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Simultaneous Direct and Indirect Channel Estimation for RIS-Assisted MIMO Communications
Daniel V. C. de Oliveira, Daniel C. Alcantara, Fazal. E Asim, Andre L. F. de Almeida, Gabor Fodor
TL;DR
RIS channel estimation must account for both the direct BS–UE path and the RIS-assisted path, while prior two-stage estimation can add delay and propagate errors. This paper proposes TenDICE, a tensor-based method that jointly estimates both links from received pilots. The reported comparisons show better performance than LS and E-TALS and similar performance to KRF while estimating the direct channel.
Problem
RIS channel estimation must account for direct and indirect paths, while prior two-stage estimation can add delay and propagate errors between estimates.
Source: §I · pp. 1–2Method
TenDICE models received pilots with an extended third-order PARAFAC tensor and estimates the involved direct and indirect channels jointly.
Source: §I · p. 2, §II · p. 2, §III · p. 3Results
TenDICE outperforms LS and E-TALS and performs similarly to KRF while simultaneously estimating the direct channel.
Source: §V · pp. 4–5Takeaways & Limitations
The reported results support simultaneous direct- and indirect-channel estimation with performance comparable to KRF and better than LS and E-TALS.
Source: §V · p. 4, §VI · p. 5The proposed model is evaluated with unstructured channels.
Source: §VI · p. 5
Abstract
from arXiv · showhide
This paper introduces an enhanced channel estimation method for reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) communications. Considering the presence of both the direct link from the base station (BS) to the user equipment (UE) and the indirect link \textit{via} the RIS, we formulate a tensor-based receiver for joint direct and indirect channel estimation called the TenDICE algorithm. The proposed receiver relies on an extended parallel factor analysis (PARAFAC) tensor model for the received pilot signals, from which the involved (direct and indirect link) channels are estimated using a simple alternating least squares scheme. The proposed TenDICE method is compared with the state-of-the-art least squares (LS) method, Khatri-Rao factorization (KRF) method, enhanced trilinear alternating least squares (E-TALS), and the theoretical Cramér-Rao lower bound (CRLB) based on normalized mean square error (NMSE) and spectral efficiency (SE). The proposed TenDICE outperforms the LS and E-TALS methods and performs similarly to the KRF methods while estimating the direct channel.
I. INTRODUCTION
Channel estimation is critical to realizing RIS benefits, yet prior RIS tensor methods often assume the direct path is blocked, while two-stage estimation can add delay and propagate errors. The paper proposes a single-stage tensor approach for jointly estimating direct and indirect links in downlink RIS-assisted MIMO.
I. INTRODUCTION
- Channel estimation is critical to obtaining the full benefits of RIS, which has been studied as a means to improve spectral efficiency and coverage.A passive RIS manipulates electromagnetic waves to improve the UE's SNR.Source: §I · p. 1
- Tensor decompositions exploit multidimensional signal structure and can provide uniqueness properties not always covered by matrix estimation methods.The introduction identifies PARAFAC and PARATUCK as examples used in wireless and MIMO communications.Source: §I · p. 1
- Prior tensor-based RIS channel-estimation studies commonly considered only transmitter-to-RIS and RIS-to-receiver paths, assuming the direct path was blocked.The cited E-TALS method estimates the direct channel with the RIS off, then estimates the indirect channel in a second stage.Source: §I · pp. 1–2
- Two-stage channel estimation can increase processing and decoding delay and propagate first-stage errors into the second-stage estimate.These drawbacks can limit performance when channel coherence times are shorter.Source: §I · p. 2
- TenDICE jointly estimates direct and indirect channels in one stage by modeling their received-signal contributions as a third-order PARAFAC tensor.The paper describes this as the first tensor model to estimate both links simultaneously in RIS-assisted MIMO.Source: §I · p. 2
Notations, definitions, and useful properties:
The section establishes notation for common matrix and tensor operations, then introduces tensor unfolding, products, and the PARAFAC representation used in the paper.
Notations, definitions, and useful properties:
- The paper uses lower case for scalars, bold lowercase for vectors, bold uppercase for matrices, and calligraphic letters for tensors.It also defines diag(x), vec(X), and unvec(x) as diagonal-matrix, vectorization, and inverse-vectorization operations.Source: Notations, definitions, and useful… · p. 2
- The n-mode unfolding converts a tensor into a matrix, and the n-mode product applies a matrix along one tensor mode.The section denotes the unfolding by [X]n and the product by ×n.Source: Notations, definitions, and useful… · p. 2
- A third-order PARAFAC tensor is represented as a sum of rank-one outer products or equivalently by factor matrices and an identity core tensor.The factor matrices are denoted A, B, and C.Source: Notations, definitions, and useful… · p. 2
II. SYSTEM MODEL
The system model combines direct BS-to-UE and RIS-assisted paths for each received pilot signal. It expresses the observations in terms of the channels, RIS phase shifts, and noise, then organizes filtered pilot signals into a tensor.
II. SYSTEM MODEL
- The model includes a direct BS-to-UE channel Hd, a BS-to-RIS channel H, and an RIS-to-UE channel G, with orthogonal pilots transmitted by the BS.The BS has M antennas, the UE has Q antennas, and the RIS has N reflecting elements.Source: §II · p. 2
- For each RIS phase pattern, the received signal contains the combined direct and indirect channel contributions plus noise.The indirect path is represented by G diag(ωp) H, while Hd represents the direct path.Source: §II · p. 2
- The compact model combines the channels in C = [G Hd] and applies a diagonal phase-shift matrix Dp for each RIS pattern.The text calls C the extended channel matrix.Source: §II · pp. 2–3
- Matched filtering and vectorization produce filtered pilot observations, which are collected across pilot sequences into a matrix and then represented as a tensor.The resulting tensor has dimensions Q × M × P.Source: §II · pp. 2–3
III. PROPOSED TENDICE APPROACH
TenDICE models the received signal as a third-order PARAFAC tensor that combines direct and indirect channels. It estimates the equivalent channel factors by alternating least squares.
III. PROPOSED TENDICE APPROACH
- TenDICE jointly estimates the equivalent channel matrices C and Hbar from the received-signal tensor.The method solves for each factor matrix in the least-squares sense and alternates the estimates.Source: §III · p. 3
- The received signal is modeled as a third-order PARAFAC tensor combining the direct and indirect channels.The noiseless tensor uses an identity core of rank N + M, with a third-order noise tensor in the noisy observation.Source: §III · p. 3
- The algorithm initializes Hbar, alternates factor updates up to a maximum iteration count, and stops when successive estimation errors differ by less than a tolerance.It returns estimates of C and Hbar.Source: §III · p. 3
IV. SCALING AMBIGUITIES AND UNIQUENESS
PARAFAC uniqueness is ensured under rank conditions, while the equivalent channel’s identity block and cancellation of diagonal scaling ambiguities yield channel estimates without scaling correction.
IV. SCALING AMBIGUITIES AND UNIQUENESS
- Uniqueness of the channel estimates is ensured under mild conditions, including full column rank of T and Ω⋄C, which is needed for the right inverses in equation (16).These rank conditions support identifiability of C and the equivalent channel.Source: §IV · p. 3
- The identity block in the equivalent channel is enforced at each ALS iteration, so the direct channel estimate has no scaling ambiguity.The cascaded-channel scaling ambiguities cancel each other, so that estimate needs no scaling-factor correction.Source: §IV · pp. 3–4
- The cascaded-channel estimate needs no scaling-factor correction because its diagonal scaling ambiguities cancel.The cancellation follows from the inverse relationship between the diagonal scaling matrices.Source: §IV · p. 4
V. SIMULATION RESULTS AND DISCUSSION
The simulations compare channel-estimation methods using NMSE and spectral efficiency (SE). TenDICE outperforms LS and E-TALS, performs similarly to KRF, and estimates direct and indirect channels simultaneously, but has higher execution time.
V. SIMULATION RESULTS AND DISCUSSION
- NMSE and SE are evaluated as functions of SNR, with SE reported in bits per channel use.The section also compares methods against the unstructured CRLB for cascaded-channel estimation.Source: §V · p. 4
- TenDICE outperforms LS and E-TALS and performs similarly to KRF when estimating the cascaded channel, while also estimating the direct channel simultaneously.The comparison uses N = 16 RIS elements, M = 4 BS antennas, and Q = 4 UE antennas; E-TALS's two-stage approach can propagate error and adds estimation delay.Source: §V · p. 4, Figure 3 · p. 4
- TenDICE substantially outperforms the first stage of E-TALS for direct-channel estimation.That E-TALS stage uses matched filtering.Source: Figure 4 · p. 4, §V · p. 5
- For RIS-assisted-link SE, LS and KRF perform similarly to TenDICE; E-TALS degrades slightly at lower SNR, and all methods match the ideal case at higher SNR.The comparison assumes precoder and combiner designs based on true channels; SE is based only on the recovered cascaded channel.Source: §V · pp. 4–5, Figure 5 · p. 4
- Execution time increases with the number of RIS elements, and TenDICE takes longer than the other methods while estimating direct and indirect channels simultaneously.Table II reports runtime for different values of N across methods.Source: §V · p. 5, Table II · p. 5
VI. CONCLUSIONS AND OUTLOOK
The conclusion presents TenDICE as promising for simultaneous direct- and indirect-channel estimation in passive RIS systems, evaluated using NMSE and SE against LS, KRF, two-stage E-TALS, and CRLB. It highlights scale-ambiguity handling and identifies extensions and complexity analysis as future work.
VI. CONCLUSIONS AND OUTLOOK
- TenDICE shows promise for simultaneous direct- and indirect-channel estimation, compared with LS, KRF, two-stage E-TALS, and CRLB using NMSE and SE.The CRLB is used as a reference bound.Source: §VI · p. 5
- The direct channel can be recovered without scale-factor correction because the model’s matrix and tensor properties and the direct-channel effect remove scaling ambiguities.Source: §VI · p. 5
- Future work includes extending TenDICE to beyond-diagonal RIS, studying hardware impairments, and evaluating their impact on receiver performance.The cited BD-RIS approach uses non-diagonal scattering matrices.Source: §VI · p. 5
- Future work can explore TenDICE’s computational complexity and compare it with other estimation methods.Source: §VI · p. 5