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Statistical Design of Multi-BD-RIS-Assisted User-Centric Cell-Free Massive MIMO

Mohammad Ezaz, Azadeh Vosoughi

arXiv:2610.33415v1eess.SP

TL;DR

Instantaneous-CSI-based BD-RIS optimization would require frequent channel acquisition and reconfiguration. The paper instead optimizes with long-term statistics; results show that group connectivity retains most of the fully connected NMSE gain, while lower NMSE does not necessarily yield higher uplink SE.

  • Problem

    Optimizing BD-RIS scattering matrices using instantaneous small-scale CSI would require frequent channel acquisition and RIS reconfiguration.

    Source: §I · p. 1

  • Method

    The paper derives aggregate-channel MMSE statistics and closed-form uplink SINR, then uses product-manifold Riemannian optimization to minimize long-term sum-NMSE under group-connected BD-RIS constraints.

    Source: §I · p. 1, §VI · p. 6

  • Results

    Group connectivity retains most of the fully connected NMSE gain, optimization gains grow as direct AP–user links weaken, and increasing BD-RIS size can improve NMSE without improving uplink SE.

    Source: §I · p. 2, §V · pp. 5–6, §VI · p. 6

  • Takeaways & Limitations

    The results highlight the need for future designs that better align channel-estimation quality with data-detection performance.

    Source: §VI · p. 6

    Because the optimization problem is nonconvex, the convergence results establish first-order stationarity rather than global optimality.

    Source: B. Convergence Analysis · p. 4

Abstract

from arXiv · show

We study the uplink of a user-centric (UC) cell-free massive MIMO system assisted by multiple beyond-diagonal reconfigurable intelligent surfaces (BD-RISs), under imperfect channel state information (CSI) and pilot reuse. The BD-RISs are modeled as reciprocal, lossless, group-connected scattering networks. We first derive the minimum mean-square error (MMSE) estimation statistics for each user's aggregate channel to its serving access points (APs). To avoid frequent CSI acquisition and RIS reconfiguration, we optimize the BD-RIS scattering matrices using long-term channel statistics, allowing their configurations to remain fixed over multiple coherence blocks. Specifically, we minimize the sum of normalized mean-square errors (NMSEs) of the MMSE aggregate channel estimators, subject to the symmetry and unitarity constraints of the group-connected architecture. We develop a product-manifold Riemannian optimization algorithm and analyze its convergence and computational complexity. To evaluate uplink spectral efficiency (SE), we separately derive a closed-form expression for the effective SINR under maximum ratio (MR) combining at the central processing unit (CPU) and establish a lower bound on the achievable ergodic SE. The SINR derivation accounts for the statistical dependence among channels associated with different APs due to shared BD-RIS-assisted propagation paths, including cross-AP second-order statistics under pilot reuse. Numerical results show that moderate group connectivity retains most of the NMSE improvements achieved by full connectivity, with larger optimization gains as BD-RIS-assisted paths strengthen relative to direct AP-user links. They also reveal that lower estimation NMSE does not necessarily lead to higher uplink SE.

I. INTRODUCTION

The paper addresses the gap between UC cell-free analysis under imperfect CSI and group-connected BD-RIS designs by deriving aggregate-channel estimation statistics and optimizing scattering matrices from long-term statistics. Its numerical results show that group connectivity retains most full-connectivity NMSE gains, while lower NMSE does not necessarily improve uplink SE.

I. INTRODUCTION

  • Existing RIS-assisted cell-free designs primarily use conventional RISs, while group-connected BD-RIS manifold formulations have focused on cellular MIMO rather than UC cell-free analysis under imperfect CSI.Earlier work studied RIS-assisted cell-free settings and BD-RIS optimization, but these strands remained largely separate.Source: §I · p. 1
  • The paper derives MMSE statistics for each user's aggregate channel to serving APs and minimizes the sum of estimator NMSEs using long-term channel statistics.The optimization respects the symmetry and unitarity constraints of the group-connected architecture and uses a product-manifold Riemannian algorithm.Source: §I · p. 1
  • Group-connected BD-RIS designs retain most of the NMSE improvement achieved by full connectivity, with larger optimization gains as BD-RIS-assisted paths strengthen relative to direct AP–user links.Lower NMSE does not necessarily translate into higher uplink SE.Source: §I · p. 2

II. SYSTEM MODEL

  • The system uses reciprocal, lossless group-connected BD-RISs with block-diagonal scattering matrices, and divides each coherence block between uplink training and data transmission.The orthogonal pilots are reused according to a predetermined assignment.Source: §II · p. 2, Figure 1 · p. 2

A. Uplink Aggregate Channel Model

The aggregate uplink channel combines the direct user–AP channel with propagation through multiple BD-RISs. The model assumes deterministic AP–RIS LoS channels and spatially correlated Rician fading for the remaining channels.

A. Uplink Aggregate Channel Model

  • AP–RIS channels are modeled as deterministic LoS channels, while the remaining channels follow spatially correlated Rician fading.The BD-RIS element correlation uses a normalized isotropic half-space model based on element spacing and operating wavelength.Source: A. Uplink Aggregate Channel Model · p. 2
  • The direct user–AP channel is parameterized by large-scale fading, a Rician factor, and the AP array response for the LoS component's angle of arrival.The AP array response is unit norm, and the channel correlation matrix has trace M.Source: A. Uplink Aggregate Channel Model · p. 2
  • The aggregate channel is the sum of the direct user–AP channel and the BD-RIS-assisted paths through the AP–RIS channels and scattering matrices.The model denotes the user–AP and user–RIS channels by f_lk and z_sk, respectively.Source: A. Uplink Aggregate Channel Model · p. 2
  • Under the stated independence assumptions, each aggregate channel is modeled as complex Gaussian with a mean and covariance determined by its direct and RIS-assisted components.The user–AP and user–RIS scattered components are mutually independent across their respective indices and between the two channel families.Source: A. Uplink Aggregate Channel Model · p. 2

B. Uplink Training Transmission and Aggregate Channel Estimation

Uplink training assigns users orthogonal pilots that may be reused across users. The received pilot signals yield MMSE estimates of aggregate channels, with estimation-error covariance used to define NMSE.

B. Uplink Training Transmission and Aggregate Channel Estimation

C. Statistical Characterization of the Aggregate Channel

The statistical characterization supplies second-order quantities needed for uplink SINR analysis. It covers cross-covariances among channels, received pilot signals, and channel estimates, including the effect of pilot reuse.

C. Statistical Characterization of the Aggregate Channel

  • The required second-order statistics include aggregate channels, received pilot signals, and channel estimates for the uplink SINR analysis.The section introduces cross-covariance quantities for these variables.Source: C. Statistical Characterization of the… · pp. 2–3
  • The analysis derives cross-covariances between received pilot signals and between channel estimates, accounting for pilot-sharing relationships.The Kronecker delta distinguishes whether pilot indices match.Source: C. Statistical Characterization of the… · p. 3
  • For a channel and its own estimate, the cross-covariance equals the estimate covariance; for users assigned different pilots, the corresponding cross-covariances are zero.These identities are stated for the matched channel and estimate and for pilot indices that differ.Source: C. Statistical Characterization of the… · p. 3

III. UPLINK DATA TRANSMISSION

The uplink data stage uses CPU-side maximum-ratio combining over each user’s serving APs and evaluates performance through an effective-SINR-based lower bound on ergodic spectral efficiency. It accounts for interference from users sharing pilots separately from other interference.

III. UPLINK DATA TRANSMISSION

  • The CPU combines received signals and channel estimates from each user’s serving APs using maximum-ratio combining.The effective-noise treatment regards all terms other than the desired signal as noise.Source: §III · p. 3
  • The analysis distinguishes aggregate interference power from pilot-sharing users and from non-pilot-sharing users.This separation is used to reveal the effect of pilot reuse.Source: §III · p. 3
  • The achievable uplink ergodic SE is lower bounded by a pre-log factor times log2(1 + γk), where γk is the effective SINR.The pre-log factor is τu/τc.Source: §III · p. 3
  • The cited passage states that the proofs of Lemmas 1–2 and Theorem 1 are omitted due to space limitations, but it does not provide the proofs themselves.Source: §III · p. 3

A. Problem Formulation and Riemannian Optimization

The paper formulates BD-RIS design over a product manifold of symmetric unitary scattering blocks and uses Riemannian steepest descent with Armijo backtracking. Its geometric updates preserve feasibility while the objective’s gradient captures coupling through aggregate-channel statistics.

A. Problem Formulation and Riemannian Optimization

  • Each feasible scattering block must be both symmetric and unitary, so an ordinary Euclidean update need not preserve feasibility.The feasible set is expressed as a product manifold across groups and BD-RISs.Source: A. Problem Formulation and Riemannian… · p. 3
  • The design optimizes the collection of scattering blocks jointly, with the Euclidean gradient capturing coupling through the statistical sum-NMSE objective.The construction covers SG blocks across the S BD-RISs.Source: A. Problem Formulation and Riemannian… · p. 3
  • The Riemannian steepest-descent method projects the Euclidean gradient onto the block tangent spaces and updates blocks with a symmetric-unitary Takagi retraction.The retraction replaces Takagi singular values by unity and is applied factorwise.Source: A. Problem Formulation and Riemannian… · p. 4
  • Armijo backtracking selects the step size, and Algorithm 1 repeats objective evaluation, gradient computation, and retracted updates until its stopping condition is met.The procedure terminates when the gradient-norm criterion is met or the iteration limit is reached.Source: A. Problem Formulation and Riemannian… · p. 4

B. Convergence Analysis

The retracted iterates remain feasible, and the objective decreases monotonically to a finite limit under the stated regularity conditions. Because the problem is nonconvex, the convergence result establishes first-order stationarity rather than global optimality.

B. Convergence Analysis

  • The objective values decrease monotonically, remain bounded below by zero, and converge to a finite value.The nonnegative lower bound follows from each NMSE being nonnegative.Source: B. Convergence Analysis · p. 4
  • Every accumulation point is critical, and the product-manifold Riemannian gradient vanishes asymptotically.The argument uses compactness of the feasible product manifold and continuity of the Riemannian gradient.Source: B. Convergence Analysis · p. 4
  • Since the objective is nonconvex, the analysis guarantees first-order stationarity rather than global optimality.The retraction preserves the symmetric-unitary constraints throughout the iterates.Source: B. Convergence Analysis · p. 4

C. Computational Complexity

The complexity analysis gives worst-case asymptotic bounds for objective evaluation, gradient computation, and Armijo search with retraction. Increasing group count lowers the manifold-related cost for fixed element count, while dense covariance multiplication remains a cost contribution.

C. Computational Complexity

  • For line 3 objective evaluation, under worst-case asymptotic upper bounds, the main BD-RIS-dependent covariance term for each AP–BD-RIS pair costs O(GN̄^2M + N^2M + NM^2), exploiting group-connected structure.These matrix-valued products can be reused across users.Source: C. Computational Complexity · p. 4
  • Each Armijo trial requires an objective evaluation and SG Takagi retractions, so iteration cost also depends on the number of backtracking reductions.The gradient computation reuses quantities from objective evaluation; the analysis also bounds pilot-dependent covariance work and AP pilot-covariance inversions.Source: C. Computational Complexity · p. 5
  • For fixed N, increasing the number of groups G reduces the manifold-related cost, while dense Γ multiplication retains an O(LSN^2M) contribution.The stated manifold-related term is SG N̄^3 = SN^3/G^2.Source: C. Computational Complexity · p. 5

V. NUMERICAL ANALYSIS

BD-RIS optimization reduces estimation NMSE across the tested designs, with G=8 retaining most full-connectivity gains. Gains increase as direct links weaken, but larger NMSE improvements do not necessarily raise uplink SE.

V. NUMERICAL ANALYSIS

  • At τp=5, NMSE reductions are 2.15%, 2.06%, and 1.44% for G=1, 8, and 64, respectively, with G=8 retaining nearly all full-connectivity benefit.Optimization reduces NMSE for every considered connectivity level and pilot length, and gains increase moderately with pilot length.Source: Figure 2 · p. 5
  • At 20 dB direct-link attenuation, ΔNMSE reaches 11.70% for G=1; under complete blockage, reductions are 90.15%, 87.39%, and 80.14% for G=1, 8, and 64.G=8 exhibits a trend nearly identical to G=1 and retains most of its gain with substantially lower inter-element connectivity.Source: §V · p. 5, Figure 3 · p. 5
  • With group size 8 fixed, ΔNMSE rises from 1.12% at N=32 to 12.34% at N=512, while sum-SE change shifts from approximately −0.04% to −0.73%.The increasing NMSE trend appeared in all 30 network realizations, whereas the sum-SE change remained small and slightly negative.Source: §V · p. 6, Figure 4 · p. 6
  • Most NMSE reduction occurs on weaker serving links, whereas MR detection is dominated by stronger links, explaining why estimation gains may yield little or no SE gain.The results therefore show that minimizing unweighted sum-NMSE does not necessarily improve uplink SE.Source: §V · p. 6
  • Most objective reduction occurs early; G=64 reaches a negligible remaining gap fastest, G=8 is intermediate, and G=1 converges most gradually.The plotted remaining gap starts at 100%, approaches zero at the final attained objective, and is averaged over network realizations.Source: Figure 5 · p. 6, §V · p. 6

VI. CONCLUSION

The paper studies multi-BD-RIS-assisted user-centric cell-free uplink under imperfect CSI and pilot reuse. It derives channel-estimation and SINR results and optimizes long-term sum-NMSE; group connectivity retains most full-connectivity gains, but lower NMSE need not improve SE.

VI. CONCLUSION

  • The study derives aggregate-channel MMSE statistics and a closed-form uplink SINR, and develops Riemannian optimization to minimize long-term sum-NMSE under group-connected symmetry and unitarity constraints.The SINR analysis accounts for dependence among AP channels through shared BD-RIS-assisted paths.Source: §VI · p. 6
  • Group connectivity retains most of the fully connected NMSE gain, and optimization becomes more effective as direct AP–user links weaken.Increasing BD-RIS size further improves NMSE, but does not necessarily increase uplink SE.Source: §VI · p. 6
  • Future designs should better align channel-estimation quality with data-detection performance.Source: §VI · p. 6
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