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Channel Estimation for RIS-Empowered Multi-User MISO Wireless Communications

Li Wei, Chongwen Huang, George C. Alexandropoulos, Chau Yuen, Zhaoyang Zhang, Mérouane Debbah

arXiv:2008.01459v2cs.ITcs.LGeess.SP

TL;DR

RIS channel estimation is difficult because large RISs require simultaneous recovery of multiple channels under hardware constraints. The paper uses PARAFAC-based ALS and VAMP algorithms, derives an ALS CRB, and evaluates downlink sum rate with estimated channels. Simulations report favorable performance over benchmarks, ALS attainment of the CRB, and sum rates reaching the perfect-channel case.

  • Problem

    RIS channel estimation is challenging because it simultaneously involves multiple channels and large numbers of RIS elements with non-linear hardware characteristics.

  • Method

    The paper unfolds the cascaded channel model with PARAFAC decomposition and uses iterative ALS and VAMP algorithms to estimate the BS-RIS and RIS-user channels, while deriving the ALS CRB.

  • Results

    Simulation results show that the proposed algorithms outperform benchmark schemes, ALS can achieve the CRB, and estimated-channel sum rates reach the perfect-channel case under various settings.

  • Takeaways & Limitations

    PARAFAC-based ALS and VAMP provide effective channel-estimation techniques for RIS-empowered multi-user MISO uplinks, with evaluated impact on downlink sum rate.

  • Takeaways & Limitations

    The algorithms require the RIS element number to satisfy feasibility constraints and incur performance reduction to remove estimation ambiguity.

Abstract

from arXiv · show

Reconfigurable Intelligent Surfaces (RISs) have been recently considered as an energy-efficient solution for future wireless networks due to their fast and low-power configuration, which has increased potential in enabling massive connectivity and low-latency communications. Accurate and low-overhead channel estimation in RIS-based systems is one of the most critical challenges due to the usually large number of RIS unit elements and their distinctive hardware constraints. In this paper, we focus on the uplink of a RIS-empowered multi-user Multiple Input Single Output (MISO) uplink communication systems and propose a channel estimation framework based on the parallel factor decomposition to unfold the resulting cascaded channel model. We present two iterative estimation algorithms for the channels between the base station and RIS, as well as the channels between RIS and users. One is based on alternating least squares (ALS), while the other uses vector approximate message passing to iteratively reconstruct two unknown channels from the estimated vectors. To theoretically assess the performance of the ALS-based algorithm, we derived its estimation Cramér-Rao Bound (CRB). We also discuss the downlink achievable sum rate computation with estimated channels and different precoding schemes for the base station. Our extensive simulation results show that our algorithms outperform benchmark schemes and that the ALS technique achieves the CRB. It is also demonstrated that the sum rate using the estimated channels always reach that of perfect channels under various settings, thus, verifying the effectiveness and robustness of the proposed estimation algorithms.

I. INTRODUCTION

RISs offer programmable, energy-efficient wireless communication, but channel estimation is difficult because multiple large channels and hardware constraints must be handled simultaneously. This paper addresses the challenge with PARAFAC-based ALS and VAMP estimation, theoretical analysis, and downlink sum-rate evaluation.

  • RISs use programmable, nearly passive reflecting elements to support energy-efficient, high-speed, massive-connectivity, and low-latency wireless communications.Each element can alter the phase of an incoming signal without requiring a dedicated power source.
  • Channel estimation is challenging because it must jointly estimate direct BS-user channels, RIS-BS channels, and RIS-user channels, often with many non-linear RIS elements.The paper focuses on the resulting multi-channel estimation burden in RIS-empowered multi-user communications.
  • The proposed framework uses PARAFAC decomposition to unfold the cascaded channel model and applies ALS and VAMP to estimate multiple large channel matrices efficiently.The high-dimensional tensor is represented through different unfolded forms that support the two iterative algorithms.
  • The paper derives feasibility conditions, computational complexity, and the estimation CRB for the ALS-based channel-estimation algorithm.These analyses theoretically assess the proposed procedures, with the CRB specifically characterizing ALS estimation performance.
  • The study evaluates downlink achievable sum rate using estimated channels with MRT, MMSE, and ZF base-station precoding schemes.Simulation comparisons include state-of-the-art techniques and the perfect-channel-estimation case.

B. Preliminaries on the PARAFAC Decomposition

PARAFAC represents the RIS channel observations as a high-dimensional tensor with alternative matrix unfoldings. These unfoldings support iterative estimation of the unknown channels using ALS and VAMP.

  • B. Preliminaries on the PARAFAC Decomposition: PARAFAC decomposes a high-dimensional tensor into rank-one components represented by factor matrices A, B, and C.Its matrix slices can be expressed through products involving one factor and the other two factor matrices.
  • C. Received Training Symbols: The received RIS training data are organized into a three-way tensor containing P postprocessed channel matrices and their noisy observations.Orthogonal pilots and P distinct RIS configurations are assumed during channel estimation.
  • C. Received Training Symbols: Three unfolded forms of the tensor provide alternative matrix representations whose Khatri-Rao products facilitate channel estimation.The mode-1, mode-2, and mode-3 unfoldings are used to expose different channel-factor relationships.
  • C. Received Training Symbols: The proposed framework estimates the BS–RIS and RIS–user channel matrices iteratively from the unfolded received signal.The paper presents ALS and VAMP algorithms for this estimation task.
  • C. Received Training Symbols: A discrete Fourier transform matrix formed from the first P rows of an N × N Fourier matrix is selected as the RIS configuration matrix.The configuration must be feasible for the PARAFAC-based estimation algorithm.

2) Iterative Update:

The iterative ALS updates estimate the two unknown channel matrices by alternating conditional least-squares problems based on different tensor unfoldings.

  • 2) Iterative Update:: The BS–RIS-related channel is updated by minimizing a conditional least-squares objective using the mode-1 unfolded observation.The update uses the estimated factor matrix formed from the current BS–RIS estimate and RIS configuration.
  • 2) Iterative Update:: The RIS–user-related channel is estimated from the mode-2 unfolding through a corresponding least-squares objective.Its measurement matrix is constructed from the RIS configuration and the current estimate of the other channel.

3) Iteration Termination Criterion:

The ALS procedure stops after reaching a maximum iteration count or when adjacent estimates change below an NMSE threshold. VAMP instead iteratively produces posterior channel estimates from observation vectors.

  • 3) Iteration Termination Criterion:: ALS terminates when the maximum number of iterations Imax is reached or adjacent estimates have NMSE below κ.The stopping test compares estimates from neighboring iterations.
  • 3) Iteration Termination Criterion:: The ALS estimates have a scaling ambiguity at convergence that can be resolved through adequate normalization.This ambiguity arises in the iterative estimation of Hs and Hr.
  • B. VAMP Channel Estimation: VAMP replaces the least-squares steps with iterative recovery from noisy measurements and is described as robust for signal reconstruction.The approach is motivated by avoiding matrix-inverse computation.
  • B. VAMP Channel Estimation: The VAMP iteration separates denoising steps from MMSE estimation steps and outputs posterior means used to construct the channel estimate.Initialization uses prior means and variances for the unknown channel vectors.

C. Feasibility Conditions

Both PARAFAC-based channel-estimation algorithms require system parameters satisfying necessary and sufficient feasibility conditions derived from system identifiability.

  • C. Feasibility Conditions: Feasibility conditions are required for both proposed channel-estimation algorithms to yield a solution.The conditions are based on the identifiability requirements of the PARAFAC model.
  • C. Feasibility Conditions: Algorithm 3 takes a feasible RIS configuration matrix Φ as an input.The VAMP procedure also requires κ and a maximum iteration count Imax.
  • C. Feasibility Conditions: The initial Hs estimate is obtained from eigenvectors associated with the N non-zero eigenvalues of (Z2)^H Z2.This initialization is used before the iterative VAMP channel-estimation procedure.

1: Initialization:

The PARAFAC model has permutation and scaling ambiguities, so identifiability requires sufficient dimensions and normalization-related feasibility conditions. When a RIS is too large, the proposed estimators partition it into feasible sub-RIS groups.

  • Feasibility and identifiability: The channel factors are identifiable only up to permutation and scaling ambiguities under the PARAFAC model.The uniqueness result concerns the triple (Φ, Hs, Hr).
  • Feasibility and identifiability: The feasibility condition requires M,K ≥ N, while the number of training RIS phase configurations P must be ≤ N.
  • RIS partitioning: When N exceeds M or K, the RIS is partitioned into non-overlapping sub-cells whose element counts satisfy the feasibility conditions.
  • RIS partitioning: For an RIS with 64 elements and M=K=8, splitting it into eight sub-RISs of 8 elements each satisfies the stated condition.

D. Computational Complexity

The section characterizes computational costs for ALS and VAMP channel estimation and derives CRBs for the ALS estimator. VAMP avoids matrix inversions, while ALS provides a theoretically optimal benchmark under the stated signal model.

  • Computational Complexity: ALS complexity is dominated by matrix inverse computations, whereas VAMP is dominated by matrix-vector multiplications.
  • Computational Complexity: The total VAMP complexity is approximated as O((M +K)(5N 2 −N)) and is lower than ALS because it avoids matrix inversion.
  • CRAMÉR-RAO BOUND ANALYSIS: ALS yields the maximum-likelihood estimate for the zero-mean AWGN model, which is asymptotically unbiased.
  • CRAMÉR-RAO BOUND ANALYSIS: To remove scaling ambiguity, Hs is fixed so that its first column equals 1N, reducing the unknown complex-parameter count to (K + M −1)N.
  • CRAMÉR-RAO BOUND ANALYSIS: The CRB is derived for the unknown channels in the trilinear model using the unbiased ALS estimator.

V. SUM RATE PERFORMANCE COMPUTATION

The paper computes downlink achievable sum rates from reciprocal uplink channel estimates using MRT, ZF, and MMSE precoding. Channel-estimation errors are incorporated into the received-signal model as interference-plus-noise.

  • SUM RATE PERFORMANCE COMPUTATION: Downlink sum rates are evaluated with MRT, ZF, and MMSE precoding using channels estimated from the proposed algorithms.
  • SUM RATE PERFORMANCE COMPUTATION: The downlink channel is obtained through uplink reciprocity, and the estimated cascade channel bH=bHrΦ bHs is used to design BS precoding vectors.
  • SUM RATE PERFORMANCE COMPUTATION: Channel-estimation errors are represented by Er and Es, with their combined effect entering the actual channel through E=ErΦ bHs+bHrΦEs−ErΦEs.
  • SUM RATE PERFORMANCE COMPUTATION: With estimated channels, the intended signal is separated from remaining terms that are treated as interference-plus-noise.

A. The estimation of phase matrix

The phase matrix is estimated from the estimated channels by solving a unit-modulus optimization problem. A fixed-point iteration produces the phase vector and diagonal RIS phase matrix with lower computational complexity.

  • The estimation of phase matrix: The phase matrix is designed using estimated channels rather than separately optimized for each precoding matrix.
  • The estimation of phase matrix: The phase-design optimization constrains every phase coefficient to unit modulus, |φn|2=1 or |vn|2=1.
  • The estimation of phase matrix: For each user, the effective channel expression is rewritten using v=[φ1,...,φN]H and Ck=diag(hrk)Hs.
  • The estimation of phase matrix: A fixed-point iteration solves the optimization problem with lower computational complexity, and the resulting vector defines Φ=diag(bv).

B. MRT Precoding

The MRT section defines the precoding vector using the estimated effective channel and gives the resulting achievable rate, alongside the perfect-channel counterpart.

  • With MRT precoding, the BS sets each precoding vector equal to the conjugate of the corresponding effective channel vector.
  • Under perfect channel estimation, the BS instead sets each precoding vector equal to the corresponding effective channel.

C. ZF Precoding

The ZF section formulates estimated-channel and perfect-channel precoding and achievable-rate expressions, then evaluates channel-estimation accuracy under varied SNR, training, and system dimensions.

  • C. ZF Precoding: ZF precoding is designed to eliminate interference among different users.
  • C. ZF Precoding: The k-th column of the ZF precoding matrix is the vector applied to user k’s symbol, with rates computed for estimated and perfect channels.
  • C. ZF Precoding: The MMSE section similarly defines a precoding matrix and achievable rates for estimated and perfect channel knowledge.
  • A. NMSE of CE: 2.5 dB separates each proposed estimator from genie-aided LS, while at SNR = 20 dB the proposed NMSE is 0.002 versus LSKRF’s 0.02 for P = 14.
  • A. NMSE of CE: Increasing N causes performance loss because larger channel matrices require more training pilots and computational complexity for ALS estimation.
  • A. NMSE of CE: Increasing P improves NMSE, with P = 40 best in the figure, but gains slow above P > 32 and complexity motivates P < 64.

B. CRB of the Proposed ALS CE

The paper evaluates ALS estimation against CRBs and studies how estimator settings affect downlink sum rate under MRT, MMSE, and ZF precoding.

  • B. CRB of the Proposed ALS CE: For M = K = T = 32, P = 8, and N = {8, 16}, ALS estimation reaches the respective CRBs when SNR≥2 dB.
  • B. CRB of the Proposed ALS CE: Increasing N degrades estimation because it increases the number of unknown variables and channel-matrix complexity, although the gap narrows at higher SNR.
  • C. Downlink Sum Rate: MMSE performs best across the tested SNR range, while MRT is close at low SNR and ZF is close at high SNR.
  • C. Downlink Sum Rate: For MRT, the estimated-channel sum-rate gap from perfect CE vanishes when SNR ≥4 dB in the tested setting.
  • C. Downlink Sum Rate: Increasing P improves achievable sum rate with ZF precoding for P = {16, 24, 32}, with narrower performance gaps at higher SNR.
  • C. Downlink Sum Rate: With varying N, ZF sum rate degrades at high SNR, whereas MRT improves before reaching a floor when SNR ≥0 dB.
  • VII. CONCLUSION: The proposed ALS and VAMP estimators have similar performance, while VAMP has lower complexity than ALS and both are constrained by RIS-element feasibility conditions.
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