## 5 Simulation Results and Discussion  
To validate the theoretical models described in the previous chapters, their capabilities were examined for digital transmission, applying binary phase-shift keying modulation (BPSK). The simulations were performed using MATLAB software. In this part, the results obtained for the comparison of the turbo codes, LDPC codes, and polar codes in the presence of an AWGN channel with ISI are presented. The code rate \(R=\frac{1}{3}\) was considered for the performance comparison of turbo codes, LDPC codes, and polar codes. Practically, the performance of all three of the previously mentioned codes was compared from the perspective of the proposed approaches, such as the perfect channel estimation, with those of the estimation with the LS (least square) algorithm, in terms of BER vs. SNR (bit error rate vs. signal to noise ration). Concerning the estimation with the LS algorithm, both iterative estimation and estimation only once were considered for the comparison of performance of the turbo codes, LDPC codes, and polar codes, in terms of BER vs. SNR. First, the case of perfect channel estimation was analyzed, meaning that the impulse response of the channel \(\mathrm{h}\) is well known by both transmitter and receiver. Therefore, the model of the equivalent discrete channel used to test the performance of the equalizers was represented by the transfer function, defined by \(\mathrm{h}= \begin{bmatrix}0.18&0.85&0.32\end{bmatrix} \). In Figures 11 and 12, the difference in the performance that the MMSE (minimum mean square error) equalization provided, in terms of BER vs. SNR, before the decoding process compared to the ZF (Zero Forcing) equalization is very obvious. In Figure 11 the number of transmitted sequences was 10,000, with 342 bits of information each, in comparison with Figure 12, where the number of transmitted sequences was 10,000, with 170 bits of information each. It was observed that the performance became better as the length of the sequence of information bits became longer. For a given transfer function, the longer the sequence of information bits, the better the performance obtained for the same SNR. Figure 11: BER vs. SNR for different codes in the case of MMSE and ZF equalization in the case of perfect channel estimation (K = 342 bits and h = (0.18 0.85 0.32)). Figure 12: BER vs. SNR for different codes in the case of MMSE and ZF equalization in the case of perfect channel estimation (K = 170 bits and h = (0.18 0.85 0.32)). In the simulation process, turbo decoding used five iterations, and LDPC decoding used 10 iterations. In Figure 12, at a given SNR of 3 dB, if the length of the transmitted sequence is 170 bits, it is observed that the performance of the LDPC codes with 10 iterations and turbo codes with five iterations were close using MMSE equalization. If the length of the transmitted sequence was 342 bits, as in Figure 11, similar performances using MMSE equalization were obtained for a given SNR of 2 dB, regarding the LDPC codes and turbo codes. Moreover, in the case of Figures 11 and 12, for a given SNR of 3 dB using MMSE equalization, the LDPC codes with 10 iterations outperformed the turbo codes with five iterations. At the same time, in Figure 11, regarding the case of ZF equalization, a convergence of the performance of LDPC codes and turbo codes could be observed with the increase of the transmitted data sequence, but for a higher SNR than in the case of MMSE equalization. Following Figure 12, if K = 170 bits, the BER value for LDPC-mmse 10-it for an SNR of 3 dB is equal to \(\frac{1}{7}\times 10^{-4}\), and the BER value for Turbo-mmse 5-it for the same SNR is \(\frac{1}{6}\times 10^{-4}\). If K = 342 bits (Figure 11), the BER value for LDPC-mmse 10-it for an SNR of 3 dB is \(\frac{1}{8}\times 10^{-5}\), and the BER value for Turbo-mmse 5-it for the same SNR is \(10^{-5}\). Thus, an improvement of BER vs. SNR was observed, for the same SNR, as K increased. In other words, BER decreased as K increased.