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Channel Estimation with Reconfigurable Intelligent Surfaces -- A General Framework

A. Lee Swindlehurst, Gui Zhou, Rang Liu, Cunhua Pan, Ming Li

arXiv:2110.00553v1cs.ITeess.SP

TL;DR

RIS channel estimation is difficult because passive surfaces require indirect CSI recovery, and individual RIS-related channel components may not be identifiable. The paper develops a general framework covering channel models, pilot training, identifiability, and estimation methods across many system conditions. It contrasts simple but training-intensive unstructured models with structured geometric models that use fewer parameters and can substantially reduce training requirements.

  • Problem

    Passive RIS elements lack active transceivers and computational resources, making indirect estimation of channels to and from the RIS necessary; the paper addresses model identifiability under available pilots and RIS training behavior.

  • Method

    The paper systematically analyzes unstructured and structured channel-estimation models across propagation, antenna, carrier, direct-link, correlation, sparsity, and prior-information assumptions.

  • Results

    Unstructured estimation is simple and robust but requires large training overhead, whereas geometric models estimate fewer parameters and can substantially reduce training requirements.

  • Takeaways & Limitations

    Composite-channel estimation is the primary target because beamforming, precoding, and RIS reflection optimization generally do not require resolving individual-channel scaling ambiguity.

Abstract

from arXiv · show

Optimally extracting the advantages available from reconfigurable intelligent surfaces (RISs) in wireless communications systems requires estimation of the channels to and from the RIS. The process of determining these channels is complicated by the fact that the RIS is typically composed of passive elements without any data processing capabilities, and thus the channels must be estimated indirectly by a non-colocated device, typically a controlling base station. In this article, we examine channel estimation for RIS-based systems from a fundamental viewpoint. We study various possible channel models and the identifiability of the models as a function of the available pilot data and behavior of the RIS during training. In particular, we consider situations with and without line-of-sight propagation, single- and multiple-antenna configurations for the users and base station, correlated and sparse channel models, single-carrier and wideband OFDM scenarios, availability of direct links between the users and base station, exploitation of prior information, as well as a number of other special cases. We further conduct numerical comparisons of achievable performance for various channel models using the relevant Cramer-Rao bounds.

I. INTRODUCTION

RIS channel estimation is challenging because passive RIS elements require indirect CSI recovery, while only composite channels are generally identifiable. The paper organizes channel models and estimation methods under a common framework spanning broad propagation, hardware, pilot, and system assumptions.

  • Motivation: Passive RIS elements require CSI to be estimated indirectly by a non-colocated device, typically the base station.UE pilots are received at the BS after RIS reflection, possibly alongside a direct path.
  • Identifiability: Only the cascaded UE–RIS–BS channel is generally identifiable; individual RIS-related channel components have a scaling ambiguity.This ambiguity usually does not prevent beamforming or RIS reflection optimization because QoS depends on the composite channel.
  • Framework: The paper systematically organizes channel-estimation approaches across fading, propagation, antenna, carrier, direct-link, and special-case assumptions.The common framework is intended to clarify advantages and disadvantages and identify future research directions.
  • Unstructured models: Unstructured models describe narrowband channels with complex coefficients or wideband channels with complex-valued impulse responses, but require large training overhead.These models are motivated by rich multipath scattering and are initially studied for narrowband single-user systems.
  • System model: The general setup considers an M-antenna BS, an N-element RIS, and a K-antenna UE with uplink pilots and synchronously varying RIS reflection coefficients.The channels are block-flat over a coherence interval sufficient for estimation and subsequent transmission.
  • Estimation: For T pilots, the composite channel can be uniquely estimated when T ≥ K(N + 1) and the resulting pilot matrix Z is full rank.Least squares is the simplest estimator; under temporally uncorrelated Gaussian noise, it is unbiased, maximum-likelihood, and attains the CRB.

2) Linear Minimum Mean Squared Error

The LMMSE estimator incorporates second-order channel statistics to estimate RIS composite channels, reducing error relative to LS when valid prior information is available. Its practical use is complicated by the difficulty of determining RIS-side correlation matrices and by the mismatch between uncorrelated fading assumptions and sparse RIS propagation.

  • LMMSE estimation uses a stochastic model and prior second-order statistics, whereas LS assumes a deterministic channel without prior information.
  • Orthogonal pilot and RIS reflection sequences simplify LMMSE computation, while covariance-dependent matrices can be computed and stored offline.
  • At high SNR or with long training, the LMMSE error covariance converges to the LS error covariance.
  • The composite-channel covariance can be constructed from correlation matrices associated with the BS, RIS, and UE sides of the constituent channels.
  • RIS-side correlation matrices are difficult to determine because the RIS is typically passive; assuming uncorrelated RIS scattering simplifies the covariance but may be unrealistic.
  • Uncorrelated fading gives LMMSE smaller error than LS through prior statistical information, but RIS environments often instead exhibit low-rank correlation and sparse propagation.

B. Wideband Single User MIMO

Wideband single-user MIMO estimation can operate in the frequency or time domain for frequency-selective channels. Frequency-domain methods estimate subcarriers individually or interpolate from pilots, but frequency-dependent RIS responses can invalidate reflection patterns designed for one subcarrier.

  • In OFDM, frequency-selective channels are represented across Nc subcarriers transmitted from the UE antennas.
  • Frequency-domain processing applies the DFT after cyclic-prefix removal and produces a channel-estimation model for each subcarrier.
  • The same estimation methods can be applied per subcarrier, while pilots on a subset of subcarriers and interpolation can reduce training overhead.
  • Frequency-independent RIS reflection properties are generally valid only over relatively narrow bandwidths.
  • Time-domain estimation directly targets the channel impulse response using L taps after cyclic-prefix removal.
  • Time-domain training requires T = ToNc ≥ KL(N + 1) pilot symbols, whereas frequency-domain methods can combine pilot and data transmission.

C. Single Antenna Scenarios

Single-antenna user scenarios simplify the notation and reduce algorithmic complexity while retaining the general composite-channel structure. Their training overhead is N + 1 samples, and grouping correlated RIS elements can reduce it further by sacrificing per-element phase flexibility.

  • C. Single Antenna Scenarios: With a single-antenna UE, G becomes a 1 × N row vector and the direct channel becomes an M × 1 vector.
  • C. Single Antenna Scenarios: The single-antenna composite channel is Hc = gH ⋄ H = H diag(g∗), and its training overhead is N + 1 samples.
  • C. Single Antenna Scenarios: Only the composite channel is identifiable in the single-antenna formulation, not the individual RIS-related channel components.
  • D. Multiple User Scenarios: For multiple UEs, orthogonal pilot sequences from all UE antennas preserve the general model, while shared RIS-BS structure can be exploited across users.
  • C. Single Antenna Scenarios: The key training challenge is overhead, motivating methods that reduce the number of pilots.
  • C. Single Antenna Scenarios: Grouping highly correlated RIS elements under identical phases reduces required training overhead by a factor of J.

2) Low-Rank Channel Covariance

Low-rank channel covariance can reduce the training and computational burden of LMMSE estimation when the covariance subspace is known or approximated. A related common-channel strategy exploits the shared RIS-BS channel across users, while geometric models reduce parameters in sparse propagation environments.

  • 2) Low-Rank Channel Covariance: If the composite-channel covariance has rank r, its column span need only lie within the training matrix span in principle, potentially reducing training below K(N + 1).Constraints on RIS reflection vectors may prevent exact realization when training is too short.
  • 2) Low-Rank Channel Covariance: Low-rank covariance also reduces LMMSE computational cost because the required matrix inverse can operate in an r-dimensional space.
  • 3) Exploiting Common Channels: Because every uplink user's composite channel shares the RIS-BS channel H, estimation can be divided into a reference-user step and a remaining-parameters step.
  • 3) Exploiting Common Channels: The reference-user step estimates H diag(g1) rather than separately identifying H and g1, requiring at least T1 = N + 1 training samples.
  • 3) Exploiting Common Channels: The second step solves for remaining channel parameters using linearly independent pilots and RIS reflection vectors.
  • 3) Exploiting Common Channels: For large M, the two-step common-channel method requires significantly less training than the standard LS requirement K(N + 1).
  • IV. ESTIMATION OF STRUCTURED CHANNELS: Geometric models describe sparse high-frequency channels with path gains, AoAs, and AoDs, often using 1–2 orders of magnitude fewer parameters than unstructured models.
  • IV. ESTIMATION OF STRUCTURED CHANNELS: For channel estimation, spatial frequencies are sufficient even when angle ambiguities remain; half-wavelength-or-smaller antenna spacing preserves angle–frequency correspondence.

A. Parametric Estimation

Parametric estimation models RIS channels through path gains and spatial frequencies, while classical array-processing methods estimate those frequencies from received observations. Compressive-sensing alternatives use overcomplete dictionaries but face grid-resolution and computational limitations.

  • Parametric channel models: Parametric RIS-BS models represent channels with path gains and steering vectors associated with BS and RIS spatial frequencies.These models are typically low-rank when the number of paths is smaller than the array dimensions.
  • Array-processing methods: Beamforming, MUSIC, ESPRIT, and maximum-likelihood methods estimate spatial frequencies from array observations using covariance or spectral criteria.MUSIC and ESPRIT require rank(S′)=d and n≥d, whereas beamforming and ML remain theoretically viable for n=1.
  • Array-processing methods: For massive arrays, all considered estimators provide asymptotically efficient angle estimates, favoring beamforming because of its low computational load and minimal assumptions.The number of paths d must be known or estimated from the data.
  • Compressive-sensing formulations: Compressive sensing represents geometric channels with sparse coefficients over overcomplete BS and RIS spatial-frequency dictionaries.The RIS dictionary requires a two-dimensional grid, increasing its size substantially.
  • Compressive-sensing formulations: Off-grid angles cause basis mismatch and energy leakage, while finer grids increase dictionary coherence and create numerical and computational issues.Angular refinement can rotate selected dictionary atoms to improve correlation with the received data.

C. Single User MIMO Single Carrier

The single-user MIMO single-carrier case uses structured representations for the RIS-UE and RIS-BS channels and analyzes the resulting composite channel. Individual component parameters remain ambiguous, but an identifiable parameterization is much smaller than a non-parametric one.

  • Structured channel representation: The RIS-UE channel is modeled with parametric and overcomplete representations analogous to those used for the RIS-BS channel.The geometric representation uses path gains and spatial-frequency parameters.
  • Composite-channel model: Training data are stacked into a linear composite-channel model, with the no-direct-link case represented through the cascaded channel.The composite channel can be decomposed using parametric models or overcomplete dictionaries.
  • Identifiability: The composite channel does not identify all underlying channel parameters or decompositions separately.This follows from ambiguities in the RIS angle parameterization and channel-gain factorization.
  • Identifiability: A common angular shift leaves the composite response unchanged, and scaling γH and γG inversely produces the same composite channel.Fixing one RIS spatial frequency and one path gain removes these ambiguities for an identifiable parameterization.
  • Parameter dimension: dim(η) = 5dH + 5dG − 4, compared with 2MKN parameters for the non-parametric composite channel.For typical path counts, the structured parameter count is much smaller than the potentially thousands of non-parametric parameters.

1) Channel Estimation for the General Case

In the general case, composite-channel estimation can be posed as structured parametric or sparse recovery, but coupled RIS angles and gains complicate estimation. Direct sparse formulations may become intractable because their dictionaries grow rapidly.

  • General model: The general composite channel admits parametric and dictionary-based decompositions built from path combinations across the RIS-UE and RIS-BS channels.The dictionary formulation uses a sparse vector whose nonzero entries encode paired path gains.
  • General model: The composite gains are nonlinear functions of the constituent channel gains, so they cannot be treated as arbitrary coefficients.This dependence prevents direct application of standard unconstrained array-estimation assumptions.
  • Identifiability: Composite steering vectors depend on differences between RIS-side angles, reflecting the fact that the corresponding RIS AoAs and AoDs are not separately identifiable.An identifiable parameterization can fix one RIS angle, after which DML estimation becomes possible under a full-rank condition.
  • Parametric estimation: DML estimation requires non-convex optimization over 3(dG + dH) − 2 spatial frequencies when the effective array response is full rank.This approach ignores the special structure of the composite gains.
  • Special case: For the single-path LoS case dH = dG = 1, only one sparse estimation problem is required.This reduces the number of sparse recovery problems but does not by itself reduce the dictionary dimension.
  • Dictionary complexity: The general sparse dictionary contains NBDNUDN 2_RD elements and reaches 10^12 elements with 100 grid points per spatial-frequency dimension.Even a reduced dictionary remains likely intractable, and LoS propagation lowers sparsity without reducing dictionary dimension.

2) A Simpler Decoupled Approach

A simpler alternative decouples estimation into two pilot stages: first estimate BS and UE-side components, then estimate the remaining RIS-related channel parameters using additional varying RIS pilots.

  • Stage 1: The decoupled approach first estimates BS and UE components with fixed RIS reflections, then removes them from the composite channel.The first stage uses orthogonal UE pilots with T1 ≥ K.
  • Stage 1: The first stage permits separate estimation of BS and UE spatial frequencies using standard angle-estimation or compressive-sensing algorithms.Estimating the UE-side frequency separately assumes K ≥ dG.
  • Stage 2: The second stage varies the RIS reflection pattern at the symbol rate while collecting T2 additional UE pilots.These measurements support estimation of the remaining RIS-side parameters.
  • Stage 2: The remaining RIS parameters can be estimated from a reduced sparse problem, but its dictionary still has N 2-dimensional scaling.The reduced formulation is smaller than the full general-case dictionary but remains computationally challenging.
  • Stage 2: Each composite observation column can be treated approximately as a single snapshot from a T2-element array with one two-dimensional spatial frequency.This yields dHdG one-dimensional angle-estimation problems when the gain and angle interdependence is ignored.
  • Special case: For LoS channels with dH = dG = 1, the approach requires only one sparse estimation problem.This is the simplest special case of the decoupled procedure.

D. Wideband Single User MIMO

The wideband geometric-channel treatment extends the estimation framework across OFDM taps, using staged processing to estimate angular and gain parameters. Single-antenna UEs enable a substantially simpler decomposition than the general multi-antenna case.

  • Wideband geometric model: The time-domain OFDM approach processes each tap separately after stacking training data and partitioning the resulting matrices into L blocks.The same estimation methods can then be applied repeatedly for paths k = 0, ..., L − 1.
  • Wideband geometric model: Wideband geometric channels are represented with L single-tap parameter blocks, increasing dictionary size and parameter dimensions by a factor of L.The model allows different angles for each tap.
  • Decoupled estimation: A decoupled two-stage procedure first estimates BS angles from fixed-RIS training, then estimates UE angles and remaining channel parameters using additional OFDM symbols.The first stage estimates BS AoAs and UE AoDs from structured blocks; the second stage solves the remaining parameter problems.
  • Single-antenna UE: For single-antenna UEs, one dG-sparse estimation followed by dH − 1 one-sparse problems replaces dHdG one-sparse AoA problems.The combined estimates recover the RIS and BS spatial frequencies and channel gains.
  • Single-antenna UE: Single-antenna UE models can retain the full geometric sparse structure or use a reduced dictionary that ignores part of that structure.The full model uses a dHdG-sparse vector, while the reduced formulation has NRDNBD terms.

2) Single Antenna UE and BS

The single-antenna UE and BS cases simplify geometric channel estimation by exploiting the resulting array structure. Multi-user extensions combine user-specific parameters in an effective dictionary, but joint optimization can become prohibitive.

  • Single-antenna UE and BS: With no BS or UE angles to estimate, the single-antenna UE-and-BS case collects T observations into one vector and forms an equivalent single-snapshot problem.The remaining spatial frequencies can be estimated with DML or sparse compressed sensing methods.
  • Multiple users: For multiple users, the composite channel groups user-specific angle, gain, and sparse-vector parameters through a permutation-based effective dictionary.Each user may have Ku antennas and dGu propagation paths, with dG equal to the sum of user path counts.
  • Multiple users: Joint estimation of all multi-user channel parameters is possible in principle, but the resulting optimization dimension is likely prohibitive except in simple cases.The paper therefore applies a decoupled approach to reduce estimation complexity.
  • Multiple users: When Ku > dGu, UE angle estimates can be obtained separately for each user by processing different block rows.This follows estimation of the BS AoAs in the decoupled procedure.
  • Multiple users: For single-antenna UEs in the multi-user setting, one dG-dimensional AoA estimate followed by dH − 1 one-dimensional problems remains sufficient.The simplification applies to the remaining channel-parameter estimation after exploiting the special structure.

G. Reducing the Complexity and Training Overhead

The paper develops geometric channel-estimation procedures that reduce training and computational demands, then evaluates their CRB across channel configurations and operating conditions. Geometric models can substantially outperform unstructured models, although additional paths and weakly observed direct links affect performance.

  • Geometric-model reductions: A staged procedure estimates Hdiag(g1), then UE angle-of-departures, and finally additional gains using fixed RIS reflections.The described sequence separates estimation of the common BS–RIS channel, UE angular parameters, and remaining channel quantities.
  • Geometric-model reductions: Geometric models exploit slowly varying spatial frequencies, allowing subsequent coherence blocks to re-estimate only complex gains.With known sparse support, least-squares gain estimation requires at least dHdG training samples.
  • CRB evaluation: The CRB is computed from the Fisher Information Matrix for unstructured and geometric channel parameterizations, with channel-level bounds obtained from geometric parameter estimates.For orthogonal training, the unstructured CRB simplifies and is identical for every composite-channel element.
  • Numerical comparisons: With M = N = 30, geometric modeling provides 10-15dB CRB gain over the unstructured case at T = 31, though calibration errors can reduce this gain.The comparison uses dH = 2, dF = dG = 5 and includes geometric models with and without the direct BS–UE channel.
  • Numerical comparisons: At SNR = 5dB, geometric modeling achieves the same performance with more than an order of magnitude fewer training samples.When a direct channel is present, geometric estimation degrades because additional parameters are observed more weakly than the RIS-assisted component.
  • Numerical comparisons: For fixed T, the best RIS-versus-BS allocation occurs near M ≃N, whereas increasing T favors a larger RIS and fewer BS antennas.As propagation paths increase, geometric CRB increases because the number of parameters to estimate offsets the received-power gain; the direct-channel path count dF has the strongest effect.
  • Numerical comparisons: The estimated direct channel Hd is essentially unchanged by whether the RIS is present in the geometric model.The “Geometric Hd” and “Geom. Hd only” results are essentially identical.

VI. ADDITIONAL TOPICS

Additional topics extend RIS channel estimation to active elements, multiple RISs, and learning-based methods, while highlighting unresolved identifiability and robustness issues. The conclusion contrasts unstructured and geometric models, emphasizing lower training burden and better performance against complexity and modeling-error costs.

  • Active RIS receivers estimate geometric channel gains and angles, then infer full H and G matrices using RIS geometry or a DNN.
  • Purely passive double-RIS estimation remains uncertain: existing work covers special cases, while geometric identifiability without active receivers is unresolved.One unstructured estimator requires at least N1N2 training samples; an active-receiver LoS approach can determine individual components.
  • Learning-based methods reduce computation after training by directly estimating or denoising channels with neural networks.Examples use synthetic Rayleigh-fading data, LS or compressive-sensing initial estimates, and reduced-dimensional composite channels.
  • Synthetic-data training leaves open whether learning-based estimators handle practical non-idealities absent from simulations.
  • Unstructured models are simple and robust but require large training overhead, whereas geometric models estimate fewer parameters and can achieve dramatically better performance.Geometric models require greater algorithmic complexity, model-order determination, calibrated structures, and can lose gains through modeling errors.
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