# Equalization Methods for Out-of-Band Nonlinearity Mitigation in Fiber-Optic Communications
## 3 Making Sense of Correlations  
Two properties of NLIN that make the time-dependent ISI model particularly useful, are the correlation times that characterize the individual ISI coefficients and the correlations between different ISI coefficients. These two properties enable the receiver to evaluate the nonlinearity and attempt to mitigate its effect. Naturally, the more types of correlations the equalization algorithm can exploit, the better it will perform. We briefly discuss the main types and origins of these correlations. 1. _Temporal correlations_ As stated previously the ISI coefficients are time-varying, because of the random data carried by the ICs. However, adjacent symbols of the COI 'feel' the interaction with the IC in a similar way, which implies that the nonlinear perturbations that are imposed upon them are correlated [23]. The correlation times are best evaluated in terms of the number of symbols, and they increase with the fiber dispersion, the baud rate and the channel spacing. In typical WDM systems implemented over standard single-mode fiber (SMF) and using baud-rates of the order of 30 Gbuad, the correlation times are of the order of tens to hundreds of symbols in metro and long haul applications, which in principle allows their tracking and mitigation. Generally, the higher the order of the ISI coefficient, the shorter its correlation time, and the more difficult is its mitigation. Indeed, the longest correlation time is that of the 0-th order ISI element, which accounts for PPRN [28; 30]. 2. _Cross-polarization correlations_ As shown in Equation (4), for dual-polarization transmission, the ISI coefficients become \(2~{}\times~{}2\) random matrices. The elements of the matrices are, however, strongly correlated to each other. For each of the ISI matrices, \(\mathbf{R}_{i}^{(n)}\), it can be shown that the two on-diagonal elements are almost identical, whereas the two off-diagonal elements are complex-conjugates of one another. A detailed explanation of this fact can be found in [8]. The fact that the various terms in the ISI matrices are strongly correlated with each other implies that there is a great advantage in joint-processing of the two polarizations. 3. _Correlations between different ISI orders_ It has been found in [28] that measurable cross-correlations exist only in the case of symmetric ISIorders, namely \(R_{l}^{(n)}\) and \(R_{-l}^{(n)}\). The correlations are fairly strong for small values of \(l\) and reduce as the value of \(l\) increases. Other pairs of ISI matrices are uncorrelated for all practical purposes. 4. _Correlations in multi-subcarrier systems_ Another relevant type of correlations relates to the scenario where multiple data-channels are jointly detected by the same receiver. This characterizes for example the cases of OFDM, or sub-carrier multiplexing. In such cases, out of band NLIN (i.e., NLIN generated by channels that are outside of the receiver's bandwidth) affects the different tributaries in a correlated manner. These correlations follow from the fact the frequency separation between the tributaries is much smaller that the separation between them and the out of band ICs, and hence they are affected by them in a similar way. This type of correlations dominates in the very low baud rate regime (as in OFDM or in symbol-rate optimization schemes [33; 34]), where the temporal correlations diminish. An example for these various correlation types is shown in Figure 2. Multi-subcarrier systems are not discussed further in this review, and remain a subject for future study. The results of Figure 2 were obtained for the 1000 km link discussed in Section 4, and calculated using the method described in [28]. The results for Figure 2d are taken from [34]. Figure 2: Examples of different auto-correlation and cross-correlations characterizing NLIN. **(a)** the auto-correlation functions of the first 4 ISI components.