I. Introduction
Any optical signal of sufficiently large power propagating in a single-mode fiber interacts with optical noise through a four-wave mixing (FWM) process; on that account, the noise experiences a frequency-dependent gain. This phenomenon is known as parametric gain (PG), and in the anomalous propagation regime it may cause modulation instability [1]. The PG process can be positively exploited to build parametric amplifiers and wavelength converters in short highly nonlinear fibers [2], [3]. However, for optical transmission systems, it is a potential source of system degradation because of the signal interaction with the amplified spontaneous emission (ASE) noise. The problem has been traditionally tackled through a small-signal approach to the nonlinear Schrödinger equation (NLSE) [1], [4], [5]. Whenever the received optical signal-to-noise ratio (OSNR) is large enough, the signal field is little affected by noise during propagation so that the propagated signal can be evaluated without ASE. The small-signal model consists of a linearization of the NLSE around the known noiseless signal field, which leads to two linear differential equations relating the real (in-phase) and imaginary (quadrature) components of the ASE noise [5]. Such a system can be solved exactly in terms of Bessel functions when the signal is a continuous wave (CW) [4], [5]. In the small-signal model, the initially white Gaussian ASE thus remains Gaussian along the propagation and changes its power spectral density (PSD), which becomes frequency dependent, i.e., colored. Starting from the small-signal model, Bosco et al. [6] evaluated the impact of PG on the bit error rate (BER) of an ON–OFF keying (OOK)-modulated signal by an exact analysis of the optical quadratic detection process [7] (thereby avoiding the usual Gaussian approximation for BER evaluation) and by calculating the decision variable statistics by expanding the PG-colored ASE on a signal-dependent Karhunen–Loéve (KL) basis. Unfortunately, such a model ignores the intersymbol interference (ISI) that inevitably affects a modulated (non-CW) signal. An improved model, which accounts for ISI while still adopting the small-signal PG model, was introduced by Holzlöhner et al. [8], where they were able to evaluate the BER through a covariance matrix method. However, as pointed out in [8], the small-signal model may fail when the noise is far from small with respect to the signal, as for instance in the tails of the noise probability density function (pdf) of the decision variable, especially when the system is operated at small OSNR. Another situation in which the small-signal model is clearly inappropriate occurs in systems operating close to the zero-dispersion wavelength [9].