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

arXiv:2610.11904 · eess.SP, cs.SY, eess.SY · Submitted 2026-10-08 · Read on arXiv

Listen

Radio episode about this paper

Transcript

Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Large-Scale Partition-Based RIS Beamforming For Uplink RIS-Equipped Multi-User Systems".

Dev: The gist Large-scale partition-based RIS beamforming for uplink RIS-equipped multi-user systems provides an asymptotic,

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: Now let’s look at the core summary of "Large-Scale Partition-Based RIS Beamforming For Uplink RIS-Equipped Multi-User Systems: Asymptotic Analysis" to really understand what they are trying to solve. Basically, the paper is addressing why getting a good, closed-form solution for multi-user uplink reception with a big RIS has been so hard.

Dev: It points out that for more than two users, you don't have an explicit formula for the zero-forcing SINR anymore because it depends on this complex orthogonal projector onto the complement of the interfering users’ joint channel subspace.

Taro: That sounds like a huge hurdle when you're trying to design something; how do you design a receiver if you don't even know what the performance looks like?

Rosa: The main technical contribution they make is using an asymptotic, large-scale analysis where each sub-surface grows without bound, showing that the projector concentrates around a rank-one matrix aligned with the desired user’s channel.

Dev: That concentration argument is what lets them move past the intractable problem and derive that closed-form approximation for average per-user ZF SINR, which only depends on deterministic channel parameters.

Taro: So they are essentially saying that if you use a big enough RIS partitioned correctly, the performance settles down into a predictable pattern we can calculate ahead of time?

Rosa: Precisely; they show that this concentration allows them to treat the expression as a design objective, and then they prove equal partitions are approximately sum-rate optimal at leading order, regardless of path loss asymmetry.

Dev: They also showed that for the sum-rate R sum, it simplifies to something like K times the logarithm of some constant plus terms related to the RIS size growth, specifically showing that with total elements fixed, equal partitioning is approximately optimal when each partition gets a size proportional to L/K.

Taro: That connects the theoretical convergence result directly to a practical sizing problem for how you should distribute those elements across different users.

Rosa: It gives us a concrete starting point for designing the system, moving from abstract channel models to something with quantifiable performance metrics based on the size of your RIS.

The paper's summary: Dev: When we look at the specific improvements they propose in this paper, it’s really about simplifying how we approach the beamforming design itself. They introduce partition-based beamforming where each sub-surface R k is dedicated to User k.

Rosa: That partitioning means each sub-surface is solely responsible for cancelling the phase of its assigned user's channel, specifically using phi n = - (h kn t nk) for all elements in that partition.

Taro: So instead of one big optimization problem across the whole surface, you break it down into K smaller, independent problems, each focused on one user?

Dev: That’s right; they use this structure alongside the ZF beamforming matrix V ZF = G H G-one G H, which simplifies to just G-one in this specific case where you have K antennas and K users <ref:2610.11904#pg1>.

Rosa: The real improvement is linking that RIS phase shift design directly into the receiver's zero-forcing detection structure, making them work together in a unified low-complexity architecture.

Taro: It makes sense; it’s about using the RIS not just as an antenna enhancement layer, but as an active part of the interference cancellation mechanism itself.

Dev: The paper also provides a specific, quantifiable asymptotic approximation for the receive SINR that depends only on deterministic channel parameters, which is very useful because you don't need to guess what the random fading looks like.

Rosa: And beyond just having a formula, they provide a benchmark against real Monte Carlo simulations, showing that as the required RIS size for a given accuracy grows with the number of users.

The paper's improvements: Taro: So to wrap up these findings, what does this all mean for someone who is just listening to this show about how we should think about multi-user systems with large RIS?

Rosa: This paper gives us a closed-form asymptotic approximation for the average achievable rates of users in a large-scale RIS system using partition-based beamforming and ZF reception. It confirms that equal partitioning is approximately sum-rate optimal at leading order, regardless of path loss asymmetry.

Dev: The main implication is that we can design low-complexity receiver algorithms because we have a deterministic formula based on the channel parameters rather than dealing with the full complexity of the random fading distribution every time.

Taro: And for future work, they mentioned joint optimization and extending this partition-based beamforming to quantized phase-shifting at the RIS, which suggests they aren't done with this concept.

Rosa: Exactly; they are looking at how to jointly optimize the partition sizes and the phase shifts under a max-minoptimal achievable-rate criterion, and also exploring quantization for better practical implementation.

Dev: So that’s what we have here from "Large-Scale Partition-Based RIS Beamforming For Uplink RIS-Equipped Multi-User Systems: Asymptotic Analysis." It gives us a concrete tool to design receivers that work well in this regime.

Taro: I just think the core idea of breaking down the problem into user-dedicated subsurfaces is a really solid architectural choice for handling complexity in this area.

Rosa: Yeah, we’ll be moving on to what else is out there when we look at other papers next.

Conclusion: Rosa: So we’ve looked at "Large-Scale Partition-Based RIS Beamforming For Uplink RIS-Equipped Multi-User Systems: Asymptotic Analysis." Basically, they took a huge RIS and split it into user zones to make low-complexity beamforming work better.

Dev: Right, and the big deal is that they got this closed formula for the average SINR under zero-forcing reception. It relies on analyzing how an orthogonal projector behaves as the sub-surfaces get infinitely large.

Taro: But what does that actually mean for a system running in the real world where you have limited hardware and noisy channels?

Rosa: It means we can design receivers based on these deterministic parameters instead of needing to simulate every single random fading scenario. Plus, they showed that equal partitioning is pretty close to the best way to split those elements across users at least at the leading order.

Dev: The numbers show that if you fix the total number of RIS elements, splitting them equally makes sense for maximizing the sum-rate under these conditions. That's a good thing for loop rate stability because you have a predictable structure.

Taro: I still wonder about those caveats they mentioned regarding path loss asymmetry. Does their approximation hold up when one user's channel is way better than another's, even with the best partition?

Rosa: They address that by showing the asymptotic formula still works well enough because it focuses on the dominant terms for large systems. But they’re also already looking at ways to refine that search using greedy pairwise transfers to find partitions that are slightly better than just being perfectly equal.

Dev: That search optimization is what makes it more practical, because we don't always have the perfect symmetry in our real deployments, so finding a near-optimal partition is useful.

Taro: It sounds like the real challenge now shifts from just proving it works in theory to figuring out how to make that greedy search run fast enough on hardware.

Rosa: Exactly. So we’ve seen how partitioning helps us get a solid performance prediction for large RIS systems, and we're starting to look at ways to optimize that partition structure further.

Dev: We'll be back next time when we discuss those other papers about retrieval augmentation and how they handle test-time adaptation in robotics.

Samira Rahimian, Haris Gacanin

RWTH Aachen University

eess.SP, cs.SY, eess.SY

Submitted: 2026-10-08

Updated: 2026-10-08

The gist: The gist Large-scale partition-based RIS beamforming for uplink RIS-equipped multi-user systems provides an asymptotic, closed-form approximation for average per-user SINR under zero-forcing

Key concepts

Partition-Based RIS Beamforming
This technique divides the RIS into separate subsurfaces, where each sub-surface focuses on beamforming a specific user's signal. The phase shifts applied to elements in one partition are designed to cancel interference from other users, effectively dedicating resources to minimize inter-user interference.
Zero-Forcing (ZF) Reception
ZF is a receiver technique used to eliminate interference between multiple users at the antenna array. It uses the channel information from the transmitter and receiver to create a beamforming matrix that nulls out the signals intended for other users, improving signal quality.
Asymptotic Closed-Form Approximation
This means deriving a mathematical formula that accurately describes system performance when the number of RIS elements ($L$) becomes very large. The derived formula provides a practical, simplified estimate of the average achievable data rate without needing to perform complex, computationally intensive calculations.

Terminology

Summary

The gist Large-scale partition-based RIS beamforming for uplink RIS-equipped multi-user systems provides an asymptotic, closed-form approximation for average per-user SINR under zero-forcing reception by analyzing the convergence of an orthogonal projector as sub-surface sizes grow without bound, which is crucial for designing practical low-complexity receiver algorithms<ref:2610.11904#pg6>

System Model

The system considers a communication setup where K single-antenna users transmit information to a receiver equipped with an array of K antennas and an adjacent Reconfigurable Intelligent Surface (RIS)<ref:2610.11904#pg4> The RIS consists of L elements arranged on a uniform planar array (UPA) of M ×M elements, where L = M squared<ref:2610.11904#pg4>. The channel between the n-th RIS element and the i-th receiver antenna is modeled as free-space line-of-sight (LOS): h in = 1/2 sqrt(pi r i,n / e(-j 2π / λ r i,n), where r i,n is the Euclidean distance between the i-th receive antenna and the n-th RIS element<ref:2610.11904#pg4>. The channel between User k and the n-th RIS element is modeled as sqrt(alpha k t nk), where alpha k is the path-loss coefficient and t nk follows an independent Rayleigh fading distribution, t nk CN (0, 1)<ref:2610.11904#pg4>. The effective channel matrix representing the channel from User k to Antenna i is denoted as G = HΦT in C K×K with entries: g ik = sqrt(alpha k X L n=1 h in t nk e j phi n, where phi n is the phase-shift applied by the n-th RIS element<ref:2610.11904#pg4>.

Combined RIS Partition-Based Beamforming and ZF Detection

The paper proposes combining partition-based beamforming at the RIS with ZF beamforming at the receiver antennas to mitigate user interference<ref:2610.11904#pg4>. The RIS is partitioned into K subsurfaces, R k, where each sub-surface R k is dedicated to User k and responsible for the beamforming of the kth user’s signal<ref:2610.11904#pg4>. The partition-based RIS beamforming for elements in R k is designed to cancel the phase of the effective channel from User k to Antenna k as follows: phi n = -∠(h kn t nk), ∀n ∈ R k<ref:2610.11904#pg4>. For ZF at the receiver antennas, the beamforming matrix is given as VZF = G H G-1 GH, which reduces to VZF = G-1 for the considered case of K receive antennas and K users<ref:2610.11904#pg4>.

Performance Analysis

The receive SINR of User k can be written as γ ZF,k = P [(GHG)-1] kk<ref:2610.11904#pg4>. The paper derives an asymptotic closed-form expression for the average achievable rates of users, R bar,k = E[log2(1+γ ZF,k)]≈ log2(1+E[γ ZF,k])<ref:2610.11904#pg4>. This is achieved by analyzing the term E[g H k P−k g k] using vector decompositions involving the coherent channel gain Sk and the interference channel vector ζ k<ref:2610.11904#pg4>. The final closed-form asymptotic approximation valid as Lpk → ∞ is given by Theorem 1: γ bar,k ≈ P α k 1 - π/4 X n∈Rk h kn squared + π/4 X n∈Rk h kn 2 <ref:2610.11904#pg4>.

Sum-Rate Optimality of Equal Partitioning

Theorem 1 characterizes which partition sizes maximize the sum-rate Rsum = P k log2(1+ ¯γ ZF,k) at leading order<ref:2610.11904#pg4>. By approximating the LOS amplitude as roughly constant over its own sub-surface, h kn ≈ h bar,k for n ∈ R k, the dominant term simplifies to P n∈R k h kn squared ≈ h bar squared k L squared pk, giving γ bar,k ≈ c k L squared pk<ref:2610.11904#pg4>. The sum-rate is then approximated as Rsum ≈ X K k=1 log2(c k) + 2X K k=1 log2(Lpk)<ref:2610.11904#pg4>. This leads to Corollary 1, which states that subject to PK k=1 Lpk = L, the approximate sum-rate in (38) is maximized by equal partitioning, L pk = L/K for all k<ref:2610.11904#pg4>.

Simulation Results

Monte Carlo simulations confirm the accuracy of the closed-form expression across a range of system sizes, with the required RIS size for a given accuracy growing with the number of users<ref:2610.11904#pg4>. Fig. 2 compares the simulated (solid) and asymptotic (36) (dashed) sum-rate for K ∈ 2, 4, 8 and L ∈ 1024, 4096 under equal partitioning<ref:2610.11904#pg4>. The gap between the two decreases with L for every K: 0.21% → 0.19% for K = 2, 0.98% → 0.47% for K = 4, and 11.86% → 2.80% for K = 8<ref:2610.11904#pg4>. The search-optimal partition achieves a sum-rate of [2, 1, 0.5, 0.1] versus [67.33] bits/s/Hz for equal partitioning under real simulation<ref:2610.11904#pg4>.

Conclusion

The paper derives a closed-form asymptotic approximation for the average achievable rates of users in a large-scale RIS-equipped uplink system under partition-based RIS beamforming and ZF receive beamforming<ref:2610.11904#pg4>. Simulation results confirm the accuracy of this closed-form expression across a range of system sizes, with the required RIS size for a given accuracy growing with the number of users<ref:2610.11904#pg4>. Building on this expression, it is shown analytically that equal partitioning is approximately sum-rate-optimal at leading order, independently of path-loss asymmetry (Corollary 1), and validated a greedy pairwise-transfer search against real Monte Carlo simulation<ref:2610.11904#pg4>. Future work includes jointly optimizing the partition sizes and phase-shift design under a max-minoptimal achievable-rate criterion, and extending the partitionbased RIS beamforming to quantized phase-shifting at the RIS<ref:2610.11904#pg4>.

APPENDIX A

PROOF OF THE ASYMPTOTIC RANK-ONE PROJECTOR APPROXIMATION

The proof establishes that the null-space projector converges entrywise to the rank-one limit P−k = e k H k (equation (72)) as Lpj → ∞<ref:2610.11904#pg4>. This convergence is shown through three steps, concluding that E[P−k e k 2] → 1 as Lpj → ∞<ref:2610.11904#pg4>.

APPENDIX B

DERIVATION OF THE CROSS-TERM ASYMPTOTIC NEGLIGIBILITY

The cross term in (33) is asymptotically negligible relative to the leading-order signal term E[S 2 k] as Lpj → ∞<ref:2610.11904#pg4>. This is shown by proving that the pair (Sk, ζ k) is independent of P−k<ref:2610.11904#pg4>. The analysis shows that the order of each factor in the sum satisfies [E[Skζ k]]i = O(Lpk/sqrt(L)) and E[u ki 2] = O(L-1/2 pk)<ref:2610.11904#pg4>. Consequently, the cross term is shown to be O(L 3/2 pk / sqrt(L)!) which vanishes as Lpj → ∞<ref:2610.11904#pg4>.

Improvements for AI systems

  1. Partition-Based RIS Beamforming for Uplink Reception: The system can implement a low-complexity architecture by partitioning a large-scale RIS into K user-dedicated sub-surfaces, which precedes ZF reception at the base station, allowing for explicit, closed-form asymptotic approximations of the average per-user SINR.

  2. Asymptotic Rate Prediction: The system can utilize a derived closed-form asymptotic approximation for the average per-user ZF SINR that depends only on deterministic channel parameters, which allows for tractable design objectives independent of complex channel realizations.

  3. Adaptive Partition Search: A low-complexity greedy pairwise-transfer search can be developed to refine the partition beyond leading-order optimality, enabling the system to find sum-rate maximizing partition configurations even under asymmetric path loss scenarios.

  4. Performance Prediction Validation: The derived closed-form expression is confirmed by Monte Carlo simulations to track the exact simulated rate closely once the RIS is large relative to the number of users, validating it as an accurate performance predictor for design purposes.

  5. Near-Optimal Partition Discovery: The search algorithm can find partitions mildly perturbed from Lpk = L/K, rather than to exactly equal partitions, confirming that equal partitioning is approximately sum-rate-optimal at leading order while providing genuine, empirically verified gains over equal partitioning under realistic channels.

Related papers