曲率域无线通信：规范固定信号空间、Fredholm容量与连续孔径的微分受限缩放
Curvature-Domain Wireless Communications: Gauge-Fixed Signal Spaces, Fredholm Capacity, and Differentiation-Limited Scaling for Continuous Apertures
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中文总结 AI 辅助
本文提出曲率域连续孔径信令框架，通过规范固定与双预算注水实现Fredholm容量最优，并证明在Shannon数以下曲率是适定且规范不变的坐标，微分受限缩放由规范主导。
中文摘要 AI 辅助
我们针对连续孔径信令提出了一种曲率域公式，其中发射相位通过其二阶空间导数表示，并在去除仿射活塞-倾斜规范自由度后进行处理。由此得到的规范固定合成算子是有界且紧致的，具有尖锐的Poincare-Wirtinger常数$C_L=L^2/β_1^2$，其模态Gram谱以闭式形式给出，$ρ_m=(L/β_m)^4$，其中$β_m$是方程$\cosβ\coshβ=1$的根。在有限支撑平方可积传播核下，切算子属于Hilbert-Schmidt类，因此无限维容量是定义良好的Fredholm行列式上确界。最优信令法则是一种双预算广义注水算法，其中一个拉格朗日乘子用于曲率功率，另一个用于相位偏移。从精确的非线性纯相位孔径法则出发，我们推导了相干切向信道，并给出了显式的Frechet余项界以及用于大偏移的多图集。在接收端，通过二阶差分从含噪相位样本推断曲率，其噪声协方差为五对角矩阵，谱范数为$Θ(Δx^{-4})$。一套确定性诊断套件将每种机制与其闭式结果进行对比，并在每项声明旁设置零运行：计算得到的谱与$(L/β_m)^4$的相对误差为$3.78\times10^{-15}$；切向余项的拟合斜率为1.0000；双预算法则在两个乘子均严格激活的情况下求解，KKT残差为$3.90\times10^{-15}$；导数噪声界被逼近至0.999981；微分受限分支在指数0.2175处被观测到，随后在模式数达到Shannon数时坍缩至0.0576，而平坦传播零运行保持在0.2019。因此，曲率是一个适定、规范不变的坐标，且在Shannon数以下，规范而非介质主导缩放行为。
英文摘要
We develop a curvature-domain formulation for continuous-aperture signaling, in which the transmit phase is represented through its second spatial derivative after quotienting out affine piston-and-tilt gauge freedom. The resulting gauge-fixed synthesis operator is bounded and compact, with sharp Poincare-Wirtinger constant $C_L=L^2/β_1^2$, and its modal Gram spectrum is available in closed form, $ρ_m=(L/β_m)^4$, with $β_m$ the roots of $\cosβ\coshβ=1$. Under a bounded-support square-integrable propagation kernel the tangent operator is Hilbert-Schmidt, so the infinite-dimensional capacity is a well-defined Fredholm-determinant supremum. The optimal signaling law is a dual-budget generalized water-filling with one Lagrange multiplier for curvature power and one for phase excursion. From the exact nonlinear phase-only aperture law we derive the coherent tangent channel with an explicit Frechet remainder bound and a multi-chart atlas for large excursions. At the receiver, curvature inferred from noisy phase samples by second differences has a pentadiagonal noise covariance with spectral norm $Θ(Δx^{-4})$. A deterministic diagnostic suite measures each mechanism against its closed form, with a null run beside every claim: the computed spectrum matches $(L/β_m)^4$ to relative error $3.78\times10^{-15}$; the tangent remainder has fitted slope 1.0000; the dual-budget law is solved with both multipliers strictly active to a KKT residual of $3.90\times10^{-15}$; the derivative-noise bound is approached to 0.999981; and the differentiation-limited branch is observed at exponent 0.2175, then collapses to 0.0576 once the mode count saturates at the Shannon number, while a flat-propagation null run holds at 0.2019. Curvature is thus a well-posed, gauge-invariant coordinate, and below the Shannon number the gauge, rather than the medium, governs the scaling.
发表机构
- NeuroBazar
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