PaperlayerView exact revision on arXiv

Source-linked AI summary

Learning to Reflect and to Beamform for Intelligent Reflecting Surface with Implicit Channel Estimation

Tao Jiang, Hei Victor Cheng, Wei Yu

arXiv:2009.14404v4eess.SPcs.IT

TL;DR

IRS systems make channel estimation difficult because their many passive elements cannot directly measure the relevant channels, while subsequent joint optimization is high-dimensional and nonconvex. The paper therefore trains a permutation-aware GNN to map received pilots and optional user locations directly to BS beamformers and IRS reflective patterns. Simulations show generalizable and interpretable optimization for sum-rate and minimum-rate objectives using substantially fewer pilots than conventional explicit channel-estimation approaches.

  • Problem

    IRS channel estimation is difficult because passive elements prevent direct measurement, the channel has many parameters, and joint beamforming and phase-shift optimization is high-dimensional and nonconvex.

  • Method

    A deep neural network with a permutation-invariant/equivariant GNN architecture maps received pilots and optional user locations directly to BS beamformers and IRS reflective patterns while maximizing network utility.

  • Results

    The proposed approach generalizes across user configurations, produces interpretable solutions, and efficiently learns both sum-rate and minimum-rate maximization with fewer pilots than conventional explicit channel-estimation methods.

  • Takeaways & Limitations

    Directly optimizing the system objective from pilots can use pilot information more efficiently than first reconstructing the channel matrix.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS), which consists of a large number of tunable reflective elements, is capable of enhancing the wireless propagation environment in a cellular network by intelligently reflecting the electromagnetic waves from the base-station (BS) toward the users. The optimal tuning of the phase shifters at the IRS is, however, a challenging problem, because due to the passive nature of reflective elements, it is difficult to directly measure the channels between the IRS, the BS, and the users. Instead of following the traditional paradigm of first estimating the channels then optimizing the system parameters, this paper advocates a machine learning approach capable of directly optimizing both the beamformers at the BS and the reflective coefficients at the IRS based on a system objective. This is achieved by using a deep neural network to parameterize the mapping from the received pilots (plus any additional information, such as the user locations) to an optimized system configuration, and by adopting a permutation invariant/equivariant graph neural network (GNN) architecture to capture the interactions among the different users in the cellular network. Simulation results show that the proposed implicit channel estimation based approach is generalizable, can be interpreted, and can efficiently learn to maximize a sum-rate or minimum-rate objective from a much fewer number of pilots than the traditional explicit channel estimation based approaches.

I. INTRODUCTION

The paper replaces explicit channel estimation with a data-driven GNN that directly configures IRS reflection and BS beamforming from pilots, optionally using user locations. The approach targets multiuser interference and supports sum-rate and minimum-rate objectives while improving pilot efficiency, generalizability, and interpretability.

  • Motivation: IRS optimization is difficult because passive elements prevent direct channel measurement, the number of channel parameters is large, and joint beamforming design is high-dimensional and nonconvex.Conventional channel estimation may also optimize an artificial criterion that does not match the ultimate system objective.
  • Approach: The proposed deep neural network maps received pilots directly to BS beamformers and IRS reflective patterns for maximizing a system-wide objective.The mapping can be parameterized and trained to optimize network utility without explicitly recovering the channel.
  • Results: The approach outperforms conventional model-based methods and can achieve higher transmission rates, especially when pilot length is limited.The paper reports that direct rate maximization extracts more pertinent information from pilots than explicit channel estimation and uses a smaller neural-network output.
  • Approach: The GNN models interactions among the IRS and users through permutation-invariant reflective patterns and permutation-equivariant user beamformers.This architecture uses shared structure across users to support generalization when the number of users changes.
  • Additional information: User locations can be incorporated as heterogeneous inputs, allowing the network to focus on small-scale fading and significantly improving utility-maximization performance.The paper notes that incorporating such heterogeneous information is difficult in conventional model-based approaches.
  • Results: The GNN reduces pilot requirements for both sum-rate and minimum-rate maximization, while producing interpretable beamforming and reflective patterns.The learned patterns visually focus electromagnetic waves toward target users and account for multiuser interference.

A. System Model

The system jointly designs BS beamformers and IRS reflection coefficients from received pilots to maximize a multiuser network utility, avoiding explicit channel recovery.

  • System model: The IRS-assisted downlink serves K single-antenna users from an M-antenna BS using N passive reflective elements.The IRS controller adjusts element phases to reflect signals toward desired directions.
  • System model: The BS beamformers satisfy a transmit-power constraint, while each IRS reflection coefficient has unit magnitude and a tunable phase.The reflection vector is parameterized by the element phase shifts.
  • System model: Users achieve rates under multiuser interference treated as noise, and beamformers and IRS phases are jointly optimized for utilities such as sum rate or minimum rate.The utility depends on all users’ achievable rates.
  • Pilot-based design: The uplink pilot phase provides received signals at the BS for learning or estimating the cascaded and direct channels.Channel reciprocity permits estimation during uplink, but the received pilots contain many unknown coefficients.
  • Pilot-based design: Because an IRS can contain hundreds of elements, short pilot sequences make explicit channel estimation difficult.The system has (M + N)K + MN unknown channel coefficients in the stated model.
  • Implicit design: The proposed formulation maps received pilots directly to beamformers and reflection phases, with a deep neural network parameterizing this mapping for expected utility maximization.The examples target sum-rate maximization and max-min fairness.

III. UPLINK PILOT TRANSMISSION AND CONVENTIONAL CHANNEL ESTIMATION

The conventional pipeline uses structured uplink pilots to estimate combined channels before downlink utility optimization, but its pilot requirements and estimator complexity can be substantial.

  • Conventional pipeline: The conventional approach separates uplink channel estimation from downlink utility maximization using estimated channels.Existing algorithms can then optimize sum rate or minimum rate under the estimated channel information.
  • Pilot transmission: Training divides L pilot symbols into τ sub-frames, each containing L0 = K symbols, with orthogonal user pilot sequences repeated across sub-frames.Users transmit simultaneously within each sub-frame.
  • Pilot transmission: The IRS keeps its phase shifts fixed within each sub-frame and changes them across sub-frames to expose both user-to-IRS and IRS-to-BS channels.The BS decorrelates received pilots by matching each user’s pilot sequence.
  • Pilot processing: After decorrelation, each user’s received pilot contribution combines its direct channel, IRS-reflected channel, and noise.The resulting observations are used to form a combined channel matrix.
  • Channel estimation: At least τ = N + 1 sub-frames, totaling (N + 1)K pilot symbols, are typically needed for a full-rank pilot matrix and successful channel recovery.For τ < N + 1, the paper constructs pilot-phase vectors randomly; other cases use a truncated DFT construction.
  • Channel estimation: MMSE estimation is optimal in principle, while LMMSE provides lower complexity but is optimal only when the unknown channel is Gaussian distributed.For general fading distributions, the optimal MMSE implementation can be computationally intensive.

IV. PROPOSED DEEP LEARNING FRAMEWORK

The proposed framework bypasses explicit channel estimation by learning a direct pilot-to-configuration mapping with a GNN that represents the IRS and users as interacting graph nodes.

  • Framework: Instead of recovering channels, the neural network learns to map received pilots directly to beamformers and IRS phase shifts for network-utility maximization.This targets the ultimate rate objective rather than an intermediate channel reconstruction metric.
  • Graph representation: The graph contains K + 1 nodes: one IRS node and K user nodes, whose representations encode the system entities.The IRS representation produces reflection phases, while user representations produce BS beamformers.
  • Graph representation: User-node updates aggregate neighboring users and the IRS so the GNN can model interactions relevant to multiuser interference.This structure is intended to learn coordinated beamforming and reflection design.
  • Permutation structure: Permuting user labels permutes the BS beamformers correspondingly while leaving the IRS reflection pattern unchanged.These permutation equivariant and invariant properties reflect the network-utility problem structure.
  • Generalization: GNN parameters are tied across users, allowing the learned architecture to generalize to different user counts without retraining.Changing the number of users requires changing graph nodes, not the learned aggregation and combination operations.
  • Architecture: The GNN uses initialization, D updating layers, and a final normalization layer to produce beamformers and reflection coefficients.User inputs begin from received pilots and can incorporate additional information such as user locations.

2) Updating Layers:

The updating layers combine node representations with permutation-aware aggregation functions designed separately for the IRS and user nodes.

  • Layer updates: Each updating layer aggregates neighboring-node representations and combines them with the node’s previous representation.The aggregate and combine functions are learned components of the GNN layer.
  • Permutation-aware aggregation: Aggregation functions are chosen to be invariant to input permutations, supporting scalable graph processing across users.Element-wise max-pooling and mean pooling are examples of such functions.
  • IRS node: The IRS update uses all user representations and an element-wise mean because its reflection pattern must serve all users.This implements permutation invariance with respect to user ordering.
  • User nodes: Each user update aggregates the other user nodes while treating multiuser interference as typically dominated by the strongest user.The proposed user aggregation uses element-wise max-pooling.
  • Design caveat: The paper notes that aggregation choices lack a general selection theory and are usually determined through empirical trials.The adopted architecture and pooling choices are reported to perform well in simulation.

3) Normalization layer:

The normalization layer converts neural-network outputs into IRS reflective coefficients and BS beamformers while enforcing their physical constraints.

  • 3) Normalization layer:: The normalization layer outputs reflective coefficients v and beamforming matrix W from neural-network representations.The reflective coefficients have 2N real and imaginary output components, while beamforming uses 2M components per user.
  • 3) Normalization layer:: The output normalization enforces unit modulus for IRS coefficients and a total power constraint for BS beamforming.These constraints ensure the generated system configuration satisfies the reflective-element and transmit-power requirements.
  • 3) Normalization layer:: The beamforming matrix W is formed by combining its first M rows with its next M rows as real and imaginary components.The resulting complex matrix is W = Zw(1 : M, :) + jZw(M + 1 : 2M, :).
  • 3) Normalization layer:: The same learned user-node update functions are reused across all user nodes, supporting operation with an arbitrary number of users.Changing the user count requires changing the graph’s number of nodes rather than relearning the combining and aggregation operations.

C. Neural Network Training

Training rewrites the achievable rate using real and imaginary variables, then optimizes the GNN end to end with an unsupervised utility-based loss. Training uses CSI offline, but deployment maps pilots directly to configurations without CSI and can require fewer pilots than the conventional pipeline.

  • C. Neural Network Training: The achievable rate is rewritten in terms of the real and imaginary parts of the beamformers and IRS coefficients for neural-network training.This real-valued representation accommodates deep-learning software that does not support complex-valued operations.
  • C. Neural Network Training: The GNN loss is the negative expected network utility computed from the users’ achievable rates.The utility can therefore be optimized directly during training rather than through an intermediate channel-estimation target.
  • C. Neural Network Training: CSI is required to generate training samples and compute utility, but the trained neural network does not require CSI during operation.Training is performed offline, so it does not affect run-time complexity.
  • C. Neural Network Training: The end-to-end method jointly designs BS beamforming and IRS phase shifts directly from received pilots.Simulation results report that it needs fewer pilots to achieve the same performance as separated channel estimation and utility maximization.

V. PERFORMANCE FOR SUM-RATE MAXIMIZATION

The sum-rate experiments study an IRS-assisted multiuser MISO system with an eight-antenna BS, a 100-element IRS, and three users. The simulations combine Rayleigh direct links with Rician BS–IRS and IRS–user links, using line-of-sight components determined by geometry.

  • V. PERFORMANCE FOR SUM-RATE MAXIMIZATION: The simulated system contains a BS with 8 antennas, an IRS with 100 passive elements, and 3 users.The IRS uses a 10 × 10 rectangular array, while the BS uses a uniform linear array.
  • V. PERFORMANCE FOR SUM-RATE MAXIMIZATION: The BS is located at (100,100,0) meters and the IRS at (0,0,0), with users distributed over × [−35,35] at z = −20.The user placement is uniform within the stated rectangular area.
  • V. PERFORMANCE FOR SUM-RATE MAXIMIZATION: Direct BS–user channels follow Rayleigh fading, whereas BS–IRS and IRS–user channels follow Rician fading.The Rician model includes line-of-sight and non-line-of-sight components, with Rician factor ε = 10 in the simulations.
  • V. PERFORMANCE FOR SUM-RATE MAXIMIZATION: The simulations use uplink pilot and downlink data transmit powers of 15 dBm and 20 dBm, with noise powers of −100 dBm and −85 dBm, respectively.These values apply unless otherwise stated.
  • V. PERFORMANCE FOR SUM-RATE MAXIMIZATION: The line-of-sight IRS–user and BS–IRS channel components are determined by array steering vectors and the corresponding arrival or departure angles.Those angles are calculated from the locations of the users, IRS, and BS, with adjacent-element spacing specified relative to the carrier wavelength.

B. Neural Network Training and Testing

The experiments train a two-layer GNN with Adam and evaluate it against perfect-CSI and channel-estimation-based BCD benchmarks, including an explicit deep-learning channel estimator.

  • B. Neural Network Training and Testing: The proposed network uses a 2-layer GNN with D = 2.The fully connected subnetwork parameters for the architecture are summarized in Table I.
  • B. Neural Network Training and Testing: Training uses TensorFlow and Adam with initial learning rate 10^-3, 100 parameter updates per epoch, and 1024 samples per gradient update.The learning rate is reduced after 300 iterations by a factor of 0.98.
  • B. Neural Network Training and Testing: Training stops when the validation loss fails to decrease for 10 consecutive epochs.This provides the stated termination rule for the neural-network training process.
  • B. Neural Network Training and Testing: Testing compares the neural network with perfect-CSI BCD, LMMSE channel estimation followed by BCD, and explicit deep-learning channel estimation followed by BCD.The explicit estimator is included to compare implicit and explicit channel-estimation strategies.
  • B. Neural Network Training and Testing: The explicit channel-estimation network outputs vectorized channel-matrix estimates for each user before beamforming and phase-shift optimization.Its final user-node representation is passed to a linear layer of size 2M(N + 1).

C. Numerical Results

The proposed GNN directly optimizes IRS reflective patterns and BS beamformers from received pilots, achieving strong rate performance with reduced training overhead and useful generalization.

  • Using only 30 pilots, the proposed approach without location information achieves about 96% of the sum rate in Benchmark 1 with perfect CSI.Benchmark 1 assumes perfect CSI.
  • With 15 pilots, the deep learning approach outperforms Benchmark 2 using 120 pilots, while Benchmark 2 needs 303 noiseless pilots for perfect channel reconstruction.
  • User location information further improves sum rate when the pilot length is insufficient, but its benefit becomes marginal as pilot length increases.The received pilots implicitly contain location information that can be learned with sufficiently many pilots.
  • Direct sum-rate maximization outperforms explicit channel-estimation learning and produces a smaller neural-network output than reconstructing the channel matrix.
  • Ten training epochs achieve more than 90% of the sum rate obtained with 80 epochs.Each epoch samples 102,400 training data.
  • The proposed GNN outperforms LMMSE with BCD, although its gain decreases as the number of BS antennas increases from 8 to 16.The authors attribute the smaller gain to increased problem dimension and suggest increasing the GNN parameter count.

D. Generalizability

The GNN generalizes across transmit powers and user counts, and it also supports minimum-rate optimization with interpretable beamforming and reflection patterns.

  • Generalization across transmit powers: A GNN trained at one downlink transmit power can significantly outperform Benchmark 2 when tested at different downlink powers.
  • Generalization across transmit powers: Training at a fixed uplink pilot power and testing at different pilot powers evaluates generalization beyond the training distribution.The reported setting trains at Pu = 15dBm with L = 75.
  • Generalization to different number of users: The GNN generalizes to different numbers of users and outperforms explicit channel-estimation Benchmark 2.It is trained with K = 6 users and tested with different user counts using L = 25K.
  • Maximum minimum-rate performance: For the max-min objective, the proposed deep learning method outperforms both LMMSE and deep-learning channel-estimation baselines.The comparison uses the empirical CDF of minimum user rate over 1000 channel realizations.
  • Generalization to different number of users: With L = 25K, the deep learning approach reaches 87%–90% of the perfect-CSI benchmark across tested user counts.
  • Pilot-length reduction: With 5K pilots, it achieves over 94% of the conventional LMMSE approach’s minimum rate using 25K pilots.
  • Interpretation of learned solutions: The learned IRS response focuses better as the number of IRS elements increases from 30 to 50 and 100.
  • Interpretation of learned solutions: For three users, the learned IRS has peaks matching their target angles, while stronger BS response compensates for a weaker IRS response for one user.The joint pattern also differentiates users and accounts for interference among them.

VIII. CONCLUSION

The paper replaces explicit CSI estimation with a generalizable GNN that maps received pilots directly to IRS configurations and per-user BS beamformers. Simulations report interpretable solutions and effective utility maximization with fewer pilots than conventional approaches.

  • The proposed method directly learns IRS phase shifts and BS beamformers from received pilots to maximize a system utility function without explicit channel estimation.
  • The trained neural network solves utility-maximization problems with much fewer pilots than conventional channel-estimation approaches and produces interpretable results.
Loading 2009.14404v4…