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Efficient Channel Estimation for Double-IRS Aided Multi-User MIMO System

Beixiong Zheng, Changsheng You, Rui Zhang

arXiv:2011.00738v3cs.IT

TL;DR

Double-IRS systems promise higher passive beamforming gains but make cascaded CSI acquisition harder because single- and double-reflection links are coupled and larger. The paper proposes always-ON, lower-dimensional channel estimation with joint IRS training design for single- and multi-user MIMO, and simulations validate its effectiveness against benchmark schemes.

  • Problem

    Double-IRS systems require efficient estimation of coupled single- and double-reflection cascaded CSI despite involving more channel coefficients than single-IRS systems.

  • Method

    The scheme keeps both IRSs always ON, tunes their training phases, and exploits lower-dimensional scaled relationships for single-user and reference-based multi-user cascaded CSI estimation.

  • Results

    Simulation results validate the proposed channel estimation scheme and joint training reflection design against benchmark schemes in training overhead and channel estimation error.

  • Takeaways & Limitations

    The proposed relationships among cascaded channels enable efficient estimation of double-IRS multi-user CSI with reduced training overhead.

Abstract

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To achieve the more significant passive beamforming gain in the double-intelligent reflecting surface (IRS) aided system over the conventional single-IRS counterpart, channel state information (CSI) is indispensable in practice but also more challenging to acquire, due to the presence of not only the single- but also double-reflection links that are intricately coupled and also entail more channel coefficients for estimation. In this paper, we propose a new and efficient channel estimation scheme for the double-IRS aided multi-user multiple-input multiple-output (MIMO) communication system to resolve the cascaded CSI of both its single- and double-reflection links. First, for the single-user case, the single- and double-reflection channels are efficiently estimated at the multi-antenna base station (BS) with both the IRSs turned ON (for maximal signal reflection), by exploiting the fact that their cascaded channel coefficients are scaled versions of their superimposed lower-dimensional CSI. Then, the proposed channel estimation scheme is extended to the multi-user case, where given an arbitrary user's cascaded channel (estimated as in the single-user case), the other users' cascaded channels can also be expressed as lower-dimensional scaled versions of it and thus efficiently estimated at the BS. Simulation results verify the effectiveness of the proposed channel estimation scheme and joint training reflection design for double IRSs, as compared to various benchmark schemes.

I. INTRODUCTION

Double-IRS systems can offer higher passive beamforming gains, but jointly estimating their coupled single- and double-reflection cascaded CSI is challenging. The paper proposes always-ON, efficiently designed training and exploits lower-dimensional channel relationships to reduce estimation overhead for single- and multi-user cases.

  • Motivation: Double-IRS links can achieve O(M^4) passive beamforming gain versus O(M^2) for conventional single-IRS systems, but require estimating more intricately coupled channel coefficients.The larger estimation burden can consume training time within a limited channel coherence interval.
  • Proposed approach: The proposed scheme jointly estimates single- and double-reflection cascaded CSI with both IRSs always ON while tuning their reflection phases over time.Always-ON reflection preserves full reflected-signal power, although the resulting channels remain superimposed during training.
  • Proposed approach: For one user, shared IRS 2→BS links make high-dimensional cascaded channels scaled versions of lower-dimensional superimposed CSI, reducing required training variables.The multi-user extension similarly represents other users’ cascaded channels as scaled versions of an estimated reference user’s channel.
  • Training design: Jointly optimized training phase shifts provide orthogonality, while the minimum training overhead decreases with BS antenna count until reaching a lower bound when N > M.The design also accounts for error propagation when allocating training time across phases.
  • Validation: Numerical results compare the proposed estimator and joint reflection design favorably with benchmark schemes in training overhead and channel estimation error.The simulations also evaluate the always-ON scheme against heuristic reflection designs.

III. DOUBLE-IRS CHANNEL ESTIMATION FOR SINGLE-USER CASE

The single-user scheme separates superimposed single- and double-reflection CSI through two training phases. It first estimates a lower-dimensional superimposed CSI, then estimates scaling factors and the remaining single-reflection channel using designed IRS phase variations.

  • Motivation: With both IRSs ON, single- and double-reflection channels are superimposed and therefore difficult to estimate separately from received pilots.The proposed two-phase procedure addresses this coupling rather than switching either IRS OFF.
  • Phase I: Phase I fixes IRS 1 and dynamically tunes IRS 2 to estimate the superimposed CSI of the channels related to IRS 2.A full-row-rank training reflection matrix enables least-squares estimation of the superimposed CSI and the IRS 1 single-reflection channel.
  • Channel relationship: Shared IRS 2→BS links make the IRS 2 single-reflection and double-reflection cascaded channels lower-dimensional scaled versions of their superimposed CSI.The resulting scaling vectors and the other single-reflection channel are the remaining quantities to estimate.
  • Dimension reduction: The original single-user model contains N(M1 + M2) + NM1M2 coefficients, whereas the reduced model requires N(M1 + M2) + M2(M1 + 1) coefficients when N ≫ 1.After Phase I, only the scaling matrix and remaining single-reflection channel need further estimation.
  • Phase II: Phase II dynamically tunes both IRS phase shifts to jointly estimate the remaining scaling factors and single-reflection channel using the superimposed CSI as an observation matrix.The required training design depends on whether that observation matrix has full column rank.

1) Case 1:

When N ≥ M2, the superimposed CSI is used to recover the remaining channels after estimating a composite CSI in the second phase. This case requires 2M1 + M2 + 2 pilot symbols.

  • 1) Case 1:: IRS 2 can use a common time-varying phase shift in Phase II while its initial reflection phases are set uniformly for simplicity.The common phase shift has unit magnitude and is applied to all IRS 2 reflecting elements or subsurfaces.
  • 1) Case 1:: For N ≥ M2, the estimated composite CSI is partitioned to recover the scaling matrix and remaining single-reflection channel through least-squares relations.The construction uses a full-column-rank training design for the Phase II reflection matrix.
  • 1) Case 1:: 2M1 + M2 + 2 pilot symbols are sufficient for the proposed single-user estimator when N ≥ M2.This overhead counts the pilots across Phases I and II.

2) Case 2: 

When N < M2, the IRS 2 observation matrix is rank-deficient, so the scheme jointly tunes both IRSs to estimate the remaining channels. The resulting overhead is higher and the larger inversion is less efficient than in the N ≥ M2 case.

  • 2) Case 2:: For N < M2, rank deficiency prevents direct estimation of the scaling matrix from the IRS 2 observation matrix alone.The scheme therefore jointly varies both IRSs’ phase shifts during Phase II.
  • 2) Case 2:: Joint Phase II training constructs a full-rank sensing matrix for estimating the scaling matrix and remaining single-reflection channel together.The design requires rank(Ξ) = M2 + M1M2 + NM1.
  • 2) Case 2:: The minimum single-user training overhead is ⌈(M2 + M1M2 + NM1)/N⌉ + M1 + M2 + 1 pilots when N < M2.The ceiling arises because the number of Phase II pilots is integer-valued.
  • 2) Case 2:: Applying the N < M2 least-squares method when N ≥ M2 is generally less efficient because it requires more phase shifts and a larger matrix inversion.The paper identifies both higher reflection-training demands and increased inversion complexity.

B. Training Reflection Phase-Shift Design

The training reflection phase-shift design minimizes channel-estimation error across the two proposed training phases. Phase I uses a DFT-based IRS 2 design, while Phase II is optimized separately according to the BS-antenna regime.

  • The training reflection phases are optimized jointly across the two proposed training phases to minimize channel-estimation error.
  • Phase I: In Phase I, a submatrix of an I1×I1 DFT matrix with its first M2+1 rows achieves the minimum MSE.
  • Phase II: Phase II uses separate optimization cases for N ≥ M2 and N < M2.
  • Phase II: For N ≥ M2, the Phase II LS-estimation MSE is expressed through the joint training reflection matrix ΩII.

1) Case 1:

When N ≥ M2, the Phase II design seeks an orthogonal joint training matrix under always-ON reflection constraints. A shifted DFT construction supplies the required orthogonality and achieves the minimum MSE.

  • Case 1: The optimal Phase II condition is ΩIIΩHII = I2I2M1+1, which minimizes the MSE under the stated case.
  • Case 1: The required always-ON joint design is nontrivial because Θ1,II and ψII must satisfy all orthogonality conditions simultaneously.
  • Case 1: Selecting M+1 rows from a DFT matrix can create perfect orthogonality in ΩII and minimize the estimation MSE.
  • Case 1: The construction moves the first row of the I2×I2 DFT matrix to the end, assigns the first M1 rows to Θ1,II, and uses the next shifted row for ψII.
  • Case 1: Row shifting makes the transformed rows pairwise orthogonal, so the conditions for the joint design are simultaneously satisfied.

2) Case 2:

The multi-user extension avoids estimating every user independently by using one user’s cascaded CSI as a reference. The remaining users are represented through lower-dimensional scaling vectors and estimated jointly.

  • Using the single-user scheme separately for K users would multiply training overhead by K, motivating the proposed reference-user extension.
  • Given one user’s cascaded CSI, the remaining users’ channels are lower-dimensional scaled versions because the users share the IRS 2→BS, IRS 1→BS, and IRS 1→IRS 2 links.
  • Extended Channel Estimation Scheme for Multiple Users: With reference CSI available, only the scaling vectors bk and b̃k need to be estimated for each remaining user.
  • Extended Channel Estimation Scheme for Multiple Users: The multi-user procedure estimates the remaining users’ scaling vectors in a subsequent Phase III after the reference user’s cascaded CSI is acquired.
  • Extended Channel Estimation Scheme for Multiple Users: The Phase III design fixes both IRS reflection phase-shifts and uses concurrent pilot symbols for joint scaling-vector estimation.
  • Extended Channel Estimation Scheme for Multiple Users: The required LS design depends on the rank of the spatial observation matrix B, with separate cases for full-column-rank and rank-deficient B.

1) Case 1:

When N ≥ M1+M2, the remaining users’ scaling vectors can be jointly estimated using designed training reflections and pilots. The design requires full relevant ranks and at least K−1 Phase III pilot periods.

  • Case 1: For N ≥ M1+M2, the observation matrix B can have full column rank, enabling joint LS estimation of the remaining users’ scaling vectors.
  • Case 1: The training reflections and pilot matrix are designed so that rank(B) = M1 + M2 and rank(X) = K −1.
  • Case 1: At least K −1 pilot periods are required to ensure rank(X) = K −1 and the existence of X†.
  • Case 1: When N < M1+M2, B is rank-deficient and the full-column-rank LS formulation cannot estimate Λ directly.

2) Case 2:

Phase III formulates least-squares estimation for the remaining cascaded channel parameters and imposes rank conditions through training reflections and pilot design.

  • 2) Case 2:: The Phase III least-squares estimate requires a full-rank training matrix, with I3N ≥ (K −1)(M1 + M2).The required rank is (K −1)(M1 + M2), and I3 is an integer training-duration parameter.

B. Training Design for Multiple Users

The multi-user training design exploits shared channel relationships to reduce overhead, while simulations compare the proposed scheme with decoupled and independently estimated benchmarks.

  • V. NUMERICAL RESULTS: The simulations use a BS ULA, two IRS UPAs with 5 × 5-element subsurfaces, and a path-loss model with γ0 = −30 dB and exponent 2.2 on nearby links.The numerical setup places the BS, IRSs, and user cluster at specified three-dimensional coordinates.
  • A. Training Overhead Comparison: The compared benchmark schemes separately estimate users or ignore common channel relationships, increasing training requirements relative to the proposed design.The decoupled scheme uses ON/OFF IRS training, while the -based benchmark estimates double-reflection channels independently across antennas and users.
  • A. Training Overhead Comparison: The proposed and decoupled schemes have much lower-order training overhead than the benchmark based on by exploiting shared channel relationships and multiple BS antennas.Table I reports the training-overhead comparison under M1 = M2 = M/2.
  • A. Training Overhead Comparison: As N increases, the proposed and decoupled schemes’ training overheads decrease dramatically, whereas the benchmark overhead is independent of N.The difference is attributed to joint estimation across BS antennas in the proposed and decoupled schemes, unlike the benchmark’s independent estimation.
  • A. Training Overhead Comparison: As K increases, overhead grows only marginally for the proposed and decoupled schemes but dramatically for, which estimates users separately over consecutive time.For the benchmark, the additional overhead with one more user is M + 1, compared with max{1, ...} for the proposed and decoupled schemes.

B. Normalized MSE Comparison for Single-User Case

The simulations compare training reflection designs, pilot-time allocation, and channel-estimation schemes using normalized MSE in the single-user setting. Results highlight gains from orthogonal DFT-based designs and trade-offs caused by error propagation between training phases.

  • Training reflection designs: The proposed DFT-based training design achieves up to 10 dB power gain over random phase shifts in Phase I and lower MSE than heuristic and random designs in Phase II.The Phase II gain is attributed to selecting DFT rows that ensure perfect orthogonality in the joint training design.
  • Pilot allocation: With fixed total overhead I1 + I2 = 1062, the MSE for some estimated channels first decreases and then increases as more pilot symbols are assigned to Phase I.More Phase I pilots reduce propagated error but leave fewer pilots for Phase II, producing a non-monotonic trade-off.
  • Channel-estimation schemes: The proposed and decoupled schemes show different normalized-MSE behavior for the two single-reflection links because only one proposed estimate is affected by Phase I-to-Phase II error propagation.The decoupled scheme exhibits symmetric performance for the two single-reflection links under equal training time.
  • Channel-estimation schemes: The proposed always-ON scheme achieves up to 3 dB power gain over the decoupled scheme for one single-reflection cascaded channel.The gain comes from exploiting the full-reflection power of both IRSs during estimation.
  • Channel-estimation schemes: For double-reflection links, the proposed scheme has lower normalized MSE than the decoupled scheme because the latter additionally suffers residual interference from imperfect signal cancellation.Both schemes experience error propagation, but the decoupled scheme also accumulates cancellation residuals.

C. Normalized MSE Comparison for Multi-User Case

The proposed always-ON channel estimation scheme achieves better normalized-MSE performance than the decoupled scheme in the multi-user case while exploiting lower-dimensional scaled CSI.

  • C. Normalized MSE Comparison for Multi-User Case: Both schemes reduce multi-user training overhead by representing other users’ cascaded channels through lower-dimensional scaling vectors relative to a reference user.The proposed scheme jointly estimates these scaling vectors with always-ON IRSs, whereas the decoupled scheme estimates them successively with one IRS turned OFF.
  • C. Normalized MSE Comparison for Multi-User Case: With always-ON IRSs, the proposed scheme achieves much better normalized MSE for the scaling vectors than the decoupled scheme using ON/OFF reflection.The comparison assumes the cascaded CSI of user 1 is perfectly available as reference CSI.
  • C. Normalized MSE Comparison for Multi-User Case: The proposed scheme has asymmetric normalized MSE across users because each b_k combines single- and double-reflection channels, unlike the decoupled scheme’s symmetric performance.The combined channel associated with b_k receives more reflection power than the single-reflection channel associated with its counterpart.
  • C. Normalized MSE Comparison for Multi-User Case: The proposed scheme significantly outperforms the decoupled scheme in normalized MSE for estimating all cascaded CSI as user transmit power varies.The comparison considers K = 10 users and comparable minimum training overheads.
  • C. Normalized MSE Comparison for Multi-User Case: The overall approach keeps both IRSs turned ON during training and uses scaled CSI relationships to reduce training overhead for single- and multi-user estimation.The paper also designs training phase shifts to minimize estimation error and accounts for error propagation across training phases.
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