## 4 Results and Discussion
Thus, the curves corresponding to \(h_{l}\) estimated using the training sequence are represented with a solid line, and the representation of the curves corresponding to \(h_{l}\) estimated based on the sequence \(\hat{b}_{k}\) from the exit of the LDPC decoder are represented with a dashed line. Figure 6 shows that the BER performance improves for the case where \(h_{l}\) is estimated using the sequence \(\hat{b}_{k}\) derived from the decoder after a single cycle compared to the case where \(h_{l}\) is assessed upon the training sequence. In the 5 th iteration (I5) at an SNR of 4 dB, a BER within the range of \(\frac{1}{3}\cdot 10^{-2}\) and \(\frac{1}{8}\cdot 10^{-2}\) was obtained when \(h_{l}\) was estimated based  
Figure 5: The schematic diagram of the proposed transmit–receive system with estimated \(h_{l}\) based on the training sequence \(\widetilde{j}_{S}\) and \(h_{l}\) re-estimated in the loop by using the \(\hat{b}_{k}\) output from an LDPC decoder. Figure 6: Performance comparison of BER vs. SNR when \(h_{l}\) is estimated using the training sequence and when \(h_{l}\) is re-estimated using the sequence \(\hat{b}_{k}\) from the output of the min-sum LDPC decoder after a single cycle. on the training sequence (It5-LS0), and a BER between \(\frac{1}{g}\cdot 10^{-2}\) and \(\frac{1}{g}\cdot 10^{-2}\) in the other case (It5-LS). Continuing this re-estimation with one more cycle, an additional improvement of the BER vs. SNR performance is obtained, as Figure 7 shows. Thus, Figure 7 depicts the BER vs. SNR after the 5 th iteration for four cases:  
* The case where we have the perfect estimate of \(h_{l}\) (It5). * \(h_{l}\) estimated using the training sequence (It5-LS0). * \(h_{l}\) estimated using the training sequence and re-estimated only once using the sequence \(\hat{b}_{k}\) from the exit of the decoder (It5-LS1). * \(h_{l}\) estimated using the training sequence and re-estimated twice using the sequence \(\hat{b}_{k}\) from the exit of the decoder (It5-LS2). Following all the results obtained from Figure 7, it can be seen that after two re-estimations of \(h_{l}\), the BER performance approaches the case where we have a perfect estimate of \(h_{l}\). This is confirmed in the It5 (meaning the fifth iteration) at an SNR of 4 dB where an approximate BER of \(\frac{1}{g}\cdot 10^{-2}\) was obtained in each of the two cases. The first case is when we assumed we had a perfect estimate of \(h_{l}\). The second one is considered when \(h_{l}\) was estimated based on the training sequence (It5-LS0) and re-estimated twice using the sequence \(\hat{b}_{k}\) from the exit of the decoder (It5-LS2). To make it easier for the reader to follow, Table 1 summarizes the results obtained from the simulations for all the previously presented cases, considering only the 5 th iteration. Also, based on the table, it seems that the BER performances achieved with \(h_{l}\) estimated based on the training sequence and re-estimated in a loop using the sequence \(\hat{b}_{k}\) from the exit of the decoder after two cycles reached the performance achieved when a perfect estimate of \(h_{l}\) was considered. Figure 7: Performance comparison of BER vs. SNR in the case of a perfect estimate of \(h_{l}\) (It5), \(h_{l}\) estimated using the testing sequence (It5-LS0), \(h_{l}\) estimated using the training sequence, and re-estimated only once using the sequence \(\hat{b}_{k}\) from the exit of the min-sum LDPC decoder (It5-LS1), and \(h_{l}\) estimated using the testing sequence and re-estimated twice using the sequence \(\hat{b}_{k}\) deriving from the exit of the min-sum LDPC decoder (It5-LS2).