Bounds in the capacity of a Poisson channel with spectral restrictions

An upper bound is derived on the capacity of a Poisson channel which 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 to upper bound the capacity of a Poisson channel under a second-moment-bandwidth constraint, a case for which simple, yet asymptotically tight, lower bounds are also presented