## 2 Related Work  
In practical communication systems, transmitted signals are subject to various distortions, such as noise, inter symbol interference due to multipath propagation, and the nonlinear effects of amplifiers. All these factors alter the original signal, so it is essential to restore the input of the channel by observing its output [25; 26; 27].  
The process of restoring the original signal uses a filter such as a finite impulse response filter as a channel equalizer [28; 29]. The role of the equalizer filter is to use the observations/estimations of the channel to restore the transmitted signal and reduce the distortions caused by the channel as much as possible [7; 10; 15; 30].  
The main issue consists of estimating the values of the coefficients of the equalizer filter, to be able to restore the transmitted signal with a low probability of error. To achieve this fact, in this work, the channel model was first estimated, and using this estimate, adjustment of the equalizer filter coefficients was then optimized.  
The optimization of these coefficients that best minimizes the probability of the appearance of errors is strongly desired. Several optimization criteria are proposed in this paper, such as ZF and MMSE.  
Channel equalization being achieved before the decoding process is considered a significant and advanced issue in wireless communication [30].  
In this respect, in [23], the authors realized a BER vs. SNR performance comparison of a turbo MMSE equalizer for uplink narrowband Internet of Things systems using BPSK and QPSK (quadrature phase shift keying) with a different number of iterations. The simulations were performed in Matlab. The analysis proved that the performance of QPSK modulation was comparable with the performance of BPSK modulation.  
In [31], the authors proposed expectation propagation (EP) as a turbo equalizer, to offer a reliable estimate that, in comparison with linear MMSE equalization, revealed a better performance. This newly proposed equalizer presented gains in the range of 1.5-5 dB in contrast to linear minimum mean square error equalization.  
A new method was presented in [32] for executing an adaptive DFE (decision feedback equalizer). Whereby, both least square and recursive least square algorithms were employed to predict the channel and adjust the equalizer vector, jointly. The performance of the proposed algorithm was close to the performance of maximum likelihood equalization.  
An investigation regarding the usage of decision feedback with a turbo equalizer to improve the limits of linear equalizers was undertaken in [33]. The turbo DFE using hard feedback with symbol-wise adaptive filters performed poorly at low spectral efficiency, whereas the turbo DFE using soft posterior feedback with symbol-wise invariant filters was outperformed by time-varying linear equalizers-interference cancellers.  
A global iteration method using feedback among LDPC and a LT decoder in the raptor code was applied in [34] to improve the BER performance. The authors proposed a joint equalization and raptor decoding algorithm, which could use the updated information from raptor decoders.  
The performance of LDPC codes and polar codes with channel equalization were evaluated in [35] using free-space optical transmission experiments, demonstrating that the BLER (block error rate) of polar codes was better than that of LDPC codes.  
Polar codes with adaptive channel equalization were adopted in [36], to decrease the feedback information. In this respect, a hybrid automatic repeat request mechanism was applied to moderate the effect of estimation errors. The results showed that the proposed scheme outperformed the turbo equalization in terms of BER.