## 2 Proposed Equalization Algorithms  
### Decision Feedback Equalizer  
A decision feedback equalizer (DFE) is an equalizer that offers superior performance compared to a linear equalizer. The use of DFE can reduce noise enhancement and improve the forward linear filter's ability to handle ISI and co-channel interference. A DFE usually has more taps than an FFE, as it requires taps in both the feedforward and feedback filters. The DFE's feedback portion takes into consideration earlier symbol decisions, which increase complexity and tap into the entire structure, but it is particularly beneficial when addressing significant channel distortion.  
The DFE structure, as mentioned above, consists of a feedback filter and a decision function (Figure 1). The feedforward filter processes the current received symbols, passing them through the decision function. The received symbols are then refined by passing through the feedback filter, which utilizes previously identified symbols that ideally have minimal or no noise, allowing for precise estimation and cancellation of ISI without enhancing the noise, provided that the previous decisions are correct. If the decisions are incorrect, the error propagates to the next decisions. One method to reduce error propagation is through decision-directed adaptation, in which the DFE updates the feedback filter solely when the decisions are accurate [34].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Literature** & **Frequency** & **Maximum Fiber** & **Modulation Format** & **Algorithm** & **Computational** \\
**Review** & **(GHz)** & **Length (km)** & & & **Complexity** \\ \hline
[13] & 28 & 10 & OFDM 16-QAM &
\begin{tabular}{c} Voltera and neural \\ network-based equalizers \\ \end{tabular} & High \\ \hline
[14] & 60 & 20 & \
\begin{tabular}{c} OFDM 16-QAM and \\ 64-QAM \\ \end{tabular} &
\begin{tabular}{c} Complex valued \\ (ANN-NLE) \\ \end{tabular} & High \\ \hline
[16] & 28 & 5–35 & \
\begin{tabular}{c} 16-PSK, 16-QAM and \\ 64-QAM \\ \end{tabular} &
\begin{tabular}{c} Built-in DSP unit in \\ simulation software \\ \end{tabular} & Low \\ \hline
[18] & 60 & 10 & BPSK & ANN-NLE & High \\ \hline
[19] & 60 & 20 & 2-PAM &
\begin{tabular}{c} Convolutional neural \\ network (CNN) and binary \\ convolutional neural \\ network (BCNN)-based \\ decision schemes \\ \end{tabular} & High \\ \hline
[28] & 60 & 25 & QPSK and 16-QAM & LMS Algorithm & Low \\ \hline
[29] & 60 & 100 & QPSK and 16-QAM &
\begin{tabular}{c} LMS, CMA, and AMF \\ Algorithms \\ \end{tabular} & Low \\ \hline
**[This work]** & **60** & **0–120** & **QPSK, 16-QAM and 128-QAM** & **DFE, MLD, and RDE Algorithms** & **Moderate** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Literature review comparisons over specific parameters.  
One notable advantage of the DFE over linear equalizers is its ability to remove ISI without enhancing noise, because a linear equalizer handles noise and signal equally, which might result in the amplification of noise [35; 36; 37]. The following equations describe the DFE [38]:  
\[\mathrm{y}[\mathrm{n}]=\sum\
olimits_{k=-N_{f+1}}^{0}f_{\mathrm{k}}(n)\cdot x[n -k]+\sum\
olimits_{k=1}^{Nb}\mathrm{b}_{\mathrm{k}}(n)\cdot\hat{s}[n-k] \tag{1}\]  
\[\mathrm{t}(n)=f(\mathrm{y}(n)) \tag{2}\]  
\[\mathrm{e}(n)=\mathrm{y}(n)-\mathrm{t}(n) \tag{3}\]  
where \(\mathrm{y}[\mathrm{n}]\) is the equalized output signal of the nth symbol.  
\(x[n-k]\): the equalizer input sequence.  
\(\mathrm{f}_{\mathrm{k}}\): the coefficients for the feedforward filter.  
\(\mathrm{b}_{\mathrm{k}}\): the coefficients for the feedback filter.  
\(N_{f}\): the order length of the feedforward filter.  
\(N_{b}\): the order length of the feedback filter.  
\(\hat{s}[n-k]\): the previously detected symbols.  
\(\mathrm{t}(n)\): the equalizer decision sequence.  
\(\mathrm{f}[n]\): the decision function.  
\(\mathrm{e}(n)\): the error signal.