## 2 Proposed Equalization Algorithms
### Maximum Likelihood Sequence Detection Algorithm  
Maximum likelihood detection (MLD) is a widely recognized and highly effective approach suitable for practical applications that offer optimal detection performance. The idea is essential in signal processing, especially when identifying and interpreting symbols despite interference from noise and distortions. MLD is founded on the fundamental concept of statistical inference. It utilizes the likelihood function of detecting the received symbol based on a given set of potentially transmitted symbols. The objective is to optimize this likelihood to ascertain the most probable transmitted symbol [39; 40; 41]. However, the main drawback of the MLD algorithm is that it needs accurate information on the potentially transmitted symbols to compute the likelihood of each possible transmitted symbol. If the potentially transmitted symbol is not correct, the performance of MLD may deteriorate dramatically, possibly leading to erroneous decisions and detection errors. Also, its complexity increases exponentially for higher-order constellations. A low-complexity MLD algorithm, referred to as simplified MLD, can be employed to approximate the MLD solution, thereby alleviating this issue [42; 43].  
In our work, we employ an optimal maximum likelihood detection algorithm that accurately estimates the transmitted data sequence \(\mathrm{t}=(\mathrm{t}^{(0)}\), \(\mathrm{t}^{(1)}\), \(\mathrm{t}^{(k-1)})^{\mathrm{T}}\). The potentially communicated data symbol vectors are \(\mathrm{t}_{\mathrm{u}}\), \(\mathrm{u}=0\),...\(\mathrm{M}^{\mathrm{k-1}}\), where \(\mathrm{M}^{\mathrm{k}}\) represents the total  
Figure 1: Block diagram of decision feedback equalizer.  
number of possible transmitted data symbol vectors and M denotes the number of potential realizations of t(k). The symbol \(\left(\cdot\right)^{T}\) denotes the transpose of the vector.  
The MLD approach employed here is the maximum likelihood sequence estimation (MLSE). MLSE reduces the sequence error probability, specifically the probability of error in the data symbol vector, which corresponds to maximizing the conditional probability \(\mathrm{P}[\mathrm{t}_{\mathrm{u}}\mid\mathrm{r}]\), indicating the likelihood that \(\mathrm{t}_{\mathrm{u}}\) was sent given the received vector \(\mathrm{r}\). The estimate of \(\mathrm{t}\) (most likely transmitted data vector) derived from MLSE is calculated as follows:  
\[\hat{t}\ =\ arg_{\mathrm{t}_{\mathrm{u}}}\max\mathrm{P}\{\mathrm{t}_{ \mathrm{u}}|\mathrm{r}\} \tag{4}\]  
with arg being the function's argument. If the noise \(\mathrm{N}_{\mathrm{t}}\) is additive white Gaussian, Equation (4) will correspond to identifying the data symbol vector \(\mathrm{t}_{\mathrm{u}}\) that minimizes the squared Euclidean distance, written as follows:  
\[\Delta^{2}(\mathrm{t}_{\mathrm{u}},\mathrm{r})=\left\lVert\mathrm{r}-\mathrm{ At}_{\mathrm{u}}\right\rVert^{2} \tag{5}\]  
between the received sequences and all the potentially transmitted sequences. The most likely transmitted data vector is calculated as follows:  
\[\hat{t}\ =\ arg_{\mathrm{t}}\min\ \Delta^{2}(\mathrm{t}_{\mathrm{u}}, \mathrm{r}) \tag{6}\]  
MLSE necessitates the assessment of the obtained Euclidean distances for the estimation of the data symbol vector \(\hat{t}\)[44].