## 3 Materials and Methods
### The System Model  
The main scope of this paper is to study the channel estimation for binary phase shift keying (BPSK)-modulated transmission in the case of the combined/joint process of equalization and decoding using min-sum LDPC codes and a Log MAP equalizer. The algorithm we used for channel estimation is LS. To compensate for the ISI that generally comes from a memory channel, in our case, coming from the AWGN channel of our system that combines equalization and decoding, we further presented the diagram of the transmission chain for reconstructing the original information. This is presented in Figure 1.  
The sequence of bits \(a_{n}\) was encoded using the 5G new radio LDPC codes of rate \(R=\dfrac{1}{2}\). The code is derived from the BG2 matrix [32] with an increase factor of \(Z_{c}=20\); thus, the information bits are 200, and the codeword totals 440 bits, where the first 40 bits were punctured, and 10,000 sequences have been sent. The process of encoding was presented in detail in [7]. The sequence of coded bits \(b_{k}\) is interleaved using an interleaver block giving the sequence \(c_{k}\) and then modulated adopting BPSK modulation, resulting therefore in the  
Figure 1: The schematic diagram for the proposed transmit–receive system with an estimated \(h_{l}\) based on the training sequence \(\widetilde{j}_{S}\). sequence \(x_{p}\). On the reception side, the signal \(y_{p}\) contains the ISI that we want to mitigate. The estimation of the received data \(y_{p}\) is carried out using an iterative receiver consisting of a Log MAP Equalizer and min-sum LDPC decoder and includes the estimator of the channel of transmission. Figure 1 shows that a training sequence \(j_{S}\) was generated when switches K1 and K2 were in the first state, which was known by the receiver's channel estimator. This routine is transmitted through the channel and changes due to the channel and noise. Taking into account the received sequence \(\widetilde{j}_{S}\) and the sequence known from the emission \(j_{S}\), the estimated value of \(h_{l}\) can be determined with the help of estimation algorithms. After its determination, the signal is processed at the receiver. We used the least squares algorithm as the simplest method to estimate the \(h_{l}\) of the transmission channel. The previously described LS method involves finding \(\hat{h}_{l}\) using the following relationship:  
\[\hat{h}_{l}=\left(J^{T}\cdot J\right)^{-1}\cdot J^{T}\cdot\widetilde{j}_{S}, \tag{5}\]  
in which \(J\) represents the Toeplitz matrix corresponding to the sent testing sequence \(j_{S}\), and \(J^{T}\) is the transpose of the \(J\) matrix. The sequence received after the memory channel of length \(L\) and Gaussian additive white noise \(w_{p}\approx N_{c}(0,\sigma^{2})\) is written as follows:  
\[y_{p}=\sum_{l=0}^{L}h_{l}x_{p-l}+w_{p}, \tag{6}\]  
where \(x_{p}\in\{-1,1\}\), \(\forall p\in[1,\ldots,N]\), with \(h_{l},l\in\{0,L-1\}\) the channel coefficients. In this case, the current symbol depends in general on the transmitted symbols, and we say that we are transmitting over a memory channel to the extent that a received symbol depends on the other \(L-1\) symbols. Equation (6) introduces a channel model representing the channel in the presence of ISI and Gaussian additive white noise. An equivalent channel model can be constructed based on Equation (6); therefore, an example of a channel owning three coefficients, \(h_{0}=0.18\), \(h_{1}=0.85\), and \(h_{2}=0.32\)[7], is presented in Figure 2.  
The architecture of the receiver consists of two parts: one for equalization and the other for decoding. The structure and operation mode of the iterative receiver that integrates equalization and decoding in a single process were presented in detail in [33]. The system's receiver involves iterative exchanges between the equalizer and decoder, with an intermediate interlacing and deinterlacing process, thus  
* Log MAP Equalizer: removes or reduces the effect of intersymbol interference (ISI) introduced by the multipath channel.