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Channel Estimation for Intelligent Reflecting Surface Assisted MIMO Systems: A Tensor Modeling Approach

Gilderlan T. de Araújo, André L. F. de Almeida, Rémy Boyer

arXiv:2008.04766v3eess.SP

TL;DR

Passive IRS operation makes receiver-side estimation of the cascaded channel from pilot signals difficult, especially with many reflecting elements. The paper develops PARAFAC-based KRF and BALS receivers that decouple the BS-IRS and IRS-user-terminal channels, with numerical results showing advantages over LS-based solutions and a trade-off between complexity, training requirements, and parameter flexibility.

  • Problem

    Passive IRS-assisted MIMO requires receiver-side estimation of cascaded channels from pilots, while phase-shift training patterns and the large number of IRS elements create CSI-acquisition challenges.

  • Method

    The paper models received pilot signals as a PARAFAC tensor and proposes closed-form KRF and iterative BALS estimators that decouple the involved channel matrices.

  • Results

    The proposed receivers outperform competing LS-based solutions; KRF provides an SNR gain of nearly 5dB over one competing method and around 7dB over an LS solution.

  • Takeaways & Limitations

    KRF offers lower complexity but more restrictive training-parameter requirements, whereas BALS is more computationally complex but supports more flexible parameter choices with lower training overhead.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is an emerging technology for future wireless communications including 5G and especially 6G. It consists of a large 2D array of (semi-)passive scattering elements that control the electromagnetic properties of radio-frequency waves so that the reflected signals add coherently at the intended receiver or destructively to reduce co-channel interference. The promised gains of IRS-assisted communications depend on the accuracy of the channel state information. In this paper, we address the receiver design for an IRS-assisted multiple-input multiple-output (MIMO) communication system via a tensor modeling approach aiming at the channel estimation problem using supervised (pilot-assisted) methods. Considering a structured time-domain pattern of pilots and IRS phase shifts, we present two channel estimation methods that rely on a parallel factor (PARAFAC) tensor modeling of the received signals. The first one has a closed-form solution based on a Khatri-Rao factorization of the cascaded MIMO channel, by solving rank-1 matrix approximation problems, while the second one is an iterative alternating estimation scheme. The common feature of both methods is the decoupling of the estimates of the involved MIMO channel matrices (base station-IRS and IRS-user terminal), which provides performance enhancements in comparison to competing methods that are based on unstructured LS estimates of the cascaded channel. Design recommendations for both methods that guide the choice of the system parameters are discussed. Numerical results show the effectiveness of the proposed receivers, highlight the involved trade-offs, and corroborate their superior performance compared to competing LS-based solutions.

I. INTRODUCTION

IRS-assisted MIMO channel estimation is challenging because passive IRS operation and many reflecting elements constrain receiver-side CSI acquisition. This paper uses tensor modeling to develop PARAFAC-based estimators that decouple the involved channels and provide design guidance.

  • Motivation: Passive IRS operation requires the receiver to estimate the cascaded transmitter-IRS-receiver channel from transmitter pilots reflected by the IRS.Training phase-shift patterns and the large number of IRS elements make CSI acquisition especially challenging.
  • Proposed methods: The proposed KRF and BALS algorithms exploit PARAFAC structure to estimate the cascaded channel by decoupling the BS-IRS and IRS-user-terminal channel matrices.KRF is closed-form and based on Khatri-Rao factorization, whereas BALS is iterative and can operate under less restrictive system-parameter conditions.
  • Expected benefit: Decoupling the channel estimates provides performance enhancement over conventional least-squares estimation of the cascaded channel.The approach targets limitations of existing passive-IRS methods and structured tensor processing is motivated by its uniqueness properties in wireless signal processing.
  • Tensor formulation: The received signals are modeled by fitting a PARAFAC decomposition to a third-order tensor.This connects IRS-assisted MIMO channel estimation with tensor-model fitting and exploits PARAFAC uniqueness properties.
  • Design and extensions: The two receivers are accompanied by system-design recommendations that ensure uniqueness of the channel estimation problem.The paper also discusses joint channel and IRS-matrix estimation under phase-shift perturbations, multi-user generalizations, and analytical CRB expressions.

II. SYSTEM MODEL

The system uses an IRS-assisted MIMO link with structured training: IRS phase shifts remain constant within blocks, while pilot signals repeat across blocks. A conventional LS baseline estimates a combined cascaded channel, whereas the proposed tensor approach decouples the BS-IRS and IRS-user channels.

  • The transmitter and receiver use M and L antennas, while the IRS contains N individually adjustable elements.
  • Signal model: The matrices H and G represent the BS-IRS and IRS-user MIMO channels, respectively, while the received signal includes additive white Gaussian noise.
  • Structured training protocol: The IRS phase-shift vector is constant within each block, and the pilot sequence repeats across all K blocks.The training duration is divided into K blocks of T time slots.
  • LS baseline: The LS baseline estimates a composite channel parameter that combines the BS-IRS and IRS-user channels.The estimate is obtained through a pseudoinverse-based least-squares solution.
  • Tensor-based estimation: The proposed tensor modeling exploits the cascaded channel’s Khatri-Rao structure to decouple estimates of H and G rather than estimating the composite channel directly.The paper presents this decoupling as an accuracy enhancement over conventional LS methods.

B. Tensor signal modeling

Under the structured training protocol, the received signals are arranged as a three-way tensor whose noiseless form follows a PARAFAC decomposition. This structure supports factor identification and enables channel estimation from tensor unfoldings.

  • The received block matrices form the frontal slices of a three-way tensor Y ∈ C^L×T×K.
  • PARAFAC representation: The noiseless received signal tensor admits a PARAFAC, or canonical polyadic, decomposition.
  • PARAFAC factors: Each tensor entry is modeled through factors associated with the IRS-user channel, pilot-related terms, and IRS phase shifts.The factors are represented using entries of G, Z, and S.
  • Tensor unfoldings: The tensor can be unfolded into three matrix forms, allowing the PARAFAC structure to be exploited algebraically.
  • Identifiability: PARAFAC factor identification relies on essential uniqueness properties rooted in the concept of Kruskal rank.

III. CHANNEL ESTIMATION METHODS

The paper estimates the BS-IRS and IRS-user channels from the noisy received tensor using designed pilot and IRS matrices. The KRF method converts the problem into rank-1 approximations and returns decoupled channel estimates.

  • The estimation goal is to recover the channel matrices H and G from the noise-corrupted received signal tensor and its matrix unfoldings.
  • Training design: Semi-unitary pilot and IRS phase-shift matrices can be designed using truncated DFT matrices.They satisfy X^H X = T I_M and S^H S = K I_N.
  • Preprocessing: Bilinear filtering produces a noisy virtual MIMO channel with Khatri-Rao structure without changing additive-noise correlation properties.The latter property follows from the semi-unitary structures of S and X.
  • KRF estimation: The KRF procedure outputs decoupled estimates of the BS-IRS and IRS-user channels after bilinear filtering and factor reconstruction.
  • KRF estimation: The KRF algorithm estimates H and G by solving N rank-1 matrix approximation subproblems.The channel vectors are obtained from dominant left and right singular vectors of the corresponding matrices.

B. BALS channel estimation

BALS estimates the two channel matrices by alternating least-squares updates while keeping the known IRS matrix fixed. It typically converges in fewer than 10 iterations, but column-orthogonal designs require K ≥ N and T ≥ M for simplified updates.

  • BALS alternates least-squares estimation of G and H using noisy tensor unfoldings.It is a simplified version of trilinear ALS because the IRS matrix S is known.
  • Iteration and convergence: The algorithm repeats the G and H updates until the reconstruction-error change falls below a threshold.
  • Iteration and convergence: BALS usually converges in fewer than 10 iterations because the known IRS matrix remains fixed during the iterations.The adopted convergence threshold is ǫ = 10^-5.
  • Complexity conditions: Column-orthogonal pilot and IRS matrices enable lower-complexity BALS updates when K ≥ N and T ≥ M.

C. Computational complexity

The methods impose different training and rank requirements, with BALS generally offering more flexible training choices while KRF is computationally simpler. KRF’s design benefits from full-rank, preferably semi-unitary pilot and IRS phase-shift matrices, whereas BALS uniqueness requires additional rank conditions.

  • Computational complexity: KRF generally has lower computational complexity than BALS because it uses closed-form rank-1 approximations instead of iterative least-squares updates.The KRF complexity is stated as O(MLN).
  • IRS perturbations: When IRS phase shifts are unknown or perturbed, TALS jointly estimates the channels and IRS matrix but requires more iterations and higher complexity.Kruskal’s condition min(L, N) + min(M, N) + min(K, N) ≥ 2N + 2 guarantees uniqueness, while convergence can be sensitive to initialization.
  • Design requirements: KRF requires full-column-rank IRS phase-shift and pilot matrices, and semi-unitary designs simplify filtering through matrix products.Semi-unitary designs also preserve the correlation properties of the filtered noise term.
  • Design requirements: BALS requires Kmin(T, L) ≥ N and T ≥ M so its alternating least-squares subproblems admit unique solutions.These conditions ensure full column rank for the relevant Khatri-Rao products and pilot matrix.
  • Training trade-offs: BALS has less restrictive training requirements than KRF in MIMO systems, operating with K < N while KRF requires K ≥ N^2.For L = 1, both methods have identical training requirements.
  • Uniqueness: The necessary BALS condition does not guarantee uniqueness; sufficient conditions depend on the rank properties of its two structured Khatri-Rao matrices.Khatri-Rao rank bounds provide conditions that guarantee full column rank and unique channel estimates.

A. The BS-IRS and IRS-UT channel matrices have full rank

For full-rank BS-IRS and IRS-user channels, uniqueness conditions trade training blocks against antenna dimensions. BALS can use fewer training blocks than KRF in MIMO settings, although this advantage disappears for single-antenna links.

  • Sufficient conditions: Under full-rank channel assumptions, the sufficient conditions are min(K, N) + min(M, N) ≥ N + 1 and min(K, N) + min(L, N) ≥ N + 1.These follow from requiring the relevant Khatri-Rao products to have full column rank.
  • Antenna regimes: When N ≥ T ≥ M and N ≥ L, uniqueness reduces to M + min(K, N) ≥ N + 1 and L + min(K, N) ≥ N + 1.The antenna dimensions and training-block count jointly determine recoverability.
  • Training–antenna trade-off: If K < N, reducing transmit or receive antennas must be compensated by more training blocks, requiring M + K ≥ N + 1 and L + K ≥ N + 1.Equivalently, min(M + K, L + K) ≥ N + 1.
  • Rank deficiency: For rank-deficient channels, KRF’s training conditions are unaffected, whereas BALS uniqueness depends on the ranks of the channel matrices.With ranks R1 and R2, BALS requires min(K, N) + R1 ≥ N + 1 and min(K, N) + R2 ≥ N + 1.
  • Rank deficiency: When K ≥ N, the rank-deficient BALS conditions are always satisfied; when K < N, they become K + R1 ≥ N + 1 and K + R2 ≥ N + 1.Thus, lower channel ranks require additional training blocks for uniqueness.
  • Method comparison: BALS can require substantially lower training overhead than KRF in MIMO systems, but the methods coincide for MISO or SIMO configurations.For M ≥ N and L ≥ N, BALS can operate with small K; for M = 1 or L = 1, both require K ≥ N.
  • Ambiguities: Channel-estimate scaling ambiguities do not affect the cascaded channel because reciprocal scaling factors cancel when the channel matrices are combined.The estimates satisfy Ĥ = Δ_HH and Ĝ = GΔ_G with Δ_HΔ_G = I_N.

V. GENERALIZATIONS TO MULTI-USER SCENARIOS

The tensor model extends from one user to multi-user IRS-assisted MIMO uplinks by combining user pilots and channel contributions. Its PARAFAC structure is preserved, allowing the same KRF and BALS algorithms to apply under stricter pilot requirements.

  • Generalization: The proposed tensor approach generalizes to multi-access and multi-user MIMO systems, with downlink adaptation obtained by exchanging BS and user-terminal roles.The supplied discussion focuses on the uplink case.
  • Uplink model: For U users communicating with one BS, the IRS-BS channel is common to all users while each user has its own pilot and uplink channel matrix.The direct user-BS link is assumed too weak or unavailable.
  • Tensor structure: The multi-user received signal retains the same PARAFAC tensor structure as the single-user model, with a modified factor matrix Z formed from U block matrices.The essential structural change is the definition of Z.
  • Training requirements: The combined pilot matrix is X = [X1, . . . , XU] ∈ C^{T×UL}, so full column rank requires T ≥ UL.This creates more restrictive choices for the pilot dimension than in the single-user case.
  • Algorithms: Because the tensor structure is unchanged, both KRF and BALS can be directly applied to the multi-user model under the resulting rank conditions.The sufficient conditions are analogous to the single-user conditions after exchanging M and L and incorporating U.

B. Multiple users communicate with multiple BSs via the IRS

The multi-user multi-BS extension increases the received tensor’s first-mode dimensionality through cooperating BSs, while modifying the uniqueness condition accordingly. Numerical results compare KRF and BALS with competing methods, showing trade-offs among accuracy, training overhead, convergence, and complexity.

  • System extension: The multi-BS model assumes P BSs with M antennas each and combines their IRS links in the composite channel H = [H_1, . . . , H_P]^T.The received signals are represented using an augmented PARAFAC signal model when the BSs cooperate.
  • System extension: Cooperating BSs increase the first mode of the received signal tensor by a factor P, changing condition (48) to min(K, N) + min(PM, N) ≥ N + 1.Condition (47) remains unchanged in this scenario.
  • Estimation accuracy: Both KRF and BALS provide satisfactory channel-estimation performance, while increasing the number of IRS elements degrades NMSE because more coefficients in G and H must be estimated.These observations are reported for the fixed Monte Carlo setup T = 4, L = 2, K = 50, M = 3, with N ∈ {50, 100}.
  • Training and complexity: BALS requires more iterations as N increases, especially at low SNR, while high SNR makes convergence less sensitive to N.With known IRS matrix S, convergence usually takes fewer than 10 iterations.
  • Comparative evaluation: KRF outperforms the competing method of by nearly 5 dB in SNR because it jointly estimates the involved channels instead of using sequential stages.The comparison uses P = 2, M = 1, and U = 1 in the uplink multi-BS scenario; the sequential method can induce error propagation.
  • Comparative evaluation: KRF gains around 7 dB over conventional LS for the equivalent channel because rank-1 approximations exploit Khatri-Rao structure and reject noise.The LS solution attains the CRB, while KRF reshapes the channel into N IRS subchannels of dimension M × L.
  • Comparative evaluation: KRF also outperforms block-LS, with a nearly 3.5 dB SNR gain for M = T = 4; its estimates become more accurate as the antenna array grows.Block-LS performance is unaffected when M increases, unlike KRF.

VII. CONCLUSION AND PERSPECTIVES

The paper proposes KRF and BALS pilot-assisted receivers that exploit tensor structure to decouple BS-IRS and IRS-UT channel estimates for passive IRS systems.

  • KRF and BALS receivers exploit the tensor structure in received signals to estimate the BS-IRS and IRS-UT channels separately.Both methods target passive IRS operation and produce decoupled channel estimates at the receiver.
  • KRF has lower complexity but requires more restrictive training-parameter choices, whereas BALS is more computationally complex but needs less training overhead.
  • The proposed design recommendations specify system-parameter conditions that guarantee uniqueness of the channel estimates.
  • Numerical results show that KRF and BALS outperform conventional LS estimation, which ignores the Khatri-Rao structure of the combined channel matrix.
  • The tensor approach also accommodates imperfectly known IRS phase shifts caused by phase perturbations or fluctuations.

APPENDIX A EXPECTED CRAM´ER RAO LOWER BOUND

The appendix derives the expected Cramér–Rao lower bound for channel estimation from the vectorized received-signal model and its Fisher information matrix.

  • The CRB lower-bounds the variance and NMSE attainable by an unbiased estimator of the channel parameter vector.
  • The CRB is expressed as the inverse of the Fisher information matrix, with block-matrix operations used to simplify the result.
  • For complex-valued parameters with nuisance parameters, the appendix uses a structured real-valued parameter representation to obtain the CRB.
  • For circular complex Gaussian observations, the Fisher information matrix is obtained using the Slepian-Bangs formula.
  • Vectorizing the 3-mode unfolding yields a linear model in the Khatri-Rao-structured channel parameters, whose noisy-observation statistics support the CRB calculation.
  • Column orthogonality of the pilot and IRS phase-shift matrices makes the relevant Fisher-information expressions simpler, eliminating the need for an expectation over parameters.

APPENDIX B SIMPLIFIED VERSION OF BALS

The appendix simplifies BALS updates under column-orthogonal pilots and IRS phase shifts, replacing inversions with lower-complexity products and enabling parallel per-element processing.

  • Column orthogonality allows BALS right pseudo-inverses to be replaced by lower-complexity matrix products.
  • The diagonal structures of ΣH and ΣG reduce the complexity of the G and H updates by replacing matrix inversions with simpler products.
  • Each update of G and H can be implemented as N independent processes, one for each IRS element, and executed in parallel.
  • BALS alternates least-squares updates of G and H by constructing updated matrices M1 and M2 from the current channel estimates.
  • The alternating updates repeat until convergence.
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