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Power-Measurement-Based Channel Autocorrelation Estimation for IRS-Assisted Wideband Communications

He Sun, Lipeng Zhu, Weidong Mei, Rui Zhang

arXiv:2502.11346v1cs.IT

TL;DR

Wideband IRS-assisted OFDM needs channel estimation without costly complex-valued pilots, especially because IRS reflections are passive and the reflected channel is high-dimensional. The paper uses RSRP-driven single-layer neural subnetworks with progressive training to recover the channel autocorrelation matrix, then optimizes discrete IRS reflections. Simulations show improved performance over existing power-measurement designs, with reduced measurement overhead and near-perfect-CSI performance in the reported settings.

  • Problem

    Wideband IRS channel estimation remains challenging because passive frequency-flat reflections and high-dimensional channels with multipath delay spread complicate efficient CSI acquisition.

  • Method

    A single-layer NN recovers the wideband channel autocorrelation matrix from RSRP measurements, while progressive subnetworks adapt model rank and estimated matrices support discrete reflection optimization.

  • Results

    The proposed methods outperform benchmark IRS reflection designs based on user power measurements, reduce measurement overhead, and approach perfect-CSI average-gain or achievable-rate upper bounds in simulations.

  • Takeaways & Limitations

    RSRP-only autocorrelation recovery provides an efficient framework for wideband OFDM IRS reflection design without requiring complex-valued pilot-based channel recovery.

Abstract

from arXiv · show

Channel state information (CSI) is essential to the performance optimization of intelligent reflecting surface (IRS)-aided wireless communication systems. However, the passive and frequency-flat reflection of IRS, as well as the high-dimensional IRS-reflected channels, have posed practical challenges for efficient IRS channel estimation, especially in wideband communication systems with significant multi-path channel delay spread. To tackle the above challenge, we propose a novel neural network (NN)-empowered IRS channel estimation and passive reflection design framework for the wideband orthogonal frequency division multiplexing (OFDM) communication system based only on the user's reference signal received power (RSRP) measurements with time-varying random IRS training reflections. In particular, we show that the average received signal power over all OFDM subcarriers at the user terminal can be represented as the prediction of a single-layer NN composed of multiple subnetworks with the same structure, such that the autocorrelation matrix of the wideband IRS channel can be recovered as their weights via supervised learning. To exploit the potential sparsity of the channel autocorrelation matrix, a progressive training method is proposed by gradually increasing the number of subnetworks until a desired accuracy is achieved, thus reducing the training complexity. Based on the estimates of IRS channel autocorrelation matrix, the IRS passive reflection is then optimized to maximize the average channel power gain over all subcarriers. Numerical results indicate the effectiveness of the proposed IRS channel autocorrelation matrix estimation and passive reflection design under wideband channels, which can achieve significant performance improvement compared to the existing IRS reflection designs based on user power measurements.

I. INTRODUCTION

IRSs offer adjustable, low-cost wireless propagation control, but wideband IRS systems still face difficult channel-estimation requirements. This work addresses that gap with RSRP-based neural estimation, progressive training, and passive-reflection design for OFDM.

  • I. INTRODUCTION: Wideband power-measurement-based IRS channel recovery had not been extended from frequency-flat narrowband systems to frequency-selective channels.Existing approaches used pilots or power measurements, but the cited power-measurement recovery methods considered only narrowband systems.
  • I. INTRODUCTION: The proposed framework estimates the wideband IRS channel autocorrelation matrix from users’ RSRP measurements using a single-layer NN with multiple same-structure subnetworks.The RSRP under each IRS reflection is represented as the sum of subnetwork outputs, and the autocorrelation matrix is recovered from the learned weights.
  • I. INTRODUCTION: Progressive training gradually increases the number of subnetworks until training loss converges, balancing channel-estimation accuracy and computational complexity without knowing the matrix rank.Larger subnetworks can improve accuracy but increase complexity, motivating adaptive rank growth.
  • I. INTRODUCTION: The estimated autocorrelation matrix is used to optimize discrete IRS reflection coefficients for maximum average channel power gain across OFDM subcarriers.The system model uses frequency-selective wideband channels and discrete phase shifts at the IRS.
  • I. INTRODUCTION: Simulations report that the proposed methods outperform benchmark power-measurement reflection designs with reduced measurement overhead and approach perfect-CSI average-gain or achievable-rate upper bounds.The evaluation spans different user numbers, IRS sizes, and power-allocation schemes.

MXHXF M

The paper formulates wideband IRS reflection optimization around the average channel power gain and addresses unavailable perfect channel knowledge with neural autocorrelation estimation.

  • MXHXF M: The channel autocorrelation matrix R is Hermitian positive semidefinite and is scaled by the transmit-power factor P/M.This structure supports its role in the power-based formulation.
  • MXHXF M: The design objective is to maximize average channel power gain across all subcarriers under discrete IRS phase-shift constraints.The optimization is formulated as problem (P1).
  • MXHXF M: Perfect IRS channel knowledge is difficult to acquire because the IRS lacks signal-processing capability and the wideband OFDM channel is high-dimensional.The proposed NN-enabled estimation method therefore uses user power measurements.

III. NN-ENABLED CHANNEL ESTIMATION BASED ON USER POWER MEASUREMENTS

The method uses RSRP measurements from randomly varying IRS reflections to characterize wideband channel power, even when reference signals occupy only a subset of OFDM subcarriers.

  • A. RSRP Measurement: Reference signals are inserted into M0 of the M subcarriers, allowing standard cellular or WiFi terminals to provide the required RSRP measurements.The transmit power of reference-signal resource elements is equally allocated within the system bandwidth.
  • A. RSRP Measurement: The IRS cycles through randomly generated phase-shift patterns, while the user measures reference-signal power across Q OFDM symbols within one coherence block.The measurements are collected for L different IRS reflection vectors.
  • A. RSRP Measurement: For sufficiently large Q, the measured RSRP approaches the average received power over the RS-bearing subcarriers.The averaging reduces the effect of receiver noise in the power measurement.
  • A. RSRP Measurement: RSRP measured on a subset of subcarriers equals the average received signal power over all OFDM subcarriers when M0 ≥ K.This relation enables wideband channel characterization without measuring every subcarrier.

B. NN-Enabled Channel Autocorrelation Matrix Estimation

The channel autocorrelation matrix is recovered by representing received power as a sum of rank-one quadratic terms implemented by a single-layer NN with multiple subnetworks.

  • B. NN-Enabled Channel Autocorrelation Matrix Estimation: The high-dimensional Hermitian PSD autocorrelation matrix is reduced to a basis-vector estimation problem by exploiting its rank, which equals the maximum number of delay taps.The rank is generally much smaller than the matrix dimension, so at least K basis vectors suffice.
  • B. NN-Enabled Channel Autocorrelation Matrix Estimation: The noiseless received power is the sum of K′ squared amplitudes |v^H a_k|^2 and can therefore be predicted by a single-layer NN.The network uses K′ subnetworks, each corresponding to one basis vector a_k.
  • B. NN-Enabled Channel Autocorrelation Matrix Estimation: The NN takes the real-valued IRS reflection vector as input, applies shared subnetwork structure, and sums squared-norm activations at the output.The complex reflection vector is represented through real and imaginary parts to avoid complex-domain NN implementation.
  • B. NN-Enabled Channel Autocorrelation Matrix Estimation: Training the subnetwork weights to match the basis matrices recovers the channel autocorrelation matrix through the relation ˆp(v) = v^H R v.If W_k = B_k for every subnetwork, the NN output equals the noiseless received power quadratic form.
  • B. NN-Enabled Channel Autocorrelation Matrix Estimation: The reconstruction requires equality over sufficiently rich IRS reflection vectors because discrete phase shifts restrict reflections to a finite subspace.The authors state that this subspace should be sufficient when the IRS has a large number of elements.

IV. NN TRAINING AND IRS REFLECTION OPTIMIZATION

A progressive supervised-training procedure estimates NN weights from RSRP labels by increasing the number of subnetworks, then uses validation error to select the model.

  • IV. NN Training and IRS Reflection Optimization: The progressive method increases the number of subnetworks in increments of τ because the channel rank and required network size are unknown.Training stops when validation error converges or the maximum allowed number of subnetworks is reached.
  • IV. NN Training and IRS Reflection Optimization: Each progressive stage reuses previously trained weights, randomly initializes only the newly added subnetworks, and trains with stochastic gradient descent.This avoids restarting the full network at every stage.
  • IV. NN Training and IRS Reflection Optimization: The NN is trained on a subset of RSRP measurements using mean squared error, while the remaining measurements form a validation set.The validation MSE determines the selected weights and convergence of progressive training.
  • IV. NN Training and IRS Reflection Optimization: The procedure outputs trained weights that are used to estimate the wideband IRS channel autocorrelation matrix.The estimated matrix then supports the subsequent passive-reflection optimization.

B. IRS Reflection Design

The estimated channel autocorrelation matrix replaces the unknown matrix in IRS reflection optimization, which is solved using semidefinite relaxation, Gaussian randomization, phase quantization, and successive refinement.

  • B. IRS Reflection Design: The reflection design substitutes the estimated autocorrelation matrix into the average-channel-power optimization problem.The original problem remains non-convex because of discrete phase-shift and rank-one constraints.
  • B. IRS Reflection Design: Semidefinite relaxation removes the rank-one constraint and replaces discrete phase constraints with diagonal constraints to obtain a convex SDP.An interior-point method solves the relaxed problem before constructing a feasible reflection vector.
  • B. IRS Reflection Design: Gaussian randomization and phase quantization produce a feasible discrete-phase reflection vector, which successive refinement then improves element by element.Each refinement step enumerates allowed phase shifts while holding the other elements fixed.
  • B. IRS Reflection Design: Successive refinement converges because each update makes the objective non-decreasing and the objective is upper-bounded by finite base-station transmit power.The resulting reflection pattern can be computed offline and updated when statistical channel characteristics change significantly.
  • B. IRS Reflection Design: The method is intended to remain useful when channel statistics are stable over time, including environments with static scatterers and fast-fading channels.The estimated autocorrelation captures statistical characteristics that persist across many coherence blocks.

V. NUMERICAL RESULTS

Simulations evaluate the proposed IRS channel autocorrelation estimation and passive reflection design under frequency-selective fading, using a 32-element IRS and a 128-subcarrier OFDM system.

  • Simulation Setup: The proposed estimation and reflection design are evaluated under frequency-selective fading channels using the stated three-dimensional deployment geometry.The BS, IRS, and receiver positions and distance-dependent path-loss parameters define the simulation geometry.
  • Simulation Setup: The simulations use a 32-element IRS, 128 OFDM subcarriers, and frequency-selective channels with 4, 4, and 3 delay taps across the links.The IRS has a 4 × 8 UPA with half-wavelength spacing; the BS-IRS and BS-user channels use exponentially decaying delay profiles, while the IRS-user link uses frequency-selective Rician fading with κ = 7 dB.
  • Simulation Setup: The evaluation assumes 30 RS-inserted OFDM symbols, 64 RSs per symbol, −90 dBm noise power, 30 dBm transmit power, and 2-bit IRS phase shifts.The OFDM power-decay factor is ε = 2 unless otherwise specified.

A. Progressive NN Training

Progressive training increases the number of NN subnetworks until validation error converges, balancing channel-estimation accuracy against training complexity.

  • Progressive NN Training: Fig. 4 tracks NN training loss against training iterations, while the implementation uses 200 iterations per NN and a learning rate initialized at 10^-3.The learning rate drops by 0.5% per iteration, and convergence is identified using ς = 0.1.
  • Progressive NN Training: The minimum validation MSE drops rapidly for K′ < 4 but changes little for K′ > 4, indicating that K′ = 4 is sufficient in this setting.The minimum MSE also decreases as L increases because the training set becomes larger.

B. IRS Channel Autocorrelation Matrix Estimation

The proposed progressive NN recovers the wideband channel autocorrelation matrix more accurately than rank-one alternatives and enables stronger IRS passive reflection gains as training data, phase resolution, or array size increases.

  • B. IRS Channel Autocorrelation Matrix Estimation: The progressive NN achieves lower NMSE than rank-one approximation and can outperform the true-rank NN when L is small.Increasing L lowers NMSE, while adaptive subnetworks and training iterations help the progressive method handle noisy, limited training data.
  • C. IRS Passive Reflection Design: IRS deployment significantly improves the power gain of each multipath tap and adds two taps after deployment, benefiting average OFDM channel power gain.The comparison uses L = 350 RSRP measurements and the proposed NN-based autocorrelation estimate for reflection optimization.
  • C. IRS Passive Reflection Design: The proposed progressive NN yields higher average channel power gain than benchmark reflection designs, including RMS, CSM, and rank-one NN methods.The comparison uses average channel power gain versus the number of random IRS reflection sets L, with perfect-CSI optimization serving as an upper bound.
  • C. IRS Passive Reflection Design: Multiple subnetworks provide higher-rank autocorrelation approximations than a single subnetwork, producing larger reflection-design gains in wideband channels.With sufficiently many random reflections, even rank-one NN approximation can outperform CSM and RMS, but progressive training performs better when L is small.
  • C. IRS Passive Reflection Design: Increasing the IRS phase-shift resolution improves average channel power gain by improving both autocorrelation recovery and reflection optimization.The reported comparison includes μ = 2 controlling bits and contrasts results across phase-resolution settings.
  • C. IRS Passive Reflection Design: Increasing the IRS size to N = 128 further improves maximum average channel power gain through increased array and beamforming gains.The N = 128 evaluation uses μ = 2 and P = 30 dBm and shows similar trends to the N = 32 case.

D. Extension to Multi-user Systems

The framework extends to multiuser systems by estimating each user’s IRS channel autocorrelation matrix and optimizing reflection from their average. Simulations show significant gains over benchmark schemes, while wideband rate performance approaches the perfect-CSI upper bound.

  • D. Extension to Multi-user Systems: The proposed framework significantly outperforms benchmark schemes in multiuser scenarios by exploiting users’ power measurements.The IRS reflection is optimized using the average of the users’ estimated channel autocorrelation matrices.
  • E. Effect of Channel Power Decaying Factor: The proposed scheme achieves higher OFDM achievable rates than other benchmarks and approaches the perfect-CSI rate upper bound.The comparison uses μ = 2 phase-shift controlling bits, CP length M_CP = 16, L = 300 random reflection sets, and 25 dBm transmit power.
  • D. Extension to Multi-user Systems: The method can also optimize the minimum channel power gain among multiple users.
  • E. Effect of Channel Power Decaying Factor: OFDM achievable rate improves as the channel power decaying factor ε increases because received power concentrates on fewer taps.This concentration lets passive reflection align with dominant paths and obtain higher beamforming gain.
  • E. Effect of Channel Power Decaying Factor: At larger ε, training with one subnetwork approaches the rate upper bound with low performance loss, reducing training complexity.

APPENDIX

The appendix establishes matrix and partial-DFT properties used in the wideband channel-autocorrelation analysis. It also proves that the proposed rank-one subnetwork decomposition requires at least as many subnetworks as the autocorrelation matrix rank.

  • APPENDIX: The partial DFT matrix is formed by selecting regularly spaced rows, making its autocorrelation cyclic symmetric.The spacing follows from uniformly inserting reference signals across selected subcarriers, with M/M_0 ∈ N.
  • APPENDIX: Zero-padding the final M−K rows of the CIR matrix reduces the analysis to its first K rows.
  • APPENDIX: Spectral decomposition expresses the Hermitian positive-semidefinite autocorrelation matrix through eigenvalues and eigenvectors.The proof then constructs rank-one terms from the eigencomponents.
  • APPENDIX: For equal or greater component counts, additional rank-one terms can be constructed to complete the decomposition.
  • APPENDIX: The autocorrelation matrix cannot be represented as a sum of fewer than its rank many rank-one components.This follows from the matrix rank inequality and yields the lower bound on the number of subnetworks.

C. Proof of Lemma 3

The proof considers the implication of equality between the quadratic form induced by the estimated and target autocorrelation matrices for every reflection vector.

  • C. Proof of Lemma 3 : If the quadratic-form equality holds for every reflection vector, the residual quadratic form must vanish for every vector.
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