Probabilistic Shaping for Nonlinearity Tolerance

Optimizing the input probability distribution of a discrete-time channel is a standard step in the information-theoretic analysis of digital communication systems. Nevertheless, many practical communication systems use transmission based on uniformly and independently distributed symbols drawn from regular constellation sets. The introduction of the probabilistic amplitude shaping architecture has helped to renew interest in using optimized probability distributions, i.e., probabilistic shaping. Traditionally, probabilistic shaping has been employed to reduce the transmit power required for a given information rate over additive noise channels. While this translates into substantive performance gains for optical fiber communication systems, the interaction of shaping and fiber nonlinearity has posed intriguing questions. At first glance, probabilistic shaping seems to exacerbate nonlinear interference noise (NLIN) due to larger higher-order standardized moments. Therefore, the optimization of shaping distributions must differ from those used for linear channels. Secondly, finite-length effects related to the memory of the nonlinear fiber channel have been observed. This suggests that not only the marginal input-symbol distribution should be looked at. In this paper, we provide a tutorial-style discussion of probabilistic shaping for optical fiber communication. Since the distinguishing property of the channel is the signal-dependent NLIN, we speak of probabilistic shaping for nonlinearity tolerance. Our analysis builds on the first-order time-domain perturbation approximation of the nonlinear fiber channel and revisits the notion of linear and nonlinear shaping gain. We largely focus on probabilistic amplitude shaping with popular types of shaping methods, and examine a linear filter model to explain several phenomena associated with probabilistic shaping for fiber nonlinearity, including the interaction between carrier phase recovery and fiber nonlinearity. The concept of shaping via sequence selection is given special consideration, as it inherently optimizes a multi-variate distribution for shaped constellations. We explore how using sign bits for sequence selection offers benefits beyond amplitude shaping, and we introduce a sign-dependent selection metric based on the perturbation model.