## 3 Turbo Equalization Concept
### Log-MAP Equalizer
It can be split into a product of two factors:  
\[\gamma_{i}\big{(}s^{\prime},s\big{)}=P(u_{i})P\big{(}y_{i}|s^{\prime},s\big{)}, \tag{9}\]  
where the term \(P(u_{i})\) represents the a priori probability of the input bit, \(u_{i}\), and \(P(y_{i}|s^{\prime},s)\) is the probability that the signal \(y_{i}\) has been received since the passing from \(s^{\prime}\) to s happened at moment \(i\). Therefore, considering an AWGN channel for receiving the estimated signal \(\hat{y}_{i}\), the term \(P(y_{i}|s^{\prime},s)\) can be written as follows:  
\[P\big{(}y_{i}|s^{\prime},s\big{)}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[- \frac{1}{2\sigma^{2}}(y_{i}-\hat{y}_{i})^{2}\right], \tag{10}\]Converting \(\alpha_{\rm i}(\rm s)\) and \(\beta_{\rm i-1}(s^{\prime})\) into the logarithmic domain [39] the recursive formulas for forward and backward recursion are as follows:  
\[\rm A_{\rm i}(\rm s)=\ln(\alpha_{\rm i}(\rm s))=\ln\left(\sum\limits_{(s^{\prime },s)}\alpha_{\rm i-1}\big{(}s^{\prime}\big{)}\,\gamma_{\rm i}\big{(}s^{\prime},s \big{)}\right)=\ln\left(\sum\limits_{(s^{\prime},s)}\exp\big{(}A_{\rm i-1} \big{(}s^{\prime}\big{)}+\Gamma_{\rm i}\big{(}s^{\prime},s\big{)}\big{)}\right), \tag{11}\]  
\[\rm B_{\rm i-1}\big{(}s^{\prime}\big{)}=\ln(\beta_{\rm i-1}\big{(}s^{\prime} \big{)})=\ln\left(\sum\limits_{(s^{\prime},s)}\gamma_{\rm i}\big{(}s^{\prime},s \big{)}\beta_{\rm i}(\rm s)\right)=\ln\left(\sum\limits_{(s^{\prime},s)}\exp \big{(}\Gamma_{\rm i}\big{(}s^{\prime},s\big{)}+B_{\rm i}(\rm s)\big{)}\right), \tag{12}\]  
where \(\Gamma_{\rm i}(s^{\prime},s)\) represents the logarithmic domain conversion of \(\gamma_{\rm i}(s^{\prime},s)\). Therefore, applying the same method for \(\gamma_{\rm i}(s^{\prime},s)\), \(\rm A_{\rm i}(\rm s)\) and \(\rm B_{\rm i-1}(s^{\prime})\) can be computed as follows:  
\[\Gamma_{\rm i}\big{(}s^{\prime},s\big{)}=\ln\big{(}\gamma_{\rm i}\big{(}s^{ \prime},s\big{)}\big{)}=\ln\big{(}P(\rm u_{\rm i})P\big{(}y_{\rm i}|s^{\prime },s\big{)}\big{)}, \tag{13}\]  
Using Equation (10), \(\Gamma_{\rm i}(s^{\prime},s)\) becomes:  
\[\Gamma_{\rm i}\big{(}s^{\prime},s\big{)}=\ln\left(\frac{1}{\sqrt{2\pi\sigma^{ 2}}}\right)-\frac{1}{2\sigma^{2}}\big{(}y_{\rm i}-\hat{y}_{\rm i}\big{)}^{2}+ \ln P(\rm u_{\rm i}), \tag{14}\]  
In Equation (14) presuming that the bits transmitted own the same probability, \(P(\rm u_{\rm i})\) is an unchanging factor and can be omitted. Also, the term \(\ln\left(\frac{1}{\sqrt{2\pi\sigma^{2}}}\right)\) could be omitted. Consequently, Equation (14) can be rewritten as follows:  
\[\Gamma_{\rm i}\big{(}s^{\prime},s\big{)}=-\frac{1}{2\sigma^{2}}(y_{\rm i}- \hat{y}_{\rm i})^{2}, \tag{15}\]  
In summary, after computing the branch metric and then performing forward and backward recursions [40], the a posteriori LLR is finally given:  
\[\rm L(u_{\rm i}\,|y)=\ln\left(\frac{\sum\limits_{(s^{\prime},s)}\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! depicts the block structure for the BG2 matrix according to the 5G standard as it mentioned in [42]. The main point of LDPC coding is the calculation of parity bits. In this paper, the parity bits p were calculated based on the double diagonal structure according to 5G standard. Taking into consideration the block structure of BG2 and Equation (18), the parity-check Equation (17) for encoding can be rewritten as follows [43]:  
\[\mathrm{H}\times\mathrm{c}^{\mathrm{T}}= \begin{bmatrix}\mathrm{A}&\mathrm{E}& 0\\ \mathrm{B}&\mathrm{C}&\mathrm{I}\end{bmatrix}\begin{bmatrix}\mathrm{m}^{ \mathrm{T}}\\ \mathrm{P}_{1}^{\mathrm{T}}\\ \mathrm{P}_{2}^{\mathrm{T}}\end{bmatrix} , \tag{19}\]  
The information bits m with a length of \(10\mathrm{Z}_{\mathrm{c}}\), the first set of parity bits \(\mathrm{p}_{1}\) with a length of \(4\mathrm{Z}_{\mathrm{c}}\), and the second set of parity bits \(\mathrm{p}_{2}\) with a length of \(38\mathrm{Z}_{\mathrm{c}}\) correspond to \( \begin{bmatrix}\mathrm{A}&\mathrm{B}\end{bmatrix} ^{\mathrm{T}}, \begin{bmatrix} \mathrm{E}&\mathrm{C}\end{bmatrix} ^{\mathrm{T}}\), and \(