A joint pilot optimization and channel estimation algorithm based on CBAM-CNN for multipath fading environments in underground coal mines
Authors: Fei Han, Chaoyang Liu, Yan Lei, Zhezhe Sun
Published: 2026

ABSTRACT
Accurate channel estimation in underground coal mine orthogonal frequency division multiplexing (OFDM) systems remains challenging due to severe multipath fading caused by confined tunnel geometry, rough surfaces, and pervasive coal dust. Conventional least squares (LS) estimators suffer from noise sensitivity, minimum mean square error (MMSE) estimators incur prohibitive O(N3) complexity, and existing deep learning approaches treat pilot design and channel estimation independently. This paper proposes a joint algorithm based on a convolutional block attention module-enhanced convolutional neural network (CBAM-CNN), in which a differentiable Concrete selector layer and a CBAM-CNN estimation network share a unified loss function for end-to-end co-optimization. The channel attention sub-module captures inter-feature correlations while the spatial attention sub-module exploits inter-subcarrier correlations along the frequency axis to restore channel continuity disrupted by multipath scattering. Simulations under a Nakagami fading model with m = 0.85 demonstrate that the proposed method achieves a normalized mean square error of 0.0089 at SNR = 10 dB, reducing estimation error by 87.3% relative to LS and 41.1% relative to FSRCNN, while approaching MMSE accuracy at O(N) complexity. Decoupled ablation experiments were conducted to determine the impact of channel attention, spatial attention, skip connection, and differential pilot selection on model performance. The full combination achieves a 48.0% improvement in NMSE compared to the baseline CNN and outperforms the linear combination, which achieves only 33.3%, demonstrating a synergistic relationship between the learned pilot pattern and the attention module. The algorithm exhibits robust behavior under various pilot densities, fading characteristics, multipath delay spreads, and SNR mismatch scenarios, allowing for a maximum pilot overhead saving of 50% in the simulated Nakagami channel. The practical test of the algorithm in mine tunnels is recommended for future work.

Introduction

The shift towards intelligent and unmanned coal mining operations demands reliable underground wireless communication, with fifth-generation (5G) technology recognized as a key enabler1. Among the various wireless technologies that have been evaluated for use in underground mining applications, orthogonal frequency division multiplexing (OFDM) is identified as a promising modulation technique due to its robustness to frequency-selective fading and spectral efficiency2. The adoption of OFDM as a core physical layer waveform for new radio standardization in 5G technology further justifies its suitability for next-generation mine communication systems3.

Although OFDM has its own advantages, the propagation environment in an underground coal mine is challenging due to its confined tunnel structure, rough surface, high humidity, and widespread coal dust, which cause severe multipath scattering, diffraction, and absorption. In contrast to common terrestrial cases where Rayleigh fading assumptions are valid, fading characteristics for the underground mine channel are more complex and can be better described by the Nakagami-m distribution, where the shape parameter m can be used to adjust fading severity due to waveguide effects and scattering within the tunnel. In such an environment with severe multipath effects, the acquisition of channel state information (CSI) at the receiver becomes a critical task, as the deterioration in channel estimation contributes to an increase in the overall bit error rates. In order to overcome the channel estimation problem, several technical directions have been explored by the research community, such as conventional estimation algorithms, deep learning-based approaches, and joint pilot-estimation optimization frameworks.

Among the traditional methods, the least squares (LS) method has low complexity but is highly sensitive to noise levels, resulting in reduced accuracy in complex subterranean environments. The minimum mean square error (MMSE) method provides high accuracy by exploiting the statistics of the channel but requires knowledge of the autocorrelation matrix and noise variance. The matrix has a complexity of O(N3) but is impractical for resource-constrained mining terminals. The DFT-LS technique provides only limited improvement by utilizing transform domain processing. Apart from the estimation algorithm used, the placement of pilots is important in the accuracy of the estimation. Pilot placement optimization via semi-definite programming (SDP)4 suffers from NP-hard computational complexity. Conventional methodologies face difficulties in achieving both high accuracy and low cost simultaneously in the context of coal mine environments.

The achievement of high accuracy in image processing and signal recovery by deep learning techniques has inspired the use of deep learning in OFDM channel estimation. This trend has been further advanced by recent developments. Cai et al.5 proposed a deep-learning channel estimator for deep-space communications capable of maintaining accuracy under high path losses and low pilot availability, while Cai, Xu, and Niyato6 introduced an adaptive Savitzky-Golay denoising approach that reformulates channel estimation as a signal denoising problem, achieving near-MMSE accuracy without requiring channel statistical knowledge. In addition to attention-oriented approaches, Fola et al.7 introduced an autoencoder model that learns compressed channel models from observed noisy pilots, resulting in high accuracy and minimizing pilot overhead. Within the mine OFDM domain, Wang et al.8 developed a framework that was specifically designed for mine OFDM communication systems. The results proved that the performance of the proposed framework is better than the traditional methods in the case of Nakagami fading. Senol et al.9 employed artificial neural networks to estimate the sparse multipath channels in OFDM communication systems. The performance of the proposed estimator is better than the compressed sensing technique. Although these studies validate the potential of estimation using neural networks, there are still two limitations. Most existing deep learning methods are designed under conventional Rayleigh or underwater acoustic channels. However, studies on the estimation of Nakagami fading characteristics in coal mine environments are limited. On the other hand, these methods assume that the pilot pattern is fixed and optimize the estimation network independently of the pilot pattern, which is suboptimal since the performance is determined by the combination of the pilot pattern and the estimation algorithm.

Considering the interaction of pilot design and channel estimation, there is a limited number of studies focusing on the joint optimization frameworks. Kang et al.10 proposed a deep learning-based scheme for MIMO-OFDM systems, where the Concrete autoencoder is utilized for pilot selection through differentiable learning and the conditional generative adversarial network is adopted for channel estimation. Although this process represents an improvement in the development of an integrated optimization technique, there remain a number of areas for improvement. For instance, the autoencoder model and the generative network are trained using a multistage process rather than a unified loss function, which may limit the degree of co-optimization that is achievable with the pilot process and the channel estimation process. Existing joint approaches have been verified under Rayleigh or Rician environments but have not been assessed under the more suitable Nakagami distribution for underground mine channels. Moreover, none of the existing joint frameworks utilize attention mechanisms that exploit the inter-subcarrier correlations inherent in the channel frequency response. These correlations are particularly valuable in severe fading environments, as they can mitigate the effects of noise on the spectral continuity of the channel response.

To address these gaps, this study presents a pilot optimization and channel estimation algorithm using a convolutional block attention module-enhanced CNN (CBAM-CNN) in the context of multipath fading channels in underground coal mines. The proposed framework has the following three major contributions. First, the study proposes an end-to-end joint framework that incorporates a differentiable Concrete selector layer for the automatic optimization of the pilot positions with a CBAM-CNN estimation network. Both the differentiable Concrete selector layer and the CBAM-CNN are jointly optimized using a shared mean square error loss function. This framework overcomes the performance bottlenecks of the traditionally independent optimization of the two modules. Second, it proposes a novel framework for channel estimation that incorporates the CBAM attention mechanism. The channel attention sub-module enables adaptive feature weighting across dimensions. The spatial attention sub-module incorporates the frequency-domain structural correlations among subcarriers to restore the continuity of the channel response degraded by noise and multipath scattering. Third, it presents a comprehensive simulation experiment for the Nakagami coal-mine channel with a shape parameter m = 0.85 via an extensive exploration of varying pilot numbers, multipath delay distributions, and SNR mismatches. The decoupled ablation analysis examines the separate and joint effects of channel attention, spatial attention, residual connections, and the differential pilot selector. The validation is restricted to statistical channel models that have been verified against measurements described in the literature; thus, the results regarding robustness and overhead reduction apply solely to these models. Measurement-based trials in actual mine tunnels are necessary before any claims regarding deployment can be made. The rest of this study is outlined as follows: Sect. “System model” introduces the system model; Sect. “Proposed CBAM-CNN based joint algorithm” describes the proposed algorithm; Sect. “Simulation results and analysis” presents the simulation results and analysis; Sect. “Discussion” discusses the results; and Sect. “Conclusion” concludes the study.

System model

Underground coal mine channel model

The selection of an appropriate channel fading model is considered a prerequisite for the development of effective channel estimation algorithms. In the context of conventional wireless research, Rayleigh and Rician distributions are commonly applied for the modeling of multipath fading channels, yet empirical measurements in underground mine tunnels revealed the inadequacy of these models for the description of fading statistics in confined environments11. Due to the irregular tunnel walls, high humidity, and entrained coal dust, complex electromagnetic scattering and waveguide propagation effects occur, and fading distributions do not follow the standard distributions, as reviewed in the comprehensive article by Hrovat et al.12. Previous studies carried out by Forooshani et al.13 also investigated wireless signal propagation inside tunnels and revealed that typical terrestrial propagation models do not accurately represent the unique scattering effects in such confined spaces. However, the Nakagami-m distribution offers a generalized and versatile description of the conditions14. The probability density function (PDF) of the Nakagami-m distribution is expressed as1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x) = \frac{{2m^{m} }}{{\Gamma (m)\Omega^{m} }}x^{2m - 1} \exp \left( { - \frac{m}{\Omega }x^{2} } \right),\quad x \ge 0$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = E[x^{2} ]$$\end{document} denotes the average power, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ( \cdot )$$\end{document} represents the Gamma function, and m ≥ 0.5 is the shape parameter. The parameter m is defined as the ratio of the squared mean power to the variance of the instantaneous power:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = \frac{{\left( {E[x^{2} ]} \right)^{2} }}{{{\mathrm{Var}}[x^{2} ]}} = \frac{{\Omega^{2} }}{{E\left[ {(x^{2} - \Omega )^{2} } \right]}}$$\end{document}

This definition reveals that m is a measure of the severity of the fluctuations of the received signal amplitude due to the superposition of the scattered and diffracted multipath signals. A lower value of m indicates greater variance of the power and hence more fading. The special case of m = 1 reduces to the well-known Rayleigh fading. The results of field measurements conducted by Boutin et al.15 in the underground mine tunnels were found to have a range of values from 0.7 to 1.2, depending on the geometry of the tunnel and the measurement distance. Based on the empirical results, this study adopts m = 0.85 to represent the severe fading conditions in confined coal mine environments, in which the propagation path is obstructed by rough wall surfaces and particulates. The multipath channel impulse response is modeled as3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\tau ) = \sum\limits_{l = 0}^{L - 1} {\alpha_{l} } \delta (\tau - \tau_{l} )$$\end{document}where L denotes the number of resolvable paths, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha_{l}$$\end{document} is the complex gain of the l-th path whose envelope follows the Nakagami-m distribution, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{l}$$\end{document} is the corresponding propagation delay. The average power of each path is governed by an exponential power delay profile (PDP):4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\left[ {|\alpha_{l} |^{2} } \right] = \sigma_{0}^{2} \exp \left( { - \frac{{\tau_{l} }}{{\tau_{rms} }}} \right),\quad l = 0,1, \ldots ,L - 1$$\end{document}is the root-mean-square delay spread characterizing the temporal dispersion of the underground channel. The overall channel model combines the Nakagami multipath fading with additive white Gaussian noise (AWGN). The simulation parameters are summarized in Table 1.Table 1Simulation parameters of the underground coal mine OFDM system.ParameterValueNumber of subcarriers (N)64FFT length64Number of pilots (default)16Modulation schemeQPSKNumber of multipath components (L)5Channel modelNakagami + AWGNNakagami fading parameter (m)0.85 / 1 / 2Cyclic prefix (CP) length16SNR range0–30 dB

Table 1Simulation parameters of the underground coal mine OFDM system.ParameterValueNumber of subcarriers (N)64FFT length64Number of pilots (default)16Modulation schemeQPSKNumber of multipath components (L)5Channel modelNakagami + AWGNNakagami fading parameter (m)0.85 / 1 / 2Cyclic prefix (CP) length16SNR range0–30 dB

Simulation parameters of the underground coal mine OFDM system.

OFDM signal model

In an OFDM system, the transmitter maps modulated data symbols onto N orthogonal subcarriers. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{X}} = {\mathrm{diag}}\{ X(0),X(1), \ldots ,X(N - 1)\}$$\end{document} denote the diagonal matrix of transmitted frequency-domain symbols. The frequency-domain channel response at the k-th subcarrier is obtained by applying the discrete Fourier transform (DFT) to the channel impulse response in Eq. (3):5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(k) = \sum\limits_{l = 0}^{L - 1} {\alpha_{l} } \exp \left( { - j\frac{{2\pi k\tau_{l} }}{{NT_{s} }}} \right),\quad k = 0,1, \ldots ,N - 1$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{s}$$\end{document} is the sampling period. After propagation through the underground multipath channel, the received signal at the k-th subcarrier can be expressed as6Y(k) = X(k)H(k) + Z(k)where Z(k) denotes AWGN with zero mean and variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma^{2}$$\end{document}. In matrix form, this becomes7Y = XH + Zwhere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Y}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Z}}$$\end{document} are the N × 1 received signal, channel frequency response, and noise vectors, respectively. In practice, pilot symbols are inserted at a subset of subcarrier positions to enable channel estimation. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}} = \{ p_{1} ,p_{2} , \ldots ,p_{{N_{p} }} \}$$\end{document} denote the set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p}$$\end{document} pilot indices. The observation at pilot positions is then8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Y}}_{p} = {\mathbf{X}}_{p} {\mathbf{H}}_{p} + {\mathbf{Z}}_{p}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Y}}_{p}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{X}}_{p}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}_{p}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Z}}_{p}$$\end{document} are the sub-vectors and sub-matrix restricted to pilot positions. The channel estimation task is to recover the full vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document} from the pilot observations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Y}}_{p}$$\end{document}.

Conventional channel estimation methods

The LS estimator presents the most straightforward approach to channel estimation by minimizing the squared difference between the received pilot signals and the reconstructed pilot signals16. The cost function is defined as9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(\widehat{{\mathbf{H}}}_{LS} ) = \left\| {{\mathbf{Y}}_{p} - {\mathbf{X}}_{p} \widehat{{\mathbf{H}}}_{LS} } \right\|^{2}$$\end{document}

Expanding and differentiating with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{LS}$$\end{document} yields10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial J}{{\partial \widehat{{\mathbf{H}}}_{LS} }} = - 2{\mathbf{X}}_{p}^{H} {\mathbf{Y}}_{p} + 2{\mathbf{X}}_{p}^{H} {\mathbf{X}}_{p} \widehat{{\mathbf{H}}}_{LS} = 0$$\end{document}

Setting the gradient to zero and solving for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{LS}$$\end{document} gives the closed-form solution11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{LS} = \left( {{\mathbf{X}}_{p}^{H} {\mathbf{X}}_{p} } \right)^{ - 1} {\mathbf{X}}_{p}^{H} {\mathbf{Y}}_{p} = {\mathbf{X}}_{p}^{ - 1} {\mathbf{Y}}_{p} = {\mathbf{H}}_{p} + {\mathbf{X}}_{p}^{ - 1} {\mathbf{Z}}_{p}$$\end{document}where the simplification exploits the diagonal structure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{X}}_{p}$$\end{document}. At each pilot subcarrier, this reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{H}_{LS} (k) = Y(k)/X(k)$$\end{document}. Although computationally simple, the LS estimator does not suppress noise, and the error term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{X}}_{p}^{ - 1} {\mathbf{Z}}_{p}$$\end{document} becomes dominant under the low SNR conditions commonly encountered in underground coal mines, leading to substantial estimation degradation.

The MMSE estimator improves upon LS by incorporating channel second-order statistics16. A correction matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{W}}$$\end{document} is introduced such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{MMSE} = {\mathbf{W}}\widehat{{\mathbf{H}}}_{LS}$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{W}}$$\end{document} is derived by enforcing the orthogonality principle, requiring the estimation error \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbf{H}}_{p} - {\mathbf{W}}\widehat{{\mathbf{H}}}_{LS} )$$\end{document} to be orthogonal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{LS}$$\end{document}. The resulting MMSE solution takes the form12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathbf{H}}}_{MMSE} = {\mathbf{R}}_{HH} \left( {{\mathbf{R}}_{HH} + \frac{{\sigma^{2} }}{{E_{s} }}{\mathbf{I}}} \right)^{ - 1} \widehat{{\mathbf{H}}}_{LS}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_{HH} = E[{\mathbf{HH}}^{H} ]$$\end{document} is the channel autocorrelation matrix, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma^{2}$$\end{document} is the noise variance, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{s}$$\end{document} denotes the average symbol energy. The MMSE estimator achieves significantly lower mean square error than LS; however, it requires prior knowledge of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_{HH}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma^{2}$$\end{document}, and the matrix inversion incurs a computational complexity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^{3} )$$\end{document}, which poses a serious implementation barrier for resource-constrained underground mining terminals.

An intermediate solution is offered by the DFT-LS method, which transforms the LS estimate into the time domain and retains only the components within the channel delay spread before transforming the result back into the frequency domain17. This reduces the out-of-band noise. A moderate improvement over the conventional LS method is obtained, however, the improvement is limited under severe Nakagami fading. This trade-off between LS simplicity and MMSE complexity motivates the data-driven CBAM-CNN approach presented in Sect. “Proposed CBAM-CNN based joint algorithm”. Clarification regarding MMSE oracle assumptions: in Eq. (12), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_{HH} = E[{\mathbf{HH}}^{H} ]$$\end{document} denotes the autocorrelation matrix of the full N × 1 frequency-domain channel response \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document} rather than the pilot-domain subchannel matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}_{p}$$\end{document}. The MMSE estimator assumes that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_{HH}$$\end{document}, the noise power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{n}^{2}$$\end{document}, and the expected energy of the pilot symbol \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{s}$$\end{document} are known prior to signal reception. In the case of the Nakagami coal-mine channel used in this study, the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{R}}_{HH}$$\end{document} is calculated beforehand based on the multipath profile shown in Table 1 for the shape parameter m = 0.85, which is considered oracle data. Therefore, the obtained MMSE is considered only a theoretical lower limit in further sections.

Proposed CBAM-CNN based joint algorithm

Overall framework design

Traditional methods treat pilot placement and channel estimation as separate optimization problems, and the majority of recent deep learning techniques follow the conventional separated model8. Although Kang et al.10 proposed the integration of a Concrete autoencoder and a conditional generative adversarial network, the two models were trained separately in a multi-stage manner rather than being optimized together in a unified manner, which may limit the potential for joint optimization of pilot placement and channel estimation. To overcome this limitation, this study proposes a joint end-to-end approach where a differential pilot optimization component and a channel estimation CBAM-CNN share a common MSE loss function, thus allowing simultaneous backpropagation of gradients across both components. Figure 1 shows that the framework consists of two stages that complement each other. Thus, the concept of “joint optimization” refers strictly to the process of co-optimization in offline training rather than per-frame online optimization. In the offline training stage, full channel realizations generated using the Nakagami distribution are fed into the Concrete selector layer to determine pilot locations, while the noisy observations are used as input to the CBAM-CNN to estimate the channel response. The MSE between the estimated channel and the true channel is back-propagated to update both the parameters of the selector and the weights of the CBAM-CNN. Once the learning process completes, the temperature parameter τ is annealed to approach zero, making the Concrete distribution converge to a deterministic one-hot pilot sequence. In the inference phase, the one-hot pilot sequence therefore becomes fixed and retains the channel-dependent structure that has been learned offline; the receiver can acquire the first LS estimate using the learned pilot sequence positions and then refine the estimate using the CBAM-CNN. The approach eliminates the need for online channel realization or frame-by-frame training, thus improving practicality in resource-constrained mining terminals. This benefit is achieved without sacrificing the gains from joint optimization, as the pilots are pre-optimized for the Nakagami coal mine channel statistics.Fig. 1Block diagram of the proposed joint pilot optimization and CBAM-CNN channel estimation framework.

Block diagram of the proposed joint pilot optimization and CBAM-CNN channel estimation framework.

Differentiable pilot optimization module

The pilot optimization problem can be formulated as selecting Np positions from N subcarrier locations such that the channel estimation MSE is minimized, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N{\mathrm{p}} \ll N$$\end{document}. The combinatorial selection problem is NP-hard and cannot be addressed through gradient-based optimization techniques, as it is impossible to compute gradients with discrete selection variables18, this study adopts a differentiable feature selection layer based on the Concrete distribution, which is a relaxation of discrete random variables19. For the l-th selector node, a N-dimensional weight vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\upalpha}}}_{l}$$\end{document} is maintained, and a relaxed selection mask \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{m}}_{l}$$\end{document} is sampled as13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{m}}_{l} = {\mathrm{softmax}}\left( {\frac{{{{\boldsymbol{\upalpha}}}_{l} + {\mathbf{g}}}}{\tau }} \right)$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{g}}$$\end{document} is a vector of independent Gumbel(0,1) noise samples and τ > 0 is the temperature parameter. The output of the l-th selector node is then computed as14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{p,l} = {\mathbf{h}}_{{{{noisy}}}}^{{ \top }} {\mathbf{m}}_{l}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{h}}_{{{\mathrm{noisy}}}}$$\end{document} is the input noisy channel vector. As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \to 0$$\end{document}, the softmax output converges to a one-hot vector, effectively selecting a single grid position. During training, the temperature is gradually annealed according to15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{t} = \tau_{0} \exp ( - \beta t)$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{0}$$\end{document} is the initial temperature, β is the decay rate, and t is the training epoch. This annealing strategy ensures a smooth transition of selection from soft to hard as training progresses, with differentiability maintained throughout the procedure. Once the selector layer determines the optimal pilot positions, the observations are passed to the estimation network described below.

Once the selector layer determines the optimal positions for the pilots, the observations associated with the pilots are passed into the next network for channel estimation, which is discussed in the following subsection.

CBAM-CNN channel estimation network

The architecture of the proposed CBAM-CNN, illustrated in Fig. 2, comprises the following sequential stages. A feature extraction layer with 32 kernels of size 3 × 3 extracts primary features from the input. A shrinking layer with 16 kernels of size 1 × 1 reduces the feature dimension to lower computational cost, following the design philosophy of the fast super-resolution CNN (FSRCNN)20. The CBAM attention module21 then refines the feature representation through two sequential sub-modules. The channel attention sub-module applies global average pooling and global max pooling to the feature map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{F}}$$\end{document}, passes both pooled descriptors through a shared two-layer MLP with a compression ratio r, and combines the outputs via element-wise addition followed by sigmoid activation to produce the channel attention weight:16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{M}}_{c} ({\mathbf{F}}) = \sigma \left( {{\mathrm{MLP}}({\mathrm{AvgPool}}({\mathbf{F}})) + {\mathrm{MLP}}({\mathrm{MaxPool}}({\mathbf{F}}))} \right)$$\end{document}Fig. 2Architecture of the proposed CBAM-CNN channel estimation network.

Architecture of the proposed CBAM-CNN channel estimation network.

The channel-refined feature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{F^{\prime}}} = {\mathbf{M}}_{c} ({\mathbf{F}}) \otimes {\mathbf{F}}$$\end{document} is then passed to the spatial attention sub-module, which computes average and max pooling along the channel dimension, concatenates the two maps, and applies a 7 × 7 convolution followed by sigmoid activation:17\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{M}}_{s} ({\mathbf{F^{\prime}}}) = \sigma \left( {{\mathrm{Conv}}_{7 \times 7} \left( {[{\mathrm{AvgPool}}({\mathbf{F^{\prime}}});\;{\mathrm{MaxPool}}({\mathbf{F^{\prime}}})]} \right)} \right)$$\end{document}

The spatially refined output is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{F^{\prime\prime}}} = {\mathbf{M}}_{s} ({\mathbf{F^{\prime}}}) \otimes {\mathbf{F^{\prime}}}$$\end{document}. Two advantages can be gained from the integration of CBAM compared to other architectures. In contrast to the SE module, which focuses solely on channel-wise dependencies22, the proposed module additionally captures spatial correlations along the frequency axis, enabling it to exploit the structural continuity of the channel frequency response. In comparison with the FSRCNN model architecture20, which is based purely on standard convolution without considering the attention mechanism, the proposed network with the CBAM mechanism can selectively highlight the informative features and suppress the noise-dominated regions, thus enhancing the feature extraction ability of the network, especially under severe fading conditions. Following the CBAM module, a mapping stage consisting of four convolutional layers with 12 kernels of size 3 × 3 each performs nonlinear feature transformation. An expanding layer with 32 kernels of size 1 × 1 restores the feature dimension, and a deconvolution layer generates the final refined channel estimate. The residual connections are incorporated after the CBAM module, mapping layers, and expanding layer to encourage the propagation of gradients and enable the network to learn improvements to the channel representations rather than completely learning the channel responses. In the entire network, the parametric rectified linear unit (PReLU) is used as the activation function23:18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{PReLU}}(x) = \max (0,x) + a \cdot \min (0,x)$$\end{document}where a is a learnable parameter. Unlike the standard ReLU, PReLU solves the dying neuron problem and permits negative output, which is essential because the entries of the channel matrix are allowed to be negative. The overall network configuration is shown in Table 2. The complementarity of the CBAM component and the learned pilot scheme arises from the task assignment performed for the solution of the inverse problem. In particular, the Concrete selector ensures efficient pilot utilization in subcarriers where channel recovery is easier because the network experiences diversity in the spectral measurements. Therefore, the LS pilot estimation becomes a sparsely and non-uniformly sampled, noisy observation of the channel frequency response. The spatial attention module operates on the frequency domain and is specialized for reconstruction with sparsely placed pilots. The high-confidence information obtained at the trained pilot locations is propagated to nearby sub-carriers using the structure of the channel frequency response, which can be thought of as a learned interpolation filter tuned to the channel statistics. The channel attention sub-network operates orthogonally on the feature maps, dynamically weighting the real and imaginary parts of the features and their respective convolutional outputs, and suppressing the channels with high Nakagami fading. Without the learned pilot pattern, the spatial attention sub-network is needed to counteract the noisy and less structured nature of the observations; similarly, without the CBAM, the trained pilots still force the network through general convolutions without frequency-axis information. Both modules function along different dimensions of the reconstruction process: pilot placement along the spectrum dimension, and feature and frequency refinement within the network. It is because of this difference that the aggregate gain in NMSE achieved by both modules together in §4.3 exceeds their sum.Table 2Hyperparameters and structural configuration of the CBAM-CNN network.ParameterValueNetwork architecture Input tensor dimensions(N_batch, 2, 1, 64) Feature extraction layer32 kernels, 3 × 3, stride 1, padding 1 Shrinking layer16 kernels, 1 × 1, stride 1 CBAM compression ratio (r)16 CBAM spatial kernel size7 × 7 Mapping layers4 layers, 12 kernels each, 3 × 3, stride 1, padding 1 Expanding layer32 kernels, 1 × 1, stride 1 Output layer (deconvolution)2 kernels, 9 × 9, stride 1, padding 4 Residual connectionsAfter CBAM module, mapping layers, and expanding layer Activation functionPReLUTraining configuration OptimizerAdam Learning rate0.001 Batch size128 Number of epochs500 (with early stopping, patience = 50) Training strategyMixed-SNR (0–30 dB, 3 dB steps, all levels combined) Training samples per SNR10,000 Validation samples per SNR2,000 Test samples per SNR2,000Pilot optimization Initial temperature (τ₀)10 Temperature decay rate (β)0.01 Annealing scheduleτ(t) = τ₀ exp(− β · t)

Table 2Hyperparameters and structural configuration of the CBAM-CNN network.ParameterValueNetwork architecture Input tensor dimensions(N_batch, 2, 1, 64) Feature extraction layer32 kernels, 3 × 3, stride 1, padding 1 Shrinking layer16 kernels, 1 × 1, stride 1 CBAM compression ratio (r)16 CBAM spatial kernel size7 × 7 Mapping layers4 layers, 12 kernels each, 3 × 3, stride 1, padding 1 Expanding layer32 kernels, 1 × 1, stride 1 Output layer (deconvolution)2 kernels, 9 × 9, stride 1, padding 4 Residual connectionsAfter CBAM module, mapping layers, and expanding layer Activation functionPReLUTraining configuration OptimizerAdam Learning rate0.001 Batch size128 Number of epochs500 (with early stopping, patience = 50) Training strategyMixed-SNR (0–30 dB, 3 dB steps, all levels combined) Training samples per SNR10,000 Validation samples per SNR2,000 Test samples per SNR2,000Pilot optimization Initial temperature (τ₀)10 Temperature decay rate (β)0.01 Annealing scheduleτ(t) = τ₀ exp(− β · t)

Hyperparameters and structural configuration of the CBAM-CNN network.

Joint loss function and training strategy

The joint loss function is defined as the MSE between the network output and the true channel over M training samples:19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}(\Theta ) = \frac{1}{M}\sum\limits_{i = 1}^{M} {\left\| {{\mathbf{H^{\prime}}}_{i} - f(\widehat{{\mathbf{H}}}^{\prime}_{LS,i} ,\;\Theta )} \right\|_{2}^{2} }$$\end{document}where Θ collectively denotes the Concrete selector parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ {{\boldsymbol{\upalpha}}}_{l} \}$$\end{document} and all CBAM-CNN weights, and f( · ) represents the end-to-end mapping from the noisy LS input to the refined estimate. The Adam optimizer24 is employed to update Θ, with the gradient flowing through both the estimation network and the selector layer in each iteration, thereby achieving genuine joint optimization. The training samples are created by simulating Nakagami fading channels for SNR levels from 0 to 30 dB. The dataset is split into training, validation, and test sets with 10,000, 2000, and 2000 samples per SNR level, respectively. The validation set is used for monitoring the training’s convergence and for using early stopping to prevent overfitting. All the hyperparameters, including the learning rate, batch size, number of epochs, and temperature annealing schedule, are given in Table 2.

Computational complexity analysis

The computational cost of each network stage is dominated by convolutional multiply-accumulate operations. For a convolutional layer with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{{\mathrm{in}}}}$$\end{document} input channels, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{{{\mathrm{out}}}}$$\end{document} output channels, kernel size K × K, and output spatial dimensions H × W, the number of floating-point operations (FLOPs) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2C_{{{\mathrm{in}}}} C_{{{\mathrm{out}}}} K^{2} HW$$\end{document}. Applying this formula, the feature extraction layer contributes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 2 \times 32 \times 9 \times HW$$\end{document} operations, the shrinking layer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 32 \times 16 \times 1 \times HW$$\end{document}, and the four mapping layers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4 \times 2 \times 16 \times 12 \times 9 \times HW$$\end{document}. The CBAM module introduces only marginal overhead: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(C^{2} /r)$$\end{document} for channel attention MLP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(2 \times 7^{2} \times HW)$$\end{document} for spatial attention convolution, both negligible relative to the main convolutional path. The overall inference complexity of the proposed CBAM-CNN is O(N), which is substantially lower than the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^{3} )$$\end{document} complexity of the MMSE estimator16 and comparable to that of FSRCNN20. A detailed comparison across all methods is presented in Table 3. Apart from the asymptotic order, four practical complexity dimensions are reported. With N = 64 subcarriers and using the channel-first input tensor shape \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N_{batch} ,2,1,64)$$\end{document} defined in Table 2, the per-layer FLOPs (formula 2C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{in} C_{out} K^{2} HW$$\end{document} are as follows: feature extraction: 0.074 MFLOPs; shrinking: 0.066 MFLOPs; channel-plus-spatial attention: 0.013 MFLOPs; four mapping layers (combined: 0.72 MFLOPs; expanding: 0.049 MFLOPs; and 9 × 9 deconvolution: 0.66 MFLOPs. This yields a total of 1.58 MFLOPs per OFDM symbol. The network consists of 13,742 trainable parameters in total, of which 12,718 belong to the CBAM-CNN backbone (feature extraction 608, shrinking 528, CBAM 148, mapping 5,664, expanding 416, deconvolution 5,186, parametric ReLU 168) and 1,024 to the Concrete selector layer (16 pilot nodes × 64-dimensional weight vector). The maximum memory usage of activation peaks in single-sample inference amounts to 290 KB in single-precision floating point arithmetic, still comfortably fitting into the available SRAM of mainstream commercial embedded processors. Average wall-clock latency, computed from 10,000 inference operations, equals 0.42 ms per OFDM symbol for an NVIDIA RTX 3090 GPU and 8.4 ms per symbol for an Intel Core i7-1185G7 CPU with single-thread ONNX execution. Both latency values are below the expected 10 ms budget at the physical layer for 5G short-symbol communication, hence confirming that channel estimation can be performed in real time on mine-compatible devices. The simplified Mashhadi-Gündüz model presented in Sect. “Simulation setup” requires 1.51 MFLOPs, 13,180 trainable parameters, 268 KB memory, and 0.36/7.4 ms latency on GPU/CPU, respectively, values that are similar to those of the proposed technique.Table 3Computational complexity comparison of different channel estimation methods.MethodFLOPs expressionOverall orderLSNO(N)DFT-LSN log NO(N log N)MMSEN3 + N2O(N3)FSRCNN202·C_in·C_out·K2·HWO(N)CNN (without CBAM)2·C_in·C_out·K2·HWO(N)CBAM-CNN (proposed)2·C_in·C_out·K2·HW + C2/r + 2·72·HWO(N)

Table 3Computational complexity comparison of different channel estimation methods.MethodFLOPs expressionOverall orderLSNO(N)DFT-LSN log NO(N log N)MMSEN3 + N2O(N3)FSRCNN202·C_in·C_out·K2·HWO(N)CNN (without CBAM)2·C_in·C_out·K2·HWO(N)CBAM-CNN (proposed)2·C_in·C_out·K2·HW + C2/r + 2·72·HWO(N)

Computational complexity comparison of different channel estimation methods.

Simulation results and analysis

Simulation setup

The performance of the proposed joint pilot optimization and CBAM-CNN channel estimation algorithm is evaluated through extensive simulation results obtained on a hybrid platform using MATLAB R2023a for signal generation and channel modeling and Python 3.9 and Keras for neural network implementation. The simulation environment adopts the parameters used in the underground coal mine OFDM system as defined in Table 1, which consists of 64 subcarriers using QPSK modulation, a cyclic prefix of 16, and a five-path Nakagami channel model with shape parameter m = 0.85 to model the severe multipath environment. For each SNR level from 0 to 30 dB with a step of 3 dB, datasets were produced with 10,000 training samples, 2000 validation samples, and 2000 test samples. The CBAM-CNN model was trained using the Adam optimizer with a learning rate of 0.001, batch size of 128, and a maximum of 500 epochs. Early stopping was enabled if the validation loss was not improved for 50 epochs. The Concrete selector layer for pilot optimization employs an initial temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{0} = 10$$\end{document} and decay rate β = 0.01 as specified in Table 2. The default pilot configuration consists of 16 pilots, and their positions are determined by the differentiable selector layer through end-to-end joint training rather than being uniformly spaced. The normalized mean square error (NMSE), defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{NMSE}} = E[\left\| {{\mathbf{H}} - \widehat{{\mathbf{H}}}} \right\|^{2} ]/E[\left\| {\mathbf{H}} \right\|^{2} ]$$\end{document}, is the primary evaluation metric. Six channel estimation methods are compared: LS, DFT-LS, MMSE (serving as the theoretical performance bound), FSRCNN20, CNN without CBAM, and the proposed joint CBAM-CNN. It should be noted that the proposed joint technique based on CAGAN presented by Kang et al.10 was not included as an experimental baseline for two reasons. The conditional generative adversarial network10 follows an adversarial learning approach that utilizes a separate discriminator network. This makes it inherently different from the regression-based approach adopted by this study. Moreover, Kang et al.10 tested their method on a Rayleigh MIMO-OFDM system with different subcarrier settings. This difference in configuration prevents direct numerical comparison with their findings. For a controlled assessment, this study uses a CNN without CBAM as the baseline. This choice isolates the effects of the proposed components. Additionally, FSRCNN20 is included as a representative example of learning-based methods for comparison. To establish a comparison between the proposed approach and a previously suggested joint learning-based pilot/channel estimation scheme, a third baseline is provided in Table 4 and in the ablation plots of §4.3. This baseline is a simplified version of the pruning-based joint design scheme of Mashhadi and Gündüz25, implemented using the MSE-regression approach. Data generation, normalization, and reproducibility procedures: For each training, validation, and testing instance, a Nakagami-m fading channel realization is created using Eq. (3) to (4) with L = 5 paths, m = 0.85, an exponential power delay profile, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau_{rms} = 50$$\end{document} ns under the default configuration. Each frequency-domain channel vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document} is normalized to unit average power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[\left\| {\mathbf{H}} \right\|^{2} /N] = 1$$\end{document}, after which AWGN of the prescribed SNR is added to the pilot observations only. Pilot symbols are QPSK reference symbols with unit modulus. The training, validation, and testing sets are produced for each SNR point using disjoint random seeds (10,000 to 19,999, 20,000 to 21,999, 30,000 to 31,999, respectively). The TensorFlow/Keras default random number generator is used with explicit numpy.random.seed and tf.random.set_seed pairing. No sample appears in more than one set. The real and imaginary parts of the noisy LS estimate are stacked as a two-channel real-valued tensor of shape \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N_{batch} ,2,1,64)$$\end{document} using Keras image_data_format = ‘channels_first’, matching the architecture in Table 2. The NMSE and BER values for all learning algorithms are reported as the average of five experiments using different initialization seeds for the networks (101, 202, 303, 404, 505). The standard deviation associated with the average is reported when it is not negligible.Table 4Performance comparison of channel estimation algorithms under default configuration.MethodMetric0 dB5 dB10 dB15 dB20 dB25 dB30 dBLSNMSE0.56230.17780.07010.03160.01410.00630.0028BER2.81 × 10⁻11.23 × 10⁻13.82 × 10⁻21.25 × 10⁻24.47 × 10⁻31.78 × 10⁻37.94 × 10⁻4DFT-LSNMSE0.44670.14130.05620.02510.01120.00500.0022BER2.51 × 10⁻11.02 × 10⁻12.95 × 10⁻29.12 × 10⁻33.24 × 10⁻31.32 × 10⁻35.37 × 10⁻4MMSENMSE0.06310.02000.00630.00200.00090.00040.0002BER1.05 × 10⁻12.14 × 10⁻23.16 × 10⁻33.55 × 10⁻42.04 × 10⁻51.12 × 10⁻⁶ < 10⁻⁷FSRCNNNMSE0.11220.03980.01510.00630.00280.00130.0006BER1.82 × 10⁻15.75 × 10⁻21.12 × 10⁻22.82 × 10⁻36.46 × 10⁻41.51 × 10⁻43.39 × 10⁻5CNN (w/o CBAM)NMSE0.08910.03160.01200.00500.00220.00100.0004BER1.51 × 10⁻14.47 × 10⁻27.94 × 10⁻31.82 × 10⁻33.89 × 10⁻47.59 × 10⁻51.58 × 10⁻5CBAM-CNN (proposed)NMSE0.07080.02240.00890.00320.00120.00050.0002BER1.20 × 10⁻13.24 × 10⁻25.25 × 10⁻31.10 × 10⁻32.04 × 10⁻43.24 × 10⁻55.13 × 10⁻⁶Default configuration: Nakagami m = 0.85, 16 pilots, QPSK modulation. All learning-based methods employ jointly optimized pilot positions; the separate contributions of pilot optimization and attention mechanism are isolated in the ablation study (Sect. “Ablation study”) using uniformly spaced pilots as the baseline. To respond to the joint-baseline comparison requested by the reviewers, additional rows for the simplified Mashhadi and Gündüz25 baseline have been computed under the same configuration and 5-seed protocol. At SNRs of 0, 5, 10, 15, 20, 25, 30 dB, the NMSE values are 0.0805 ± 0.0017, 0.0285 ± 0.0008, 0.0108 ± 0.0005, 0.0044 ± 0.0002, 0.0019 ± 0.0001, 0.0008, and 0.0004, respectively, with corresponding BERs of 1.42 × 10⁻1, 3.95 × 10⁻2, 6.85 × 10⁻3, 1.52 × 10⁻3, 3.21 × 10⁻4, 6.18 × 10⁻5, and 1.24 × 10⁻5. The mean ± 1σ values of the learning algorithms (FSRCNN, CNN without CBAM, M-G simplified, and CBAM-CNN) are presented for SNR levels ranging from 0 to 20 dB, where the variance is relatively large; however, when the SNR exceeds this range, the variance becomes negligible relative to the last reported significant figure.

Table 4Performance comparison of channel estimation algorithms under default configuration.MethodMetric0 dB5 dB10 dB15 dB20 dB25 dB30 dBLSNMSE0.56230.17780.07010.03160.01410.00630.0028BER2.81 × 10⁻11.23 × 10⁻13.82 × 10⁻21.25 × 10⁻24.47 × 10⁻31.78 × 10⁻37.94 × 10⁻4DFT-LSNMSE0.44670.14130.05620.02510.01120.00500.0022BER2.51 × 10⁻11.02 × 10⁻12.95 × 10⁻29.12 × 10⁻33.24 × 10⁻31.32 × 10⁻35.37 × 10⁻4MMSENMSE0.06310.02000.00630.00200.00090.00040.0002BER1.05 × 10⁻12.14 × 10⁻23.16 × 10⁻33.55 × 10⁻42.04 × 10⁻51.12 × 10⁻⁶ < 10⁻⁷FSRCNNNMSE0.11220.03980.01510.00630.00280.00130.0006BER1.82 × 10⁻15.75 × 10⁻21.12 × 10⁻22.82 × 10⁻36.46 × 10⁻41.51 × 10⁻43.39 × 10⁻5CNN (w/o CBAM)NMSE0.08910.03160.01200.00500.00220.00100.0004BER1.51 × 10⁻14.47 × 10⁻27.94 × 10⁻31.82 × 10⁻33.89 × 10⁻47.59 × 10⁻51.58 × 10⁻5CBAM-CNN (proposed)NMSE0.07080.02240.00890.00320.00120.00050.0002BER1.20 × 10⁻13.24 × 10⁻25.25 × 10⁻31.10 × 10⁻32.04 × 10⁻43.24 × 10⁻55.13 × 10⁻⁶Default configuration: Nakagami m = 0.85, 16 pilots, QPSK modulation. All learning-based methods employ jointly optimized pilot positions; the separate contributions of pilot optimization and attention mechanism are isolated in the ablation study (Sect. “Ablation study”) using uniformly spaced pilots as the baseline. To respond to the joint-baseline comparison requested by the reviewers, additional rows for the simplified Mashhadi and Gündüz25 baseline have been computed under the same configuration and 5-seed protocol. At SNRs of 0, 5, 10, 15, 20, 25, 30 dB, the NMSE values are 0.0805 ± 0.0017, 0.0285 ± 0.0008, 0.0108 ± 0.0005, 0.0044 ± 0.0002, 0.0019 ± 0.0001, 0.0008, and 0.0004, respectively, with corresponding BERs of 1.42 × 10⁻1, 3.95 × 10⁻2, 6.85 × 10⁻3, 1.52 × 10⁻3, 3.21 × 10⁻4, 6.18 × 10⁻5, and 1.24 × 10⁻5. The mean ± 1σ values of the learning algorithms (FSRCNN, CNN without CBAM, M-G simplified, and CBAM-CNN) are presented for SNR levels ranging from 0 to 20 dB, where the variance is relatively large; however, when the SNR exceeds this range, the variance becomes negligible relative to the last reported significant figure.

Performance comparison of channel estimation algorithms under default configuration.

Default configuration: Nakagami m = 0.85, 16 pilots, QPSK modulation. All learning-based methods employ jointly optimized pilot positions; the separate contributions of pilot optimization and attention mechanism are isolated in the ablation study (Sect. “Ablation study”) using uniformly spaced pilots as the baseline. To respond to the joint-baseline comparison requested by the reviewers, additional rows for the simplified Mashhadi and Gündüz25 baseline have been computed under the same configuration and 5-seed protocol. At SNRs of 0, 5, 10, 15, 20, 25, 30 dB, the NMSE values are 0.0805 ± 0.0017, 0.0285 ± 0.0008, 0.0108 ± 0.0005, 0.0044 ± 0.0002, 0.0019 ± 0.0001, 0.0008, and 0.0004, respectively, with corresponding BERs of 1.42 × 10⁻1, 3.95 × 10⁻2, 6.85 × 10⁻3, 1.52 × 10⁻3, 3.21 × 10⁻4, 6.18 × 10⁻5, and 1.24 × 10⁻5. The mean ± 1σ values of the learning algorithms (FSRCNN, CNN without CBAM, M-G simplified, and CBAM-CNN) are presented for SNR levels ranging from 0 to 20 dB, where the variance is relatively large; however, when the SNR exceeds this range, the variance becomes negligible relative to the last reported significant figure.

Comprehensive comparison with existing methods

Table 4 summarizes the NMSE results of all algorithms at selected SNR points. Tests were conducted under the Nakagami fading setup (m = 0.85, 16 pilots). The outcomes show a consistent ranking across methods. LS estimation performs worst throughout all SNR levels, as it lacks noise suppression. This reflects the general difficulty of linear estimation in challenging underground channels. DFT-LS offers slight improvement by removing out-of-band noise in the transform domain. However, its gains stay small under strong Nakagami fading, where channel amplitude fluctuates considerably.

The results show that the learning-based methods significantly outperform the traditional methods. The FSRCNN architecture achieves a significant reduction in NMSE by learning non-linear mappings from noisy LS estimates to improved channel responses. The CNN architecture without CBAM further outperforms FSRCNN, as it has a deeper network architecture and benefits from residual connections, as explained in Sect. “CBAM-CNN channel estimation network”. The proposed joint CBAM-CNN architecture achieves the best performance among all learning-based methods, and its performance is close to the theoretical MMSE bound, especially in the medium-to-high SNR regime. This superiority stems from two factors: the CBAM attention mechanism selectively enhances informative features while suppressing noise (Sect. “CBAM-CNN channel estimation network”), and the end-to-end joint framework enables co-optimization toward a common MSE loss (Sect. “Overall framework design”).

Quantitative analysis reveals that with SNR = 10 dB, the NMSE value for the proposed method is 0.0089. This value is 87.3% lower than the value obtained by the LS (0.0701) and 41.1% lower than the value obtained by the FSRCNN (0.0151). With SNR = 20 dB, the NMSE value for the proposed method is 0.0012, which is in close proximity to the value achieved by the MMSE method (0.0009) but with much reduced complexity of O(N) as opposed to the O(N3) complexity of the MMSE. This complexity advantage, as discussed in the complexity analysis of the proposed method in Sect. “Computational complexity analysis”, makes the proposed method applicable for implementation in resource-constrained underground mining terminals where the full MMSE implementation is not feasible. Under the low SNR regime (0–5 dB), there is still a significant performance gap between the proposed method and the MMSE due to the inherent challenge in learning the channel representation from severely corrupted observations. Nevertheless, the proposed joint CBAM-CNN still outperforms all other learning-based methods by employing the differentiable pilot optimization, which can adaptively select the pilot positions conditioned on the channel conditions. Above 15 dB, the proposed method converges closely with the MMSE bound, which assumes perfect knowledge of channel statistics and represents an idealized upper bound unattainable in practice. The contribution of individual modules, including the proposed CBAM attention mechanism and pilot optimization layer, is also quantitatively evaluated through ablation studies as described in Sect. “Ablation study”. The BER performance as shown in Panel B of Table 4 corroborates the communication quality of the proposed approach. For an SNR of 10 dB, the joint CBAM-CNN achieves a BER of 5.25 × 10–3 at QPSK modulation, whereas the BERs achieved by the LS and FSRCNN methods are 3.82 × 10–2 and 1.12 × 10–2, respectively, with reductions of 86.3% and 53.1%. At 20 dB, the BER reaches 2.04 × 10–4, outperforming LS (4.47 × 10–3) and FSRCNN (6.46 × 10–4), confirming that NMSE gains translate directly to improved communication quality. As illustrated by the simplified Mashhadi-Gündüz joint baseline25, joint pilot-estimation learning comes with inherent advantages. For instance, it outperforms the CNN without CBAM by 10.0% in NMSE at SNR = 10 dB (0.0108 vs 0.0120) and by 13.8% in BER (6.85 × 10⁻3 vs 7.94 × 10⁻3), demonstrating the value of jointly designed pilots even without an attention mechanism. The proposed CBAM-CNN can reduce NMSE and BER by 17.6% and 23.4%, respectively, compared to the M-G method under the same SNR value. This difference reflects the extra advantage gained through the channel and spatial attention modules, in addition to the joint pilot optimization discussed in §4.3. Across the full SNR sweep, the M-G baseline NMSE values are 0.0805 ± 0.0017 (0 dB), 0.0285 ± 0.0008 (5 dB), 0.0108 ± 0.0005 (10 dB), 0.0044 ± 0.0002 (15 dB), 0.0019 ± 0.0001 (20 dB), 0.0008 (25 dB), and 0.0004 (30 dB). The corresponding BERs are 1.42 × 10⁻1, 3.95 × 10⁻2, 6.85 × 10⁻3, 1.52 × 10⁻3, 3.21 × 10⁻4, 6.18 × 10⁻5, and 1.24 × 10⁻5, respectively. Mean ± 1σ values across five independent training seeds (101, 202, 303, 404, 505 are reported for learning-based methods at 0–20 dB; the variance falls below the last reported significant digit at higher SNRs.

Ablation study

To evaluate the individual impact of the CBAM attention mechanism and the differentiable pilot optimization module in the context of underground coal mine OFDM systems, a systematic ablation study is performed for the default Nakagami fading configuration (m = 0.85). Four experimental configurations are defined to be used in comparative analysis: (i) a baseline configuration based on a standard CNN without the use of the CBAM and with uniformly spaced pilots, (ii) Experiment A, which extends the baseline CNN by incorporating the pilot optimization module, (iii) Experiment B, which extends the baseline CNN by incorporating the CBAM attention module, and (iv) Experiment C, which represents the proposed method in its entirety by activating both modules. The NMSE performance of these configurations across the SNR range is illustrated in Fig. 3A.Fig. 3Ablation study and training behavior of the proposed CBAM-CNN under Nakagami fading (m = 0.85). (A) NMSE performance of different module combinations versus SNR; (B) Training and validation MSE loss convergence curves averaged over 5 seeds (early stopping at epoch 287 on average).

Ablation study and training behavior of the proposed CBAM-CNN under Nakagami fading (m = 0.85). (A) NMSE performance of different module combinations versus SNR; (B) Training and validation MSE loss convergence curves averaged over 5 seeds (early stopping at epoch 287 on average).

The results show that both modules make significant contributions to the overall improvement in performance. A comparison between Experiment A and the baseline results reveals that the differentiable pilot optimization module alone reduces the NMSE from 0.0171 (baseline) to 0.0145 at SNR = 10 dB, a relative reduction of 15.2%, showing the effectiveness of the learned pilot positions in capturing the channel characteristics. In Experiment B, the improvement is more significant, as the CBAM attention mechanism can reduce the NMSE by around 25.7% compared to the baseline at the same SNR level. The main reason for the greater improvement is the ability of the attention module to make use of the inter-subcarrier correlations in the channel frequency response, as well as the ability of the module to suppress the features that are dominated by noise, as discussed in Sect. “CBAM-CNN channel estimation network”.

The complete method (Experiment C) results in a cumulative NMSE reduction of 48.0% compared to the baseline, exceeding the cumulative improvements of Experiments A and B. This synergistic effect proves that the proposed joint optimization approach enables the two modules to achieve mutual complementarity, where the optimization of the pilots improves the accuracy of the input observations, and the CBAM attention mechanism more effectively uses the channel information from the optimized observations. The synergistic effect is clearly found for the low SNR range (0–5 dB), where the precise location of the pilots is important for providing accurate initial observations to the network. Decoupled ablation. To comply with the reviewer’s comment on decoupling the contributions of channel attention (CA), spatial attention (SA), residual connection (RC), and pilot optimization (PO), seven variants are tested with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{SNR}} = 10$$\end{document} dB under the default Nakagami fading configuration (m = 0.85, 16 pilots, same training settings and seeds). The configurations and corresponding NMSE results are: (i) plain CNN baseline with uniform pilots, NMSE = 0.0171; (ii) + CA only, NMSE = 0.0163, reduction of 4.7%; (iii) + SA only, NMSE = 0.0152, reduction of 11.1%; (iv) + RC only, NMSE = 0.0167, reduction of 2.3%; (v) + CBAM (CA + SA + RC) with uniform pilots, NMSE = 0.0127, reduction of 25.7%; (vi) + PO only, NMSE = 0.0145, reduction of 15.2%; (vii) full method (CBAM + PO), NMSE = 0.0089, reduction of 48.0%.

Some conclusions can be drawn. Pilot optimization provides the strongest individual contribution, with a reduction of 15.2% by replacing uniform spacing with channel-aware learned pilot positions, and is closely followed by the spatial attention sub-module, which achieves an 11.1% reduction by exploiting inter-subcarrier continuity of the channel frequency response. The channel attention sub-module provides a smaller improvement (4.7%) by adaptively weighting the real and imaginary feature branches, whereas the residual connection gives the least improvement (2.3%) because it acts as a gradient flow enabler rather than a representation learning module. In the CBAM approach, the combined CA + SA + RC strategy results in a decrease of 25.7%, higher than the 18.1% linear addition of all three parts, suggesting an internal synergy of 7.6% within CBAM. The full method (CBAM + PO) provides a reduction of 48.0%, which is better than both the 33.3% linear addition of all four components and the 40.9% linear addition of CBAM and PO alone. The extra 7.1% represents true synergy between the learned pilot pattern and attention module, given that the two complement each other in solving the reconstruction problem from orthogonal angles (pilot placement in the spectrum versus feature-wise and frequency-wise refinement in the network), as explained in §3.3. The synergy effect is most noticeable at low-to-medium SNR, where both pilot information value and noise reduction become equally important.

Training convergence. As depicted in Fig. 3B, the MSE loss functions for the training and validation sets of the CBAM-CNN model under consideration, trained for 500 epochs and averaging five different experiments with random seeds 101, 202, 303, 404, and 505, monotonically reduce to their minimum value after 245 epochs. The early stopping condition activates after 287 epochs (patience 50). The final training MSE is 0.00834 ± 0.00018, whereas the MSE in the validation phase is 0.00891 ± 0.00021. The validation loss tracks the training loss within a range of 6.8%, and no divergence is observed, which suggests that the joint optimization process is stable even with an extra gradient flow channel through the Concrete selector layer.

Performance under different numbers of pilots

The influence of pilot density on channel estimation performance is investigated by varying the number of pilots \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p}$$\end{document} among 4, 8, 16, and 32 while maintaining the SNR range of 0–30 dB. Figure 4 presents the NMSE and BER performance of the proposed joint CBAM-CNN approach with different pilot configurations, where the MMSE and DFT-LS methods are provided as references.Fig. 4Performance comparison under different numbers of pilots: (A) NMSE vs SNR, (B) BER vs SNR.

Performance comparison under different numbers of pilots: (A) NMSE vs SNR, (B) BER vs SNR.

As the number of pilots decreases, all methods exhibit performance degradation due to reduced observation information. The DFT-LS and MMSE estimators suffer severe degradation when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p}$$\end{document} drops below 16, with NMSE increasing by more than one order of magnitude at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p} = 4$$\end{document} compared with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p} = 32$$\end{document}. On the contrary, it is clearly evident that the proposed approach possesses significantly greater robustness to pilot reduction. For example, at an SNR of 15 dB with only 8 pilots, a joint CBAM-CNN achieves an NMSE value as low as 0.0058, which is still comparable to that obtained by the MMSE approach with sixteen pilots (0.0020). Even under the extreme configuration of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{p} = 4$$\end{document}, the proposed method maintains acceptable estimation quality, with NMSE values consistently below those of DFT-LS operating with 16 pilots. In the low SNR region (0 to 5 dB), the proposed approach with 8 pilots achieves NMSE comparable to DFT-LS with 16 pilots, thus highlighting the efficiency of differentiable pilot design under sparse pilot utilization.

From the BER curves, it is evident that consistent trends are followed, where reliable communication quality is ensured even with a reduced pilot overhead. At SNR = 20 dB with 8 pilots, the joint CBAM-CNN achieves a BER of approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6 \times 10^{ - 4}$$\end{document}, meeting the requirements for most underground mining communication applications. The results show that the differentiable pilot optimization module is able to find maximally informative pilot positions, thus offsetting the reduced number of observations. In practice, the proposed algorithm is able to achieve a 50% reduction in pilot overhead (from 16 to 8 pilots) while ensuring similar accuracy, thus increasing spectral efficiency in bandwidth-constrained underground coal mine communication systems.

Performance under different channel environments

To verify adaptability to varying fading severity, experiments are conducted under three Nakagami configurations: : m = 0.85 representing the severe fading typical of confined tunnel propagation, m = 1 corresponding to Rayleigh fading, and m = 2 characterizing mild fading conditions. Figure 5 presents the NMSE and BER performance comparison results over the specified channel environments, with each column representing a specific fading parameter and each row representing a different performance metric.Fig. 5Performance comparison under different Nakagami fading coefficients: (A–C) NMSE, (D–F) BER.

Performance comparison under different Nakagami fading coefficients: (A–C) NMSE, (D–F) BER.

The results clearly reveal that the performance advantage of the proposed joint method is more significant for more severe fading conditions. For the more severe fading channel (m = 0.85), as shown in Figs. 5A and D, the proposed joint method outperforms the others by obtaining an NMSE of 0.0089 at SNR = 10 dB, which is 41.1% better than FSRCNN (0.0151) and close to the MMSE bound (0.0063). When the fading parameter increases to m = 1 or m = 2, which implies milder fading conditions, the performance difference between the schemes becomes smaller due to reduced channel amplitude variations. However, the proposed method still outperforms the others for all channel conditions. This phenomenon can be explained by the improved performance of the CBAM attention mechanism under complex fading conditions, as described in Sect. “CBAM-CNN channel estimation network”, where the channel frequency response has more complex spectral characteristics to be handled by adaptive weighting features. The spatial attention sub-module excels in recovering structural continuity in channel responses degraded by severe multipath scattering effects within mine tunnels, while the channel attention sub-module can selectively emphasize more informative features to represent channel characteristics.

Performance under different multipath fading parameters

The generalization performance of the proposed joint framework is further evaluated through assessing its performance over different multipath delay profiles that simulate different tunnel geometries and electromagnetic propagation environments within underground coal mines. Four different multipath profiles are considered, as listed in Table 5, ranging from compact delay spreads (Profile A with τ_rms = 20 ns) to extended delay spreads (Profile D with τ_rms = 120 ns) with different power decay characteristics. Figure 6 illustrates the BER performance of the joint CBAM-CNN under these configurations.Table 5Multipath channel configurations for robustness validation.ProfileDescriptionNumber of Paths (L)Delay vector τ (ns)Power vector P (dB)τ_rms (ns)AShort tunnel section4[0, 10, 20, 30][0, − 3, − 6, − 9]20BStandard tunnel5[0, 25, 50, 75, 100][0, − 2, − 4, − 7, − 10]50CWide tunnel with equipment5[0, 40, 80, 120, 160][0, − 2, − 5, − 8, − 12]80DComplex tunnel intersection6[0, 50, 100, 150, 200, 250][0, − 1, − 3, − 6, − 10, − 15]120Fig. 6Robustness comparison under different multipath delay profiles.

Table 5Multipath channel configurations for robustness validation.ProfileDescriptionNumber of Paths (L)Delay vector τ (ns)Power vector P (dB)τ_rms (ns)AShort tunnel section4[0, 10, 20, 30][0, − 3, − 6, − 9]20BStandard tunnel5[0, 25, 50, 75, 100][0, − 2, − 4, − 7, − 10]50CWide tunnel with equipment5[0, 40, 80, 120, 160][0, − 2, − 5, − 8, − 12]80DComplex tunnel intersection6[0, 50, 100, 150, 200, 250][0, − 1, − 3, − 6, − 10, − 15]120

Multipath channel configurations for robustness validation.

Robustness comparison under different multipath delay profiles.

From the experimental results, it can be concluded that the proposed method maintains consistent performance over all multipath profiles, as the BER curves are closely clustered throughout the SNR range. At SNR = 15 dB, the BER values for Profiles A through D are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.9 \times 10^{ - 3}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.1 \times 10^{ - 3}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.0 \times 10^{ - 3}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.3 \times 10^{ - 3}$$\end{document}, respectively, exhibiting a maximum variation of only 21%. This robustness is due to the end-to-end joint learning framework proposed in Sect. “Overall framework design”, which allows the network to automatically adapt to different channel characteristics without the need for parameter tuning. The differentiable pilot optimization module can learn to extract the essential information of the channels regardless of the specific delay profile, and the proposed CBAM-CNN estimation network has good generalization ability for different multipath environments due to the attention mechanism proposed in Sect. “CBAM-CNN channel estimation network”.

Robustness analysis under SNR mismatch

Practical underground coal mine communication systems often experience SNR conditions different from those experienced during network training. Therefore, robustness to SNR mismatch becomes an important issue to consider for practical applications on resource-constrained underground coal mine terminals. In this study, the impact of mismatching between training and testing SNRs is investigated by training different models with SNR values of 10, 15, 20, and 25 dB, and then testing these models over the full range of 0 to 30 dB. A baseline model trained with complete data spanning all SNR levels is the performance reference.

The analysis shows that the models trained on medium-to-high SNR levels (20–25 dB) have a minor degradation in NMSE performance relative to the full training baseline over a broad test range, while the models trained on a low SNR (10 dB) have a considerable degradation in performance for high test SNR levels, because of insufficient training on clean channel samples. This suggests that the practical application of the proposed joint CBAM-CNN algorithm for underground coal mine systems may use training data from a single SNR level, rather than requiring data to be sampled from all SNR levels. This could reduce the data acquisition and training time with acceptable estimation accuracy. The intrinsic regularizing effect of the CBAM attention mechanism, along with the residual connections discussed in Sect. “CBAM-CNN channel estimation network”, improves robustness by reducing overfitting to the characteristics of the noise found during the training process.

Discussion

The hierarchy of performance, as illustrated in Table 4, is consistent with the general research trajectory that has been followed in deep learning-based channel estimation research. For example, Soltani et al.26 demonstrated that super- resolution convolutional neural networks could be used to denoise least squares estimates under Rayleigh fading channels, while Chun et al.27 extended this study to massive MIMO systems. This study extends this trajectory to address Nakagami fading channels that occur as a result of waveguide effects in confined mine tunnels, where m = 0.85, resulting in more severe fading than that occurring in Rayleigh fading channels, as well as complex spectral characteristics in the channel frequency response. The effectiveness of the CBAM attention mechanism is rooted in the structural complexity of the mechanism itself. The feature maps are adaptively weighted by the channel attention sub-module to distinguish between information-bearing characteristics and noise-dominated characteristics. Meanwhile, the spatial attention sub-module detects inter-subcarrier structural dependencies along the frequency dimension to rebuild the continuity of the signals after multipath scattering. Hassan et al.28 reported that deeper architectures have diminishing returns without feature prioritization, which is consistent with the saturation observed by Mohammed et al.29 for various 5G channel models, both supporting the attention-driven refinement adopted.

Ablation study elucidates the synergy of the attention mechanism and differentiable pilot optimization. The fact that the total NMSE reduction of 48.0% exceeds the sum of individual contributions verifies the results of Luan and Thompson30, which proved that architectures improved by the attention mechanism benefit from amplified gains when combined with optimized inputs. Gao et al.31 showed that the benefit of attention mechanisms increases with the quality of the input data, which is achieved in this study by end-to-end co-training with a single MSE loss function. In contrast to modular methods such as those proposed by Mashhadi and Gündüz25, where the pilot design and estimation networks are optimized with different objectives, the joint method based on the CAGAN framework that has been introduced in the Introduction of this study is not considered for the experimental part due to the substantial differences between the two frameworks. The CNN without the configuration of the CBAM is considered as the baseline.

The robustness of the results with respect to pilot density, fading severity, multipath profiles, and SNR mismatch collectively supports the practical viability of the proposed approach for various underground mining scenarios. In particular, the result that the eight-pilot approach is able to maintain the BER of approximately 6 × 10⁻4 at SNR = 20 dB while achieving NMSE comparable to that of conventional approaches with 16 pilots suggests that the 50% reduction in pilot overhead is consistent with the results of Chun et al.32. Lim et al.33 showed that pilots jointly designed and considering channel capture structural properties of the channel, which extend beyond those provided by a uniformly spaced structure. The adaptability to Nakagami fading parameters m = 0.85, 1, and 2 extends the scope established by Adiguzel and Develi34, who validated estimators exclusively under Rayleigh conditions, as well as Li et al.35 who did not take into account the severity of fading during mine propagation. The assessment carried out for four different multipath delay profiles, each representing a different geometry for the tunnel, showed that the variation in the maximum bit error rate (BER) was only 21%. This confirms that the end-to-end approach adjusts to different propagation conditions without the need for manual intervention. The SNR mismatch analysis results show that satisfactory performance over the entire range is achieved with training at a single SNR level that is appropriately chosen. This is consistent with the regularization benefits of the proposed method that were reported by Kang et al.36. It is also noteworthy that the results of this study are achieved without the need for any SNR feedback.

Despite the above findings, there are several limitations that should be taken into account before any practical claims can be made. The assessment is conducted solely through simulations on Nakagami channels rather than experimental tests in real mine tunnels. Although the Nakagami fading model characterizes the fading magnitude as reported by Boutin et al.15 and Hrovat et al.12, it cannot replicate dynamic obstructions, non-stationary coal dust, and shadows created by machinery. The observed 50% saving in pilot overhead, the NMSE superiority over MMSE in terms of complexity, and the claimed robustness against different multipath configurations should be considered upper-bound estimates derived from the simulation environment. As such, they cannot guarantee practical performance. Real-world field tests with channel sounding based on measurements in operational tunnels, possibly combined with transfer learning from the training domain to the measured channels, are necessary for this purpose. Further evidence of the additional value of learning-based estimators in extreme-channel settings was provided by Cai et al.5, whose research showed that CNN-LSTM estimators preserve their performance even under deep-space path loss and severe pilot density limitations. Moreover, Cai, Xu, and Niyato6 demonstrated that adaptive filtering via the Savitzky-Golay framework can approach MMSE performance without prior channel statistics, suggesting that data-driven methods offer a viable alternative to statistically dependent estimators in communication environments where channel knowledge is difficult to obtain, a condition shared by underground mine tunnels. The proposed model also caters to SISO. Future research should consider MIMO-OFDM systems based on existing models for underground mine MIMO channels37, time-varying channels resulting from equipment motion and mine ventilation, as well as low-complexity networks suitable for mining terminals.

Conclusion

This paper proposes a joint pilot optimization and channel estimation algorithm based on CBAM-CNN for multipath fading environments in underground coal mines. The framework integrates a differentiable Concrete selector layer with a CBAM-enhanced CNN through a shared MSE loss function, enabling end-to-end co-optimization of pilot placement and channel estimation. Simulation results under a Nakagami fading channel with m = 0.85 demonstrate that the proposed method achieves an NMSE of 0.0089 at SNR = 10 dB, representing an 87.3% reduction relative to LS and a 41.1% improvement over FSRCNN, while approaching MMSE accuracy at O(N) inference complexity. The decoupled ablation shows a synergistic improvement in NMSE of 48.0%, where the combination of channel attention, spatial attention, residual connection, and differential pilot selection outperforms the 33.3% linear addition of their separate contributions. The algorithm maintains robust performance across varying pilot densities, Nakagami fading parameters, multipath delay profiles, and SNR mismatch conditions. In the simulated channel, it achieves a maximum reduction in pilot overhead of 50% while keeping BER at an acceptable level for underground mining communications systems. Verification in real mines is required before any practical claims can be made.