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Channel Estimation and Passive Beamforming for Intelligent Reflecting Surface: Discrete Phase Shift and Progressive Refinement

Changsheng You, Beixiong Zheng, Rui Zhang

arXiv:1912.10646v2cs.ITcs.NI

TL;DR

The paper tackles IRS implementation when CSI is difficult to estimate, training symbols are limited per block, and phase shifts are discrete. It proposes hierarchical progressive channel estimation and rate-oriented passive-beamforming refinement, with numerical results demonstrating effectiveness and gains over benchmark schemes.

  • Problem

    Existing IRS studies often assume perfect CSI and continuous phase shifts, while all-at-once estimation can require excessive pilots and conflict with short communication blocks.

  • Method

    The paper jointly designs discrete-phase training reflection matrices and progressive passive beamforming using IRS grouping, subgroup partitioning, and successive refinement across blocks.

  • Results

    Numerical results show that the proposed progressive channel-estimation and passive-beamforming designs improve achievable rates over blocks and significantly outperform benchmark schemes under practical setups.

  • Takeaways & Limitations

    Discrete-phase IRS designs should account for correlated channel-estimation error and exploit progressive refinement when each block has insufficient training symbols.

Abstract

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Prior studies on Intelligent Reflecting Surface (IRS) have mostly assumed perfect channel state information (CSI) available for designing the IRS passive beamforming as well as the continuously adjustable phase shift at each of its reflecting elements, which, however, have simplified two challenging issues for implementing IRS in practice, namely, its channel estimation and passive beamforming designs both under the constraint of discrete phase shifts. To address them, we consider in this paper an IRS-aided single-user communication system with discrete phase shifts and design the IRS training reflection matrix for channel estimation as well as the passive beamforming for data transmission, both subject to the constraint of discrete phase shifts. We show that the training reflection matrix design for discrete phase shifts greatly differs from that for continuous phase shifts, and thus the corresponding passive beamforming should be optimized by taking into account the correlated channel estimation error due to discrete phase shifts. Specifically, we consider a practical block-based transmission, where each block has a finite (insufficient) number of training symbols for channel estimation. A novel hierarchical training reflection design is proposed to progressively estimate IRS elements' channels over multiple blocks by exploiting IRS-elements grouping and partition. Based on the resolved IRS channels in each block, we further design the progressive passive beamforming at the IRS with discrete phase shifts to improve the achievable rate for data transmission over the blocks.

I. INTRODUCTION

The paper addresses practical IRS channel-estimation and passive-beamforming challenges caused by finite per-block training, discrete phase shifts, and imperfect CSI. It progressively estimates IRS channels across blocks and refines discrete passive beamforming accordingly.

  • Motivation: Existing IRS designs commonly assume perfect CSI and continuous phase shifts, although estimating individual IRS channels is practically difficult.Only cascaded user–IRS–AP channels can be estimated directly at the AP or user using pilots.
  • Motivation: All-at-once estimation requires pilot length that grows with the number of IRS elements and conflicts with blocks containing few pilot symbols.Grouping IRS elements reduces the required pilots from the number of elements to the number of groups, but does not resolve individual elements immediately.
  • Proposed approach: The paper proposes hierarchical training that progressively estimates IRS-element channels over consecutive blocks using grouping and subgroup partitioning.Training vectors are decomposed into group-wise basis and intra-group components, while subgroup sizes decrease over blocks.
  • Proposed approach: The basis training reflection matrix minimizes per-group channel-estimation MSE under unit-modulus, discrete-phase, and full-rank constraints.DFT or Hadamard matrices are optimal in special cases; otherwise, a DFT-Hadamard-based construction provides a low-complexity near-orthogonal design.
  • Proposed approach: Progressive passive beamforming maximizes each block’s achievable rate while accounting for training overhead and correlated estimation error.A successive-refinement algorithm with three initializations obtains low-complexity suboptimal solutions, and numerical results show rate improvement over blocks and benchmark schemes.
  • System setting: The system assumes a single-user, single-antenna IRS-assisted uplink with quasi-static channels remaining constant across I0 blocks.The IRS has a large number of passive reflecting elements and operates under discrete phase shifts for both training and data transmission.

B. Proposed Progressive Channel Estimation and Passive Beamforming Design

The proposed design progressively estimates IRS channels across finite-symbol blocks through hierarchical training, then uses the resolved channels for block-wise data transmission. Grouping, partitioning, and decomposed training reflections allow increasingly fine channel resolution over successive blocks.

  • Block-Based Protocol: Each transmission block divides M0 symbols between M training symbols and M0 −M data symbols for progressive channel estimation and transmission.The estimated channels from each block support the corresponding passive beamforming design.
  • Estimation Procedure: The AP estimates per-group effective channels from received training signals using the basis matrix, then estimates subgroup channels from effective channels collected across blocks.The two-stage procedure separates per-group estimation from intra-group refinement.
  • Hierarchical Training: IRS elements are divided into M groups of L ≜ N/M adjacent elements, enabling M per-group effective channels to be estimated in each block when M ≪N.This exploits potential channel correlation among adjacent elements to reduce training overhead.
  • Hierarchical Training: The element-wise training reflection vector is decomposed into basis and intra-group components through a Hadamard-product structure.The basis component is shared across blocks, while the intra-group component varies across blocks.
  • Hierarchical Training: As blocks progress, groups are recursively partitioned into subgroups, yielding i subgroups per group in block i and resolving additional subgroup aggregated channels.The intra-group reflection design applies common coefficients within each subgroup so its aggregated channel can be resolved.
  • Progressive Resolution: For I0 = L, all IRS elements’ individual channels can be resolved by block I0 = L because ML = N.When I0 < L, the first I0 block designs are used; when I0 > L, element-wise channels are already estimated by block L.

2) Passive Beamforming:

The AP designs discrete-phase passive beamforming in each block from the subgroup aggregated-channel estimates. As more subgroup channels are resolved, the achievable rate is progressively improved over the blocks.

  • Beamforming Optimization: In each block, the AP optimizes the IRS passive beamforming vector from estimated subgroup aggregated channels to maximize the achievable rate.The resulting phase-shift values are sent to the IRS controller for implementation.
  • Beamforming Structure: Because every subgroup uses one common reflection coefficient, the block-i passive beamforming vector has iM entries.Its dimension equals the total number of subgroups with resolved aggregated channels.
  • Rate Model: Channel estimation error creates additional interference whose power depends on both the passive beamforming vector and the estimation error.The achievable rate is determined by the resulting SINR and includes a practical modulation-and-coding rate gap Γ.
  • Feedback: The design assumes error-free, zero-delay feedback, while a one-block feedback delay can be accommodated by using the preceding block’s beamforming design.The delayed-feedback modification applies the block i −1 design in block i for i = 2, · · · , L.
  • Algorithm: The progressive procedure repeats estimation, beamforming optimization, feedback, and block updating until i > L.Algorithm 1 summarizes this sequence across blocks.

III. PER-GROUP EFFECTIVE CHANNEL ESTIMATION

This section formulates per-group effective channel estimation as an IRS basis-training design problem under a given intra-group reflection design. The objective is to minimize the least-squares estimation MSE.

  • Objective: The basis training reflection matrix is designed to minimize the MSE of least-squares per-group effective channel estimation.The intra-group training reflection design is treated as given in this section.

A. Problem Formulation

The problem formulation uses a full-rank basis matrix for least-squares estimation and imposes unit-modulus and discrete-phase constraints on its entries. These constraints define the feasible training-matrix design.

  • Least-Squares Estimation: If Θs is full-rank, the least-squares estimate of the per-group effective channels is available from the received training signals.The full-rank condition ensures feasibility of the least-squares estimation.
  • Constrained Design: The optimization minimizes the per-group effective channel estimation MSE while enforcing unit-modulus and discrete phase shifts on Θs.The discrete-phase constraints apply to every entry of the basis training reflection matrix.

B. Proposed Basis Training Reflection Matrix Design

The paper designs discrete-phase basis training reflection matrices for channel estimation, combining special-case optimal orthogonal constructions with a low-complexity near-orthogonal method for general cases.

  • Optimization problem: Problem (P1) is always feasible, but its discrete phase-shift, unit-modulus, and full-rank constraints make it non-convex and NP-hard.Exhaustive search has complexity O(2^bM^2), which can be practically prohibitive as M and b increase.
  • Special cases: For M ∈ {2^c | c = 1, 2, ···, b}, the DFT matrix is optimal, while for M ∈ U, the Hadamard matrix is optimal.Here U = {u | u = 2 or u = 4r, r ∈ Z+}.
  • Special cases: An orthogonal basis training reflection matrix satisfying all constraints is optimal for (P1).The optimality condition is Θ_s^HΘ_s = MI.
  • General cases: For general cases, the DFT-Hadamard-based design constructs a near-orthogonal matrix using DFT quantization when b ≥ 2 and Hadamard truncation when b = 1.The construction addresses cases where the existence of a feasible orthogonal matrix is unknown.
  • General cases: The quantized DFT matrix is generally invertible for b ≥ 2 and can achieve MSE close to continuous-phase shifts when b is sufficiently large.For 1-bit phase shifters, the quantized-DFT matrix is mostly noninvertible for different M.
  • MSE implication: Unlike continuous phase shifts, where MMSE is σ^2/P, the proposed discrete-phase MSE depends on the designed basis training reflection matrix because it is generally non-orthogonal.The dependence arises from the non-orthogonality of the discrete-phase design.

IV. PROGRESSIVE INTRA-GROUP CHANNEL ESTIMATION

The intra-group estimation stage progressively resolves subgroup aggregated channels by designing subgroup partitions and training reflection matrices across blocks, then derives the resulting MSE.

  • Progressive estimation: The design first selects subgroup partitions and subgroup training reflection matrices to resolve subgroup aggregated channels for different groups over consecutive blocks.It then derives the MSE while accounting for errors from both per-group effective-channel and intra-group estimations.

A. Subgroup Partition and Training Reflection Matrix Design

The paper progressively refines IRS channel resolution by splitting subgroups across blocks and designing full-rank subgroup training matrices for the resulting partitions.

  • Channel representation: The subgroup training reflection vector is identical across groups and training symbols within each block, while subgroup aggregated channels are formed from indexed element sets.Effective group channels use the per-group estimation from the earlier stage.
  • Subgroup partition: In block i + 1, one parent subgroup containing multiple elements is partitioned into two child subgroups, while other subgroups remain unchanged.This increases the number of resolved subgroup aggregated channels by one.
  • Symmetric subgroup partition: Symmetric partition selects the largest subgroup and divides it into two as-equal-as-possible children.This scheme is illustrated as the left case in Fig. 4.
  • Asymmetric subgroup partition: Asymmetric partition divides the largest subgroup into unequal children, including one child containing a single element.This scheme is illustrated as the right case in Fig. 4.
  • Training reflection matrix: The extended subgroup training matrix replicates parent-subgroup training coefficients for the newly created child subgroups.This extends the previous block’s matrix before adding the new training reflection vector.
  • Training reflection matrix: Subgroup training reflection matrices are designed iteratively across blocks to be feasible and full rank, reducing reliance on exhaustive search.The proposed construction produces full-rank matrices for the illustrated symmetric case across blocks 1 ≤ i ≤ 4.

B. MSE of Intra-Group Channel Estimation

The paper obtains subgroup aggregated-channel estimates by successive per-group and intra-group estimation, and shows that estimation MSE depends on both training matrices and accumulates across blocks.

  • Estimation procedure: The estimated subgroup aggregated channels are obtained through successive per-group effective-channel and intra-group channel estimations.Their dimension increases with the block index i.
  • Estimation representation: The all-group estimate is assembled from the group-wise estimates through a permutation-based representation.The derivation invokes permutation-matrix properties when simplifying the MSE expression.
  • MSE characterization: The MSE increases with block index because errors from per-group effective-channel and intra-group estimations accumulate and propagate.The derivation uses independent effective-channel estimation errors across blocks and the block-diagonal structure of E^(i).

V. PROGRESSIVE PASSIVE BEAMFORMING OPTIMIZATION

The paper optimizes progressive IRS passive beamforming in each block using estimated group or subgroup aggregated channels while accounting for channel estimation error.

  • V. PROGRESSIVE PASSIVE BEAMFORMING OPTIMIZATION: Progressive passive beamforming is optimized block by block from estimated group or subgroup aggregated channels.The objective is to maximize achievable data-transmission rate while incorporating channel estimation error.

A. Problem Formulation

The formulation uses estimated aggregated channels and their error covariance to express block-dependent SINR and maximize achievable rate under discrete, unit-modulus IRS phases.

  • A. Problem Formulation: The formulation begins with estimated group or subgroup aggregated channels.These estimates are used to construct the effective channel representation for each block.
  • A. Problem Formulation: Channel estimation error is represented through an error covariance matrix.The covariance enters the subsequent SINR formulation.
  • A. Problem Formulation: The SINR-related coefficient depends on both the basis training reflection matrix and the subgroup training reflection matrix.Thus, training-reflection design affects the subsequent beamforming formulation.
  • A. Problem Formulation: The SINRs share a common form across blocks, while the passive beamforming vector grows with the number of resolved IRS elements.The block-varying estimated channel matrix changes with the training stage but does not alter the optimization method.
  • A. Problem Formulation: Average-rate maximization is converted to SINR maximization after dropping the constant P/σ2 term.The resulting problem retains unit-modulus and discrete-phase constraints.

B. Proposed Algorithm for Problem (P2)

Problem (P2) is non-convex under discrete phase constraints, so the paper combines several initialization strategies with successive refinement to obtain practical suboptimal solutions.

  • B. Proposed Algorithm for Problem (P2): Exhaustive search solves (P2) but has complexity O(2^biM), which grows exponentially with the beamforming dimension.The paper therefore proposes an efficient successive refinement algorithm for suboptimal optimization.
  • 1) Initialization Methods:: SDR initialization relaxes the discrete-phase constraint and reformulates the problem using Φ = φφ^H.Dropping the rank-one constraint produces a semidefinite relaxation whose solution can provide an upper bound.
  • 1) Initialization Methods:: The initialization choice determines both the computational complexity and solution quality of successive refinement.A low-complexity alternative has order O(iM2^b) when b is small.
  • 1) Initialization Methods:: The relaxed SDP solution is converted into a feasible discrete-phase beamformer through Gaussian randomization when the relaxed matrix has rank greater than one.The resulting vector is a high-quality suboptimal solution for (P3).
  • 1) Initialization Methods:: Phase-quantization initialization maps each continuous phase to its nearest value in the discrete set F.This directly constructs a discrete-phase starting beamformer from the continuous-phase solution.
  • 1) Initialization Methods:: Replication-based initialization reuses the previous block's passive beamforming to reduce the current block's initialization complexity.The method is motivated by the high complexity of SDR initialization for large iM.

2) Successive Refinement:

Successive refinement searches discrete subgroup phases iteratively, reducing complexity while progressively improving beamforming across blocks; simulations evaluate estimation, rate, initialization, and practical-scope effects.

  • 2) Successive Refinement:: Each refinement iteration searches the finite phase set for every subgroup while fixing the other subgroup phases.The process continues until the fractional decrease in γ(φ) falls below a small threshold.
  • 2) Successive Refinement:: O(log(1/ϵ)iM2^b) complexity is achieved for any feasible initialization, compared with exhaustive search.The stated order depends on the target solution accuracy ϵ.
  • 2) Successive Refinement:: The achievable-rate evaluation uses 30-symbol blocks, with Rician factors K_UI = 3 dB and K_IA = −20 dB.These are part of the default simulation settings.
  • A. Per-Group Effective Channel Estimation: The proposed basis training reflection matrix outperforms the naive and random-selection benchmarks in achievable rate.The study also reports a tradeoff between channel-estimation accuracy and training overhead.
  • A. Per-Group Effective Channel Estimation: Increasing discrete phase resolution from 1-bit to 2-bit significantly improves the proposed design's rate, unlike the marginal improvement of random selection.With 1-bit phases, the naive scheme can perform worse than random phase-shift selection.
  • B. Intra-Group Channel Estimation: Symmetric subgroup partition initially grows faster, but asymmetric partition overtakes it after block 14 because its lower MSE eventually improves beamforming gain.Symmetric partition is more suitable for smaller I0 because of faster rate convergence.
  • B. Intra-Group Channel Estimation: Both proposed partition schemes outperform progressive random phase-shift selection, indicating that progressive CSI refinement is more effective than random reflection selection.All-at-once estimation provides an achievable-rate upper bound because it uses fully resolved CSI without intra-group estimation error.
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