# Equalization Methods for Out-of-Band Nonlinearity Mitigation in Fiber-Optic Communications
## 4 Equalization Algorithms
### Turbo Equalization  
As mentioned earlier, all of the equalization algorithms that we employed are decision-directed, meaning that they rely on the estimation of the transmitted symbols. In most of the works dealing with equalizing NLIN, this estimation was based on a simple minimum-distance decision regarding symbol values. A significant improvement to the algorithm performance can be achieved by combining the process of equalization with forward error correction (FEC) coding--a procedure that is usually referred to as _turbo-equalization_[38; 39]. Indeed, including a decoder in the equalization loop results in more accurate symbol estimation, and thus better equalization performance. For example, in [20; 21; 35], a maximum-a posteriori (MAP) was used to provide improved symbol estimation for an LMS equalizer and a window-averaging equalizer, resulting in improved detection. A low-density parity check (LDPC) decoder was used to improve nonlinear phase noise mitigation in [15], an LDPC code and an RLS equalizer were used in [40], while [19] used an LDPC code together with a Kalman equalizer. The structure of a generic Turbo-equalizer is shown in Figure 5. The main idea is that the symbol estimation is performed using the post-FEC decoded bits. These include much fewer errors than the pre-FEC bits, and so the symbol estimation is much better. The way in which the symbols are estimated  
Figure 4: SNR improvement of different equalization algorithms. (**a**) SNR vs. power for each algorithm. The curves represent results obtained using the Virtual Lab tool [8], whereas the dots correspond to split-step simulations. (**b**) Peak SNR gain vs. computational cost, obtained by measuring the time that was needed to process a single set of symbols for each algorithm. The SNR gain is defined as the difference between the peak SNR of the unequalized case to those of the different algorithms. varies between different decoding schemes. For example, one may want to use soft symbol estimates instead of hard decisions, as done in [15; 19], or include bit interleaving. The decoding-equalization loop is usually iterated several times, until the algorithm converges. In order to compare the performance of the different equalizer, we employed a Turbo-equalization scheme based on soft-decision LDPC encoder, as done in [19]. The code rate was \(R=0.85\), corresponding to a FEC overhead of 15%, while all other system parameters where kept as in the previous section. The rate was chosen so that the post-FEC BER of the unequalized signal was significant (\(5\times 10^{-3}\)), which enabled to observe the differences between equalizers. The post-FEC BER is shown in Figure 6, as a function of Turbo-equalization iterations, using the same equalizers discusses above. It is evident that each equalizer converges to a different error-floor. As in the case of SNR gain estimation, the best performance is achieved by the 3-tap Kalman filter, where the BER was 10 times lower than that of the unequalized case. For the reader's convenience, Table 1 summarizes the gain and cost of the various equalization schemes. The results shown are for the conditions studied at Sections 4.2 and 4.3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Title 1** & **Normalized Cost** & **SNR Gain** &
\begin{tabular}{c} **Minimal BER** \\ **(with Turbo equ.)** \\ \end{tabular} \\ \hline \(1\) tap LMS & 1 & 0.2 dB & \(3.4\times 10^{-3}\) \\ \(1\) tap RLS & 0.88 & 0.19 dB & \(3.5\times 10^{-3}\) \\ \(1\) tap Window averaging & 1.88 & 0.45 dB & \(2.4\times 10^{-3}\) \\ \(3\) tap Window averaging & 5.5 & 0.9 dB & \(1\times 10^{-3}\) \\ \(1\) tap Kalman filter & 22 & 0.7 dB & \(1.4\times 10^{-3}\) \\ \(3\) tap Kalman filter & 45 & 1.6 dB & \(2.6\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of equalization algorithms performance. Note that the minimal BER for the unequalized case is \(4.5\times 10^{-3}\). Figure 5: A block-diagram description of a Turbo equalizer.