## 4 Design of the Equalizers
### Least Square Method for Channel Estimation  
The vast majority of the existing methods for estimating the channel with ISI are based on the least square (LS) method or the minimum mean square error (MMSE). The LS estimation method consists of finding the \(\rm\hat{h}\) that minimizes the squared error by applying the following equation [45]:  
\[\rm\hat{h}=\underset{h}{arg\,min}\|\rm\widetilde{s}-Sh\|^{2}, \tag{23}\]  
Supposing that the noise is white and Gaussian, the solution to Equation (23) can be represented as follows:  
\[\rm\hat{h}_{LS}=(S^{T}S)^{-1}S^{T}S, \tag{24}\]  
where \(S= \begin{bmatrix}s_{L}&\ldots&s_{1}&s_{0}\\ s_{L+1}&\ldots&s_{2}&s_{1}\\ \vdots&\vdots&\vdots&\vdots\\ s_{L+N-1}&\ldots&s_{N}&s_{N-1}\end{bmatrix} \) is the Toeplitz matrix that corresponds to the transmitted training sequence \(\rm s= \begin{bmatrix}s_{0}&s_{1}&\ldots&s_{N+L-1}\end{bmatrix} \), and \(\rm S^{T}\) represents the transposition of the matrix \(S\). Figure 7 depicts a block diagram of the transmission-reception system, wherein the estimation and the equalization of the channel are performed on the reception side, in the case of using turbo codes. Related to Figure 7, \(\rm\Pi\) represents the inter-leaver block; \(c_{m}^{\prime}\) represents the parity bits and the bits of information; \(\rm c_{m}\) is the parity bits, the bits of information, and the parity bits of the interleaved sequence; \(\rm x_{p}\) is the sequence obtained after BPSK modulation; \(\rm y_{p}\) denotes the received sequence; \(\rm z_{p}\) denotes the equalized sequence; \(\rm\hat{b}_{k}\) represents the estimated information bits; and \(\rm\widetilde{s}\) is the received training sequence. On the other hand, in the case of adopting other encoding and decoding techniques regarding LDPC codes and polar codes, the transmission-reception system, shown in Figure 8, is different from the previous one, because of the lack of an inter-leaver at the transmitter. The two block diagrams shown in Figures 7 and 8 works in the same manner. Thus, when the switches are in position 1, a training sequence s is transmitted; this is known as the channel estimator. Based on the received sequence and using an estimation algorithms such as LS, the estimated value of h(f) can be determined. When the switches are in position 2, data are transmitted. Figure 8: Block diagram of a transmission–reception system that performs the estimation and the equalization of the channel at the reception, in the case of applying LDPC codes and polar codes [46]. Figure 7: Block diagram of the transmission–reception system that performs the estimation and the equalization of the channel at the reception, in the case of applying turbo codes [46]. An improvement of the turbo channel estimation, and implicitly of the turbo decoding, can be obtained using the block diagram depicted in Figure 9. It depicts the iterative estimation in the case of applying turbo codes. In addition, in Figure 10 is presented a block diagram of performing the iterative estimation and the equalization of the channel at the reception, in case of applying LDPC codes and polar codes. Figure 10: Block diagram of a transmission–reception system that performs the iterative estimation and the equalization of the channel at the reception, in the case of applying LDPC codes [46]. Figure 9: Block diagram of a transmission–reception system that performs the iterative estimation and the equalization of the channel at the reception, in the case of applying turbo codes [46]. Practically, comparing the schemes depicted in Figures 7 and 8 with the ones presented in Figures 9 and 10 shows that the difference resides in the fact that the channel estimation is done iteratively in the latter Figures, in contrast to the first ones, where the channel estimation is performed once.