Achievable Cutoff Rates of the Additive Exponential Noise Channels

We derive closed-form expressions for the Gallager error function of the additive exponential noise channels and provide theoretical upper bounds on its maximum over the input probability distribution space (the cutoff rate), thus extending the previous results derived for the additive white Gaussian noise channels. Our theoretical results allow us to design constellations using simple optimization techniques that directly maximize the Gallager error function of the additive exponential noise channels. We perform simulations and show that projected gradient ascent optimization yields constellations competitive to existing ones that are optimized for Shannon information metrics, which require numerical estimation.