Intelligent Reflecting Surface with Discrete Phase Shifts: Channel Estimation and Passive Beamforming

I. INTRODUCTION

• Existing IRS studies often assume perfect CSI, although obtaining it is practically difficult because IRSs have many reflecting elements and lack signal-processing capabilities.Source: §I · p. 1
• With discrete phase shifts, it remains unknown whether the continuous-phase DFT training pattern applies and how discrete shifts affect estimation error and beamforming performance.Source: §I · p. 1
• The paper proposes a practical successive protocol for channel estimation and passive beamforming with discrete-phase reflecting elements.The training design targets channel-estimation MSE, while the data-transmission design accounts for training overhead and correlated estimation error.Source: §I · pp. 1–2
• The proposed DFT-Hadamard-based training pattern uses DFT quantization or Hadamard truncation to construct a near-orthogonal design, and numerical results verify the designs’ effectiveness.Source: §I · pp. 1–2

II. SYSTEM MODEL

• A single-antenna user communicates uplink to a single-antenna AP with assistance from an IRS placed near the user.Source: §II · p. 2
• The IRS’s N reflecting elements are divided into M groups of adjacent elements, with each sub-surface sharing a common reflection coefficient.Source: §II · p. 2
• Each sub-surface uses full-on reflection and one of K = 2^b uniformly quantized phase shifts; the frame assigns M + 1 symbols to training and the remainder to data.The AP estimates the extended channel from training, then optimizes discrete-phase passive beamforming and feeds the coefficients back to the IRS controller.Source: §II · p. 2

III. CHANNEL ESTIMATION

• The channel-estimation section formulates and solves an optimization problem that designs the IRS reflection pattern to minimize LS channel-estimation MSE.Source: §III · p. 2

A. Problem Formulation

• The extended reflection pattern Θp stacks the M + 1 extended reflection vectors, and the channel estimate requires Θp to have full rank.Source: A. Problem Formulation · p. 3
• The feasible pattern’s first column must be all ones to estimate the direct link without phase shift.Source: A. Problem Formulation · p. 3
• The remaining pattern entries must satisfy unit-modulus and discrete-phase constraints, and the optimization is formulated under these feasibility conditions.Source: A. Problem Formulation · p. 3

B. Proposed Reflection Pattern Design

• A naive reflection pattern makes the optimization problem feasible regardless of phase-shifter resolution and channel training time.Its entries satisfy the stated feasibility constraints.Source: B. Proposed Reflection Pattern Design · p. 3
• The reflection-pattern problem is NP-hard, and exhaustive search is practically prohibitive because its complexity increases exponentially with M and/or b.The problem includes non-convex objective and feasibility constraints, including full rank and unit modulus.Source: B. Proposed Reflection Pattern Design · p. 3
• For IRS with equally spaced discrete phase shifts, the DFT matrix is an optimal reflection pattern when M+1 ∈ {2^c | c=1,…,b}, and the Hadamard matrix is optimal when M+1 ∈ U.U = {u | u = 2 or u = 4r, r ∈ Z+}.Source: B. Proposed Reflection Pattern Design · p. 3
• For other cases, the DFT–Hadamard design constructs near-orthogonal patterns using DFT-matrix quantization for b ≥ 2 and Hadamard-matrix truncation for b = 1.For 1-bit phase shifters, the quantized DFT pattern is often non-invertible, motivating the truncated Hadamard construction.Source: B. Proposed Reflection Pattern Design · pp. 3–4
• The proposed DFT–Hadamard pattern has much smaller MSE than the naive scheme, reaching the continuous-phase lower bound when M + 1 ∈ U.Its MSE fluctuates with the number of IRS groups, while the naive scheme’s MSE increases monotonically; the quantized DFT pattern also reaches the lower bound for 2-bit phase shifters.Source: Figure 2 · p. 4, B. Proposed Reflection Pattern Design · p. 4

• The estimated channel provides the basis for optimizing the IRS passive beamforming.Source: §IV · p. 4
• The optimization targets the achievable rate for data transmission.Source: §IV · p. 4
• The rate optimization accounts for reflection-pattern-dependent channel-estimation error.Source: §IV · p. 4

• Channel-estimation error contributes additional interference to the received signal.Source: A. Problem Formulation · p. 4
• The average achievable rate is expressed using an SINR that includes a practical modulation-and-coding rate gap Γ ≥ 1.Source: A. Problem Formulation · p. 4
• The SINR, and therefore the achievable rate, depends on the IRS training reflection pattern through Rp.Source: A. Problem Formulation · p. 4
• Maximizing average achievable rate is reformulated as SINR maximization subject to unit-modulus and discrete-phase constraints.Source: A. Problem Formulation · p. 4

B. Proposed Algorithms for Problem (P2)

• Exhaustive search for the non-convex discrete-phase problem has complexity O(2^bM), motivating a successive-refinement solver with a designed initialization.Source: B. Proposed Algorithms for Problem (P2) · p. 4
• The initialization relaxes the discrete-phase and rank-one constraints, applies a Charnes-Cooper transformation, and solves the resulting semidefinite program.If the relaxed solution is not rank one, Gaussian randomization produces a feasible suboptimal initialization.Source: B. Proposed Algorithms for Problem (P2) · p. 5
• The continuous-phase initialization is rotated without changing its SINR, then each remaining phase is quantized to the nearest allowed discrete value.Source: B. Proposed Algorithms for Problem (P2) · p. 5
• Each refinement step searches one sub-surface’s discrete phases while fixing the others, and the objective is non-decreasing until the stopping threshold is reached.Source: B. Proposed Algorithms for Problem (P2) · p. 5
• The total complexity is O(log(1/ϵ)2^bM +(M +1)^3.5), which the authors describe as practically affordable for moderate M and b.The bound combines the refinement and SDP-initialization costs.Source: B. Proposed Algorithms for Problem (P2) · p. 5

V. NUMERICAL RESULTS

• The simulations use a single-antenna user and access point, with the IRS placed near the user and equipped with a half-wavelength-spaced uniform rectangular array.The user is at (20, 50, 0), the access point at (20, 0, 0), and the IRS is centered at (18, 50, 0).Source: §V · p. 5
• Increasing discrete phase-shifter resolution from 1-bit to 2-bit significantly improves the proposed scheme's rate, while random phase shifts yield similar rates at both resolutions.The naive pattern ranks below random phase shifts at 1-bit but above them at 2-bit.Source: §V · pp. 5–6
• With N = 80, the proposed training pattern outperforms random and naive patterns as the number of IRS groups varies.The comparison reflects a tradeoff: too few groups reduce CSI accuracy, while too many leave less time for data transmission.Source: §V · p. 5, Figure 3 · p. 6
• With M = 4, the 2-bit successive refinement algorithm achieves near-optimal rates versus exhaustive search and only a small loss versus the continuous-phase upper bound.It significantly outperforms random phase shifts and channel-gain maximization; the small gap to quantization supports the proposed initialization.Source: Figure 4 · p. 6, §V · p. 6

VI. CONCLUSION

• The proposed DFT-Hadamard-based reflection pattern minimizes channel-estimation MSE under unit-modulus, discrete-phase, and full-rank constraints.The rate-maximization design uses reflection-pattern-dependent channel-estimation error, and successive refinement provides a high-quality suboptimal beamforming solution.Source: §VI · p. 6