静态多阈值采样的非均匀采样理论
A Nonuniform-Sampling Theory for Static Multi-Threshold Sampling
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- Huazhong University of Science and Technology（华中科技大学）
- Wuhan National Laboratory for Optoelectronics（武汉光电国家研究中心）
- University of Science and Technology of China（中国科学技术大学）
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中文总结 AI 辅助
本文为静态均匀多阈值采样建立非均匀采样理论，给出稳定采样的充要条件、可计算准则及扰动界，并证明无限制有限能量情形无法实现通用无先验采样定理。
中文摘要 AI 辅助
多阈值（MT）采样固定电压电平，并记录波形穿越这些电平的时刻。由此产生的由信号生成的事件流并非由预设时钟控制，而是由实际穿越集的非均匀几何结构所支配。在经典非均匀采样框架内，我们为静态、均匀间隔的阈值发展了一套保留事件MT理论。对于实际实现的保留MT时间集，Beurling弱极限定理给出了对$PW_\Omega$进行稳定采样的精确充要条件。对于固定信号类别，同一原理给出了先验准则：均匀稳定MT采样等价于该类别弱极限包中每个集合的Bernstein唯一性。在结构化先验下，该准则可简化为可计算形式。周期模板给出有限纤维/秩检验，周期间隙流给出Shannon型完全插值边界和稳定采样密度规则，有限状态间隙先验将非临界认证简化为最大循环均值条件。Kadec型局部证书、保留密度界和有限矩阵检验为有限活动记录提供了实用的充分条件，而条件扰动界量化了抖动、前端噪声和阈值误差如何在事件对应下消耗干净采样裕度。该理论还识别了该模型的不可避免边界：无限制的有限能量保留MT记录具有零全线低Beurling密度，因此无法为$PW_\Omega$提供通用的无先验采样定理。在保留事件点采样中，MT恢复问题被精确解决：精确恢复由事件集几何决定，可计算保证需要结构先验或有限维/尾部假设，而无限制的静态有限能量问题是不可能的。
英文摘要
Multi-Threshold (MT) sampling fixes voltage levels and records when a waveform crosses them. The resulting signal-generated event stream is governed not by a prescribed clock but by the nonuniform geometry of the realized crossing set. Within classical nonuniform sampling, we develop a retained-event MT theory for static, uniformly spaced thresholds. For a realized retained MT time set, Beurling's weak-limit theorem gives the exact necessary-and-sufficient condition for stable sampling of $PW_Ω$. For a fixed signal class, the same principle gives the a priori criterion: uniform stable MT sampling is equivalent to Bernstein uniqueness for every set in the class weak-limit hull. This criterion reduces to computable forms under structured priors. Periodic templates give finite fiber/rank tests, periodic-gap streams give a Shannon-type complete-interpolation boundary and a stable-sampling density rule, and finite-state gap priors reduce noncritical certification to a maximum-cycle-mean condition. Kadec-type local certificates, retained-density bounds, and finite-matrix tests provide practical sufficient conditions for finite active records, while conditional perturbation bounds quantify how jitter, front-end noise, and threshold error consume a clean sampling margin under event correspondence. The theory also identifies the unavoidable boundary of this model: unrestricted finite-energy retained MT records have zero full-line lower Beurling density and therefore cannot provide a universal prior-free sampling theorem for $PW_Ω$. Within retained-event point sampling, the MT recovery question is resolved precisely: exact recovery is governed by event-set geometry, computable guarantees require structural priors or finite-dimensional/tail assumptions, and the unrestricted static finite-energy problem is impossible.
