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Parallel Factor Decomposition Channel Estimation in RIS-Assisted Multi-User MISO Communication
Li Wei, Chongwen Huang, George C. Alexandropoulos, Chau Yuen
TL;DR
RIS channel estimation is difficult because large passive surfaces create substantial training demands and hardware constraints. The paper applies PARAFAC decomposition with ALS to iteratively estimate the BS–RIS and RIS–user channels. Simulations report better performance than a benchmark LS scheme and show sensitivity to RIS size and training-symbol count.
Problem
Channel estimation in RIS-assisted multi-user communications is challenging because of massive training overhead and nonlinear hardware constraints.
Source: §I · p. 1Method
The method uses PARAFAC decomposition and an ALS algorithm to iteratively estimate the BS–RIS and RIS–user channels.
Source: §I · p. 1, B. Decomposition of the Received… · p. 2, A. Proposed Algorithm · p. 3Results
Simulation results show that the proposed channel estimation outperforms a benchmark LS scheme.
Source: §V · p. 4Takeaways & Limitations
RIS unit-element count and training-symbol count significantly affect the proposed channel-estimation algorithm.
Source: §V · p. 4
Abstract
from arXiv · showhide
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 enabling massive connectivity and low latency communications. Channel estimation in RIS-based systems is one of the most critical challenges due to the large number of reflecting unit elements and their distinctive hardware constraints. In this paper, we focus on the downlink of a RIS-assisted multi-user Multiple Input Single Output (MISO) communication system and present a method based on the PARAllel FACtor (PARAFAC) decomposition to unfold the resulting cascaded channel model. The proposed method includes an alternating least squares algorithm to iteratively estimate the channel between the base station and RIS, as well as the channels between RIS and users. Our selective simulation results show that the proposed iterative channel estimation method outperforms a benchmark scheme using genie-aided information. We also provide insights on the impact of different RIS settings on the proposed algorithm.
I. INTRODUCTION
RISs are presented as low-power metasurfaces for future wireless communication, while channel estimation remains difficult because of large-scale passive hardware and training requirements. The paper proposes PARAFAC-based iterative estimation for the cascaded channels in a RIS-assisted multi-user MISO system.
I. INTRODUCTION
- RISs use passive reflecting elements to adjust incoming-signal phases without dedicated power amplifiers, supporting energy-efficient and low-latency communication –[9].Source: §I · p. 1
- Channel estimation must recover BS–user, BS–RIS, and RIS–user channels despite massive training overhead and nonlinear RIS hardware constraints –.Source: §I · p. 1
- Existing approaches either require extensive offline training, active RIS elements, or additional energy consumption, while some frequency-domain methods do not explicitly estimate separate RIS-based channels [19]–[21].Source: §I · p. 1
- The proposed method uses PARAFAC decomposition and ALS to iteratively estimate the BS–RIS channel and RIS–user channels without a priori channel information.Source: §I · p. 1
II. SYSTEM AND SIGNAL MODELS
- The system model considers a base station serving multiple single-antenna users through a RIS-assisted downlink.Source: Figure 1 · p. 2, §II · p. 2
A. System Model
The system model represents RIS-assisted downlink observations across users, antennas, time slots, and RIS configurations. PARAFAC unfolds the cascaded channel into matrix products that facilitate estimation of the unknown channel factors.
A. System Model
- A discrete-element RIS mediates downlink communication from an M-antenna base station to K single-antenna users, while the direct path is neglected.The RIS contains N configurable unit cells and observations span T time slots and P phase configurations.Source: A. System Model · p. 2
- H1 and H2 denote the RIS–BS and users–RIS channel matrices, while X carries the transmitted pilots and Wp represents postulated AWGN.Channel estimation requires T ≥ M, and the RIS phase configurations are stored in Φ.Source: A. System Model · p. 2
B. Decomposition of the Received Training Signal
- After pilot removal, each noiseless end-to-end channel entry is a sum of products involving H2, H1, and the corresponding RIS phase coefficient.The factorization couples user, RIS-element, base-station, and configuration indices.Source: B. Decomposition of the Received… · p. 2
- PARAFAC represents the received channel slices as tensor unfoldings whose matrix products combine Khatri–Rao factors with individual channel matrices.These forms create the structure used by the subsequent ALS estimator.Source: B. Decomposition of the Received… · p. 2
III. PROPOSED ITERATIVE CHANNEL ESTIMATION
The paper introduces an iterative algorithm to estimate all involved channel matrices at the receiver and analyzes when the procedure is feasible.
III. PROPOSED ITERATIVE CHANNEL ESTIMATION
- The proposed procedure iteratively estimates all involved channel matrices at the receiver.Source: §III · p. 2
- The section also discusses feasibility conditions for applying the iterative estimation algorithm.Source: §III · p. 2
- The algorithm is framed as a receiver-side channel-estimation method rather than a phase-optimization procedure.Source: §III · p. 2
A. Proposed Algorithm
The proposed ALS method stacks the pilot-processed observations into a three-dimensional tensor and uses its noiseless unfolded forms for iterative channel estimation.
A. Proposed Algorithm
- The observation tensor eZ ∈ C^{K×M×P} stacks the P pilot-processed matrices across its third dimension.A corresponding tensor fW stacks the postprocessed noise matrices.Source: A. Proposed Algorithm · p. 3
- The noiseless tensor Z is obtained through the unfolded forms and supplies the structure used by the ALS-based iterative estimator.Source: A. Proposed Algorithm · p. 3
- Algorithm 1 summarizes the subsequent steps for estimating the channel matrices from the stacked observations.Source: A. Proposed Algorithm · p. 3
1) First Step (Initialization):
The algorithm initializes channel estimates from eigenvector matrices associated with non-zero eigenvalues of noisy tensor unfoldings.
1) First Step (Initialization):
- The initialization relies on eigenvector matrices derived from noisy versions of the tensor’s mode-1 and mode-2 forms.Source: 1) First Step (Initialization) · p. 3
- The initial estimate of H2 is obtained from the eigenvector matrix associated with the N non-zero eigenvalues of the noisy mode-2 unfolding.Source: 1) First Step (Initialization) · p. 3
- The initial estimate of H1 uses the eigenvector matrix associated with the N non-zero eigenvalues of the noisy mode-1 unfolding.Source: 1) First Step (Initialization) · p. 3
2) Second and Third Steps (Iterative Update):
The second and third steps alternately estimate H2 and H1 by minimizing conditional least-squares objectives based on tensor unfoldings.
2) Second and Third Steps (Iterative Update):
- The algorithm estimates H2 by minimizing a conditional least-squares objective using the mode-1 unfolding and A1 = H1^T ◦ Φ.Source: 2) Second and Third Steps (Iterative… · p. 3
- The two channel estimates are obtained iteratively by alternating between the conditional least-squares updates.Source: 2) Second and Third Steps (Iterative… · p. 3
- It then estimates H1 using the mode-2 unfolding and A2 = Φ ◦ H2 in a corresponding least-squares objective.Source: 2) Second and Third Steps (Iterative… · p. 3
3) Fourth Step (Iteration Stop Criterion):
After initialization and alternating updates, the algorithm stops when the estimate change is sufficiently small or the maximum iteration count is reached.
3) Fourth Step (Iteration Stop Criterion):
- The iterative channel-estimation algorithm terminates when the change between successive H2 estimates satisfies the specified Frobenius-norm threshold.Source: 3) Fourth Step (Iteration Stop… · p. 3
- If the stopping condition is not met, the algorithm increments the iteration index and repeats the update steps until Imax is reached.Source: 3) Fourth Step (Iteration Stop… · p. 3
- Algorithm 1 initializes a feasible phase matrix and an eigenvector-based H1 estimate before alternating H2 and H1 updates.Source: 3) Fourth Step (Iteration Stop… · p. 3
B. Feasibility Conditions
PARAFAC-based estimation requires identifiable system dimensions; when the RIS exceeds those dimensions, the proposed procedure must be applied to partitioned groups of cells.
B. Feasibility Conditions
- The ALS solution requires normalization to resolve scaling ambiguity and system parameters satisfying necessary and sufficient identifiability conditions.Source: B. Feasibility Conditions · p. 3
- For the stated PARAFAC model, M and K must be at least N, while the number of training configurations P must not exceed N.Source: B. Feasibility Conditions · p. 3
- When N exceeds M or K, the RIS is partitioned into non-overlapping cell groups whose sizes satisfy the feasibility conditions.Source: B. Feasibility Conditions · p. 3
- The algorithm is then applied separately to each group, and the resulting channel portions are combined; the general case is deferred to an extended version.Source: B. Feasibility Conditions · p. 3
IV. PERFORMANCE EVALUATION RESULTS
The simulations evaluate NMSE versus SNR for the proposed ALS-based estimator, comparing it with LS benchmarks and varying RIS size and training configurations.
Simulation setup
- NMSE curves are averaged over 2000 independent Monte Carlo channel realizations using a DFT training matrix and fixed ALS parameters.The simulations use ϵ = 10^-5 and Imax = 20 for the presented curves.Source: §IV · p. 4
Benchmark comparison
- The proposed algorithm achieves performance comparable to LS estimation with genie-aided channel knowledge, with approximately 4 dB and 2.5 dB gaps for H1 and H2, respectively.The smaller H2 gap is attributed to H2 being square with fewer unknown elements than H1.Source: §IV · p. 4
RIS size
- Increasing the number N of RIS unit elements causes increasing NMSE loss and higher ALS computational complexity.Larger N increases the numbers of rows and columns in H1 and H2.Source: §IV · p. 4, Figure 3 · p. 4
Training configurations
- Increasing the number P of RIS training configurations improves NMSE, with P = 40 performing best under the evaluated setting.The gains between adjacent curves decrease as P increases, while larger P also raises computational complexity and must satisfy P ≤64 here.Source: §IV · p. 4, Figure 4 · p. 4
V. CONCLUSION
The paper concludes with an iterative ALS-based channel estimator that exploits PARAFAC decomposition to estimate the BS–RIS and RIS–user channels. Simulations show improved performance over LS, while RIS elements and training symbols materially affect performance.
Conclusion
- The proposed iterative ALS method estimates BS–RIS and RIS–user channels in closed form at each step and outperforms a benchmark LS scheme.The method's feasibility conditions and the effects of RIS unit elements and training symbols are also investigated.Source: §V · p. 4