## 2 Related Work  
Firstly, a summary of existing SISO equalization algorithms derived from classical equalization methods according to the maximum a posteriori (MAP) or minimum mean-square error (MMSE) criterion is presented. In this respect, we referred to [4; 5; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. Using soft decisions provided by the decoder at the entry of an equalizer was demonstrated to considerably reduce the BER at the reception [16; 17]. From this point of view, different SISO equalization techniques were built in the literature to improve the information from the decoding levels within the iterative decoding approach. The idea of using a priori information from the channel decoder to improve equalization was proposed in [4]. The decoder and the equalizer improve each other the estimate of the transmitted data in an iterative process similar to turbo codes. This first version of turbo equalization uses equalization based on the maximum-likelihood sequence detection (MLSD) criterion using the soft-output Viterbi algorithm (SOVA). The soft-output Viterbi equalizer used in [4] represented the first soft-in soft-out equalizer within the iterative decoding method at the reception. The soft-output Viterbi equalizer consists of trellis construction of the channel being easily decoded by applying the Viterbi algorithm. Even so, it did not succeed in decreasing the BER. Then, an adaptive turbo-equalization structure was proposed in [18]. The equalizer created on the MMSE criterion, including a priori information, was provided in [19], employing a matrix presentation of the channel. In [20], the principle for a turbo equalization technique whose coefficients are computed in the frequency domain was adopted. Further, enhancements to the soft-output Viterbi equalizer were performed by the MAP algorithm to succeed in reducing the BER [5]. In contrast to the soft-output Viterbi equalizer, the better efficiency of the MAP equalizer comes from the higher computational complexity of the MAP equalization algorithm. Max-Log MAP, described in [21], presents a lower computational complexity in contrast to MAP equalizer but also a lower performance in terms of BER. In this sense, another method (delayed-decision-feedback sequence estimator) employed to decrease the difficulty of the MAP equalizer consisted of reducing the number of states of the trellis. The soft-output sequential algorithm, a trellis-based equalizer presented in [7], used the sequential estimation method within the turbo equalization process. In [22], a soft-output Viterbi algorithm built on soft-output decision-feedback sequence estimation was detailed. The minimum mean-square error linear equalizer (MMSE LE) was applied in [23] to decrease computation complexity in the case of an equalizer using a priori information. Also, the minimum mean-square error decision-feedback equalizers (MMSE DFE), including or not a priori information in the finite impulse filters, supplying hard or soft decisions, were presented in [24; 25]. The iterative techniques created on the MAP or MMSE principle, including a priori information, facilitate the removal of interferences almost totally and obtain the full benefit from the diversity of the channel. These techniques can also be used for other sources of interference. Code division multiple access (CDMA) systems need to regard the interference caused by multiple users. In this respect, in [26], the iterative handling for CDMA is described. In addition, the principles of turbo equalization can be used in the issue of linear precoding or precoding multiantenna systems [27]. On the other hand, we referred to other papers [28; 29; 30; 31] where forward-error-correction decoders (such as spatially coupled low-density parity-check codes, protograph codes, repeat-accumulate codes, spatially-coupled codes, or Bose-Chaudhuri-Hocquenghem codes) were employed in connection with the Bahl-Cock-Jelinek-Raviv (BCJR) detector or minimum mean-square error linear equalizer (MMSE LE) to mitigate the impact of ISI.