Parametric Holographic MIMO Channel Modeling and Its Bayesian Estimation
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摘要: 全息多输入多输出(HMIMO)技术因其高空间复用效率和信道容量被视为6G通信系统的关键技术之一，但电磁传播模型复杂、用户角度随机给电磁信道建模和估计带来较大困难。现有方法依赖简化假设或统计模型，存在模型失配问题，且难以同时解耦信道、位置与角度。针对上述挑战，该文提出一种融合神经网络、凸优化和因子图的混合信道建模与估计方法，该方法首先学习信道与坐标的非线性映射关系，构建参数化信道模型；其次基于欧拉角旋转理论描述用户角度，并将其嵌入因子图实现信道、坐标及角度的全局建模；最后利用消息传递算法完成参数联合解耦与信道估计。仿真结果表明，所提方法的信道估计误差较现有近似方法降低3 dB以上。该研究突破了现有方法对天线平行假设的依赖，为复杂电磁环境下的高精度信道估计与位置感知提供了新的解决方案。Abstract:
Objective Holographic Multiple-Input Multiple-Output (HMIMO), based on continuous-aperture antennas and programmable metasurfaces, is regarded as a cornerstone of 6G wireless communication. Its potential to overcome the limitations of conventional massive MIMO is critically dependent on accurate channel modeling and estimation. Three major challenges remain: (1) oversimplified electromagnetic propagation models, such as far-field approximations, cause severe mismatches in near-field scenarios; (2) statistical models fail to characterize the coupling between channel coefficients, user positions, and random orientations; and (3) the high dimensionality of parameter spaces results in prohibitive computational complexity. To address these challenges, a hybrid parametric-Bayesian framework is proposed in which neural networks, factor graphs, and convex optimization are integrated. Precise channel estimation, user position sensing, and angle decoupling in near-field HMIMO systems are thereby achieved. The methodology provides a pathway toward high-capacity 6G applications, including Integrated Sensing And Communication (ISAC). Methods A hybrid channel estimation method is proposed to decouple the “channel-coordinate-angle” parameters and to enable joint estimation of channel coefficients, coordinates, and angles under random user orientations. A neural network is first employed to capture the nonlinear relationship between holographic channel characteristics and the relative coordinates of the base station and user. The trained network is then embedded into a factor graph, where global optimization is performed. The neural network is dynamically approximated through Taylor expansion, allowing bidirectional message propagation and iterative refinement of parameter estimates. To address random user orientations, Euler angle rotation theory is introduced. Finally, convex optimization is applied to estimate the rotation mapping matrix, resulting in the decoupling of coordinate and angle parameters and accurate channel estimation. Results and Discussions The simulations evaluate the performance of different algorithms under varying key parameters, including Signal-to-Noise Ratio (SNR), pilot length L, and base station antenna number M. Two performance metrics are considered: Normalized Mean Square Error (NMSE) of channel estimation and user positioning accuracy, with the Cramér–Rao Lower Bound (CRLB) serving as the theoretical benchmark. At an SNR of 10 dB, the proposed method achieves a channel NMSE below –40 dB, outperforming Least Squares (LS) estimation and approximate model-based approaches. Under high SNR conditions, the NMSE converges toward the CRLB, confirming near-optimal performance ( Fig. 5a ). The proposed channel model demonstrates superior performance over “approximate methods” due to its enhanced characterization of real-world channels. Moreover, the positioning error gap between the proposed method and the “parallel bound” narrows to nearly 3 dB at high SNR, confirming the accuracy of angle estimation and the effectiveness of parameter decoupling (Fig. 5b ). Moreover, the proposed method maintains performance close to the theoretical bounds when system parameters, such as user antenna number N, base station antenna number M, and pilot length L, are varied, demonstrating strong robustness (Figs. 6 ～8 ). These results also show that the Euler angle rotation-based estimation effectively compensates for coordinate offsets induced by random user orientations.Conclusions This study proposes a framework for HMIMO channel estimation by integrating neural networks, factor graphs, and convex optimization. The main contributions are threefold. First, Euler angles and coordinate mapping are incorporated into the parameterized channel model through factorization and factor graphs, enabling channel modeling under arbitrary user antenna orientations. Second, neural networks and convex optimization are embedded as factor nodes in the graph, allowing nonlinear function approximation and global optimization. Third, bidirectional message passing between neural network and convex optimization nodes is realized through Taylor expansion, thereby achieving joint decoupling and estimation of channel parameters, coordinates, and angles. Simulation results confirm that the proposed framework achieves higher accuracy—exceeding benchmarks by more than 3 dB, and demonstrates strong robustness across a range of scenarios. Future work will extend the method to multi-user environments, incorporate polarization diversity, and address hardware impairments such as phase noise, with the aim of supporting practical deployment in 6G systems. -
表 1 概率分布和对应函数表达式
概率分布和物理意义 函数表达式 概率分布和物理意义 函数表达式 似然函数$ p\left( {{{\boldsymbol{Y}}}|{{\boldsymbol{H}}},\gamma } \right) $ $ {f_{\boldsymbol{Y}}}({{\boldsymbol{Y}}},{{\boldsymbol{H}}},\gamma ) = {\text{CN}}({{\boldsymbol{Y}}};{{\boldsymbol{\varPhi}} {\boldsymbol{H}}},{\gamma ^{ - 1}}{\mathbf{I}}) $ 旋转参数化$ {f_{\boldsymbol{A}}}({{\boldsymbol{A}}},{{\boldsymbol{\theta }}}) $ 如式(7)所示 坐标映射$ p({{\boldsymbol{R}}}|{{\boldsymbol{B}}},{{{\boldsymbol{r}}}^t}) $ $ {f_{\boldsymbol{R}}} = \prod\nolimits_n {{f_{{{\boldsymbol{r}}_n}}} ({{{\boldsymbol{b}}}_n},{{\boldsymbol{r}}}_n^{\text{t}},{{\boldsymbol{{r}}}^{\text{t}}})} = \prod\nolimits_n {\delta ({{{\boldsymbol{r}}}_n} - {{\boldsymbol{b}}}_n^{\text{t}} - {{{\boldsymbol{r}}}^{\text{t}}})} $ 参数化信道$ p({{\boldsymbol{H}}}|{{\boldsymbol{R}}}) $ $ {f_{{{{h}}_{mn}}}} ({{{h}}_{mn}},{{{\boldsymbol{r}}}_n}) = \delta ({{h}}_{mn}^{} - {\boldsymbol{\phi}} _{mn}^{}\exp ({\text{i}}{k_0}{{{r}}_{mn}})) $ 旋转$ p({{\boldsymbol{B}}}|{{\boldsymbol{A}}}({{\boldsymbol{\theta}} })) $ $ {f_{\boldsymbol{B}}} = \prod\nolimits_n {{f_{{{\boldsymbol{b}}_n}}}({{{\boldsymbol{b}}}_n},{{\boldsymbol{A}}}(\theta ))} = \prod\nolimits_n {\delta ({{{\boldsymbol{b}}}_n} - {{\boldsymbol{A}}}{{{\bar b}}_n})} $ 噪声先验$ p(\gamma ) $ $ {f_\gamma } = 1/\gamma $ 
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1 基于消息传递的HMIMO信道估计算法
输入：观测$ {{\boldsymbol{Y}}} $，导频$ {{\boldsymbol{S}}} $；输出：用户坐标$ {{\hat{\boldsymbol{ r}}}^{\mathrm{t}}} = {({\hat x^{\text{t}}},{\hat y^{\text{t}}},{\hat z^{\text{t}}})^{\text{T}}} $，信道$ {\hat {\boldsymbol{H}}} $。 (1) 初始化噪声精度$ \hat \gamma $、偏置矩阵矩阵$ {\hat {\boldsymbol{B}} = \bar {\boldsymbol{B}}} $, $ {{{\boldsymbol{S}}}_{\boldsymbol{H}}}{\text{ = }}{{{{\textit{0}}}}_{N \times M}} $, $ {\hat {\boldsymbol{H}}}{\text{ = }}{{{{\textit{0}}}}_{N \times M}} $和$ {{\boldsymbol{V}}_{\boldsymbol{H}}} = {{{{\textit{1}}}}_{N \times M}} $。 (2) For iter=1:T (3) $ {{\boldsymbol{V}}_{\boldsymbol P}} = |{{\boldsymbol{\varPhi}} }{|^2}{{\boldsymbol{V}}_H} $和$ {{\boldsymbol{P}}} = {{\boldsymbol{\varPhi}} \hat {\boldsymbol{H}}} - {{\boldsymbol{V}}_{\boldsymbol P}} \cdot {{{\boldsymbol{S}}}_{\boldsymbol{H}}} $， (4) $ {{\boldsymbol{V}}_{\boldsymbol Z}} = {{\boldsymbol{V}}_{\boldsymbol P}}/(\hat \gamma {{\boldsymbol{V}}_{\boldsymbol P}} + 1) $和$ {{\boldsymbol{Z}}} = (\hat \gamma {{\boldsymbol{Y}}} + {{\boldsymbol{P}}}./{{\boldsymbol{V}}_{\boldsymbol P}}) \cdot {{\boldsymbol{V}}_{\boldsymbol Z}} $， (5) $ \widehat{\gamma }={M}\times {L}/\left(\Vert \boldsymbol{Y}-\boldsymbol{Z}{\Vert }^{2}+\Vert {\boldsymbol{V}}_{{\boldsymbol{Z}}}\Vert \right) $, (6) $ {{\boldsymbol{V}}_{{\boldsymbol{{S}}_{\boldsymbol{H}}}}} = 1/({{\boldsymbol{V}}_{\boldsymbol P}} + {\hat \gamma ^{ - 1}}) $和$ {{{\boldsymbol{S}}}_{\boldsymbol{H}}} = {{\boldsymbol{V}}_{{{\boldsymbol{S}}_{\boldsymbol{H}}}}} \cdot ({{\boldsymbol{Y}}} - {{\boldsymbol{P}}}) $， (7) $ {{\boldsymbol{V}}_{{{\boldsymbol{Q}}_{\boldsymbol{H}}}}} = 1/(|{{{\boldsymbol{\varPhi}} }^{\text{H}}}{|^2}{{\boldsymbol{V}}_{{{\boldsymbol{S}}_{\boldsymbol{H}}}}}) $和$ {{{\boldsymbol{Q}}}_{\boldsymbol{H}}} = {\hat {\boldsymbol{H}}} + {{\boldsymbol{V}}_{{{\boldsymbol{Q}}_{\boldsymbol{H}}}}} \cdot ({{{\boldsymbol{\varPhi}} }^{\text{H}}}{{{\boldsymbol{S}}}_{\boldsymbol{H}}}) $ (8) 利用式(19)计算$ {f_{{h_{mn}}}} $到$ x_n^{\rm t} $消息$ {m_{{f_{{h_{mn}}}} \to x_n^{\rm t}}}(x_n^{\rm t}),\forall m,n $， (9) 利用式(20)和式(21)分别计算消息$ {m_{x_n^{\rm t} \to {f_{x_n^{\rm t}}}}}(x_n^{\rm t}),\forall n $和$ {m_{{f_{x_n^{\rm t}}} \to {x^{\rm t}}}}({x^{\rm t}}),\forall n $， (10) 利用式(22)和式(23)分别计算$ {x^{\rm t}} $的置信$ b({x^{\rm t}}) $和消息$ {m_{{x^{\rm t}} \to {f_{x_n^{\rm t}}}}}({x^{\rm t}}) $， (11) 利用式(24)和式(25)分别计算消息$ {m_{{f_{x_n^{\rm t}}} \to x_n^{\rm t}}}(x_n^{\rm t}),\forall n $和置信$ b(x_n^{\rm t}),\forall n $， (12) 利用式(26)和式(27)分别计算消息$ {m_{x_n^{\rm t} \to {f_{{h_{mn}}}}}}(x_n^{\rm t}) $和$ {m_{{f_{{h_{mn}}}} \to {h_{mn}}}}({h_{mn}}),\forall m,n $， (13) 利用式(28)计算$ {h_{mn}} $的置信$ b({h_{mn}}),\forall m,n $，利用(29)堆叠$ b({h_{mn}}) $，得到$ {\hat {\boldsymbol{H}}} $和$ {{\boldsymbol{V}}_{\boldsymbol{H}}} $， (14) 利用式(30)计算欧拉角$ {{\hat {\boldsymbol{\theta}} }^{{\text{iter}} + 1}} $，更新$ {{\boldsymbol{A}}}({{\boldsymbol{\theta}} }) $,$ {\hat {\boldsymbol{B}}} = {({{\hat {\boldsymbol{b}}}_1},{{\hat {\boldsymbol{b}}}_2}, \cdots ,{{\hat {\boldsymbol{b}}}_N})^{\text{T}}} $。 End For
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