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Large-Scale Partition-Based RIS Beamforming For Uplink RIS-Equipped Multi-User Systems: Asymptotic Analysis

Samira Rahimian, Haris Gacanin

arXiv:2610.11904v1eess.SPcs.SYeess.SY

TL;DR

For more than two users, ZF SINR lacks an explicit low-dimensional closed form, limiting tractable analysis for RIS-assisted uplinks. The paper analyzes a K-user system asymptotically, deriving an average-SINR approximation and showing equal partitioning is approximately sum-rate-optimal at leading order regardless of path-loss asymmetry.

  • Problem

    For more than two users, the ZF SINR lacks an explicit low-dimensional closed-form expression, and its distribution is analytically intractable for design.

    Source: §I · p. 2

  • Method

    The paper analyzes the large-RIS regime, where partition sizes grow without bound, to derive a closed-form asymptotic approximation for average per-user ZF SINR.

    Source: §I · p. 2, §VI · p. 7

  • Results

    Equal partitioning is approximately sum-rate-optimal at leading order regardless of path-loss asymmetry, and simulations confirm the closed-form rate approximation's accuracy across system sizes.

    Source: §I · p. 2, §VI · p. 7

  • Takeaways & Limitations

    Equal partitioning provides a leading-order sum-rate benchmark independent of users' path-loss asymmetry.

    Source: §I · p. 2

    The RIS size needed for a given approximation accuracy grows with the number of users.

    Source: §VI · p. 7

Abstract

from arXiv · show

Combining a reconfigurable intelligent surface (RIS) with a receive antenna array is a promising low-complexity architecture for multi-user uplink reception, but its performance analysis for more than two users has remained an open problem: the zero-forcing (ZF) signal-to-interference-plus-noise ratio (SINR) no longer admits an explicit, low-dimensional closed-form expression, and its distribution is analytically intractable for design purposes. This paper addresses this gap for a K-user, K-antenna uplink system in which a large-scale, L-element RIS, partitioned into K user-dedicated sub-surfaces, precedes ZF reception at the base station. Through an asymptotic analysis in which the sub-surface sizes grow without bound, we show that the orthogonal projector underlying the ZF SINR converges to a rank-one matrix aligned with the desired user's channel. We use this convergence to derive a closed-form asymptotic approximation for the average per-user SINR that depends only on deterministic channel parameters. Treating this expression as a tractable design objective, we prove that equal partitioning is approximately sum-rate-optimal at leading order regardless of path-loss asymmetry across users. We also develop a low-complexity greedy pairwise-transfer search that refines the partition beyond this leading-order optimum. Monte Carlo simulations across a range of system and RIS sizes confirm that the closed-form SINR tracks the exact simulated rate closely once the RIS is large relative to the number of users. The greedy search also yields consistent, if modest, sum-rate gains over equal partitioning, validating the theory as both an accurate performance predictor and a practical design tool.

I. INTRODUCTION

The paper extends partition-based RIS beamforming with ZF reception from a two-user setting to K users, addressing the lack of a tractable general-K SINR analysis. It uses a large-scale RIS asymptotic analysis to support performance prediction and partition design.

I. INTRODUCTION

  • Prior receiver-centric RIS studies include fully digital surfaces and passive RIS-plus-array designs –, while the authors' partitioned ZF analysis covered only two users.For general K, the ZF SINR depends on a projector onto the interfering users' joint channel subspace, without the explicit low-dimensional expression available in the 2 × 2 case.Source: §I · pp. 1–2

II. SYSTEM MODEL

  • The system has K single-antenna users, K receive antennas, and an L-element RIS; user-to-RIS links have path-loss coefficients and independent Rayleigh fading.The RIS is a uniform planar array, and the receiver antennas form a uniform linear array.Source: §II · p. 2

III. COMBINED RIS PARTITION-BASED BEAMFORMING AND ZF DETECTION

  • The proposed receiver combines user-dedicated RIS partitions with ZF detection to address multi-user interference in the large-RIS regime.Each partition is assigned to a specific user and shapes that user's incident signal.Source: §III · p. 2, A. RIS Phase Shift and Receiver… · p. 3

A. RIS Phase Shift and Receiver Antenna Beamforming Designs

  • Each partition's RIS phases cancel the assigned user's effective-channel phase at its designated antenna, while the receiver uses ZF with computational complexity O(L).For K users and K receive antennas, the ZF beamforming matrix reduces to G^-1.Source: A. RIS Phase Shift and Receiver… · p. 3

B. Performance Analysis

The performance analysis expresses the ZF SINR through a projector onto the null space of the interfering channels. Under a far-field model, channel concentration supports an asymptotic projector approximation used in the rate derivation.

B. Performance Analysis

  • The ZF SINR analysis uses the rank-one projector onto the null space of the interfering users' channel matrix, whose direction is determined by the null vector.The derivation characterizes that vector through the mean and covariance of the interfering effective-channel columns.Source: B. Performance Analysis · pp. 3–4
  • The desired-user channel is decomposed into a coherent gain along the user's coordinate and an interference vector, providing terms for the average-rate derivation.The analysis applies the law of large numbers in deriving the average achievable rates.Source: B. Performance Analysis · p. 3
  • Under the far-field planar phase-progression assumption [22], the interfering-channel mean matrix is diagonally dominant, with diagonal magnitudes O(L_pk) and off-diagonal magnitudes O(L_pk/L).The RIS partitions are contiguous blocks of flattened indices, so their phase progression is treated as two-dimensional.Source: B. Performance Analysis · p. 4
  • As partition sizes grow, the fluctuation-to-mean energy ratio vanishes and the interfering channel converges in mean square to its mean matrix.The null-space projector then converges entrywise to a rank-one limit.Source: B. Performance Analysis · p. 4

E. Asymptotic

As partition sizes grow, the cross term in the expected ZF-SINR expression becomes negligible relative to the leading signal term. This yields a closed-form asymptotic approximation.

E. Asymptotic

  • The cross-term contribution for i = k vanishes exactly; the remaining contributions are jointly O(L_pk), below the leading O(L^2/K^2) signal term for fixed K.The cross term vanishes at rate O(1/L).Source: E. Asymptotic · p. 5
  • The relative negligibility of the cross term becomes arbitrarily strong as partition size grows.The stated convergence is for fixed K.Source: E. Asymptotic · p. 5
  • The closed-form asymptotic approximation follows after the cross term becomes negligible as L_pk grows without bound.The derivation also uses the asymptotic projector result.Source: E. Asymptotic · p. 5

IV. ASYMPTOTIC ZF SINR

For the stated K-user uplink system, the paper gives a closed-form asymptotic expression for each user's average ZF SINR. Its validity is established as every RIS partition grows without bound under the stated channel assumptions.

IV. ASYMPTOTIC ZF SINR

  • The analyzed setup uses K single-antenna users, K receive antennas, and an L-element RIS divided into contiguous sub-surfaces with per-element phase design and ZF reception.The result is derived under the LOS channel model between the RIS and base-station antennas.Source: §IV · p. 5
  • The asymptotic result assumes far-field conditions and requires every partition size L_pk to grow without bound.The derivation of the result is given in Section III.Source: §IV · p. 5
  • As L_pk grows without bound for all users, Theorem 1 gives a closed-form asymptotic expression for each user's average ZF SINR.Source: §IV · p. 5

A. Sum-Rate Optimality of Equal Partitioning

The asymptotic sum-rate objective is maximized by equal RIS partitioning, independently of users’ path losses and channel strengths. This is a leading-order result based on high-SINR and within-sub-surface amplitude approximations, not a guarantee of exact optimality.

A. Sum-Rate Optimality of Equal Partitioning

  • The coherent term in the rate expression scales as O(L_pk^2), dominating the O(L_pk) incoherent term for large sub-surfaces.Approximating LOS amplitude as roughly constant within each physically localized sub-surface simplifies the dominant term.Source: A. Sum-Rate Optimality of Equal… · p. 5
  • Equal partitioning, L*_pk = L/K for every user, maximizes the approximate sum-rate independently of path loss and channel strength.The result is a leading-order optimum, not an exact guarantee.Source: A. Sum-Rate Optimality of Equal… · pp. 5–6
  • Under the high-SINR approximation, the partition-dependent objective reduces to maximizing the sum of log2(L_pk) terms subject to the total-element constraint.The channel-dependent coefficient c_k does not affect the maximizing allocation.Source: A. Sum-Rate Optimality of Equal… · pp. 5–6
  • The approximation relies on high SINR and treating each user’s LOS amplitude as roughly constant within its sub-surface.The true LOS amplitudes vary across the array; the simulation setting reports a within-partition coefficient of variation of roughly 18–19%.Source: A. Sum-Rate Optimality of Equal… · p. 6
  • The greedy pairwise-transfer search finds partitions mildly perturbed from equal allocation rather than exactly equal partitions.This empirical refinement is consistent with equal partitioning being a leading-order rather than exact optimum.Source: A. Sum-Rate Optimality of Equal… · p. 6

V. SIMULATION RESULTS

Simulations compare exact Monte Carlo rates with the asymptotic objective and evaluate greedy partition search against equal partitioning. The search finds small simulated gains, while the paper concludes that equal partitioning is approximately sum-rate-optimal at leading order.

V. SIMULATION RESULTS

  • Each configuration uses 1000 Monte Carlo realizations; exact rates use the full random effective channel matrix, while asymptotic rates apply Theorem 1’s average SINR directly.The simulations use λ=0.1 m, dA=2λ, dB=5λ, and RIS element spacing λ/2.Source: §V · p. 6

A. Validation of the Average Asymptotic Rate

  • Average simulated–asymptotic sum-rate gaps fell with L: 0.21%→0.19% (K=2), 0.98%→0.47% (K=4), and 11.86%→2.80% (K=8).The comparison uses equal partitioning and αk=1; the paper reports the asymptotic expression is accurate once the RIS is sufficiently large relative to K.Source: A. Validation of the Average… · p. 6

B. Sum-Rate Maximisation: Search vs. Equal Partitioning

  • The greedy search starts from equal partitioning and repeatedly transfers one element between the user pair that most increases the deterministic asymptotic sum-rate, stopping when no transfer improves it.The integer partition objective is neither convex nor concave, and exhaustive search is computationally infeasible for the RIS sizes considered.Source: B. Sum-Rate Maximisation: Search vs.… · p. 6
  • For K=8, L=4096 at P=20 dB, search-optimal partitioning achieved 150.15 bits/s/Hz versus 149.41 for equal partitioning in real simulation (+0.492%), consistent in direction and magnitude with asymptotic prediction (+0.353%).Source: B. Sum-Rate Maximisation: Search vs.… · p. 7, Figure 4 · p. 7

VI. CONCLUSION

  • Equal partitioning is approximately sum-rate-optimal at leading order independently of path-loss asymmetry; future work targets joint partition and phase-shift optimization under a max-min rate criterion.The conclusion also identifies quantized RIS phase shifting as a future direction.Source: §VI · p. 7

APPENDIX A PROOF OF THE ASYMPTOTIC RANK-ONE PROJECTOR

The proof establishes that the interfering-channel mean matrix becomes diagonally dominant and that its random fluctuations vanish relative to the mean as sub-surface sizes grow. It then shows that the desired coordinate of the ZF null vector concentrates at unit magnitude, yielding the rank-one projector limit.

APPENDIX A PROOF OF THE ASYMPTOTIC RANK-ONE PROJECTOR

  • For the coplanar BS array, off-diagonal mean terms vanish at O(1/L) under the remaining non-resonance condition; this is slower than O(1/L_pj).The condition fails at exact integer y-direction spatial frequency, including grazing incidence for half-wavelength spacing.Source: Appendix A · p. 8
  • The mean interfering-channel matrix is diagonally dominant: each column’s mean magnitude concentrates on its own diagonal entry.The proof bounds off-diagonal mean entries using a decomposition of each contiguous RIS partition into at most three pieces.Source: Appendix A · pp. 7–8
  • When partition sizes grow without bound while L/L_pj remains bounded, fluctuation energy divided by squared mean norm vanishes, so each channel column approaches its mean in mean square.This relative concentration is used to show that the interfering-channel null-space equation concentrates.Source: Appendix A · pp. 8–9
  • As each sub-surface grows, the expected squared magnitude of the desired coordinate of the unit null vector converges to 1, implying convergence in probability and the rank-one projector limit.The argument uses the null vector’s unit-norm bound to pass from convergence in probability to convergence of its expectation.Source: Appendix A · pp. 9–10

1) Sources of randomness:

The analysis separates the desired user’s fading-dependent signal term from the interfering-channel projector and establishes their independence. It then shows that the cross term is asymptotically negligible relative to the leading signal term as the sub-surface grows.

1) Sources of randomness:

  • The desired-user signal term depends only on that user’s fading magnitudes on its assigned RIS partition, while the interfering-channel projector depends on the other users’ channel columns.The remaining component of the desired channel also carries randomness from phase shifts designed for other users.Source: 1) Sources of randomness: • · p. 10
  • The desired-user signal term is independent of the interfering-channel projector, including the channel component whose phase shifts depend on other users’ fading.The proof conditions on the other users’ fading variables to establish that this component’s conditional distribution does not depend on them.Source: 1) Sources of randomness: • · p. 10
  • As sub-surface size grows, the cross term is asymptotically negligible relative to the leading-order signal term.The first two proof steps are exact; the final step uses the large-sub-surface approximation established earlier.Source: 1) Sources of randomness: • · p. 10
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