## 3 Materials and Methods
### The System Model
The role of the equalizer placed in front of the decoder will be to compensate for this ISI, by trying to filter the output so that it finds the shape of the input pulse or, in any case, to reduce the ISI by identifying the channel parameters and taking this information into account in the decision-making process. Figure 2: Schematic of the memory transmission channel with three coefficients and AWGN. * Data interleaving/de-interleaving: whose role here consists essentially in separating the error at the equalizer's output (to prevent the spread of errors) and decoupling the LLRs transferred among the decoder and the equalizer. Then, the iterative receiver can completely counteract the damage owing to the intersymbol interference, as long as the channel interleaver is large enough and judiciously built. * Min-sum LDPC decoder: the achievement of the iterative decoding in the reception creates an LLR transfer among the decoder and the equalizer in a certain number of iterations. Various types of equalizers can be used to eliminate ISI. Equalization algorithms developed on trellis searches, such as ML or MAP, ensure good receiver performance and reduced complexity. Optimal equalizers based on algorithms exploiting the channel trellis, namely, the Bahl, Cock, Jelinek, and Raviv, or BCJR, algorithm and the Viterbi algorithm (SOVA), give fine results provided that the channel is well known. The BCJR algorithm represents a technique applying trellis structures to decode each bit or symbol of information. It uses the MAP criterion in a recursive form. This algorithm is different from the SOVA, which makes a maximum likelihood (ML) decision upon the complete sequence. In this paper, a Log MAP equalizer was adopted, which is a version of the BCJR algorithm. Figure 3 illustrates the trellis diagram of the equalizer employed in this paper. In Figure 3, the parameter \(\alpha_{i}(s)\) represents the probability at the time \(i\) of state \(s\), taking into account the sequence of past events. This means \(\alpha_{i}(s)\) can be computed iteratively for each new symbol received. This is why the algorithm is called \"forward\". The factor \(\beta_{i}(s)\) reflects the probability of future observations knowing the state \(s\) at the time \(i\). Also, \(\beta_{i}(s)\) can be recursively calculated, hence the name \"backward\" regarding the algorithm. The parameter \(\gamma_{i}(s^{\prime},s)\) is nothing but the passing probability in the state \(s\) deriving from \(s^{\prime}\), depending on the channel outputs. The computation of all these parameters was presented in [7]. The equalization part is made up of two modules: a channel estimator and a Log MAP equalizer. The Log MAP equalizer demands, excluding the received signal \(y_{p}\), the knowledge about the estimated value of \(h_{l}\) (\(\hat{h}_{l}\)), which the channel estimator supplies. The decoding of the data (initially encoded by the LDPC code) is ensured by the min-sum LDPC decoder. In this respect, we are interested in the min-sum algorithm (message passing algorithm) which is explicated in [34; 35]. For this type of algorithm, the probabilities are expressed as a maximum likelihood.