## 3 Turbo Equalization Concept
### The Proposed System Model  
In this paper, a transmission scheme on AWGN channels with ISI was proposed according to Figure 4.  
The proposed transmission scheme described in Figure 4 follows the basic architecture of the iterative receiver using the turbo equalizer presented in Figure 1 in Section 3.1. The differences consist of the LDPC encoding of the input binary data sequence \(\mathrm{u}_{\mathrm{i}}\) used for transmission, and the Log-MAP equalizer and the min-sum LDPC decoder were used for reception in the iterative and decoding equalization process. Figure 3: Block structure of BG2 according to 5G standard. sequence (data) u\({}_{\rm i}\) is LDPC encoded, and then it is punctured to obtain a rate of \(\dot{\gamma}_{\rm i}\) interlaced, and BPSK modulated. Next, the signal is transmitted through the AWGN channel with ISI, and the reception follows the classic turbo equalization scheme, but in this case, using a Log-MAP equalizer and an LDPC decoder. In the research process related to LDPC coding, the BG2 base matrix, according to the standard [41] with the expansion factor \(Z_{\rm c}=20\), was used. A total of 10,000 sequences of 200 bits of information each (K = 200) were transmitted. After the coding process, the length of the code words is N = 440 bits. To obtain a rate of \(\dot{\gamma}_{\rm i}\) the first \(2Z_{\rm c}\) bits were punctured according to the standard. The N-\(2Z_{\rm c}\) length message is then interleaved, modulated, and transmitted on an AWGN channel with ISI. In this paper, three impulse responses' h functions used in the literature were taken, each having three different weights. They are as follows: h1 = [0.18 0.85 0.32] [13], h2 = [0.302 0.725 0.456] [47], and h3 = [0.407 0.815 0.407] [48]. Following the basic turbo-equalizer concept, the received signal y\({}_{\rm i}\) enters the Log-MAP equalizer, as presented in Figure 4. The equalizer knows the impulse response h, making a first estimate of the transmitted sequence y\({}_{\rm i}\). In the iterative process of equalization and decoding, the equalizer works with the interlaced sequence and the decoder with the deinterlaced sequence. The data from the Log-MAP equalizer are deinterlaced and sent to the decoder, which makes a first estimate of the transmitted information bits. During the decoding process, the decoder also makes a first estimate of the first \(2Z_{\rm c}\) bits that were punctured after coding. The min-sum LDPC decoder works iteratively, as was mentioned in Section 4.2. Considering that the equalizer works with N-\(2Z_{\rm c}\) bits (in our case, with 400 bits), information about only these bits will be transmitted from the decoder to the Log-MAP equalizer. That is, from \(2Z_{\rm c}+1\) to N. This information is interlaced not before subtracting the information from the input of the decoder from the previous iteration so that it does not receive its information as a priori information. Based on the a priori information from the decoder and the information at the exit of the channel, the equalizer improves the estimate of the transmitted sequence with each iteration. This information from the Log-MAP equalizer is deinterlaced and sent to the decoder, not before subtracting the a priori information from the input of the equalizer from the previous iteration; thus, it does not receive its information as a priori information. At the output of the decoder, the first K bits estimated values from the N values represent estimates of the transmitted information bits. In this work, all LLRs transmitted from the decoder to the equalizer and vice versa are weighted with their maximum absolute value. LDPC decoders offer a low-complexity implementation compared to other forward error-correcting codes, according to [15; 49; 50]. Figure 4: LDPC coded system achieving equalization and channel decoding iteratively.