# Equalization Methods for Out-of-Band Nonlinearity Mitigation in Fiber-Optic Communications
## 2 The Time-Varying ISI Model of NLIN  
We begin the discussion from the \"Time-varying ISI\" model of NLIN, which describes the effect of two-channel nonlinear interference [5; 8; 22]. Our goal is to provide an intuition for the consequences of this model, without repeating the rigorous derivations given in [8]. We label the two interacting channels, co-propagating along the fiber, by \(A\) and \(B\), where channel \(A\) is regarded as the channel of interest (COI) and channel \(B\) is an interfering channel (IC). At the link's output, the channels are de-multiplexed, detected, sampled, and dispersion compensation is applied to them. It is assumed that the receivers are optimized _for the linear channel_, i.e., that, without the effects of nonlinearity, the received symbols are corrupted only by amplified spontaneous emission (ASE) noise. The two channels carry a series of pulses, modulated by the transmitted data. For the sake of simplicity, we consider only single-polarization transmission at this point, whereas dual polarization transmission will be discussed later. We denote the symbols at time slot \(n\) by \(a_{n}\) and \(b_{n}\), for channels \(A\) and \(B\), respectively. Suppose that we want to evaluate the effect of NLIN on a certain data symbol, \(a_{n}\). As shown in [23], under first-order perturbation approximation the nonlinear interactions induced by two-channel interference take the form  
\[\Delta a_{n}=\sum_{h,k,l}iX_{h,k,l}b_{n-h}b_{n-k}^{*}a_{n-l}. \tag{1}\]  
where \(X_{h,k,l}\) is an interaction coefficient. Each term in the summation can be viewed as representing a collision between pulses [24], and it accounts for the changes imposed by the interaction of these particular pulses on the data symbol \(a_{n}\). The interaction coefficients \(X_{h,k,l}\) are determined by many parameters, such as the link length, fiber properties, frequency difference between the channels, and their spectral shapes. The derivation of the expression in Equation (1) and the exact calculation of these interaction coefficients are described in [23]. In the case of a WDM system with more than two channels this process is performed separately for each of the ICs, and their contributions must be summed in order to obtain the overall effect. A simpler description can be obtained by characterizing the collective effect of the nonlinear interactions, rather than the contributions of each individual pulse collision. By rearranging the indices in Equation (1), one obtains  
\[\Delta a_{n}=i\sum_{l}\left(\sum_{h,k}X_{h,k,l}b_{n-h}b_{n-k}^{*}\right)a_{n- l}=i\sum_{l}R_{l}^{(n)}a_{n-l}. \tag{2}\]  
The idea behind this formulation is to separate between the IC's data symbols, which are not available to the receiver, and those of the COI, which need to be estimated. The coefficients \(R_{l}^{(n)}\) encapsulate the effect of the interfering channel. The fact that the IC carries random data causes the coefficients \(R_{l}^{(n)}\) to change over time, as is indicated by the superscript \((n)\). The key strength of this approach is that it is sufficient to characterize the behavior of the \(R_{l}^{(n)}\) coefficients in order to fully describe the effect of nonlinearity. The full derivation of this concept is described in [7; 8; 25]. Using Equation (2), we can express the received signal after dispersion compensation, matched filtering, and standard carrier-phase and frequency recovery, as  
\[s_{n}=a_{n}+\Delta a_{n}+w_{n}=a_{n}+i\sum_{l}R_{l}^{(n)}a_{n-l}+w_{n}, \tag{3}\]  
where \(w_{n}\) is a noise term containing the contribution of ASE, as well as all of the nonlinear effects that are not treated by the model (such as three or four-channel interactions). This equation describes a _linear_, time-varying ISI channel, which has been extensively investigated in the field of RF communication, and is commonly referred to as a \"doubly-selective channel\" [26; 27]. In the case of polarization multiplexed transmission the transmitted data symbols become two-element vectors, \(\tilde{a}_{n}\) and \(\tilde{b}_{n}\).