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Multi-Threshold Sampling: Signal Space, Sampling Operators, and First Recorded Crossing-Time Distributions
Ao Qiu, Qingguo Xie
TL;DR
MT sampling records threshold-crossing times for parameter estimation and waveform reconstruction, but describing their distribution requires accounting for variable counts, quantization, and random sampling stages. The paper defines a framework for that process and derives an exact first recorded crossing-time CDF for nonhomogeneous Poisson arrivals; numerical results for a scintillation pulse model agree with independent Monte Carlo simulations.
Problem
Connecting a signal distribution to recorded crossings requires definitions that handle variable crossing counts, multiplicity after time quantization, and the combined effects of random signal paths, timing, and selection.
Source: §I · p. 1Method
The paper defines signal spaces and crossing, timing, and selection operators, using finite point measures and associated Markov kernels to model recorded crossings.
Source: §I · p. 1, §VI · p. 5, §IX · p. 9Results
For nonhomogeneous Poisson photon arrivals and deterministic MT sampling, the exact first recorded crossing-time CDF has a one-dimensional Fourier representation under additional regularity and local monotonicity conditions, and scintillation-model CDFs agree with independent Monte Carlo simulations.
Source: A. Implications of the Framework and… · p. 8, §IX · p. 9Takeaways & Limitations
The framework provides a mathematical foundation for MT performance evaluation, parameter estimation, waveform reconstruction, and sampler design.
Source: §IX · p. 9
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
MT sampling represents signals through threshold-crossing times, which can support parameter estimation and waveform reconstruction. This section motivates a framework for defining the signal and sampling operators and tracing signal distributions to recorded crossings.
I. INTRODUCTION
- MT sampling records crossing times rather than amplitudes on a fixed grid and can reduce data volume, hardware cost, and power for high-speed signals.Recorded crossings are used for parameter estimation and waveform reconstruction.Source: §I · p. 1
- Continuous paths, jumps, equality intervals, variable crossing counts, and randomized timing and selection make crossing definitions and recorded-crossing distributions challenging.The observation space must preserve multiplicity when distinct analog crossings quantize to the same recorded coordinates.Source: §I · p. 1
- The framework maps signal distributions to recorded-crossing distributions while distinguishing model mismatch, signal noise, threshold mismatch, and threshold noise.Finite point measures accommodate variable crossing counts and preserve multiplicity after time quantization.Source: §I · p. 1
II. SIGNAL SPACE
- Signals are modeled as paths with continuous evolution and jumps, and random signals are decomposed into nominal signal, model mismatch, and signal noise.The mean signal and pointwise zero-mean signal noise determine this decomposition for a fixed nominal model.Source: §II · p. 2, be a fixed compact observation window… · pp. 2–3
III. CROSSING OPERATOR
The crossing operator represents threshold encounters as finite point measures and assigns crossing times and direction marks. An admissible path domain ensures finitely many crossings, while threshold paths can include systematic and random deviations.
A. Representation of Crossings
- Finite point measures represent crossings by time, threshold index, and direction, with mass recording multiplicity and total mass giving the crossing count.This representation accommodates repeated crossing locations after timing.Source: A. Representation of Crossings · p. 3
B. Threshold Model and Crossing Operator
- Threshold paths combine nominal thresholds, threshold mismatch, and zero-mean threshold noise.Mismatch describes systematic displacement; threshold noise describes random deviations from the mean threshold paths.Source: B. Threshold Model and Crossing Operator · p. 3
- Comparison states indicate whether a signal is below, at, or above a threshold, and admissibility ensures finitely many analog crossings in the observation window.The admissible domain imposes a finite partition condition on path pairs.Source: B. Threshold Model and Crossing Operator · p. 3
- The crossing operator uses comparison states before, at, and after each time to identify crossings and assign direction marks, and it is measurable on its admissible domain.Marks distinguish transitions across a threshold, transitions into or out of equality, and isolated contacts.Source: B. Threshold Model and Crossing Operator · pp. 3–4
C. Stochastic Crossing Operator and Kernel
For a fixed signal path, random threshold paths induce random analog crossings through the stochastic crossing operator. Its induced Markov kernel describes the output crossing distribution for that input.
C. Stochastic Crossing Operator and Kernel
- The stochastic crossing operator maps a fixed admissible signal path and random threshold paths to random finite measures of analog crossings.The operator is defined almost surely for each admissible signal path.Source: C. Stochastic Crossing Operator and… · p. 4
- The induced crossing kernel assigns an output probability distribution to each admissible signal path.Markov kernels describe sampling-stage output distributions for given inputs.Source: C. Stochastic Crossing Operator and… · p. 4
- Joint measurability and probability-one sections ensure that the stochastic crossing construction defines a Markov kernel.Partially defined sampling maps are evaluated on their probability-one domains.Source: C. Stochastic Crossing Operator and… · p. 4
IV. TIMING OPERATOR
The timing operator assigns continuous or quantized times to analog crossings while preserving counts by threshold and direction. Its stochastic version and induced kernel describe random timing outcomes.
IV. TIMING OPERATOR
- The timing operator assigns continuous-valued or quantized time coordinates to analog crossings and preserves their counts for each threshold and direction.The operator is defined as a measurable map that preserves the corresponding count measure.Source: §IV · p. 4
- A stochastic timing operator is jointly measurable, and its induced timing kernel is a Markov kernel.The kernel describes timing outcomes for a given crossing measure.Source: §IV · p. 4
- Finite point measures retain multiplicity when timing maps distinct analog crossings to the same time, threshold index, and direction mark.The crossing operator itself produces a simple point measure.Source: §IV · p. 4
V. SELECTION OPERATOR
The selection operator retains or removes timed crossings to produce recorded crossings. It preserves their times, threshold indices, and direction marks while never increasing multiplicity at any location.
V. SELECTION OPERATOR
- Selection produces recorded crossings by retaining or removing timed crossings without changing their times, threshold indices, or direction marks.The output multiplicity at each location is no greater than the input multiplicity.Source: §V · p. 4
- A selection operator is defined as a measurable map, with a stochastic version also introduced.The stochastic selection operator is defined on a probability space.Source: §V · pp. 4–5
selection operator is a jointly measurable map
The selection stage maps finite point measures to selected measures, and its joint measurability makes the induced selection kernel a Markov kernel. The full MT sampler composes crossing, timing, and selection stages to describe recorded-crossing distributions.
selection operator is a jointly measurable map
- Joint measurability of the selection map ensures that its induced selection kernel is a Markov kernel.The map takes a finite point measure and selection randomness as inputs; the kernel gives the selection distribution.Source: selection operator is a jointly… · p. 5
VI. MT SAMPLING OPERATOR
- The stochastic MT sampling operator composes stochastic crossing, timing, and selection operators to produce recorded crossings.For a fixed signal path, the MT sampling kernel gives the distribution of recorded crossings.Source: §VI · p. 5
- For a deterministic input signal, the sampling distribution reduces to the deterministic-signal case.The recorded-crossings measure is defined for a random signal independent of sampler randomness, with admissibility ensuring the defining domain has probability one.Source: §VI · p. 5
VII. FIRST RECORDED CROSSING-TIME DISTRIBUTIONS UNDER NONHOMOGENEOUS POISSON ARRIVALS
For nonhomogeneous Poisson photon arrivals and deterministic MT sampling, the paper develops a signal model and admissible sampler. It establishes conditions for the modeled paths and gives the framework needed to analyze recorded crossings.
VII. FIRST RECORDED CROSSING-TIME DISTRIBUTIONS UNDER NONHOMOGENEOUS POISSON ARRIVALS
- For this model, the analysis derives the exact first recorded crossing-time CDF and validates numerical predictions against Monte Carlo simulations.A one-dimensional Fourier representation is obtained under additional regularity and local monotonicity conditions.Source: §VII · p. 5
1) Signal Model and Sampler:
- The Poisson-arrival signal uses a bounded, right-continuous single-photon response with finite left limits and finitely many analytic pieces.The deterministic sampler uses nominal threshold paths and a fixed positive time-quantization step.Source: 1) Signal Model and Sampler · p. 5
- The sampler's timing and selection maps are measurable; quantization preserves count, and selection commutes with quantization because it depends only on direction marks.Selection does not increase the measure, while quantization changes only crossing times.Source: 1) Signal Model and Sampler · p. 5
- The constructed signal paths belong to an admissible domain, and the Poisson signal lies in that domain almost surely.The path-space argument partitions time at shifted breakpoints and verifies finitely many crossings; the resulting path family is Borel.Source: 1) Signal Model and Sampler · p. 6
2) Exact First Recorded Crossing-Time Distributions:
The section defines the first recorded crossing time at a selected threshold within a recorded-time interval and derives its exact CDF by conditioning on photon count. The count-truncated CDF has a uniform-in-time error bound, while additional conditions yield an endpoint-based Fourier representation.
2) Exact First Recorded Crossing-Time Distributions:
- The first recorded crossing time is the earliest selected-threshold crossing inside the open recorded-time interval, with boundary crossings excluded.If no such crossing is recorded, the value is infinity; its probability of being finite is the recording probability.Source: 2) Exact First Recorded Crossing-Time… · p. 6
- Conditioning on total photon count gives the exact first recorded crossing-time CDF, with a truncation error bound uniform in time.The formula sums contributions over photon counts and arrival configurations for which the first recorded crossing occurs by time t.Source: 2) Exact First Recorded Crossing-Time… · p. 6
- Under additional regularity and local monotonicity conditions, the CDF reduces to signal-threshold exceedance probabilities at the corresponding analog-time interval endpoints.The monotonicity condition requires a fixed direction in which the shifted response is continuous and nondecreasing over the effective window.Source: 2) Exact First Recorded Crossing-Time… · p. 6
2) At each fixed time, the probability that the signal equals the threshold is zero:
For the rising-edge scintillation-pulse case, the section establishes that the signal equals the threshold with probability zero at each fixed time and gives conditions for evaluating the first-crossing CDF.
2) At each fixed time, the probability that the signal equals the threshold is zero:
- On the effective analog window, continuity and local monotonicity imply at most one analog crossing at the threshold, with the specified direction mark.The result uses condition (56) to exclude threshold-valued intervals and crossings at the endpoints.Source: 2) At each fixed time, the probability… · p. 7
- At each fixed time, the signal has no probability mass at the threshold; its only possible atom is at zero.The argument conditions on photon-arrival counts and uses finite level sets of the single-photon response.Source: 2) At each fixed time, the probability… · p. 7
- The first recorded crossing-time CDF is zero when its corresponding analog time interval is empty; otherwise, Theorem 1 yields its Fourier representation.The Fourier integrand is defined at zero by continuous extension.Source: 2) At each fixed time, the probability… · pp. 6–7
B. Numerical Experiments
The numerical first-crossing CDF closely matches independent Monte Carlo results, and the calculated crossing-time standard deviations also agree with simulation.
B. Numerical Experiments
- The CDF validation used τd/τr = 4, τs/τr = 2, and T = [0, 2τr], with settings satisfying the effective analog time-window condition.Case (b) checks exclusion of recorded crossings at or before the lower endpoint of I.Source: B. Numerical Experiments · p. 7
- The characteristic-function calculation was checked for numerical convergence by refining the Fourier cutoff, quadrature resolution, and crossing-time tolerance.The characteristic function was evaluated using (57).Source: B. Numerical Experiments · p. 7
- 9.24 × 10^-4 maximum absolute CDF discrepancy across three settings falls within the 95% simultaneous Monte Carlo band of half-width 1.55 × 10^-3.Each setting used 10^6 independent photon-arrival realizations.Source: B. Numerical Experiments · p. 7
- 0.04839τr to 0.11257τr is the standard-deviation increase across Vn/A = 1, 6, 12, with independent Monte Carlo agreement at those thresholds.The minimum recording probability was approximately 0.9999989.Source: B. Numerical Experiments · p. 7
VIII. DISCUSSION
The paper defines a framework for mapping signal distributions to recorded crossings and derives and validates a first-crossing CDF for a specified Poisson model and deterministic sampler. Its discussion also identifies conditions affecting threshold comparisons and directions for further work.
A. Implications of the Framework and Crossing-Time Distributions
- Separating timing from selection distinguishes changes in crossing time coordinates from reductions in crossing counts.The framework provides a common description for parameter estimation, waveform reconstruction, and sampler comparisons.Source: A. Implications of the Framework and… · pp. 7–8
- For admissible signal paths, the exact CDF has a truncation error bound uniform in time, including for nonmonotone paths.Under additional local monotonicity conditions, the CDF reduces to endpoint threshold-exceedance probabilities, enabling the Fourier representation.Source: A. Implications of the Framework and… · p. 8
- Lower thresholds give smaller first-crossing-time standard deviations over the tested range when paths are nondecreasing on the effective window and recording probabilities are near one.At low signal-to-noise ratios, excluding late high-threshold crossings outside the recorded-time interval can reduce the standard deviation; Fig. 3(b) plots standard deviation versus threshold.Source: A. Implications of the Framework and… · p. 8, Figure 3 · p. 8
B. Scope and Further Research
- The framework assumes sampling-stage randomness is mutually independent and independent of the signal; shared randomness or additional signal information requires extending the operator inputs and joint probability model.Source: B. Scope and Further Research · pp. 8–9
- Further work should establish threshold requirements for identifiability and necessary and sufficient conditions for exact recovery of individual signal paths.The discussion also calls for joint-crossing analyses and optimization of thresholds together with estimators or reconstructors.Source: B. Scope and Further Research · p. 9
IX. CONCLUSION
- The framework maps signal distributions to recorded-crossing distributions using sampling operators and Markov kernels, while finite point measures preserve crossing counts and multiplicity after time quantization.It distinguishes model mismatch, signal noise, threshold mismatch, and threshold noise.Source: §IX · p. 9
- For nonhomogeneous Poisson photon arrivals and deterministic MT sampling, the exact first recorded crossing-time CDF has a one-dimensional Fourier representation under additional regularity and local monotonicity conditions, and numerical results agree with independent Monte Carlo simulations.The CDF is for a specified threshold within a recorded-time interval.Source: §IX · p. 9