Improved Upper Bounds and Structural Results on the Capacity of the Discrete-Time Poisson Channel

Improved Upper Bounds and Structural Results on the Capacity of the Discrete-Time Poisson Channel. Cheraghchi, M. & Ribeiro, J. IEEE Transactions on Information Theory, 65(7):4052–4068, 2019. Preliminary version in Proceedings of ISIT 2018.

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New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint must only be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be an unbounded set.

@ARTICLE{ref:CR19:poi,
  author =	 {Mahdi Cheraghchi and Jo\~{a}o Ribeiro},
  title =	 {Improved Upper Bounds and Structural Results on the
                  Capacity of the Discrete-Time {Poisson} Channel},
  year =	 2019,
  journal =	 {IEEE Transactions on Information Theory},
  volume =	 65,
  number =	 7,
  pages =	 {4052--4068},
  note =	 {Preliminary version in Proceedings of {ISIT 2018}.},
  doi =		 {10.1109/TIT.2019.2896931},
  url_Link =	 {https://ieeexplore.ieee.org/document/8632953},
  url_Paper =	 {https://arxiv.org/abs/1801.02745},
  abstract =	 {New capacity upper bounds are presented for the
                  discrete-time Poisson channel with no dark current
                  and an average-power constraint. These bounds are a
                  consequence of techniques developed for the
                  seemingly unrelated problem of upper bounding the
                  capacity of binary deletion and repetition
                  channels. Previously, the best known capacity upper
                  bound in the regime where the average-power
                  constraint does not approach zero was due to
                  Martinez (JOSA B, 2007), which is re-derived as a
                  special case of the framework developed in this
                  paper. Furthermore, this framework is carefully
                  instantiated in order to obtain a closed-form bound
                  that improves the result of Martinez
                  everywhere. Finally, capacity-achieving
                  distributions for the discrete-time Poisson channel
                  are studied under an average-power constraint and/or
                  a peak-power constraint and arbitrary dark
                  current. In particular, it is shown that the support
                  of the capacity-achieving distribution under an
                  average-power constraint must only be countably
                  infinite. This settles a conjecture of Shamai (IEE
                  Proceedings I, 1990) in the affirmative. Previously,
                  it was only known that the support must be an
                  unbounded set.  }
}

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