## 3 Turbo Equalization Concept
### Log-MAP Equalizer  
On the other hand, different optimization criteria are distinguished for realizing the SISO equalizer, leading to distinct families of turbo equalizers. The MAP turbo equalization is an optimal equalizer in the sense of maximum a posteriori likelihood [37]. The SISO equalizer is then typically implemented using the MAP algorithm, which creates LLR for each entry bit, \(u_{i}\). This equalizer is ideal, especially for reducing the symbol error probability. To calculate the LLR a posteriori \(L(u_{i}\,|\,y)\), it was used the trellis representation associated with the transmission on the selective channel in frequency [38]. By applying the Bayes relation, the relation for LLR can be written in the sense of Equation (1). Among the set of possible transmitted sequences, each candidate sequence draws a unique path in the trellis. It can then be evaluated the joint probability \(P(u_{i}=0\)_or_\(1|y\)_) by summing the probability \(P(s^{\prime},s,y)\) of passing through a particular transition of the trellis connecting a state \(s^{\prime}\) at the time \(i-1\) to a state \(s\) at the time \(i\), over the set of transitions between times \(i-1\) and \(i\) for which the bit associated with these transitions is 0 or 1. In consequence:  
\[L(u_{i}|y)=\ln\left(\frac{\sum\limits_{(s^{\prime},s)/u_{i}=+1}P(s^{\prime},s, y)}{\sum\limits_{(s^{\prime},s)/u_{i}=-1}P(s^{\prime},s,y)}\right), \tag{5}\]  
where the \((s^{\prime},s)/u_{i}=+1\) represents the group of transitions from state \(s^{\prime}\) to state \(s\) as a consequence of the bit \(u_{i}=+1\), and in the same manner, \((s^{\prime},s)/u_{i}=-1\) designates the group of transitions from state \(s^{\prime}\) to state \(s\) assignable to the bit \(u_{i}=-1\). Considering a trellis example for a binary system, at each time interval, there are M = 2 transitions quitifing each state and \(\lambda=4\) transitions caused by the bit \(u_{i}=+1\), which belong to the set of \((s^{\prime},s)/u_{i}=+1\). In the same way, there will be \(\lambda=4\) transitions assignable to the bit \(u_{i}=-1\), belonging to the group of \((s^{\prime},s)/u_{i}=-1\). Considering a modulation with M states and a discrete channel with L coefficients, the total number of branches per section of the trellis amounts to MxML-1 = ML [1]. By adopting an approach similar to that presented in the original article by [PERSON] et al. [10], it can be shown that the joint probability \(P(s^{\prime},s,y)\) associated with each transition considered breaks down into a product of three terms:  
\[P(s^{\prime},s,y)=\alpha_{i-1}\big{(}s^{\prime}\big{)}\gamma_{i}\big{(}s^{ \prime},s\big{)}\beta_{i}(s), \tag{6}\]  
The probabilities \(\alpha\) and \(\beta\) are computable recursively for each state and each moment in the trellis, by applying the following recursive formulas:  
\[\alpha_{i}(s)=\sum\limits_{(s^{\prime},s)}\alpha_{i-1}\big{(}s^{\prime}\big{)} \gamma_{i}\big{(}s^{\prime},s\big{)},\text{ Initial values : }\alpha_{0}(s)= \begin{cases}1&s=0\\ 0&s\
eq 0\end{cases}  \tag{7}\]  
\[\beta_{i-1}\big{(}s^{\prime}\big{)}=\sum\limits_{(s^{\prime},s)}\gamma_{i} \big{(}s^{\prime},s\big{)}\beta_{i}(s),\text{ Initial values : }\beta_{N}(s)=\frac{1}{M^{-1}},\quad\forall s \tag{8}\]  
These two probabilities are named \"forward probability\" and \"backward probability\", respectively. The summations relate to the set of pairs of states with indices \((s^{\prime},s)\) for which there is a valid transition between two consecutive moments in the trellis. The initial condition or the starting state (index 0 by convention) is perfectly known. To assign equal weight to each state in back recursion is a common practice, as the state of arrival is usually unknown. To complete the description of the algorithm, the term \(\gamma_{i}(s^{\prime},s)\) in Equation (6) remains to be developed further. This term is assimilated to the transition probability.