AI 中文总结
该研究针对仅报告信道索引的带符号子频带频率估计问题，建立了临界尺度下的极小极大极限，推导了傅里叶-柯西零空间谱，分析了不同M值下的极小极大特性，得出了曲率可见性定律等结论。
AI 中文摘要
M通道离散傅里叶变换（DFT）信道化器将栅上正弦信号路由至单个输出。当每一帧仅报告一个与能量成比例的信道索引时，带符号子频带频率估计会变得非正则：暗信道概率是偏移量的二次函数，而方向则以三次方形式进入。我们研究在已知均匀替换概率ε_N（服从N次独立带标签报告）、所有频率与帧共享单个确定性酉矩阵、且受路由缺陷τ/N约束的情况下的问题。若√N·ε_N→λ<∞，我们在临界尺度下建立了达到的全局频率极小极大极限：频率偏移为N^{-1/4}，酉矩阵扰动与替换为N^{-1/2}，路由缺陷为N^{-1}。有效酉矩阵切空间是对称完全图边场模去中心化干扰项，其第一变分是依赖于半径的加权散度。对于λ>0，在k个不同归一化偏移幅值处求值得到精确的傅里叶-柯西零空间谱：对于任意不同幅值的选择，需要且仅需要⌊(M-1)/2⌋次求值来证明所有幅值处的中立性。终端零空间具有最大公约数维度公式和正定聚合曲率。因此，对于M=3，精确DFT路由在完整临界切空间类内是唯一的极小极大方法；而对于M≥4，每个正的临界缺陷预算都会严格改善极小极大常数。该分析还得出最小素数曲率可见性定律，且对于M≥5，存在不连续的相容性几何，但在零临界替换下限λ=0处具有连续的极小极大值。
英文摘要
An $M$-channel discrete Fourier transform (DFT) channelizer routes an on-grid sinusoid to a single output. When each frame reports only one energy-proportional channel index, signed sub-bin frequency estimation becomes nonregular: dark-channel probability is quadratic in the offset, whereas orientation enters cubically. We study $N$ independent labeled reports under a known uniform-replacement probability $ε_N$ and a single deterministic unitary shared by all frequencies and frames, constrained to routing defect $τ/N$. If $\sqrt{N}ε_N \to λ< \infty$, we establish an attained global-in-frequency minimax limit at the critical scales $N^{-1/4}$ for frequency offset, $N^{-1/2}$ for unitary perturbation and replacement, and $N^{-1}$ for routing defect. The effective unitary tangent is a symmetric complete-graph edge field modulo one centering nuisance, and its first variation is a radius-dependent weighted divergence. For $λ>0$, evaluation at $k$ distinct normalized offset magnitudes yields an exact Fourier-Cauchy nullity spectrum: $\lfloor(M-1)/2\rfloor$ evaluations are necessary and sufficient, for every choice of distinct magnitudes, to certify neutrality at all magnitudes. The terminal nullspace has a greatest-common-divisor dimension formula and positive-definite aggregate curvature. Consequently, exact DFT routing is uniquely minimax within the complete critical tangent class for $M=3$, whereas every positive critical defect budget strictly improves the minimax constant for $M\geq4$. The analysis also yields a smallest-prime curvature-visibility law and, for $M\geq5$, discontinuous compatibility geometry but a continuous minimax value at the zero critical replacement floor $λ=0$.
