Probabilistic Shaping Distributions for Optical Communications

Probabilistic shaping is widely employed in local oscillator-based coherent optical systems to improve receiver sensitivity and provide rate adaptation. This widespread adoption has been enabled, in part, by simple closed-form solutions for the optimal input distribution and channel capacity for these standard coherent channels. By contrast, the optimal input distributions and channel capacities for many direct-detection optical channels remain open problems. The lack of non-negative root-Nyquist pulses, signal-dependent noise, and the possible discreteness of the capacity-achieving input distribution have historically prevented standard information-theoretic techniques from obtaining simple closed-form solutions for these channels. In this tutorial, we review a high-rate continuous approximation (HCA) for analytically approximating the optimal input distribution. HCA, which was first developed for source coding, approximates the input constellation by a dense high-dimensional coset code that can be approximated well by a continuum, transforming the problem of computing the optimal input distribution subject to an average-power constraint to a problem of finding a minimum-energy shaping region in a high-dimensional continuous space. HCA yields closed-form continuous approximations to the capacity-achieving input distributions and shaping gains at high signal-to-noise ratio. We explain how enumerating a coset code in natural coordinates enables extension of HCA to direct-detection optical channels, allowing one to obtain closed-form approximations for the capacity-achieving input distributions and shaping gains for a variety of direct-detection systems that detect the intensity or Stokes vector and are limited by thermal or optical amplifier noise. We also discuss the implementation of probabilistic shaping in direct-detection systems.