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Capacity Bounds for the Poisson-Repeat Channel

Mohammad Kazemi, Tolga M. Duman

TL;DR

Borders on the capacity of Poisson-repeat channels for which each input bit is independently repeated according to a Poisson distribution are developed and a way of obtaining capacity lower bounds is described using information rates of the auxiliary channel and the entropy rate of the provided side information.

Abstract

We develop bounds on the capacity of Poisson-repeat channels (PRCs) for which each input bit is independently repeated according to a Poisson distribution. The upper bounds are obtained by considering an auxiliary channel where the output lengths corresponding to input blocks of a given length are provided as side information at the receiver. Numerical results show that the resulting upper bounds are significantly tighter than the best known one for a large range of the PRC parameter $λ$ (specifically, for $λ\ge 0.35$). We also describe a way of obtaining capacity lower bounds using information rates of the auxiliary channel and the entropy rate of the provided side information.

Capacity Bounds for the Poisson-Repeat Channel

TL;DR

Borders on the capacity of Poisson-repeat channels for which each input bit is independently repeated according to a Poisson distribution are developed and a way of obtaining capacity lower bounds is described using information rates of the auxiliary channel and the entropy rate of the provided side information.

Abstract

We develop bounds on the capacity of Poisson-repeat channels (PRCs) for which each input bit is independently repeated according to a Poisson distribution. The upper bounds are obtained by considering an auxiliary channel where the output lengths corresponding to input blocks of a given length are provided as side information at the receiver. Numerical results show that the resulting upper bounds are significantly tighter than the best known one for a large range of the PRC parameter (specifically, for ). We also describe a way of obtaining capacity lower bounds using information rates of the auxiliary channel and the entropy rate of the provided side information.
Paper Structure (11 sections, 1 theorem, 13 equations, 3 figures)

This paper contains 11 sections, 1 theorem, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Poisson-Repeat Channel
  3. Genie-Aided Upper Bounds on the PRC Capacity
  4. Bounds on the Capacity of the Auxiliary Channel
  5. An Alternative Explanation
  6. Upper Bound on the PRC Capacity
  7. Bounds with Limitation on the Number of Repetitions of Each Bit
  8. On Lower Bounding on the PRC Capacity
  9. Numerical Results
  10. Conclusion
  11. Acknowledgments

Key Result

Lemma 1

(DMC output partitioning) Let $Ch\left(L\right)$ and $\mathcal{B}_k$'s be a DMC with input length $L$ and the non-overlapping partitions of its outputs (as depicted in Fig. fig0), respectively. We define $Ch\left(L,\mathcal{B}_k\right)$ as $Ch\left(L\right)$ with outputs limited to $\mathcal{B}_k$ ( where $I\left(\cdot;\cdot\right)$ denotes the mutual information, $P\left({\mathcal{B}_k} \right)$

Figures (3)

  • Figure 1: Partitioning of the DMC outputs.
  • Figure 2: Concatenation of the auxiliary channel with another one.
  • Figure 3: Upper bounds for different values of the parameter $\lambda$.

Theorems & Definitions (1)

  • Lemma 1