Multi-Threshold Sampling: Signal Space, Sampling Operators, and First Recorded Crossing-Time Distributions

Abstract

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.