[Paper Review] Statistical Non-linear Model, Achievable Rates and Signal Detection for Photon-level Photomultiplier Receiver

Zhimeng Jiang, Chen Gong+1 more|arXiv (Cornell University)|Mar 30, 2018

Quantum Information and Cryptography25 references3 citations

TL;DR

This paper proposes a statistical non-linear model for photon-level photomultiplier tube (PMT) receivers in optical wireless communications, accounting for multi-stage amplification and Poisson photon statistics. It derives optimal and suboptimal duty cycles under power constraints, introduces a threshold-based classifier to distinguish PMT operating regimes, and evaluates mean power detection and photon counting detection performance across finite sampling rates, showing that sampling at intervals equivalent to dead time yields near-optimal performance.

ABSTRACT

We characterize the practical receiver in a wide range of signal intensity for optical wireless communication, from discrete pulse regime to continuous waveform regime. We first propose a statistical non-linear model based on the photomultiplier tube (PMT) multi-stage amplification and Poisson channel, and then derive the optimal and tractable suboptimal duty cycle with peak-power and average-power constraints for on-off key (OOK) modulation in linear regime. Subsequently, a threshold-based classifier is proposed to distinguish the PMT working regimes based on the non-linear model. Moreover, we derive the approximate performance of mean power detection with infinite sampling rate and finite over-sampling rate in the linear regime based on small dead time and central-limit theorem. We also fomulate a signal model in the non-linar regime. Furthermore, the performance of mean power detection and photon counting detection with maximum likelihood (ML) detection for different sampling rates is evaluated from both theoretical and numerical perspectives. We can conclude that the sample interval equivalent to dead time is a good choice, and lower sampling rate would significantly degrade the performance.

Motivation & Objective

• To develop a practical statistical non-linear model for PMT receivers that captures multi-stage amplification and Poisson photon statistics across a wide dynamic range of signal intensities.
• To address the lack of models incorporating internal PMT characteristics such as single photoelectron spectrum and non-linearity due to space charge effects.
• To optimize duty cycle for on-off keying (OOK) modulation under peak and average power constraints in the linear regime.
• To classify PMT operating regimes—discrete pulse, continuous waveform, and transition—using a threshold-based classifier derived from the non-linear model.
• To evaluate and compare the performance of mean power detection (MPD) and photon counting detection (PCD) under maximum likelihood (ML) criterion across different sampling rates.

Proposed method

• Develops a statistical non-linear PMT model based on Poisson-distributed photon arrivals and asymmetric shot noise, incorporating multi-stage amplification and dead time effects.
• Derives the moment generating function (MGF) of the PMT output under the non-linear model, enabling performance analysis in the linear regime via central limit theorem and short dead time approximation.
• Proposes a threshold-based classifier that distinguishes three PMT working regimes using the mean and variance of output samples as functions of input optical power.
• Optimizes duty cycle for OOK modulation using analytical derivations under peak and average power constraints, with a tractable suboptimal solution derived for practical implementation.
• Evaluates detection error probability for MPD with both infinite and finite sampling rates, using the central limit theorem and short dead time assumption.
• Employs a discrete optimization framework using Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to fit the non-linear function from experimental data on mean and variance of output samples.

Experimental results

Research questions

• RQ1 How can a statistical non-linear model be constructed to accurately represent PMT behavior across discrete pulse, continuous waveform, and transition regimes?
• RQ2 What is the optimal and suboptimal duty cycle for OOK modulation under peak and average power constraints in the PMT linear regime?
• RQ3 How can the PMT operating regime (pulse, waveform, transition) be reliably classified based on statistical characteristics of the output?
• RQ4 What is the theoretical performance of mean power detection (MPD) with infinite and finite sampling rates in the linear regime, under short dead time and central limit theorem assumptions?
• RQ5 How does sampling rate affect the error probability of MPD and PCD under maximum likelihood detection, and what sampling interval yields near-optimal performance?

Key findings

• The sample interval equivalent to the PMT dead time provides near-optimal performance for mean power detection, significantly outperforming lower sampling rates.
• Lower sampling rates lead to a significant performance degradation in both mean power detection and photon counting detection, as confirmed by theoretical and numerical evaluations.
• The proposed threshold-based classifier effectively distinguishes the three PMT operating regimes using measured mean and variance of output samples.
• Theoretical error probability expressions for MPD with infinite and finite sampling rates are derived and validated through simulations, showing strong agreement with numerical results.
• The non-linear function modeling the PMT output is successfully fitted using experimental data via a BFGS-based discrete optimization method, enabling accurate performance prediction.
• The asymmetric shot noise model outperforms the Gaussian model in capturing real PMT behavior, particularly in the non-linear regime due to space charge effects at the last dynode-anode interface.