## 4 Performance Assessment
### Simulation Analysis
#### 4.1.2 Simulation Results  
In simulation analysis, bit error rate (BER), symbol error rate (SER), and output mutual information (MI) are used as performance indicators. The mutual information (MI) can be computed using  
\[MI(L_{o};X)=\frac{1}{2}\sum\limits_{x\in[-1,1]}\int_{-\infty}^{+\infty}f_{L_{o}} (|x\rangle\cdot\log_{2}\frac{2f_{L_{o}}(|x\rangle}{f_{L_{o}}(|l\!+\!1)+f_{L_{o}} (|l\!-\!1)}\mathrm{d}l \tag{24}\]  
where \(L_{o}\) denotes the output LLRs of the equalizer or the decoder in the proposed method, and \(f_{L_{o}}(|x\rangle)\) denotes the conditional probability distribution function that can be estimated through observing the histogram of the output LLRs. The value of MI ranges from 0 to 1. MI = 0 denotes no a priori information, and MI = 1 denotes perfect a priori information. MI not only can reflect the reliability of the output LLR, but also can be used to determine the iteration number of turbo equalization. When the output LLR of the decoder is close to 1, the performance of turbo equalization reaches convergence, and the iterative process can be terminated.  
The SER and BER of the proposed method and the output MI after equalization and decoding in the proposed method for \(E_{b}/N_{0}=2\)dB are shown in Figures 5 and 6. The performance of the method using traditional MMSE adaptive equalization is also shown in Figure 5 for comparison with the proposed method. In Figure 5, the signal to noise ratio (SNR) can be computed by \(E_{b}/N_{0}=\sum\limits_{k=0}^{L-1}\left|h_{k}^{m}\right|^{2}/2\delta_{\eta}^{ 2}Q\).  
Several observations can be obtained through Figures 5 and 6. Firstly, when the same \(E_{b}/N_{0}\) is used, the performance of the method using traditional MMSE adaptive equalization is equal to that of the first equalization of the proposed method. The reason for this observation is that in the first equalization of the proposed method, the average \(\overline{v}^{m}=\frac{1}{N_{x}}\sum\limits_{n=1}^{N_{x}}v_{n}^{m}\) equals 1 due to the a priori LLR of the equalizer \(L_{a}^{c}\!\left(\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!  
is that after the second equalization and decoding, the output MI of the decoder is very close to 1 as shown in Figure 6, and therefore the additional iterative process cannot improve the performance since the communication performance has reached convergence. Fourthly, when mapping patterns are the same, compared with the method using traditional MMSE adaptive equalization, the proposed method achieves better performance after convergence with bit rate reduction by half. The reason for this observation is that the 1/2 rate convolutional code used in turbo equalization can improve the communication performance, but it also leads to the reduction of transmission efficiency. Lastly, when different mapping patterns are adopted, the proposed method with 8 phase shift keying (8 PSK) mapping achieves better communication performance at the higher bit rate than the method using traditional MMSE adaptive equalization with BPSK mapping. The reason for this observation is that the iteration process of turbo equalization and the extrinsic information exchanging in the proposed method and can further suppress interference and improve communication performance.  
Figure 5: Simulation results of the proposed method and the method using traditional MMSE adaptive equalization: (**a**) and (**b**) the symbol error rate (SER) of low-complexity channel estimation (CE)-based MMSE equalization in the proposed method and the method using traditional MMSE adaptive equalization; (**c**) and (**d**) the bit error rate (BER) after decoding in the proposed method and the method using traditional MMSE adaptive equalization.  
The above simulation results illustrate that the performance of the proposed method is superior to that of the method using traditional MMSE adaptive equalization.