The Lattice-Input Discrete-Time Poisson Channel

We consider the lattice-input discrete-time Poisson (LIDTP) channel, which is a discrete-time Poisson (DTP) channel where the input is constrained to a lattice $\alpha \mathbb{Z}^{\geq 0}$, for some fixed parameter $\alpha> 0$, and where $\mathbb{Z}^{\geq 0}$ denotes the set of non-negative integers. The LIDTP channel arises in the analysis of some channel models proposed in the DNA storage literature, which motivated our study. We show that the difference in capacity between the LIDTP channel and the standard DTP channel is upper-bounded by $K(\alpha)+R(\mathcal{E})+o(1)$, where $\mathcal{E}$ is a mean power constraint on the input and the asymptotics are with respect to the ratio $\mathcal{E} / \alpha$ going to infinity. The terms $K(\alpha)$ and $R(\mathcal{E})$ are given explicitly and tend to 0 with $\alpha \rightarrow 0$ and $\mathcal{E} \rightarrow \infty$, as $O\left(\alpha \log \frac{1}{\alpha}\right)$ and $O\left(\mathcal{E}^{-1 / 2}\right)$, respectively. Thus, for fixed $\alpha$ and $\mathcal{E} \rightarrow \infty$, the term $K(\alpha)$ bounds the gap in capacity incurred by enforcing a regular discrete input, and, as $\alpha$ gets ever smaller, the asymptotic capacity of the LIDTP with $\mathcal{E} \rightarrow \infty$ matches that of the DTP. We also show a non-asymptotic bound on the difference between the capacities of the LIDTP and DTP channels, of the form $K(\alpha)+R^{\prime}(\mathcal{E})+\frac{1}{2} \log 2 \pi e$. The term $R^{\prime}(\mathcal{E})$ is also given explicitly and vanishes as $O(1 / \mathcal{E})$ when $\mathcal{E} \rightarrow \infty$.