AI 中文总结
本文针对空间-波数域失配的HMIMO信道，将连续算子投影到一维PSWFs的张量积子空间，推导了谱近似误差与遍历容量差距的超指数衰减界，开发了求积规则，指出常规截断会遗漏关键模式。
AI 中文摘要
本文针对具有方形孔径且物理上给定圆形波数支撑的连续全息多输入多输出（HMIMO）信道，建立了有限维谱近似与遍历容量收敛的定量结果。由此产生的空间-波数域失配会导致一个不可分的方形圆盘集中问题，而基于长球波函数（PSWFs）的经典可分构造无法直接应用。我们将连续算子投影到一维（1D）PSWFs的张量积子空间，同时保留圆形波数支撑，得到一个通常非对角但高度稀疏的有限维矩阵。我们表明，尽管失去了可分性并引入了非对角耦合，考虑保留特征值扰动和残余谱尾的全谱近似误差仍由一维PSWF特征值尾包络控制。在显式的一维截断阈值之外，该误差呈超指数衰减。该分析进一步给出了扁平化二维特征值排序下特征谱的渐近上包络。我们还在各自的传输协方差优化下，建立了连续信道与张量-PSWF截断信道的实际遍历容量之间差距的非渐近上界。结合谱结果，该容量差距界继承了与截断阶数相同的超指数依赖关系。最后，开发了具有显式径向和角节点阈值的求积规则，用于评估投影矩阵。数值结果表明，基于空间自由度的常规截断会遗漏与性能相关的模式，尤其是对于紧凑孔径。
英文摘要
We establish quantitative results on finite-dimensional spectral approximation and ergodic-capacity convergence for continuous Holographic Multiple-Input Multiple-Output (HMIMO) channels with square apertures and physically prescribed circular wavenumber support. The resulting spatial-wavenumber domain mismatch leads to a non-separable square-disk concentration problem for which the classical separable construction based on prolate spheroidal wave functions (PSWFs) cannot be directly applied. We project the continuous operator onto a tensor-product subspace of one-dimensional (1D) PSWFs while preserving the circular wavenumber support, yielding a generally non-diagonal but highly sparse finite-dimensional matrix. We show that the whole-spectrum approximation error, accounting for retained-eigenvalue perturbations and the residual spectral tail, remains controlled by a 1D PSWF eigenvalue-tail envelope despite the loss of separability and induced off-diagonal coupling. Beyond an explicit 1D truncation threshold, this error decays super-exponentially. This analysis further yields an asymptotic upper envelope for the eigenspectrum under the flattened two-dimensional eigenvalue ordering. We further establish a non-asymptotic upper bound on the gap between the actual ergodic capacities of the continuous and tensor-PSWF-truncated channels under their respective transmit-covariance optimizations. Combined with the spectral result, this capacity-gap bound inherits the same super-exponential dependence on the truncation order. Finally, quadrature rules with explicit radial and angular node thresholds are developed for evaluating the projected matrix. Numerical results show that conventional truncation based on spatial degrees of freedom can omit performance-relevant modes, particularly for compact apertures.
