## 4 Results and Discussion  
We performed the simulations using the Matlab programming environment. Figure 4 shows the BER against SNR characteristics regarding two cases: the first case is when the estimate of \(h_{l}\) (\(\hat{h}_{l}\)) is the same as \(h_{l}\) (perfect estimate of \(h_{l}\), meaning that  
Figure 3: Trellis diagram of the applied Log MAP equalizer. \(\hat{h}_{l}=h_{l}\)), and the other case is when \(h_{l}\) is estimated based on a training sequence according to Figure 1. In this paper, we considered \(h_{l}= \begin{bmatrix}0.18&0.85&0.32\end{bmatrix} \)[7]. The repetitive system regarding the equalization and decoding makes five iterations, and the min-sum LDPC decoder makes twenty iterations. In Figure 4, only the results related to iterations 0 (It0, It0-LS0) and 5 (It5, It5-LS0) are represented. We refer to the previous paper [7], where It0 was the iteration in which the decoder sent no a priori information to the equalizer. In Figure 4, concerning the perfect estimation of \(h_{l}\), the results are better than in the case when \(h_{l}\) was estimated on a training sequence. According to the legend of Figure 4, the solid line represents the BER curves for the perfect estimation of \(h_{l}\), both in the case of It0 and in the case of It5. The dashed line represents the BER curves for the other case when \(h_{l}\) was assessed on a testing sequence (with a length L = 128), both for It0-LS0 and for It5-LS0. In Figure 4, it can be seen that the 5 th iteration makes improvements in both analyzed cases. For example, at an SNR of 4 dB, the BER reaches the value of \(\dfrac{1}{9}\cdot 10^{-2}\) in the case of the perfect estimation of \(h_{l}\) and a value within the range of \(\dfrac{1}{7}\cdot 10^{-2}\) and \(\dfrac{1}{8}\cdot 10^{-2}\) when \(\hat{h}_{l}\) was assessed on the testing sequence. To improve the estimation of \(h_{l}\), a more extended training sequence can be used, or the schematic diagram in Figure 5 can be used. The operation of this scheme presented in Figure 5 is similar to the one shown in Figure 1, only that from the sequence of bits obtained at the exit of the min-sum LDPC decoder \(\hat{b}_{k}\) (\(\hat{a}^{\prime}_{n}\) represents the estimated information bits (\(\hat{a}_{n}\)) less the first 40 information bits that were punctuated according to [32], and (\(\hat{p}_{b}\)) represents the parity bits estimated) and from the sequence (\(y_{p}\)) considered at the exit of the channel could be re-estimated the value of \(h_{l}\), which was first determined based on the training sequence \(\widetilde{j}_{S}\). At the exit, using the extrinsic information, the decoder determines the estimated values of the information bits (\(\hat{a}^{\prime}_{n}\)) and the parity bits (\(\hat{p}_{b}\)). Thus, the estimated sequence of \(\hat{b}_{k}=(\hat{a}^{\prime}_{n};\hat{p}_{b})\) is further interlaced (\(\hat{c}_{k}\)), BPSK modulated (\(\hat{x}_{p}\)), and finally entered into the channel estimator. The channel estimator calculates the estimated value of (\(h_{l}\)) by using both the sequence from the exit of the channel (\(y_{p}\)) and the sequence estimated from the exit of the decoder (\(\hat{b}_{k}\)). Figure 4: Performance comparison of BER vs. SNR when we have a perfect estimation of \(h_{l}\) and when \(h_{l}\) is estimated on a testing sequence with a length of 128 bits. In the case of Figure 5, three cycles of re-estimation of \(h_{l}\) were carried out: the first one based on the training sequence \(\widetilde{j}_{S}\) and the other two based on the estimated values of \(\hat{b}_{k}\) from the exit of the decoder. Figure 6 shows the BER vs. SNR performance for \(h_{l}\) estimated using the testing sequence compared to \(h_{l}\), re-estimated using the sequence \(\hat{b}_{k}\) derived from an LDPC decoder after a single cycle.