## 2 System Model
### Direct Adaptive Multi-Resampling Turbo Equalizer
#### 2.2.2 Parallel Turbo Equalizer  
The second-stage equalizer is the turbo equalizer, which is different from the first-stage equalizer in that it uses soft symbols to achieve decision feedback equalization, and the equalized symbols are passed between the equalizer and the decoder in the form of extrinsic information [34]. Due to the lack of accuracy of the Doppler factor coarsely estimated by the first-stage equalizer, the second-stage equalizer also adopts the method of parallel multi-resampling, and the M-resampled branches selected by the last structure are combined as inputs, and these branches near the correct Doppler factor provide more frequency shift bases for the equalizer [35]. The result of the equalizer output is  
\[\hat{x}_{n}=\sum_{m=1}^{M}\epsilon_{m,n}^{H}r_{m,n}e^{-j\theta_{m}}-\mathbf{b}^{H} \overline{x}_{n} \tag{15}\]  
where \(\overline{x}_{n}=\left[\overline{x}_{n-1},\dots,\overline{x}_{n-N_{3}} \right]^{\prime}\), and \(\overline{x}_{n}\) is the prior mean at time \(n\), which is expressed as  
\[\overline{x}_{n}=\sum_{\alpha_{n,i}\in S}\alpha_{n,i}P(x_{n}=\alpha_{n,i}) \tag{16}\]  
where \(S\) is the corresponding set of symbols of the symbol \(x_{n}\) and \(\alpha_{n,i}=1\) or \(-1\) for BPSK modulation. In BPSK modulation, each bit is mapped to a symbol, where symbols \(1\) and \(-1\) correspond to bits \(1\) and \(0\). In detail, \(1\), \(0\) to \(1\), \(-1\) is achieved through mapping, and \(1\), \(-1\) to \(1\), \(0\) is achieved through demapping. In addition to the judgment result, the equalizer also outputs the extrinsic information \(L_{e}^{E}(\alpha_{n,i})\), which is used for transmission and iteration with the decoder, and it is expressed as  
\[L_{e}^{E}(\alpha_{n,i}) =\underbrace{ln\frac{P(\alpha_{n,i}=1|\hat{x}_{n})}{P(\alpha_{n, i}=-1|\hat{x}_{n})}}_{\text{Posterior LLR}}-\underbrace{ln\frac{P(\alpha_{n,i}=1)}{P(\alpha_{n,i}=-1)}}_{ \text{Prior LLR}} \tag{17}\] \[=ln\frac{P\left(\hat{x}_{n}|\alpha_{n,i}=1\right)}{p\left(\hat{x} _{n}|\alpha_{n,i}=-1\right)}\] \[p(\hat{x}_{n}|\alpha_{n,i}) =\frac{1}{\pi\delta_{n}^{2}}exp\left(-\frac{|\hat{x}_{n}-\mu_{n} \alpha_{n,i}|^{2}}{\delta_{n}^{2}}\right),\alpha_{n,i}=1\;or\;-1 \tag{18}\]  
In general, assuming that the channel impulse response remains unchanged within the length of a frame of data, the length is \(N_{s}\), and the channel parameters \({\mu^{\prime}}_{n}\) and \(\delta_{n}^{2}\) are approximated by time averaging:  
\[{\mu^{\prime}}_{n}=\frac{1}{N_{s}}\sum_{n=1}^{N_{s}}\frac{\hat{x}_{n}}{Q(\hat {x}_{n})} \tag{19}\]  
\[\delta_{n}^{2}=\frac{1}{N_{s}}\sum_{n=1}^{N_{s}}\left|\hat{x}_{n}-{\mu^{ \prime}}_{n}Q(\hat{x}_{n})\right|^{2} \tag{20}\]  
where \(Q(\hat{x}_{n})\) is the decided value of \(\hat{x}_{n}\). The extrinsic information \(L_{e}^{E}(\alpha_{n,i})\) output by the equalizer is obtained by (15)-(20), and the prior information \(L_{a}^{E}(\alpha_{n,i})\) input to the decoder is obtained after demapping and deinterleaving. The decoder described in [36] is used for decoding, and the process is not described in detail. The decoder obtains two outputs: the decision information bit result \(\hat{d}_{n}\) and the decoder extrinsic information \(L_{e}^{D}(\alpha_{n,i})\). The extrinsic information passes through the S16 interleaver and mapping to obtain the prior information \(L_{a}^{D}(\alpha_{n,i})\) input to the equalizer, which is used to calculate the prior probability \(P(x_{n}=\alpha_{n,i})\) in (16):  
\[P(x_{n}=\alpha_{n,i})=\frac{1}{2}\Bigg{[}1+\alpha_{n,i}tanh\frac{L_{a}^{D}( \alpha_{n,i})}{2}\Bigg{]} \tag{21}\]  
With the iteration, the accuracy of the prior information gradually improves, and finally the joint decoding effect of the turbo equalizer is achieved. The Doppler estimation, compensation, and communication reception and decoding process are realized through the cascading of first- and second-stage equalizers.