## 5 Simulation Results and Discussion
Furthermore, according to [47], LDPC codes were the best performing, even outperforming turbo codes. Figures 13 and 14 present the results obtained for the ZF and MMSE equalization in the case when the channel response to the impulse was estimated once only using the LS algorithm, based on a learning sequence with a length of N = 16 (Figure 13), respectively N = 8 (Figure 14). At this point, the specifications from the previous section related to Figures 7 and 8, when the channel estimation was performed only one time, could be taken into account. At the same time, the length of the data sequence transmitted was considered the same (K = 342 bits) in both cases. Both figures show that the MMSE equalization was more efficient than the ZF equalization with respect to BER vs. SNR. In the case of a larger training sequence (Figure 13), the bit error rate at a signal to noise ratio of 3 dB for all three types of codes analyzed (Turbo codes, LDPC codes, and Polar codes) was lower compared to the case in which the training sequence was considered N = 8 (Figure 14), both for MMSE equalization and ZF equalization. The estimation of h became better as the training sequence was lengthened. Turbo codes had a better performance than LDPC codes, both for MMSE equalization and ZF equalization, meaning that at the same value of signal to noise ratio, turbo codes provide a lower level of BER. Finally, Figures 15 and 16 illustrate the results obtained in the case of the iterative LS estimation of the channel response to the impulse h.  
Figure 15 shows the results related to the performances of turbo codes, LDPC codes, and polar codes obtained when the first estimation of h was made based on a training sequence of N = 16. Figure 16 presents the same elements in the condition where the length of the training sequence was N = 8. In both situations, the following estimates of h were achieved based on the sequence obtained at the output of the decoder \(\hat{\mathrm{b}}_{\mathrm{k}}\) using the same LS algorithm. In addition, the specifications from the previous section related to Figures 9 and 10, in the case where the channel estimation was performed iteratively, could be taken into consideration. As a matter of fact, in both situations, only MMSE equalization was used, because it proved to be more efficient than ZF equalization. Figure 14: BER vs. SNR for different codes in the case of MMSE and ZF equalization applying LS estimation (K = 342 bits h = (0.18 0.85 0.32), N = 8). It was observed that the iterative LS estimation of h offered substantial improvements in the case of all three codes than its estimation only once. Figure 16: BER vs. SNR for different codes in the case of MMSE equalization applying iterative LS estimation (K = 342 bits h = (0.18 0.85 0.32), N = 8). Figure 15: BER vs. SNR for different codes in the case of MMSE equalization applying iterative LS estimation (K = 342 bits h = (0.18 0.85 0.32), N = 16). The better the initial estimation of h (Figure 15), the lower level of BER was obtained for a certain value of SNR, compared to the case presented in Figure 16, where the training sequence was smaller and involved a higher number of iterations to reach the same level of BER, as in the situation depicted in Figure 15.  
This paper is limited to the performance analysis of turbo codes, LDPC codes, and polar codes on an AWGN channel in the presence of inter symbol interference. As was mentioned in the Section 1, to mitigate the negative effect of interference it is necessary to apply channel equalization. Therefore, to perform this analysis, only the BER vs. SNR curves for turbo codes, LDPC codes, and polar codes, on an AWGN channel with ISI, were drawn in Figures 11-16.