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One-Cold Poisson Channel: A Simple Continuous-Time Channel with Zero Dispersion

Cheuk Ting Li

TL;DR

The paper introduces the One-Cold Poisson Channel (OCPC), a continuous-time Poisson channel in which the transmitter attenuates one of multiple bands at a time, achieving a simple, infinitely divisible information carrier in the perfect case ($ abla=$, $ abla=0$) with capacity $C=1$ nat per unit time and dispersion $V=0$. It derives exact capacity, information spectrum, and dispersion expressions for the general OCPC, provides a closed-form non-asymptotic error probability for the perfect OCPC, and shows absence of dispersion eliminates the usual $O(^{1/2}T)$-type penalties; it also develops non-asymptotic coding results under sufficient diversity and a perfect-feedback, one-col d tree framework that links to prefix codes and Huffman coding. Additionally, the paper analyzes channel simulation for OCPC, giving bounds on the required shared randomness and showing how to convert between OCPC and bits with negligible overhead in the perfect case, and provides general bounds with Gaussian-approximation techniques for finite diversity. These results establish OCPC as a simple, foundational channel primitive for non-asymptotic coding, information-s spectrum analysis, and channel simulation with potential optical and passive communication applications.

Abstract

We introduce the one-cold Poisson channel (OCPC), where the transmitter chooses one of several frequency bands to attenuate at a time. In particular, the perfect OCPC, where the number of bands is unlimited, is an extremely simple continuous-time memoryless channel. It has a capacity 1, zero channel dispersion, and an information spectrum being the degenerate distribution at 1. It is the only known nontrivial (discrete or continuous-time) memoryless channel with a closed-form formula for its optimal non-asymptotic error probability, making it the simplest channel in this sense. A potential application is optical communication with a tunable band rejection filter. Due to its simplicity, we may use it as a basic currency of information that is infinitely divisible, as an alternative to bits which are not infinitely divisible. OCPC with perfect feedback gives a generalization of prefix codes. We also study non-asymptotic coding and channel simulation results for the general OCPC.

One-Cold Poisson Channel: A Simple Continuous-Time Channel with Zero Dispersion

TL;DR

The paper introduces the One-Cold Poisson Channel (OCPC), a continuous-time Poisson channel in which the transmitter attenuates one of multiple bands at a time, achieving a simple, infinitely divisible information carrier in the perfect case (, ) with capacity nat per unit time and dispersion . It derives exact capacity, information spectrum, and dispersion expressions for the general OCPC, provides a closed-form non-asymptotic error probability for the perfect OCPC, and shows absence of dispersion eliminates the usual -type penalties; it also develops non-asymptotic coding results under sufficient diversity and a perfect-feedback, one-col d tree framework that links to prefix codes and Huffman coding. Additionally, the paper analyzes channel simulation for OCPC, giving bounds on the required shared randomness and showing how to convert between OCPC and bits with negligible overhead in the perfect case, and provides general bounds with Gaussian-approximation techniques for finite diversity. These results establish OCPC as a simple, foundational channel primitive for non-asymptotic coding, information-s spectrum analysis, and channel simulation with potential optical and passive communication applications.

Abstract

We introduce the one-cold Poisson channel (OCPC), where the transmitter chooses one of several frequency bands to attenuate at a time. In particular, the perfect OCPC, where the number of bands is unlimited, is an extremely simple continuous-time memoryless channel. It has a capacity 1, zero channel dispersion, and an information spectrum being the degenerate distribution at 1. It is the only known nontrivial (discrete or continuous-time) memoryless channel with a closed-form formula for its optimal non-asymptotic error probability, making it the simplest channel in this sense. A potential application is optical communication with a tunable band rejection filter. Due to its simplicity, we may use it as a basic currency of information that is infinitely divisible, as an alternative to bits which are not infinitely divisible. OCPC with perfect feedback gives a generalization of prefix codes. We also study non-asymptotic coding and channel simulation results for the general OCPC.
Paper Structure (6 sections, 6 theorems, 26 equations, 2 figures)

This paper contains 6 sections, 6 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The One-Cold Poisson Channel
  3. Optimal Non-Asymptotic Coding with Sufficient Diversity
  4. Perfect Feedback and One-Cold Tree
  5. Channel Simulation for OCPC
  6. Acknowledgement

Key Result

Theorem 3

The capacity of the $(\mathsf{L},\alpha)$-OCPC is An information spectrum is the distribution of where $Z_{1}\sim\mathrm{Poi}(\alpha)$ is independent of $Z_{2}\sim\mathrm{Poi}(\mathsf{L}-1)$, and a channel dispersion is In particular, for $\mathsf{L}=\infty$, the capacity is $\alpha\ln(\alpha/e)+1$, an information spectrum is the distribution of $Z_{1}\ln\alpha-\alpha+1$ (degenerates to $1$ if

Figures (2)

  • Figure 1: One-cold Poisson channel with $\mathsf{L}=4$ bands and $\alpha=0$, where the encoder blocks one band. Detection events at each band form a Poisson process with rate $1$, except the blocked band which has rate $0$.
  • Figure 2: Two $3$-ary one-cold trees for $\mathcal{S}=\{1,2,3\}$ (assume $P_{S}(1)$ is the largest). The first tree corresponds to sending $t\mapsto s$ initially, which takes an expected $1/2$ unit time to eliminate one value of $s$, and $1$ unit time to eliminate another value, resulting in $3/2$ unit time. The second tree corresponds to sending $t\mapsto\min\{s,2\}$ initially (grouping $2,3$ together), using an expected $2-P_{S}(1)$ unit time in total. The optimal tree is one of these two trees.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Corollary 8
  • Remark 9
  • Definition 10
  • ...and 4 more