## 3 Adaptive Resampling Method  
### Adaptive Process  
The proposed DAM-TEQ realizes the adaptive process by using the resampling rate update method based on equalization coefficient detection, as shown in the coefficient detection and branch update blocks in Figure 2. The adaptive process consists of initialization, an adaptive update of the center resampling rate and branch interval, and the selection of an appropriate resampling rate interval, and the implementation process is shown in Figure 3.  
According to (10) and (14), it can be seen that in the first stage of the DAM-TEQ, the equalizer output is composed of the M parallel FFF and FBF outputs, where M FFF provides received signal branches with different Doppler shifts. In the process of parallel equalization, the content of each branch in the final output \(\hat{x}_{n}\) is determined by the equalization coefficient after convergence. The Doppler Effect may worsen when the resampling factor \(u_{m}\) deviates from the correct Doppler factor, so this branch should occupy a small proportion of \(\hat{x}_{n}\) to ensure a small estimation error and a good equilibrium effect. Based on this fact, it is possible to determine the detection method of the first-stage of the equalizer: the most suitable resampling branch is selected by judging the amplitude of the FFF coefficient, which is expressed as  
\[[|c_{max}|,m_{max}]=max(|c_{1}|,\ldots,|c_{M}|) \tag{22}\]  
where \(|c_{max}|\) is the FFF coefficient with the largest amplitude, \(m_{max}\) is the branch's subscript, and the branch's resampling rate is \(f_{d}\). Let \(f_{d}\) be the central resampling rate, and then select the estimated Doppler factor \(\mu_{max}\) by updating \(f_{d}\) and narrowing the sampling rate interval \(\Delta f\), continuously changing the resampling factor and retaining the resampling branch with a large coefficient amplitude. To facilitate the study, the number of resampled branches \(M=3\). The specific implementation steps are as follows:  
Step 1: Initialize the resampling branch. Each resampling factor \(\mu_{m}\) corresponds to a resampling rate \(f_{m}\), and firstly, the initial central sampling rate \(f_{d0}\) (generally using the initial sampling rate \(f_{s}\)) and the initial resampling rate interval \(\Delta f_{0}\) are set, then the resampling rates of the three branches are \(f_{d0}-\Delta f_{d0},f_{d0},f_{d0}+\Delta f_{d0}\);  
Figure 3: Adaptive process.  
Step 2: Search and update the central resampling rate. The first-stage equalizer processes the parallel input to detect the equalization coefficient, and selects the one with the largest amplitude of the FFF coefficient in the three branches as the central branch, and its sampling rate is updated to the central sampling rate \(f_{d}\);  
Step 3: Update the adaptive resampling rate interval \(\Delta f\). If the center frequency is the same as the previous search, the \(\Delta f\) is reduced. If it is different, the \(\Delta f\) is kept the same and the resampling rates for the three updated branches are \(f_{d}-\Delta f,f_{d},f_{d}+\Delta f\). Repeat steps 2 and 3 until the \(\Delta f\) is narrowed down to a preset minimum to achieve a coarse Doppler estimate;  
Step 4: Select the resampling rate interval \(\Delta f_{d}\). In order to provide a frequency-shift basis for the parallel input of the second-stage equalizer, re-select the appropriate sampling rate interval \(\Delta f_{d}\) and send the three branches of \(f_{d}-\Delta f_{d},f_{d},f_{d}+\Delta f_{d}\) to the multi-resampling turbo equalizer for joint equalization decoding.