AI 中文总结
该研究为空间平稳 HMIMO 信道建模建立理论框架，用 NGLQ 方法，证明求积误差超指数衰减，推导出 cDoF 和 eDoF，解决 EVD 病态问题，经数值评估和模拟验证了相关结论及频谱收敛特性。
AI 中文摘要
本文利用带高斯 - 勒让德求积的奈斯特罗姆方法（NGLQ），为空间平稳全息 MIMO（HMIMO）信道的连续到离散建模建立了一个全面的理论框架。从对 NGLQ 方法的算子理论分析开始，证明其求积误差呈现超指数衰减。推导出空间采样阈值即计算自由度（cDoF），揭示了 1D 阵列在物理自由度上有\(\pi/2\)的过采样惩罚，2D 可分离网格有 68%的计算冗余。为解决奈斯特罗姆离散化固有的特征值分解（EVD）问题的病态性，调用多维塞戈 - 维德姆渐近展开，得到 2D 矩形孔径有效自由度（eDoF）的物理基础半解析表达式，捕获各向异性边界截断效应以指导部分 EVD 并降低计算复杂度。数值评估证实了 cDoF 阈值在最坏情况端射条件下的紧密性，模拟验证了导出的 eDoF 作为精确渐近近似的准确性，通过部署精确的非均匀离散傅里叶变换消除插值误差底限，证明了非各向同性散射环境下的频谱收敛到机器精度水平。
英文摘要
This paper establishes a comprehensive theoretical framework for the continuous-to-discrete modeling of spatially stationary holographic MIMO (HMIMO) channels utilizing the Nystrom method with Gauss-Legendre quadrature (NGLQ). Starting with an operator-theoretic analysis of the NGLQ method, we prove that its quadrature error exhibits a super-exponential decay. Furthermore, we derive a spatial sampling threshold, termed computational degrees of freedom (cDoF), which reveals a π/2 oversampling penalty over the physical DoF for 1D arrays, compounding to a 68% computational redundancy for 2D separable grids. To address the ill-conditioning of the eigenvalue decomposition (EVD) problem inherent to the Nystrom discretization, we invoke the multidimensional Szego-Widom asymptotic expansion. This analysis yields a physically grounded semi-analytical expression for the effective DoF (eDoF) of 2D rectangular apertures, capturing the anisotropic boundary truncation effects to guide partial EVD and reduce computational complexity. Numerical evaluations confirm the tightness of the cDoF threshold under worst-case end-fire conditions. Moreover, simulations utilizing closed-form kernels for isotropic scattering verify that the derived eDoF acts as an accurate asymptotic approximation. Finally, by deploying the exact non-uniform discrete Fourier transform to eliminate interpolation error floors, we demonstrate spectral convergence down to the machine-precision level for non-isotropic scattering environments.
