## 4 Simulation
### Channel Coding  
In previous subsection, we have discussed the detector based on SIF together with SVM classifier, when we compare performances at low BER. In this section, we focus on the performance after the sequence has been corrected by an LDPC decoder and the ability of SVM detector to provide accurate posterior probability estimates instead of measuring the performance of the demodulator at low BER,  
Figure 11: The BER performance comparison of SVM and threshold decision for general system and narrow band system with N = 5. We use dashed-dotted lines for the SVM BER, solid lines for the threshold decision BER. We represent the BER of narrow band system with 0 and general system with \(\circ\), respectively. because the channel decoder can achieve those BER values at significantly lower signal power. Earlier this year a study has been undertaken to give the approximate LLR for LDPC decoding [29], so we now modify the method to provide posterior probability estimates and the method is referred to as modified approximate posterior probability estimates (MAPPE) in this paper. In Figure 12(a) we have depicted the BER for the MAPPE and SVM receiver. The BER can be significantly reduced by the SVM-K2N5-LDPC method compared to the MAPPE-K2N5-LDPC method. Also, we can appreciate that the SVM-K2N5-LDPC is even better than MAPPE-K2N20-LDPC. Therefore, the SVM method significantly reduces the BER at lower SNR, because SVM posterior probability estimates are more accurate than MAPPE and the LDPC decoder can rely on these trustworthy predictions. Although the SVM-K2N5-LDPC has a higher data rate than SVM-K2N20-LDPC, their performance are almost identical; this is a benefit of the SVM features extraction method which makes full use of the signal characteristics and narrows the difference between the long bit duration system and the short one. In what follows we only report results for the SVM-K2N5-LDPC for clarity purposes. In addition, we plot the BER performance of narrowband system that applied LDPC codes by using the soft-input of SVM detector output. The BER performance of NB-SVM-K2N5 outperforms the MAPPE-K2N5-LDPC by 2.5 dB and is less than 1 dB from the MAPPE-K2N20. For completeness, we have also depicted the FER in Figure 12(b). The FER performance is typically used when we are only interested in error-free frames, which is a more relevant measure in data-package networks. The results in FER are similar to the BER ones. Figure 12: We plot the BER performance at the output of LDPC decoder with soft-inputs using a SVM and MAPPE detector with N = 5, N = 20 and narrowband system in (**a**) and its corresponding FER in (**b**). We represent the SVM method for N = 20 with *, N = 5 with \(\circ\) and narrow band system with \(\Box\), respectively; the MAPPE method for N = 5 with \(\lx@sectionsign\)X and N = 20 with \(\lozenge\), respectively. To understand the difference in PPEs, we have plotted the curves for the SVM and the MAPPE in Figures 13(a,b), respectively, with bit duration N = 5 and SNR = \(-\)5 dB. We depict the estimated probabilities \(P\big{(}y\!=\!1\!\mid\!x\big{)}\) with the true ones and \(P\big{(}y\!=\!0\!\mid\!x\big{)}\) with \(1\!-\!P\big{(}y\!=\!0\!\mid\!x\big{)}\) for clarity purposes. Figure 13(a) shows that the SVM PPEs are closer to \"1\" (or \"0\") and less spread, most of the values of SVM detector output are between 0.9 and 1 for symbol \"1\" (or between 0 and 0.1 for symbol \"0\"). Thereby, SVM estimates are closer to the true posterior probability, which explains its improved performance with respect to the MAPPE, when we measure the BER after the LDPC decoder. In the last experiment we will compare the ability to provide PPEs between SVM and MAPPE with different sampling rates. In Figure 14 we have depicted the BER for the two receivers. The BER performance of the two receivers follows the same lines that we have seen in the previous detection case: the SVM performs well and outperforms the MAPPE.