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Intelligent Reflecting Surface Assisted Multi-User OFDMA: Channel Estimation and Training Design

Beixiong Zheng, Changsheng You, Rui Zhang

arXiv:2003.00648v3cs.IT

TL;DR

Accurate CSI is difficult to acquire in IRS-assisted multi-user OFDMA systems because passive elements cannot transmit pilots and channel parameters grow with IRS size and user count. The paper proposes parallel and sequential estimation schemes with optimized pilot allocations and IRS reflection patterns, deriving training and user-support limits. Simulations verify the designs and report up to 13 dB SNR gain for the proposed SeUCE pilot allocation over a permuted benchmark.

  • Problem

    Acquiring accurate CSI for IRS passive beamforming is challenging because passive elements cannot transmit pilots and the number of channel parameters increases with reflecting elements and users.

  • Method

    The paper proposes SiUCE for arbitrary frequency-selective channels and SeUCE for LoS-dominant user-IRS channels, optimizing pilot tone allocations and time-varying IRS reflection patterns.

  • Results

    Up to 13 dB SNR gain is achieved by the proposed SeUCE pilot tone allocation over the permuted pilot tone allocation benchmark.

  • Takeaways & Limitations

    SeUCE supports more users than SiUCE by exploiting the common IRS-AP channel, at the expense of higher complexity and some degraded channel-estimation performance.

Abstract

from arXiv · show

To achieve the full passive beamforming gains of intelligent reflecting surface (IRS), accurate channel state information (CSI) is indispensable but practically challenging to acquire, due to the excessive amount of channel parameters to be estimated which increases with the number of IRS reflecting elements as well as that of IRS-served users. To tackle this challenge, we propose in this paper two efficient channel estimation schemes for different channel setups in an IRS-assisted multi-user broadband communication system employing the orthogonal frequency division multiple access (OFDMA). The first channel estimation scheme, which estimates the CSI of all users in parallel simultaneously at the access point (AP), is applicable for arbitrary frequency-selective fading channels. In contrast, the second channel estimation scheme, which exploits a key property that all users share the same (common) IRS-AP channel to enhance the training efficiency and support more users, is proposed for the typical scenario with line-of-sight (LoS) dominant user-IRS channels. For the two proposed channel estimation schemes, we further optimize their corresponding training designs (including pilot tone allocations for all users and IRS time-varying reflection pattern) to minimize the channel estimation error. Moreover, we derive and compare the fundamental limits on the minimum training overhead and the maximum number of supportable users of these two schemes. Simulation results verify the effectiveness of the proposed channel estimation schemes and training designs, and show their significant performance improvement over various benchmark schemes.

I. INTRODUCTION

The paper addresses the difficulty of acquiring accurate CSI in IRS-assisted multi-user OFDMA systems, where passive IRS elements cannot transmit pilots and training overhead grows with users and reflecting elements. It proposes two channel-estimation schemes with optimized pilot allocations and IRS reflection patterns, and characterizes their training and user-support limits.

  • Motivation: Accurate CSI is essential for realizing IRS passive beamforming gains, but passive elements cannot transmit or receive pilot signals for conventional channel estimation.The AP therefore estimates cascaded user-IRS-AP channels using user pilots and time-varying IRS reflections.
  • Motivation: User-by-user successive estimation can incur prohibitive training overhead that scales with the number of users and may exceed finite channel coherence time.This motivates multi-user channel estimation methods tailored to OFDMA and frequency-selective fading.
  • Proposed schemes: The SiUCE scheme estimates all users’ CSI in parallel at the AP under arbitrary frequency-selective fading channels.Its training design jointly optimizes user pilot tone allocations and the IRS reflection pattern to minimize channel estimation error, with a closed-form optimum.
  • Proposed schemes: The SeUCE scheme targets LoS-dominant user-IRS channels by exploiting the common IRS-AP channel shared by all users.It estimates a reference user first, then recovers other users through normalized effective user-IRS channels; its design separates reference-user optimization from subsequent pilot allocation.
  • Comparison: Under equal channel training time, SeUCE supports more users than SiUCE but has higher estimation complexity and somewhat degraded channel-estimation performance.The paper compares both schemes using complexity, supportable users, and per-user minimum training overhead, while evaluating optimized designs against benchmarks.

III. SIMULTANEOUS-USER CHANNEL ESTIMATION AND TRAINING DESIGN FOR ARBITRARY CHANNELS

For arbitrary channels, the paper introduces SiUCE, which estimates all users’ CSI simultaneously at the AP and derives its training limits and optimal joint training design.

  • Simultaneous-user channel estimation: SiUCE estimates the CSI of all users in parallel at the AP for arbitrary channels in an IRS-assisted multi-user OFDMA system.The scheme’s minimum training time, maximum supportable users, and optimal pilot-allocation and IRS-reflection design are derived.

A. Channel Estimation and Maximum Number of Supportable Users

The SiUCE scheme separates users through disjoint pilot tones and estimates their cascaded channels using time-varying IRS reflections. Unique recovery requires sufficient pilot sub-carriers and training symbols, yielding a minimum training time and a user-support limit.

  • SiUCE channel estimation: Disjoint pilot-tone allocations decouple the received signals of different users at the AP.Each user’s selected sub-carriers isolate its signal from the others.
  • SiUCE channel estimation: The LS channel estimates are obtained using pseudo-inverses of the pilot-design matrix and IRS reflection-pattern matrix.The required CSI can be recovered when the pilot matrix has full column rank and the reflection matrix has full row rank.
  • Training requirements: Each user requires at least L assigned sub-carriers, while the training duration must cover the direct link and M reflecting links.These rank conditions ensure a unique channel-estimation solution when suitable designs exist.
  • Training requirements: The minimum training time is τmin = M + 1, and the maximum supportable-user count is determined by the available N sub-carriers and per-user pilot allocation.The SiUCE limit K1 is obtained from the disjoint pilot-tone constraint.

B. Optimal Training Design

The SiUCE training design minimizes estimation error through an orthogonal unit-modulus IRS reflection pattern and equispaced, disjoint pilot tones. This design also avoids matrix inversion and achieves the illustrated support limit under the minimum training duration.

  • Optimization formulation: The joint pilot-allocation and IRS-reflection optimization is non-convex because of binary, unit-modulus, and objective-function constraints.The paper nevertheless derives an optimal solution for the SiUCE design problem.
  • Optimal design: The optimal IRS reflection pattern is orthogonal and satisfies ΞΞH = (M + 1)I_M+1.Every reflection coefficient also satisfies the unit-modulus constraint.
  • Optimal design: The optimal pilot tones for each user are equispaced, use at least L tones, and remain disjoint across users.Equal pilot-tone counts are used across users for fairness.
  • Implementation: The proposed design uses an (M + 1) × (M + 1) DFT reflection matrix and equispaced pilot tones.The resulting structure dispenses with matrix inversion and reduces implementation complexity.
  • Illustrative example: For N = 9, M = 3, and Lp = L = 3, the minimum training time is τmin = 4 and the SiUCE scheme supports K1 = 3 users.This example illustrates the equispaced pilot-tone allocation shown in Fig. 2.

IV. SEQUENTIAL-USER CHANNEL ESTIMATION AND TRAINING DESIGN FOR LOS DOMINANT USER-IRS CHANNELS

For LoS-dominant user-IRS channels, SeUCE exploits the common IRS-AP channel by estimating one reference user and recovering the remaining users through normalized user-IRS channels. This reduces the channel parameters requiring estimation and can support more users than SiUCE.

  • Scheme overview: SeUCE targets LoS-dominant user-IRS channels and estimates one reference user before recovering the remaining users.Its minimum training overhead, supportable-user limit, and training design are derived for this setting.
  • Channel assumption: For L2 = 1, or by retaining only the strongest LoS path when L2 > 1, SeUCE ignores NLoS paths and treats them as noise to reduce complexity.The approximation is motivated by typically short user-to-IRS distances.
  • Common-channel exploitation: The normalized channel relation Qk = G diag(uk) = G diag(u1)(diag(u1))−1diag(uk) enables recovery of each user’s cascaded channel from the reference channel.The normalized user-IRS vector is represented by ak, with a1 = 1M×1.
  • Channel-parameter reduction: When Ld ≥ Lr, SeUCE reduces the required channel coefficients to LM + (K − 1)M + KL by exploiting the common IRS-AP channel.SiUCE instead estimates the cascaded reflecting channels for all users.
  • Estimation procedure: The SeUCE pipeline estimates Q1 and d1 for the reference user, estimates ak for other users, and reconstructs each Qk.This follows from the representation hk = Q1Θak + dk.
  • Training design: The SeUCE training design is optimized in two stages: first for the reference user and IRS pattern, then jointly for the remaining users.The staged design addresses the coupling between reference-user and non-reference-user estimation.

A. Channel Estimation and Optimal Training Design for Reference User

The supplied passage identifies the pilot-tone index set for the reference user and assumes that its allocation is identical across time slots.

  • Reference-user pilot allocation: The reference user’s pilot tones are indexed by J1, which remains identical across different time slots.This defines the pilot allocation used for reference-user channel estimation.

1) Channel Estimation:

The reference user's received pilot signals are collected across training slots and used to obtain least-squares estimates of its direct and cascaded channels. Full-column-rank conditions and an orthogonal, unit-modulus IRS reflection pattern support minimum-MSE estimation.

  • Channel Estimation: The reference user's received pilot signals are collected over the training slots to form the observation matrix used for channel estimation.
  • Channel Estimation: Left- and right-multiplying the collected observations yields least-squares estimates of the direct channel and cascaded channels for the reference user.
  • Channel Estimation: The reference-user estimator requires at least L pilot tones so that its estimation matrix has full column rank.
  • Channel Estimation: An orthogonal IRS reflection matrix with unit-modulus entries and equispaced pilot tones achieves the minimum reference-user channel-estimation MSE.
  • Channel Estimation: The corresponding minimum MSE is given by εref = σ2N.

B. Channel Estimation for Non-reference Users and Maximum Number of Supportable Users

For non-reference users, the SeUCE procedure estimates normalized user-IRS channels after estimating a reference user's CSI, using disjoint pilot allocations and least-squares recovery. Its training requirements determine the supportable-user limit, which is no smaller than that of SiUCE.

  • Channel Estimation for Non-reference Users: After estimating the reference user's CSI, SeUCE estimates each non-reference user's normalized user-IRS channel to recover its cascaded reflecting channel.
  • Channel Estimation for Non-reference Users: Disjoint pilot-tone allocations isolate each non-reference user's received signal across M + 1 training slots for subsequent estimation.
  • Channel Estimation for Non-reference Users: Least-squares estimates of the normalized user-IRS and direct channels are obtained by left-multiplying the received vector with the left pseudo-inverse of Ck.
  • Maximum Number of Supportable Users: Each non-reference user requires at least M + L pilot tones, while full-column-rank recovery additionally depends on the stated rank condition for Ck.
  • Maximum Number of Supportable Users: The SeUCE supportable-user limit follows by combining the pilot-tone and reference-user requirements, with equality under ζk = PM+1 and |J1| = L.
  • Maximum Number of Supportable Users: SeUCE supports at least as many users as SiUCE; equality occurs when L = 1 or L = N, with L = 1 corresponding to frequency-flat single-path links.
  • Limitation: With non-negligible user-IRS multipath delay spread, exact recovery of non-reference channels through the common IRS-AP channel remains challenging because of channel convolution.

C. Pilot Tone Allocation for Non-reference Users

The SeUCE design optimizes pilot tone allocations for non-reference users under rank and training constraints, using an MSE-based combinatorial formulation and a general allocation rule. In an example, SeUCE supports more users than SiUCE at the same minimum training time, while its proposed allocation performs better as user count grows.

  • Rank and feasibility: The pilot allocation must ensure each non-reference user's matrix D_k is full rank so normalized user-IRS and direct user-AP channels can be jointly estimated.The stated conditions separately support estimating each channel component without interference and jointly estimating both components.
  • Optimization formulation: Problem (P2) is a non-convex combinatorial optimization because pilot allocations are binary, and its objective lacks a tractable closed form.The difficulty also reflects unavailable distribution knowledge of Q_1 and matrix inversion in the MSE objective.
  • Allocation design: The design first uses brute-force searches in small systems to identify minimum-MSE allocation patterns, then provides an allocation applicable to arbitrary N and M.The general construction distributes pilot tones across M + 1 time slots and remaining unassigned sub-carriers.
  • Performance comparison: For N = 9, M = 3, and L = 3, SeUCE supports K2 = 5 users with minimum training time τ_min = 4, versus K1 = 3 for SiUCE.The comparison uses the same minimum training time, τ_min = M + 1 = 4.
  • Scheme selection: SiUCE is preferred for 1 ≤ K ≤ K1, whereas SeUCE supports K1 + 1 ≤ K ≤ K2 at the cost of higher complexity.This is the paper's stated selection rule for the two proposed schemes.

V. SIMULATION RESULTS

Simulations evaluate channel-estimation performance under specified OFDMA settings and compare proposed training designs with heuristic pilot-allocation and IRS-reflection benchmarks. The proposed designs generally reduce normalized MSE, with performance depending on SNR, user count, and Rician interference.

  • Simulation setup: The simulations use an IRS with 128 reflecting elements divided into 8 sub-surfaces and evaluate uplink training over 9 OFDM symbols.Each OFDM symbol has 16 sub-carriers.
  • Benchmark designs: The benchmark pilot allocations use adjacent tones for SiUCE and permuted tones for SeUCE, while reflection benchmarks use ON/OFF or random phase patterns.The proposed alternative is equispaced pilot allocation with a DFT-based IRS reflection pattern.
  • SiUCE results: The proposed equispaced pilot allocation and DFT-based reflection pattern jointly achieve the minimum SiUCE MSE among the compared designs.The theoretical MSE analysis agrees with simulation results.
  • SeUCE results: 13 dB SNR gain is achieved by the proposed SeUCE pilot allocation over the permuted benchmark under the stated settings.The passage attributes this gain to a smaller matrix condition number and correspondingly lower MSE.
  • SeUCE results: As the Rician factor increases from 0 to 20 dB, SeUCE normalized MSE decreases drastically before approaching an error floor from 20 to 40 dB.The error is NLoS-interference-limited below 20 dB and noise-limited above 20 dB; the proposed allocation outperforms the permuted benchmark throughout.
  • User-scaling results: With SiUCE, proposed equispaced-allocation MSE is invariant to user count, whereas adjacent-allocation MSE increases dramatically; with SeUCE, both increase but the proposed design performs better for larger user counts.These comparisons use SNR = 10 dB, L1 = 4, and L2 = 1.

VI. CONCLUSIONS

The conclusion presents two efficient uplink channel-estimation schemes for IRS-assisted multi-user OFDMA and compares their supportability, complexity, overhead, and estimation performance. It reports optimized training designs, derived limits, and simulation-based validation against heuristic benchmarks.

  • VI. CONCLUSIONS: The paper proposes SiUCE and SeUCE for different channel setups in IRS-assisted multi-user OFDMA.SeUCE exploits the common IRS-AP channel shared by users.
  • VI. CONCLUSIONS: SeUCE supports more users than SiUCE at the expense of higher channel-estimation complexity and some degraded channel-estimation performance.The comparison is made under the schemes' respective channel setups.
  • VI. CONCLUSIONS: The paper optimizes user pilot-tone allocations and IRS reflection patterns to minimize channel-estimation error.It also derives minimum training overhead and maximum supportable-user limits.
  • VI. CONCLUSIONS: Simulation results demonstrate the effectiveness of the proposed schemes and training designs compared with heuristic benchmark schemes.The conclusion also states that the essential approaches and design methods can be extended to downlink estimation.

APPENDIX

The appendix derives the optimal joint training design for SiUCE by separating IRS reflection-pattern and pilot-tone allocation optimization. It identifies an orthogonal DFT-based reflection pattern and equispaced pilot tones as optimal choices under the stated formulation.

  • SiUCE training optimization: The SiUCE training-design objective can be decomposed into separate IRS reflection-pattern and pilot-tone allocation subproblems.This follows because the joint optimization decouples under the derived objective.
  • IRS reflection pattern: An IRS reflection pattern satisfying ΞΞH = (M + 1)I_M+1 minimizes the reflection-pattern subproblem.The required pattern is orthogonal with unit-modulus entries.
  • IRS reflection pattern: The (M + 1) × (M + 1) DFT matrix is an optimal IRS reflection pattern for the subproblem.It meets the orthogonality and unit-modulus requirements.
  • Pilot tone allocation: The pilot-allocation subproblem is equivalent to MSE minimization for traditional multi-user OFDMA.Its minimum MSE is achieved with equispaced pilot tones and |J_k| ≥ L.
  • Optimality result: Combining the optimal solutions of the two subproblems yields the results stated in Proposition 1.The appendix concludes the corresponding proof after combining both solutions.
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