## 5 Conclusions  
In conclusion, new methods were proposed for estimating the AWGN channel with ISI in the equalization and decoding system using LDPC codes. This study explored the use of the least squares algorithm in various scenarios and compared their performance. The main focus was on how the bit error rate changes with the signal-to-noise ratio. The best performance was achieved when the estimation was based on the training sequence and then re-estimated twice from the output of the LDPC decoder.  
The proposed schemes for estimating the AWGN channel with ISI in the equalization and iterative decoding system that uses LDPC codes can also use an alternative estimator, such as LMMSE, ML, or MMSE, rather than LS.  
After simulating the results, we came closer with the BER performances to the ideal case (\(h_{l}=\hat{h}_{l}\)) when we estimated \(h_{l}\) based on the received training sequence and re-estimated \(h_{l}\) twice in the loop according to the estimated sequence from the output of the min-sum LDPC decoder, in compliance with Figure 5, in Section 4.  
As can be seen from Figures 4, 6 and 7, after the 5 th iteration, no more errors were found after an SNR of 4 dB, considering that 10,000 sequences were transmitted.  
As we specified in [7], the performances of BER vs. SNR depend on the \(h\) used.  
If we compare with the specialized literature, the results we obtained in this work are better than those obtained using turbo equalization [36], where at iteration 5, after an SNR of 4 dB, some errors were still detected.  
However, using turbo equalization [36], the BER vs. SNR performance is slightly higher up to an SNR of 4 dB.  
Conceptualization, F.L.M., A.-M.C. and C.G.; methodology, F.L.M., A.-M.C. and A.-M.G.; software, A.-M.C. and F.L.M.; validation, A.-M.C., F.L.M., A.-M.G. and C.G.; formal analysis, A.-M.G. and C.G.; investigation, F.L.M. and A.-M.C.; resources, F.L.M.; data curation, A.-M.C. and A.-M.G.; writing--original draft preparation, A.-M.C. and F.L.M.; writing--review and editing, A.-M.C., A.-M.G. and C.G.; visualization, A.-M.C. and A.-M.G.; supervision, A.-M.G. and C.G. All authors have read and agreed to the published version of the manuscript.  
The APC was funded by the University of Oradea, Romania.  
**Institutional Review Board Statement:** Not applicable.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Perfect impulse response estimation, when \(\hat{h}_{l}=h_{l}\)** & \\ \hline
**Figure** & **h** & **BER (Range)** \\ \hline Figure 4 & \(h_{l}\) = [0.18 0.85 0.32] & \(1/9\times 10^{-2}\) \\ \hline
**Estimation of \(h_{l}\) based on the training sequence** & \\ \hline
**Figure** & **h** & **BER (Range)** \\ \hline Figure 4 & \(h_{l}\) = [0.18 0.85 0.32] & \(1/7\times 10^{-2}\)–\(1/8\times 10^{-2}\) \\ \hline \(h_{l}\) **is re-estimated based on the sequence \(\hat{b}_{k}\) from the min-sum LDPC decoder output after** \\
**one cycle.** & \\ \hline
**Figure** & **h** & **BER (Range)** \\ \hline Figure 6 & \(h_{l}\) = [0.18 0.85 0.32] & \(1/8\times 10^{-2}\)–\(1/9\times 10^{-2}\) \\ \hline \(h_{l}\) **is re-estimated based on the sequence \(\hat{b}_{k}\) from the min-sum LDPC decoder output after** \\
**two cycles.** & \\ \hline
**Figure** & **h** & **BER (Range)** \\ \hline Figure 7 & \(h_{l}\) = [0.18 0.85 0.32] & \(1/9\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bit error rate (BER) versus signal-to-noise ratio (SNR) for the iterative equalization and decoding system using low-density parity check (LDPC) codes considering the 5 th iteration and an SNR of 4 dB when estimating the impulse response of the additive white Gaussian noise (AWGN) channel with intersymbol interference (ISI).  
Data Availability Statement: The data presented in this article are available on request from the corresponding author.  
The authors declare no conflicts of interest.