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Deep Residual Learning for Channel Estimation in Intelligent Reflecting Surface-Assisted Multi-User Communications

Chang Liu, Xuemeng Liu, Derrick Wing Kwan Ng, Jinhong Yuan

arXiv:2009.01423v4eess.SP

TL;DR

IRS-MC channel estimation involves cascaded channels whose optimal MMSE estimation is difficult to implement. The paper formulates estimation as denoising and develops a DReL-based CDRN, achieving almost the same accuracy as the optimal MMSE estimator.

  • Problem

    IRS-MC channel estimation involves cascaded channels, while the optimal MMSE estimator requires intractable multidimensional integration and perfect channel-state information availability.

  • Method

    The paper formulates channel estimation as denoising, develops a Bayesian DReL-based framework, and designs a CNN-based CDRN to denoise noisy channel matrices.

  • Results

    The proposed method achieves almost the same estimation accuracy as the optimal MMSE estimator based on a prior channel PDF.

  • Takeaways & Limitations

    The CDRN provides a data-driven approach for improving channel-estimation accuracy in IRS-MUC systems.

Abstract

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Channel estimation is one of the main tasks in realizing practical intelligent reflecting surface-assisted multi-user communication (IRS-MC) systems. However, different from traditional communication systems, an IRS-MC system generally involves a cascaded channel with a sophisticated statistical distribution. In this case, the optimal minimum mean square error (MMSE) estimator requires the calculation of a multidimensional integration which is intractable to be implemented in practice. To further improve the channel estimation performance, in this paper, we model the channel estimation as a denoising problem and adopt a deep residual learning (DReL) approach to implicitly learn the residual noise for recovering the channel coefficients from the noisy pilot-based observations. To this end, we first develop a versatile DReL-based channel estimation framework where a deep residual network (DRN)-based MMSE estimator is derived in terms of Bayesian philosophy. As a realization of the developed DReL framework, a convolutional neural network (CNN)-based DRN (CDRN) is then proposed for channel estimation in IRS-MC systems, in which a CNN denoising block equipped with an element-wise subtraction structure is specifically designed to exploit both the spatial features of the noisy channel matrices and the additive nature of the noise simultaneously. In particular, an explicit expression of the proposed CDRN is derived and analyzed in terms of Bayesian estimation to characterize its properties theoretically. Finally, simulation results demonstrate that the performance of the proposed method approaches that of the optimal MMSE estimator requiring the availability of the prior probability density function of channel.

I. INTRODUCTION

IRS-assisted multi-user communication requires accurate cascaded-channel estimation, but the channel's nonconventional statistics make optimal MMSE estimation impractical. Existing methods face accuracy and training-overhead challenges, motivating a denoising-based deep residual approach that approaches MMSE performance without requiring the channel prior PDF.

  • Motivation: Accurate CSI is essential for IRS beamforming, yet perfect CSI is generally unavailable in practical systems.The IRS's promised performance gain relies on accurate channel-state information.
  • Challenges: Because the IRS is passive, its user-to-IRS and IRS-to-BS channels are generally unavailable separately; only the cascaded channel can be estimated.This cascaded channel introduces the central estimation difficulty in IRS-assisted systems.
  • Challenges: The cascaded channel does not follow conventional Rayleigh fading, making optimal MMSE estimation computationally intensive through multidimensional integration.LMMSE and LS estimators retain a substantial performance gap relative to optimal MMSE estimation.
  • Challenges: Large IRSs create high-dimensional cascaded channels, increasing the training overhead required by conventional LS and LMMSE estimation.The paper notes that training overhead is a separate practical challenge from estimation accuracy.
  • Prior Approaches: Prior schemes address estimation accuracy or training overhead through reflection control, DFT-based training, factorization, subsurfaces, and matrix-completion techniques.Binary reflection can reduce interference but yields small received SNR and potentially long delays when many elements are present.
  • Proposed Approach: The proposed work models channel estimation as denoising and uses a DReL framework with a CNN-based DRN to learn residual noise from noisy pilot observations.The CDRN uses element-wise subtraction and exploits spatial channel features and additive noise without requiring prior statistical information.
  • Results: Simulation results show that the proposed method approaches optimal MMSE accuracy while avoiding the need for the channel's prior probability density function.The analysis also establishes that the CDRN can achieve the same performance as the LMMSE estimator under the stated framework.
  • Scope: The paper focuses on improving CSI estimation accuracy rather than reducing training overhead.This scope is stated explicitly in the paper's note on training overhead.

II. SYSTEM MODEL

The IRS-MUC uplink protocol estimates cascaded channels using multiple IRS reflection patterns and orthogonal user pilots. Accurate estimation is needed for downlink beamforming, but conventional MMSE estimation is computationally intractable for the non-Rayleigh cascaded channel.

  • System configuration: The system comprises one base station, one IRS with passive reflecting elements, and K single-antenna users operating under TDD.The BS has M antennas and the IRS has N passive reflecting elements.
  • Cascaded channel: The user-to-IRS-to-BS reflecting link is a cascaded channel whose response is controlled by the IRS phase-shift matrix.The reflecting response can be written as Gdiag(r)f_k, equivalently Gdiag(f_k)r.
  • Estimation protocol: Each estimation frame contains channel-estimation and data-transmission phases, with the estimation phase using C ≥ N + 1 IRS patterns and L ≥ K orthogonal pilot symbols per sub-frame.The IRS holds one phase-shift matrix during each sub-frame and switches matrices across sub-frames.
  • Observation model: The received pilot observations are separated user by user, stacked across sub-frames, and used with the reflection-pattern matrix P to recover H_k.The patterns are designed using a discrete Fourier transform to improve received signal power at the BS.
  • Estimator trade-off: For the cascaded IRS channel, optimal MMSE estimation requires multidimensional integration, while LMMSE remains practical but has a performance gap from MMSE.The difficulty arises because H_k generally does not follow a Rayleigh fading model.

III. DEEP RESIDUAL LEARNING-BASED CHANNEL ESTIMATION FRAMEWORK

The framework recasts IRS-MUC channel estimation as denoising: a deep residual network learns additive residual noise from noisy pilot-based observations. Its subtraction architecture combines received-observation features with the additive structure of noise.

  • Denoising formulation: Additive noise is treated as the key obstruction to recovering channel coefficients from the system observations.The framework therefore models channel estimation as a denoising problem.
  • Residual learning: The DReL framework trains a deep residual network to learn residual noise and recover channel coefficients from noisy observations.The method is organized around a denoising model and a DReL-based estimation framework.
  • Network structure: The DRN uses an element-wise subtraction architecture to exploit both received-observation features and the additive nature of noise.This design is intended to improve estimation performance.

A. Denoising Model for Channel Estimation

The denoising model uses the LS estimate as a coarse noisy observation of the channel, then applies data-driven learning because the cascaded-channel prior is non-Gaussian and lacks an explicit Bayesian estimator.

  • LS baseline: The LS estimation error depends on noise power and decreases as the number of reflection patterns C increases.The reported expression identifies the error as a function of the noise term and C.
  • Denoising interpretation: This makes channel estimation equivalent to recovering H_k from the noisy LS observation.The LS result provides the coarse estimate used as the denoising input.
  • Denoising model: The LS-based observation ˜X_k is decomposed into the channel matrix H_k plus a noise component ˜Z_k.The observation and noise are defined through the LS estimator and the transformed pilot noise.
  • Motivation for DReL: Because the cascaded channel has a non-Gaussian prior, the model is not a Bayesian general linear model with an explicit estimator expression.The paper therefore adopts a data-driven DReL framework for IRS-MUC channel estimation.

B. The Developed DReL-based Channel Estimation Framework

The DReL framework trains a residual network offline on LS-based noisy observations and ground-truth channels, then applies the trained estimator online. With sufficiently large training data, its estimator converges to optimal MMSE performance.

  • III. DEEP RESIDUAL LEARNING-BASED CHANNEL ESTIMATION FRAMEWORK: The framework has offline training and online estimation phases, producing a trained DRN that directly estimates channel coefficients from test data.The online phase sends test inputs to the well-trained DRN.
  • 1) Offline Training:: Training examples pair LS-based inputs with ground-truth channel matrices, and the DRN contains a DNN followed by element-wise subtraction.The resulting network is trained to map noisy observations toward channel estimates.
  • 1) Offline Training:: The DRN is trained by minimizing empirical MSE because the Bayesian MSE is unavailable with a finite training set.Backpropagation updates the network parameters to obtain the DRN-based MMSE estimator.
  • Bayesian interpretation: The DRN-based estimator is interpreted as a data-driven counterpart to the model-driven MMSE estimator, using empirical rather than Bayesian MSE.The distinction is framed in Bayesian-estimation terms.
  • Bayesian interpretation: The model-driven MMSE approach remains difficult for cascaded channels because it lacks a closed-form expression and has high computational cost.This limitation motivates the data-driven alternative.
  • Bayesian interpretation: As the training-set size tends to infinity, the DRN estimator converges to the MMSE estimator; with a sufficiently large training set, performance converges to optimal MMSE.This theoretical result links empirical-risk training with the optimal Bayesian estimator.

2) Online Estimation:

Online estimation first applies LS estimation to obtain a coarse channel, then uses the trained DRN to refine it. The framework is universal and can balance performance and complexity through DRN selection.

  • 2) Online Estimation:: The test observation is first processed by an LS estimator to obtain a coarse channel estimate, then refined by the trained DRN estimator.The refined estimate is produced from the LS-based input.
  • 2) Online Estimation:: The DReL framework can use different DNN architectures, including dense networks and CNNs.This makes the framework applicable beyond a single network realization.
  • 2) Online Estimation:: The universal framework balances system performance and computational complexity through the selection of the DRN.The DRN uses an element-wise subtraction operator to exploit channel features and the additive nature of noise.
  • 2) Online Estimation:: The proposed DRN-based method is intended to improve channel estimation performance for IRS-MUC systems.This is the stated application-level objective of the framework.

IV. CNN-BASED DEEP RESIDUAL NETWORK FOR CHANNEL ESTIMATION

The CDRN realizes deep residual learning with a CNN that extracts spatial features from noisy complex channel matrices while learning additive residual noise. Repeated subtraction-based denoising blocks produce the channel estimate, and the architecture is scalable to different channel sizes.

  • IV. CNN-BASED DEEP RESIDUAL NETWORK FOR CHANNEL ESTIMATION: The developed DReL framework is implemented with a CNN because CNNs extract features from noisy matrices and subtraction exploits additive noise.This realization is called the CDRN.
  • A. CDRN Architecture: The CDRN contains an input layer, D denoising blocks, and an output layer.Each denoising block uses a residual subnetwork together with an element-wise subtraction operator.
  • A. CDRN Architecture: Each denoising block learns residual noise from the noisy channel matrix and subtracts it element-wise from the block input.The subtraction structure matches the additive nature of the received-signal noise.
  • A. CDRN Architecture: The complex-valued input is divided into real and imaginary parts, which are processed through separate real-valued neural-network channels.The division is used to facilitate feature extraction from the complex channel matrix.
  • A. CDRN Architecture: The residual subnetworks use convolutional layers, with Conv+BN+ReLU operations in the first N_l−1 layers and a final convolution for residual-noise estimation.Batch normalization is included to improve network stability and accelerate training.
  • A. CDRN Architecture: The CDRN architecture remains scalable when the channel size changes with the IRS-MUC system settings.The paper states that it can be extended to different shapes.

B. CDRN-based Channel Estimation Algorithm

The CDRN-based algorithm trains offline on LS-based examples and performs online refinement of LS estimates. Its residual architecture is analyzed theoretically, including conditions under which it matches or approximates LMMSE behavior.

  • B. CDRN-based Channel Estimation Algorithm: The CDRN-based channel estimation algorithm consists of offline training followed by online estimation.Algorithm 1 uses a training set of input-output pairs and then performs channel estimation with the trained CDRN.
  • B. CDRN-based Channel Estimation Algorithm: During online estimation, an LS estimator produces a coarse channel estimate that the trained CDRN then refines.The input is transformed into real-valued representations before passing through the denoising blocks.
  • C. Theoretical Analysis: The d-th denoising block subtracts the residual subnetwork output from its input, and D blocks progressively produce the denoised channel matrix.The block recurrence is expressed through A_d = A_d−1 − R_θd(A_d−1).
  • B. CDRN-based Channel Estimation Algorithm: The CDRN is trained with LS-based examples to learn distinguishable features for denoising and improve estimation accuracy relative to LS.The estimator is described as a data-driven nonlinear estimator with a trained parameter vector.
  • C. Theoretical Analysis: When the channel prior PDF is Gaussian, the proposed CDRN estimator is equivalent to the LMMSE estimator and may achieve optimal performance for Rayleigh fading.For non-Gaussian priors, the optimal MMSE estimator lacks a closed form, while CDRN can learn features from training examples.
  • C. Theoretical Analysis: The CDRN can approximate LMMSE behavior through learned features, whereas LMMSE relies only on a second-order channel correlation matrix.The paper states that CDRN can achieve the same performance as LMMSE in the analyzed setting.

V. NUMERICAL RESULTS

The numerical study evaluates CDRN channel estimation with NMSE across SNR and channel dimensionality, using Rician fading, path loss, and Monte Carlo simulations. It compares CDRN with MMSE, LMMSE, LS, BALS, and B-LMMSE methods and also examines denoising visualizations and complexity.

  • V. NUMERICAL RESULTS: The simulations use an IRS-MUC system with M = 8 antennas, N = 32 reflecting elements, K = 6 users, and C = 33 by default.The IRS operates with continuous phase shifts unless otherwise specified.
  • V. NUMERICAL RESULTS: The channel model combines path loss with Rician fading and includes LOS and Rayleigh components.The Rician model includes the LOS and Rayleigh channel models as limiting cases.
  • V. NUMERICAL RESULTS: Channel-estimation performance is measured with normalized mean square error (NMSE).The estimated channel is compared with the ground-truth channel.
  • V. NUMERICAL RESULTS: The comparison methods include optimal MMSE, LMMSE, LS, BALS, and B-LMMSE, alongside CDRN with D = 3 denoising blocks.Each simulation point averages performance over 100,000 Monte Carlo realizations.
  • V. NUMERICAL RESULTS: The study examines NMSE versus SNR and channel dimensionality, and visualizes the proposed CDRN denoising process.The simulations also investigate computational complexity.

A. NMSE versus SNR

Across SNR conditions, NMSE generally decreases as SNR increases, and the proposed CDRN achieves the strongest estimation performance among the compared methods. CDRN also approaches optimal-MMSE performance for the reflection channel.

  • NMSEs of all algorithms decrease as SNR increases because higher transmit power mitigates noise.
  • CDRN further improves estimation performance over LMMSE and achieves the best performance among the considered algorithms.
  • CDRN achieves a 5 dB SNR gain over E-LMMSE at NMSE ≈10^-1.
  • With 2-bit quantized IRS phase shifts, CDRN almost matches its performance with continuous phase shifts.
  • CDRN outperforms LS and achieves almost the same estimation performance as the optimal MMSE method for the reflection channel.

B. NMSE versus Channel Dimensionality

The study evaluates how channel dimensionality and pilot count affect NMSE. CDRN remains the strongest method, while its performance benefits from larger IRS-related input matrices and additional pilots but is insensitive to BS antenna count.

  • IRS reflecting elements: CDRN achieves the best performance among the considered algorithms as the number of IRS reflecting elements increases.DFT-controlled methods improve with N, whereas the binary-controlled MMSE method remains nearly constant.
  • BS antennas: NMSE is insensitive to the number of BS antennas M.The increase in array gain is neutralized between the numerator and denominator of NMSE.
  • Number of pilots: NMSEs of all algorithms decrease as the number of pilots C increases, but their slopes differ.B-LMMSE decreases gently, whereas LS, E-LMMSE, and CDRN decrease sharply.
  • Number of pilots: CDRN achieves the best performance with increasing pilot count by exploiting spatial features from larger input matrices.
  • Denoising visualization: The denoising visualization shows noise becoming progressively tidier through intermediate blocks and largely eliminated at the final CDRN output.The displayed stages are the input A and denoising-block outputs A1, A2, and A3.

D. Complexity Analysis

CDRN delivers the best system performance among the compared estimators but has higher computational complexity. GPU parallelization can nevertheless greatly reduce its actual online estimation time.

  • CDRN complexity includes the CDRN computations, LS-based input processing, and additional offline training complexity.
  • CDRN achieves the best system performance at the expense of higher computational complexity than LS and LMMSE.
  • GPU parallelization can greatly reduce CDRN's actual online estimation time despite its large number of parameters.The timing comparison is reported using desktop CPU and GPU hardware.
  • The paper's estimator is built by modeling channel estimation as denoising and using a DRN-based MMSE framework.
  • The CDRN uses a CNN denoising block with element-wise subtraction to exploit spatial channel features and additive noise.
  • Simulations show CDRN achieves almost the same estimation accuracy as the optimal MMSE estimator based on a channel prior PDF.
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