多阈值采样：信号空间、采样算子与穿越时间分布
Multi-Threshold Sampling: Signal Space, Sampling Operators, and First Recorded Crossing-Time Distributions
- 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）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出统一数学框架，将信号分布映射至记录穿越分布，分析多阈值采样中失配、噪声、量化等效应，并给出首次穿越时间精确分布，为性能极限评估奠定基础。
中文摘要 AI 辅助
多阈值（MT）采样在选定的阈值处记录穿越时间，用于参数估计和波形重建。对于合适的高速信号，与高速率均匀时域采样相比，它可以减少数据量、硬件成本和功耗。核科学中的应用包括正电子发射断层扫描、石油测井和光子计数X射线成像。然而，其理论基础需要进一步发展，以支持性能评估和系统设计。必要的第一步是严格定义信号空间和采样算子，但为此目的的一般数学框架仍然缺失。我们提出了一个统一的数学框架，通过采样算子将信号分布映射到记录的穿越分布。它提供了信号和阈值中的失配与噪声、时间量化和选择如何塑造记录数据的共同描述，使得它们各自及组合效应能够被分析。对于非齐次泊松光子到达和确定性MT采样，该框架在给定的记录时间间隔内，在指定阈值处产生首次记录穿越时间的精确分布。闪烁脉冲模型的预测与独立的蒙特卡洛模拟一致，证明了该框架的预测能力。该框架为确定MT采样的基本性能极限和设计接近这些极限的系统奠定了基础。
英文摘要
Multi-threshold (MT) sampling records crossing times at selected thresholds for parameter estimation and waveform reconstruction. We develop a mathematical framework that defines the signal space and sampling operators and maps the signal distribution to the distribution of recorded crossings. The framework distinguishes model mismatch, signal noise, threshold mismatch, and threshold noise. Finite point measures accommodate variable crossing counts and preserve multiplicity after time quantization. Stochastic crossing, timing, and selection operators and their associated Markov kernels describe the sampling process. For nonhomogeneous Poisson photon arrivals and deterministic MT sampling, we derive the exact cumulative distribution function (CDF) of the first recorded crossing time at a specified threshold within a recorded-time interval. Under additional regularity and local monotonicity conditions, this CDF admits a one-dimensional Fourier representation. For a scintillation pulse model, numerical CDFs and standard deviations of the first recorded crossing time agree with independent Monte Carlo simulations, providing a quantitative basis for comparing threshold settings. The framework provides a foundation for MT performance evaluation, parameter estimation, waveform reconstruction, and sampler design.
