# Turbo Decoder Design based on an LUT-Normalized Log-MAP Algorithm
## 3 The Normalized Log-MAP Algorithm
However, the greatest advantage of the proposed normalization factor lies in greatly reducing the logic resources in the hardware implementation, while maintaining a similar performance or having a negligible loss of performance. In this view, the factor is independent of code length or code rate to some extent. Figure 5 shows the effect of decoding performance under the normalization factor \(\eta=1.125\) with different \(|\mathrm{a}-\mathrm{b}|\) ranges, where we define \(|\mathrm{a}-\mathrm{b}|<\mathrm{range}\). In Figure 5, the decoding performance is better when \(\mathrm{range}<9\) and \(\mathrm{range}<10\), and the decoding performance with \(\mathrm{range}<9\) is better than that \(\mathrm{range}<10\) with a signal noise ratio (SNR) from 1.5 to 2.8 dB. Considering decoding performance with a medium to high SNR, the range \(|\mathrm{a}-\mathrm{b}|\) can be determined as \(|\mathrm{a}-\mathrm{b}|<9\). Therefore, the Nor-Log-MAP algorithm can be rewritten as Equation (13). In the hardware simulation, Equation (13) can be realized by Equation (14) after binary conversion, where \(\mathrm{max}\left(\mathrm{a},\mathrm{b}\right)\times 0.001\) can be computed by intercepting the corresponding data based on selected bit width and the number of decimal digits. During the hardware simulation, 7-bit channel information is used, where 5 bits are integers and 2 bits are decimals. When computing, the data part that exceeds two decimal places is removed, however this results in performance loss. As can be seen from Figure 6, there is a little gap between the removed extra decimal places and the complete data in decoding performance. We also know that hard decision in turbo decoding relates to the sign bit of information directly rather than the absolute value of the data. In this way, the algorithm can replace multiplication with an addition operation, and will reduce some logic resource consumption compared with the lookup table method. \[\mathrm{max}^{*}(\mathrm{a},\mathrm{b})\approx\left\{ \begin{array}{cc}1.125 \cdot\mathrm{max}(\mathrm{a},\mathrm{b})&,if\big{|}\mathrm{a}-\mathrm{b} \big{|}<9\\ \mathrm{max}(\mathrm{a},\mathrm{b})&,\mathrm{otherwise}\end{array} \right. \tag{13}\]  
\[\mathrm{max}^{*}(\mathrm{a},\mathrm{b})\approx\left\{ \begin{array}{cc}\mathrm{ max}(\mathrm{a},\mathrm{b})+\mathrm{max}(\mathrm{a},\mathrm{b})*0.001&,if\big{|} \mathrm{a}-\mathrm{b}\big{|}<9\\ \mathrm{max}(\mathrm{a},\mathrm{b})&,\mathrm{otherwise}\end{array} \right. \tag{14}\]  
The comparison between the proposed Nor-Log-MAP algorithm and other approximate algorithms in decoding performance is shown in Figure 7. As can be seen, the decoding performance of the proposed Nor-Log-MAP algorithm is superior to the Max-Log-MAP algorithm, and has a maximum gain of 0.25 dB in decoding performance. Compared with the Log-MAP algorithm and LUT-Log-MAP algorithm, the Nor-Log-MAP algorithm has some disadvantages, but the decoding performance of the Nor-Log-MAP algorithm is close to (or even surpasses) the LUT-Log-MAP algorithm with a high SNR. Considering that the logic complexity of the Nor-Log-MAP algorithm is close to that of the  
Figure 5: Decoding performance with same \(\eta\) and different ranges. Max-Log-MAP algorithm theoretically, it can be concluded that the Nor-Log-MAP algorithm has a considerable decoding performance and less logic complexity.