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Identification Capacity of the Discrete-Time Poisson Channel

Wafa Labidi, Christian Deppe, Holger Boche

TL;DR

The paper addresses event-driven molecular communications where Shannon capacity may be inadequate and studies the randomized identification (RI) capacity of the discrete-time Poisson channel (DTPC) under peak and average power constraints. Using information-spectrum methods, it proves a strong converse and shows that the RI capacity equals the channel's Shannon capacity, i.e., $C_{ID}(W,P_{max},P_{avg}) = C(W,P_{max},P_{avg})$, with a corollary for the finite-state (i.i.d.) case via averaging over states. It also proves that the capacity-achieving input distribution is discrete with a finite number of mass points and that the strong converse holds, extending the results to state-dependent DTPCs through the averaged channel $W^a$. These results justify randomized identification as a viable performance metric for DTPC-based event detection and provide exact capacity formulas, enabling design insights for RI codebooks that scale as $N = 2^{2^{nR}}$. The work thus lays a theoretical foundation for RI in molecular communications and points to future work on deterministic identification with randomness resources and finite-blocklength encoding analyses.

Abstract

Numerous applications in the field of molecular communications (MC) such as healthcare systems are often event-driven. The conventional Shannon capacity may not be the appropriate metric for assessing performance in such cases. We propose the identification (ID) capacity as an alternative metric. Particularly, we consider randomized identification (RI) over the discrete-time Poisson channel (DTPC), which is typically used as a model for MC systems that utilize molecule-counting receivers. In the ID paradigm, the receiver's focus is not on decoding the message sent. However, he wants to determine whether a message of particular significance to him has been sent or not. In contrast to Shannon transmission codes, the size of ID codes for a Discrete Memoryless Channel (DMC) grows doubly exponentially fast with the blocklength, if randomized encoding is used. In this paper, we derive the capacity formula for RI over the DTPC subject to some peak and average power constraints. Furthermore, we analyze the case of state-dependent DTPC.

Identification Capacity of the Discrete-Time Poisson Channel

TL;DR

The paper addresses event-driven molecular communications where Shannon capacity may be inadequate and studies the randomized identification (RI) capacity of the discrete-time Poisson channel (DTPC) under peak and average power constraints. Using information-spectrum methods, it proves a strong converse and shows that the RI capacity equals the channel's Shannon capacity, i.e., , with a corollary for the finite-state (i.i.d.) case via averaging over states. It also proves that the capacity-achieving input distribution is discrete with a finite number of mass points and that the strong converse holds, extending the results to state-dependent DTPCs through the averaged channel . These results justify randomized identification as a viable performance metric for DTPC-based event detection and provide exact capacity formulas, enabling design insights for RI codebooks that scale as . The work thus lays a theoretical foundation for RI in molecular communications and points to future work on deterministic identification with randomness resources and finite-blocklength encoding analyses.

Abstract

Numerous applications in the field of molecular communications (MC) such as healthcare systems are often event-driven. The conventional Shannon capacity may not be the appropriate metric for assessing performance in such cases. We propose the identification (ID) capacity as an alternative metric. Particularly, we consider randomized identification (RI) over the discrete-time Poisson channel (DTPC), which is typically used as a model for MC systems that utilize molecule-counting receivers. In the ID paradigm, the receiver's focus is not on decoding the message sent. However, he wants to determine whether a message of particular significance to him has been sent or not. In contrast to Shannon transmission codes, the size of ID codes for a Discrete Memoryless Channel (DMC) grows doubly exponentially fast with the blocklength, if randomized encoding is used. In this paper, we derive the capacity formula for RI over the DTPC subject to some peak and average power constraints. Furthermore, we analyze the case of state-dependent DTPC.
Paper Structure (4 sections, 3 theorems, 48 equations, 1 figure)

This paper contains 4 sections, 3 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Models and Main Results
  3. Proof of Theorem \ref{['theorem:randomizedID']}
  4. Conclusions

Key Result

Theorem 3

The RI capacity $C_{ID}(W,P_{\text{max}},P_{\text{avg}})$ of the channel $W$ under peak and average power constraints $P_{\text{max}}$ and $P_{\text{avg}}$, respectively, is given by

Figures (1)

  • Figure 1: Discrete-time memoryless Poisson channel with random state

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Definition 5
  • Corollary 6
  • proof
  • Definition 7
  • Definition 8
  • Theorem 9: capacityAchievingDistribution