## 2 Related Work  
Communication systems with high data rates must handle impairments determined by AWGN and ISI. Iterative equalization and decoding methods, also known as turbo equalization, have gained significant popularity in recent years. These methods are used to raise the performance of the digital communication systems affected by intersymbol interference or multiple access interference. By utilizing the power of forward error correction, turbo equalization enhances the system's ability to correct errors and maintain efficient communication. Turbo equalization is a highly effective method for reaching the optimal bit error rate (BER), and in wireless communication, this is used to recover original information bits [9]. Having complete knowledge of the communication channel and using MAP (maximum a posteriori probability) equalization in parallel with decoding provides a reliable solution for fixing the problems concerning the combined process of equalization and decoding [10]. In [11], turbo equalization was introduced for optimal detection, while in [12], the turbo equalization concept was presented for interference repeal. The idea of turbo equalization for linear equalization and decision feedback was discussed in [13] in the context of the minimum mean square error. These architectures are suitable for eliminating ISI over communication channels where the receiver knows the channel coefficients, offering good performance, particularly when combined with powerful encoders, such as low parity density check codes or turbo codes for data coding. Turbo equalizers' fundamental structures [14] have undergone several improvements, including modifications and substitutions of decoding or equalization modules. These modifications have been optimized for channels already known to the receiver. However, some researchers have assumed that the channel factors are unknown and have proposed methods to estimate them. Notable examples are presented further to depict this idea. To drop the complexity of the turbo equalizer, in [15], the authors considered a minimum mean square error decision-aided equalizer (MMSE DAE). In [16], the authors proposed reducing the number of channel paths considered by the trellis of the MAP equalizer to lower the complexity. In the same respect, the authors suggested in [17] the application of the simplified LOG MAP equalization. The authors of [18] analyzed the limitations of the turbo equalizer presented in [5] and suggested the utilization of the MAP equalizer instead of the interference canceler in the first iteration to enhance its performance. In [19], the authors supposed that the channel coefficients are unknown and proposed their estimation using the training sequences. They demonstrated by simulation that the calculation of the interferences canceler's filters from the estimation of the channel makes it possible to reduce the quantum of iterations. In [20], the authors proposed an architecture of the turbo equalizer using a channel equalizer based on the Kalman filter. The authors recommended truncating the quantum of states in the trellis representation for the applied equalizer to lower the computational difficulty. The authors of [21] proposed an iterative architecture that uses a new training sequence for Rayleigh fading channels with frequency selectivity. They used the Valenti and Woerner estimator to estimate the channel impulse response in a multipath context. In [22], the authors used a nonlinear Kalman filter for the channel estimation, which was modeled by an auto-recursive process. The author of [23] proposed a linear prediction method with interpolation of the channel during the first iteration, followed by refinement using LS in the non-recursive and recursive versions over subsequent iterations. In [24], the authors recommended a turbo equalizer including an RLS channel estimator for selective Rayleigh channels in the frequency.