • Finite-system throughput: The uplink ergodic net throughput is an achievable lower bound on channel capacity with a closed-form expression under MR combining.The bound is formulated for finite numbers of APs and RIS elements.
  • SINR structure: Joint processing at the CPU makes the SINR numerator increase with the square of the sum of channel-estimate variances across APs.The APs send their received signals to the CPU for centralized data detection.
  • RIS and interference: RIS assistance increases desired-signal strength but introduces additional interference for users sharing the surface.The RIS-related interference appears alongside mutual interference and noise in the SINR denominator.
  • Pilot contamination: With sufficiently long coherence time and orthogonal pilots for every user, coherent pilot-contamination interference can be completely suppressed.Under this condition, the effective SINR simplifies to the expression given for orthogonal pilot allocation.

A. Downlink Data Transmission Phase

The downlink uses uplink channel estimates for cooperative MR precoding by all APs, and the paper derives asymptotic received-signal results and an achievable closed-form net-throughput bound. The analysis highlights the roles of coherent transmission, pilot contamination, channel statistics, and RIS phase shifts.

  • Downlink transmission: All APs cooperatively transmit the same data symbol to each user using MR precoding based on uplink channel estimates.Channel reciprocity allows the APs to treat the uplink estimates as the channels used to construct downlink beamformers.
  • Uplink–downlink distinction: Unlike uplink processing, downlink precoding depends on channel estimates for all users because the APs jointly transmit data to the network.This dependence makes the uplink and downlink analyses different and preserves user coexistence under pilot reuse.
  • Asymptotic analysis: As the number of APs, or both APs and RIS elements, grows large, the received signal converges to a deterministic equivalent.The asymptotic regimes considered are M→∞ and M,N→∞.
  • Downlink throughput: The downlink ergodic net throughput has an achievable lower-bound closed form under MR precoding, with an effective SINR incorporating desired signal, beamforming uncertainty, and interference.The bound follows from a channel-capacity bounding technique.
  • Statistics-based optimization: The uplink and downlink throughput expressions depend only on large-scale fading statistics and channel covariance matrices, not instantaneous CSI.This permits long-term RIS phase-shift optimization; the paper specifically optimizes phase shifts to minimize channel-estimation error.
  • Scope boundary: The paper postpones optimizing RIS phase shifts directly from the closed-form throughput expressions to future research.The reported phase-shift optimization instead targets channel-estimation error.

V. NUMERICAL RESULTS

The numerical study evaluates RIS-assisted Cell-Free Massive MIMO under blocked direct links, varying network size, users, RIS elements, spatial correlation, and capacity bounds. Analytical predictions closely match simulations, while RIS assistance is most beneficial when direct links are unreliable.

  • Simulation setup: The simulations use a 1.5 × 1.5 km^2 area with APs and users placed in separated sub-regions, a RIS at the origin, 1.9 GHz carrier frequency, and 20 MHz bandwidth.Each coherence interval contains τ_c = 200 symbols; pilot power is 100 mW and each AP has a 200 mW power budget.
  • Compared systems: The evaluated benchmarks include RIS-CellFree, conventional CellFree without a RIS, and RIS-assisted Cell-Free Massive MIMO with completely blocked direct links.The blocked-direct-link case relies only on RIS-assisted transmission.
  • Validation: The analytical frameworks closely overlap with Monte Carlo simulations for the net-throughput CDF, supporting the accuracy of the proposed analysis.The comparison uses Theorems 1 and 2 for the analytical results and the corresponding SINR-based simulations.
  • Direct-link reliability: As the direct-link unblocked probability decreases, conventional Cell-Free Massive MIMO performs worst and its average net throughput tends to zero as p̃ → 0.RIS-assisted Cell-Free Massive MIMO offers the best average net throughput, particularly for p̃ < 0.2.
  • Throughput comparisons: At p̃ = 0.2, RIS-assisted Cell-Free Massive MIMO has a net advantage, especially in the downlink; reported gains reach 1.7× and 2.6× in the considered setups.Average sum net throughput also increases with the number of APs, users, and orthonormal pilot signals.
  • Spatial correlation and RIS configuration: Uniform phase shifts provide much higher average throughput under the reported spatially correlated fading conditions, whereas random and uniform shifts differ little without spatial correlation.The phase-shift design is optimized using statistical CSI, and its objective is to improve aggregated-channel estimation quality.
  • RIS size: With fixed total RIS size, no significant average-net-throughput difference is observed for element sizes no smaller than λ/3.Further study is identified as necessary for deep sub-wavelength structures, alternative phase-shift criteria, and mutual coupling.
  • Capacity bounds: Use-and-then-forget and hardening bounds provide closed-form net-throughput expressions, but reduced channel hardening with an RIS can make other bounds better estimates of actual capacity.The statistical bound produces better ergodic net throughput per user in the reported comparison.

VI. CONCLUSION

The paper integrates Cell-Free Massive MIMO and RIS as complementary technologies and analyzes their performance under spatial correlation and unreliable direct links. It concludes that RIS assistance is especially useful when AP–user direct links are unreliable, while identifying several directions for extending the analysis.

  • VI. CONCLUSION: The proposed RIS-assisted Cell-Free Massive MIMO system combines complementary technologies for improving performance in harsh communication environments.The system is analyzed in TDD mode under fading spatial correlation and blockage of direct AP–user links.
  • VI. CONCLUSION: An aggregated channel estimation scheme reduces the overhead associated with estimating individual RIS-element channels.The approach supports closed-form uplink and downlink ergodic net-throughput analysis.
  • VI. CONCLUSION: The RIS phase shifts are designed to minimize channel estimation error before analyzing system performance.The resulting framework studies the effects of fading spatial correlation and direct-link blocking probability.
  • VI. CONCLUSION: RIS assistance is particularly useful when AP–user direct links are unreliable with high probability.The numerical results support this conclusion, and the RIS can overcome direct-link unreliability.
  • VI. CONCLUSION: Future extensions include throughput-oriented RIS phase-shift optimization, non-compact and deep sub-wavelength RIS structures, and mutual coupling.These topics are identified as possible generalizations of the presented results.

APPENDIX

The appendix develops Gaussian-moment and matrix-trace tools used in the asymptotic analysis and evaluates moments and correlations of aggregated channels. It also derives expressions by exploiting independence among direct and cascaded channels.

  • APPENDIX: The appendix introduces lemmas used for asymptotic analysis, including a trace bound for arbitrary and positive-semidefinite matrices.For positive-semidefinite X and Y, the trace product is bounded using the spectral norm of X.
  • APPENDIX: Gaussian vector moments are evaluated by whitening the vector and applying fourth- and second-moment identities of standard complex Gaussian variables.The derivation distinguishes equal and unequal index combinations in the resulting expectations.
  • APPENDIX: The appendix completes intermediate derivations by algebraic simplification after substituting the stated moment expressions.Equations (66) and (68) are connected to earlier expressions through these manipulations.
  • APPENDIX: Second moments and correlations of aggregated channels are obtained using propagation-channel independence and standard expectation identities.The appendix also uses the trace-of-product property and applies the resulting expressions to channel-estimation quantities.
  • APPENDIX: The fourth moment of an aggregated channel is decomposed into direct, indirect, and cross terms before applying Gaussian moment identities.Independence between direct and RIS-assisted links simplifies several expectations.

C. Proof of Corollary 1

The proof of Corollary 1 reformulates channel-estimation correlation quantities using centered-moment identities and the closed-form projected training signal and channel estimate. Substitution then yields the corollary.

  • C. Proof of Corollary 1: The proof computes covariance-related expectations by applying centered-moment identities to the relevant aggregated-channel variables.The identities express centered products and variances through ordinary moments and means.
  • C. Proof of Corollary 1: The quantity Q_mm′k is written in closed form using the projected training signal and channel estimate.The derivation retains only terms with nonzero expectations and uses an earlier moment expression.
  • C. Proof of Corollary 1: The proof concludes by inserting the closed-form expression for Q_mm′k into the preceding result.This substitution directly establishes the corollary.

D. Proof of Corollary 2

The proof of Corollary 2 analyzes the channel-estimation objective when direct links are negligible. It shows that the objective decreases with a trace quantity and characterizes equality in the resulting bound through aligned phase vectors.

  • D. Proof of Corollary 2: When direct links are negligible, the channel-estimate NMSE is reformulated in terms of the RIS-related quantities.This provides the starting point for optimizing the estimation objective.
  • D. Proof of Corollary 2: The NMSE objective is a monotonically decreasing function of tr(Φ^H R Φ R) because its derivative is nonpositive.The proof uses b_mk ≥ 0 for all APs and users.
  • D. Proof of Corollary 2: The trace expression is bounded by applying Cauchy–Schwarz to two vectors formed from correlation entries and RIS phase differences.The vectors contain r_nn′ and r_nn′e^j(θ_n−θ_n′), respectively.
  • D. Proof of Corollary 2: Equality holds only when the two constructed vectors are parallel, which implies equal RIS phases θ_n = θ_n′ for all element pairs.Combining this equality condition with the monotonicity establishes the corollary.

E. Proof of Theorem 1

The proof derives a closed-form uplink SINR by decomposing desired-signal, interference, and noise terms and evaluating their expectations using channel-estimation identities and correlation properties.

  • The uplink SINR derivation begins by computing the desired-signal term from the definition of the processed channel.
  • The denominator separates interference according to the pilot-reuse pattern into non-coherent and coherent components.
  • Expectations are evaluated using aggregated-channel properties, channel estimates, estimation errors, and identities for uncorrelated random variables.
  • The mutual interference between users sharing a pilot sequence is combined into a closed-form expression.
  • The additive-noise contribution is obtained from independence between the channel estimate and noise, completing the SINR proof.

F. Proof of Theorem 2

The proof derives a closed-form downlink SINR by simplifying the desired signal, beamforming uncertainty, mutual interference, and noise-related terms.

  • The downlink SINR numerator is simplified using the uncorrelation between the channel estimate and channel estimation error.
  • The beamforming uncertainty term is expanded through channel-estimation relations and second-moment identities.
  • The mutual interference term is rewritten using pilot-sequence orthogonality and expectations of interference contributions.
  • The remaining interference component is further analyzed through its non-coherent contribution and noise symmetry properties.
  • Substituting the derived terms into the downlink SINR yields the stated closed-form result.
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