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Cascaded Channel Estimation for Intelligent Reflecting Surface Assisted Multiuser MISO Systems

Huayan Guo, Vincent K. N. Lau

arXiv:2108.09002v1eess.SP

TL;DR

The paper addresses uplink cascaded channel estimation for sub-6 GHz IRS-assisted MU-MISO systems with nonsparse propagation and potentially more IRS elements than BS antennas. It proposes an always-ON, common-link-based joint estimation framework with optimized training phases, achieving strong performance and more than 3 dB gain over random phase configurations.

  • Problem

    Accurate cascaded-channel estimation is challenging in sub-6 GHz IRS-assisted MU-MISO systems because propagation is not sparse, IRS elements may outnumber BS antennas, and common-link variables are highly coupled.

  • Method

    The paper combines an always-ON protocol, common-link decomposition, covariance-based joint estimation with alternating optimization, and SCA-based training phase-shift optimization.

  • Results

    The proposed protocol reduces pilot overhead, avoids reflection power loss and reference-user selection, while optimized phase shifting achieves more than 3 dB gain over random configuration.

  • Takeaways & Limitations

    Exploiting the shared BS-IRS link yields a lower-overhead estimation framework that is robust to user locations and reference-user channel quality.

Abstract

from arXiv · show

This paper investigates the uplink cascaded channel estimation for intelligent-reflecting-surface (IRS)-assisted multi-user multiple-input-single-output systems. We focus on a sub-6 GHz scenario where the channel propagation is not sparse and the number of IRS elements can be larger than the number of BS antennas. A novel channel estimation protocol without the need of on-off amplitude control to avoid the reflection power loss is proposed. In addition, the pilot overhead is substantially reduced by exploiting the common-link structure to decompose the cascaded channel coefficients by the multiplication of the common-link variables and the user-specific variables. However, these two types of variables are highly coupled, which makes them difficult to estimate. To address this issue, we formulate an optimization-based joint channel estimation problem, which only utilizes the covariance of the cascaded channel. Then, we design a low-complexity alternating optimization algorithm with efficient initialization for the non-convex optimization problem, which achieves a local optimum solution. To further enhance the estimation accuracy, we propose a new formulation to optimize the training phase shifting configuration for the proposed protocol, and then solve it using the successive convex approximation algorithm. Comprehensive simulations verify that the proposed algorithm has supreme performance compared to various state-of-the-art baseline schemes.

I. INTRODUCTION

The paper addresses uplink cascaded channel estimation for IRS-assisted MU-MISO systems in sub-6 GHz settings, where propagation is non-sparse and IRS size may exceed BS antenna count. It proposes always-ON training, common-link exploitation, joint optimization, and phase-shifting design to reduce overhead and improve estimation.

  • Motivation: IRS-assisted channel estimation is challenging because IRS elements lack sensing elements, radio-frequency chains, and baseband processing capability.The system therefore requires channel estimation through the BS and user transmissions.
  • Motivation: Directly extending single-user protocols to multiuser systems ignores the common BS-IRS link and requires estimating MNK variables instead of MN + NK independent variables.The shared BS-IRS link creates exploitable structure across user cascaded channels.
  • Limitations of Existing Methods: Existing approaches can incur error propagation, reflection power loss, unavailable statistical priors, ambiguity when M < N, or heavy-tailed relative-channel distributions.Reference-channel inaccuracies can jeopardize relative-channel estimates, while switching IRS elements off reduces received SNR.
  • Contributions: The paper proposes an always-ON protocol that exploits the common-link structure, avoids on-off amplitude control, and reduces pilot overhead to K + N + ⌈NThe supplied passage truncates the final overhead expression, but states that the protocol avoids reflection power loss and supports any IRS size, including N ≤ M.
  • Contributions: An optimization-based joint estimator decomposes cascaded coefficients into common-link and user-specific variables and uses a MAP formulation with practical cascaded-channel statistical information.The variables are highly coupled because the common-link structure is exploited without on-off amplitude control.
  • Contributions: The proposed phase-shifting optimization uses successive convex approximation and achieves a more than 3 dB gain over state-of-the-art baselines.The formulation maximizes average IRS reflection gain for the multiuser setting.

B. IRS Cascaded Channel Model

The cascaded channel is modeled through a shared BS–IRS link and user-specific IRS–user links, reducing the independent structure across users. The selected on-off protocol estimates a reference cascaded channel before recovering other users’ channels, but requires IRS element switching and incurs substantial training overhead.

  • Channel structure: Each user’s cascaded channel is formed from the common BS–IRS channel and a user-specific IRS–user channel.The common-link structure means the cascaded channels are not independent across users.
  • Operating assumptions: The proposed estimation framework assumes sub-6 GHz operation, where propagation is generally non-sparse and the IRS may have more elements than BS antennas.The model focuses on the case N > M and uses cascaded-channel covariance information as a weaker requirement than individual-link covariances.
  • Selected on-off protocol: The selected on-off protocol estimates BS–user channels first, then a reference user’s cascaded channel, followed by the remaining users’ cascaded channels.The reference channel is estimated using a single-user MISO procedure, while later stages exploit the common-link property.
  • Training overhead: Estimating relative channels with switched IRS subsets requires ⌈N/M⌉ timeslots and an overall pilot overhead of K + N + ⌈N/M⌉(K −1).The protocol activates different groups of M IRS elements across timeslots until all relative-channel coefficients are estimated.

B. Always-ON Channel Estimation Protocol

The always-ON protocol uses two training stages with orthogonal multiuser pilots followed by a shared pilot, keeping all IRS elements active while exploiting the common-link structure. Its total overhead is K + N + ⌈N/M⌉(K −1), about M times lower than protocols requiring NK pilots.

  • Protocol structure: The proposed protocol keeps all IRS elements ON and consists of Stage I with L1+1 timeslots and Stage II with L2 = N −L1 timeslots.Here L1 = ⌈N/M⌉, and the two stages use different sample patterns.
  • Stage I: In Stage I, all K users transmit orthogonal pilot sequences while the IRS applies a phase-shifting vector in each timeslot.Each Stage I timeslot contains K received samples.
  • Stage II: In Stage II, all users transmit the first column of the orthogonal pilot matrix while the IRS changes its phase configuration across timeslots.Each Stage II timeslot contains one received sample.
  • Pilot overhead: K + N + ⌈N/M⌉(K −1) is the proposed protocol’s overall pilot overhead.The stage-wise overheads are (⌈N/M⌉+1)K and N −⌈N/M⌉, respectively.
  • Pilot overhead: The proposed overhead is reduced by about M times relative to schemes requiring NK pilots.The reduction comes from exploiting the common-link structure in the multiuser cascaded channels.

2) Signal Pre-processing:

Signal preprocessing removes the BS–user contribution from received observations before cascaded-channel estimation. The protocol uses paired phase configurations to separate channel components and can recover all users’ cascaded channels under the stated rank and phase-shifting conditions.

  • Channel decoupling: Setting θ0 = −θ1 decouples estimation of the BS–user channel from the cascaded IRS channel.The BS–user channel can then be estimated from the corresponding received observations using an LMMSE estimator.
  • Signal preprocessing: The preprocessing stage removes the BS–user channel from received signals to facilitate cascaded IRS-channel estimation.Separate preprocessing operations are applied to the Stage I and Stage II observations.
  • Recovery condition: All K cascaded channels can be perfectly recovered with probability one in the noiseless case under an orthogonal, non-zero phase-shifting configuration and the stated angular-domain channel model.The recovery result assumes the specified factorization of G and hr,k.
  • Virtual reference channel: The proposed protocol constructs a virtual reference channel that is fair to all users and can be reconstructed from N observations.Relative channels are then estimated using multiple phase-shifting vectors rather than a fixed measurement configuration.
  • Measurement design: With a proper phase-shifting configuration, the measurement matrix can reach rank N when L1 ≥ ⌈N/M⌉.This condition supports reasonable estimation of the relative channels in the intuitive two-step explanation.

IV. OPTIMIZATION-BASED MU-CASCADED IRS CHANNEL ESTIMATION

The paper formulates joint MAP estimation of common-link and user-specific variables for cascaded channels using covariance information, while recognizing that individual channel factors are ambiguous although the cascaded channel remains identifiable.

  • MAP formulation: The MAP problem uses pre-processed observations and the prior distribution of the cascaded channel to estimate all cascaded channels jointly.The formulation can also use maximum likelihood when channel prior knowledge is unavailable.
  • MAP formulation: Individual common-link and user-specific channels are difficult to estimate because their prior covariances cannot be readily obtained from cascaded-channel covariance.This motivates introducing a more general auxiliary variable set for the decomposition.
  • MAP formulation: The cascaded channel is decomposed into common-link and user-specific variables, enabling an optimization formulation that exploits shared structure across users.The common-link matrix Hg contains columns hg,m, while Hu contains columns hu,k.
  • MAP formulation: The proposed protocol can include the earlier on-off protocol as a special case, while avoiding the need to treat it as the only training configuration.The special case uses a pilot vector with one nonzero entry and different IRS elements selected as ON or OFF.
  • Identifiability: P(A) has non-unique factor solutions, but all equivalent solutions produce the same cascaded channel estimate.Thus, individual factors are ambiguous even though the estimated cascaded channel is unique.

B. Channel Estimation Algorithm based on Alternative Optimization

The coupled optimization is solved by alternating between convex quadratic subproblems for Hu and Hg, with each update obtained from its first-order optimality condition.

  • Alternative optimization: The objective is difficult because Hu and Hg are coupled in the likelihood functions, but the problem is bi-convex and admits alternating optimization.The method decomposes the original problem into two convex subproblems.
  • Update Hu: With Hg fixed, optimizing Hu is a convex quadratic problem in vec(Hu), whose solution is obtained from the root of the first-order derivative.The update is represented by subproblem P(Au).
  • Update Hg: With Hu fixed, optimizing Hg is a convex quadratic problem in the common-link columns, and each hg,m is obtained from its first-order derivative.The update is represented by subproblem P(Ag).

3) Initial Estimation on Hg:

The algorithm initializes the common-link variable with a least-squares estimator that uses observations from all N + 1 training timeslots, then alternates variable updates until convergence.

  • Initial estimation: An efficient estimator for Hg is constructed to initialize alternating optimization, whose solution quality depends strongly on the initial point.The initialization forms a special feasible pair of common-link and user-specific variables.
  • Initial estimation: The common-link variable Hg is initialized by a least-squares estimator based on the available training observations.The estimator is described as unbiased and uses observations from all N + 1 training timeslots.
  • Convergence: The objective is non-increasing at every update, and the alternating iterations converge to a local optimum of P(A).This guarantee applies to the unconstrained convex-quadratic subproblems used by the algorithm.
  • Algorithm: The algorithm repeats updates of Hu and every hg,m until the objective fA(Hg, Hu) converges, then outputs the reconstructed cascaded channels.The reconstruction is hI,k,m = diag(hu,k)hg,m.
  • Complexity: The overall computational complexity is O(IK3N 3 + IKMN 3), where I is the number of alternating-optimization iterations.The update costs are O(K3N 3 + KMN 2) for Hu and O(KMN 3) for Hg.

V. TRAINING PHASE SHIFTING CONFIGURATION

The training phase-shifting design scans N spatial directions and optimizes an additional unit-modulus steering vector using cascaded-channel covariance information.

  • Phase-shifting design: The protocol scans the cascaded channel across N spatial directions in N training timeslots, so phase shifts must preserve information across all directions.The paper illustrates the impact of different phase-shifting configurations in Fig. 5.
  • Constraints: The phase-shifting design is constrained by unit modulus for every element of the steering vector and phase-shifting matrix.These constraints preserve the IRS phase-only configuration.
  • Baseline configuration: For the single-user case, the MSE-minimizing phase-shifting matrix is the DFT matrix, whose columns may be permuted without changing the MSE.The unit-modulus constraint applies to every matrix entry.
  • Proposed configuration: The proposed protocol retains DFT-based phase shifts and introduces an additional unit-modulus steering direction ϑ for more flexible design.The training vector combines the DFT column fℓ with the additional steering direction.
  • Optimization objective: Because the proposed algorithm’s Hu MSE is complicated, the steering vector is designed indirectly using known cascaded-channel covariances and effective received power.The resulting objective maximizes fB(ϑ).

C. Solution for P(B)

The phase-shifting design problem P(B) is non-convex because it maximizes a convex objective under unit-modulus constraints. The paper solves it iteratively with successive convex approximation and evaluates the resulting estimator against several baselines.

  • P(B) is non-convex because it maximizes a convex objective subject to unit-modulus constraints.
  • Successive convex approximation replaces P(B) with an iteratively solved surrogate problem based on a first-order approximation.
  • The SCA algorithm has a convergence proof referenced by the paper.
  • Simulation baselines: The simulations benchmark the proposed scheme against LMMSE, BALS, MAP-modified BALS, and a selected on-off protocol.
  • Simulation baselines: The proposed protocol and Baselines 3–4 use the same pilot overhead, while the selected on-off baseline uses a reference user.
  • BS-user channel estimation: The proposed preprocessing makes BS-user channel estimation independent of cascaded-channel estimation and provides a theoretical 3 dB gain over shutting down the IRS.

B. Simulation Results

The simulations show that the proposed optimization-based protocol improves cascaded-channel estimation across transmit power, antenna number, IRS size, and user-location conditions. Optimized phase shifts provide additional gains, while the method remains more robust than the reference-user baseline.

  • Transmit power: The proposed optimization-based estimator achieves significant NMSE gains over all baselines as transmit power varies.
  • Transmit power: More than 3 dB gain is obtained by optimized phase shifts over the random configuration baseline.
  • BS antenna number: NMSE increases with BS antenna number, while the phase-shifting gain becomes larger as M increases.
  • IRS size: As IRS size increases, the proposed scheme’s NMSE decreases while Baselines 1 and 4 change little.
  • User locations: The proposed scheme’s performance gains are insensitive to user locations and remain more robust than the selected on-off protocol.
  • Overall performance: The conclusion reports more than 15 dB gain over the benchmark and more than 3 dB gain from optimized versus random phase shifting.

APPENDIX A PROOF OF LEMMA 1

The appendix proves identifiability of the virtual reference channel and relative channels through rank arguments. It also gives the direct-channel LMMSE estimator and notes a theoretical 3 dB gain from doubled observations.

  • The virtual reference channel Hv is defined from the common BS-IRS link and a training phase vector.
  • The cascaded channels factor into the virtual reference channel and user-specific relative channels, reducing the remaining task to estimating HA.
  • The proof establishes rank(Ψ) = min{N, ML1} with probability one using independent full-rank channel statistics.
  • The permuted sensing matrix retains rank min{N, ML1} with probability one by induction over the BS antennas.
  • The BS-user direct channel is estimated using an LMMSE estimator.
  • The proposed protocol may achieve a 3 dB gain for BS-user channel estimation by exploiting doubled observation samples.

APPENDIX C PROOF OF LEMMA 2

The appendix analyzes the optimization objective underlying the joint channel estimation problem. It shows convexity in the relevant variable blocks, supporting alternating optimization steps.

  • The objective function of P(A) is expressed directly in terms of the cascaded channel coefficients.
  • Any feasible pair in the specified solution set is an optimal solution of P(A).
  • The user-variable objective is a quadratic function with a Hermitian positive semidefinite Hessian.
  • Each component objective fg,m is a convex quadratic function of the corresponding BS-IRS channel variable.
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