# Equalization Methods for Out-of-Band Nonlinearity Mitigation in Fiber-Optic Communications
## 1 Introduction  
The challenge of mitigating nonlinear interference in wavelength division multiplexed (WDM) networks has attracted significant efforts. Receiver-side digital nonlinearity compensation is of particular interest, as it requires no additional optical hardware and can be easily adapted to many cases. Nonlinear interference caused by Kerr nonlinearity is a deterministic process--given the input waveform and fiber link structure, one can predict the effect of nonlinearity at the fiber's output with an accuracy limited only by the nonlinear noise-signal and noise-noise interference that kick in only at very high signal powers and low signal to noise ratios. Furthermore, neglecting nonlinearity involving the noise, the nonlinear distortions can be canceled by digitally back-propagating the signal so as to invert the nonlinearity accumulated along the fiber. The problem is that in realistic WDM systems, the receiver has access only to a small fraction of the overall signal bandwidth. Namely, the nonlinear interference that is experienced by the detected signal and which was induced by signals that are outside the bandwidth of the receiver, cannot be inverted by such means. Broadly, the sources of nonlinearity can be divided into two groups, as illustrated in Figure 1: in-band nonlinearity, which involves interactions of frequency components that are within the receiver's bandwidth, and out-of-band nonlinearity, which is produced by interaction with WDM channels that are outside the received bandwidth. The treatment of these two nonlinearity sources is very different; in-band nonlinearity can be viewed as a deterministic signal-dependent effect, and hence it can be effectively reduced by means of digital back propagation [1], sequence detection [2], or Volterra-series equalization [3]. In contrast, out-of-band nonlinearity must be considered as noise, commonly referred to as \"nonlinearinterference noise\" (NLIN). In most systems, the dominant source of out-of-band NLIN is two-channel interference, namely cases where one out-of-band WDM channel interacts with one in-band channel. Interactions involving three or four channels are also present, but their effects in standard fibers are usually negligible compared to two-channel interactions (Three and four channels effects can be neglected in systems using standard single-mode fiber, when the baud rate is lager than 10 GBaud. In dispersion shifted fibers, or if the symbol rate is significantly lower and the number of channels is higher, these effects may becomes the dominant noise source [4]. As such systems are not common, we do not consider this regime in the paper). It has been well established that two-channel interference is manifested predominantly in the form of linear time-varying inter-symbol interference (ISI) [5; 6]. Therefore, out-of-band NLIN can be mitigated using adaptive equalization techniques. Nonlinearity mitigation using adaptive equalization has been demonstrated in many reported studies. A wide variety of equalization algorithms has been used in different transmission scenarios. The vast majority of these efforts have been directed towards phase and polarization-rotation noise (PPRN) mitigation [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16], which is the most significant NLIN contribution, as we elaborate in what follows. Yet, as discussed bellow, PPRN is only one component of the interference, and it is accounted for by the zeroth-order term in the time-varying ISI model. The contribution of higher order terms is non-negligible and, as we will show, its mitigation has the potential of notably improving performance. Various aspects of this issue have been dealt with in [17; 18; 19; 20; 21]. In this paper, we review the recent advances in the application of equalization techniques for out-of-band NLIN mitigation, discuss the differences and similarities between different approaches, and assess the performance of several equalization schemes.