Joint source-channel coding of one random variable over the Poisson channel

We study the transmission of a single random variable across the Poisson channel, which takes a continuous-time waveform {λ_t : 0 ≤ λ_t ≤ T} as an input, where 0 ≤ λ_t ≤ A, for all 0 ≤ t ≤ T. The output of the channel is a non-homogeneous Poisson arrival process with rate λ^T. We explore the class of schemes that are optimal in the distortion exponent sense under mean squared loss. We determine a family of optimal encoders for this channel which achieves the minimum mean squared error. In addition, we characterize the distortion exponent for `separation´ based schemes, and also provide an upper bound for the maximal distortion exponent across all joint source channel strategies under this setting.