Bounds on the capacity of a spectrally constrained Poisson channel

An upper bound is derived on the capacity of a Poisson channel that has a stationary input process of a given spectrum and is subjected to peak and average power constraints. The bound is shown to be asymptotically tight with the relaxation of the spectral constraints. Its maximization over a given set of admissible spectra is closely related to an analogous problem in the AWGN regime. The results are used for bounding the capacity of a Poisson channel under a second-moment-bandwidth constraint, as well as the capacity under a strict bandwidth constraint. Asymptotically tight lower bounds on the channel capacity for the above two cases are also presented. The approach for lower bounding the capacity for the latter case yields, as a by-product, improved bounds on the bit-error probability in uncoded amplitude shift keying (and on-off modulation as a special case) operating over a Poisson channel impaired by intersymbol interference