## 3 Methods
### Multiple Targets Detection Doppler Compensation
where \({}^{1}fd^{i}_{\text{B1I,r}^{j}}\) is the compensation Doppler frequency of method 1 and \(j^{\prime}\) is the compensation number. Using the estimated information of targets, the compensated Doppler frequencies are set as Doppler frequency estimation values in method 2. \[{}^{2}fd^{i,j^{\prime}}_{\text{B1I,r}}={}^{2}fd^{i,j^{\prime}}_{\text{B1I,e}} \tag{11}\]  
where \({}^{2}fd^{i,j^{\prime}}_{\text{B1I,r}}\) is the compensation Doppler frequency of method 2. After acquiring the corresponding Doppler frequency, the compensation phase is expressed as  
\[ \begin{split}{}^{Q}CH^{i,j^{\prime}}_{\text{B3I,r}}& =\sum\limits_{q=1}^{Q}\sum\limits_{n=1}^{N_{C}}u_{T_{c}}(t-qT_{P}-nT _{c})\cdot 2\pi fd^{i,j^{\prime}}_{\text{B3I,r}}(t-qT_{P}-nT_{c})\\ &={}^{Q}CPh^{i,j^{\prime}}_{\text{B3I,r}}(fd^{i,j^{\prime}}_{ \text{B3I,r}})\end{split}  \tag{12}\]  
To solve the multiple Doppler phase influences and restore the weakened target peak, we propose the multiple target Doppler compensation (MTDC) method, which compensates for the echoes with multiple Doppler phase sequences at the corresponding range code period position of targets and fetches the maximum value of the filter outputs as the compensation result. Next, (13) expresses the compensated echo. \[{}^{Q}CR^{i,j,j^{\prime}}_{\text{B3I,r}}={}^{Q}R^{i,j}_{\text{B3I,r}}({}^{Q} PH^{i,j}_{\text{B3I,r}}+{}^{Q}CPh^{i,j^{\prime}}_{\text{B3I,r}}) \tag{13}\]  
where \({}^{Q}CR^{i,j,j^{\prime}}_{\text{B3I,r}}\) denotes the echoes of satellite \(i\) and target \(j\) with compensation \(j^{\prime}\). The echoes from satellite \(i\) with multiple-target Doppler compensation can be expressed as  
\[{}^{Q}CR^{i,j^{\prime}}_{\text{B3I}}={}^{Q}CR^{i}_{\text{B3I,d}}+\sum\limits_{ j=1}^{N}{}^{Q}CR^{i,j,j^{\prime}}_{\text{B3I,r}} \tag{14}\]  
Next, we matched echoes with the corresponding satellite signal as  
\[{}^{Q}CM^{i,j^{\prime}}_{\text{B1I}}={}^{Q}CR^{i,j^{\prime}}_{\text{B3I}} \otimes s^{i}_{\text{B3I}}(t) \tag{15}\]  
Finally, (16) shows the selected maximum of each compensation as the result. \[{}^{Q}CM^{i}_{\text{B1I}}{}^{c}=\max({}^{Q}CM^{i,1}_{\text{B1I}},\dots,{}^{Q} CM^{i,j^{\prime}}_{\text{B1I}},\dots,{}^{Q}CM^{i,N^{\prime}}_{\text{B1I}}) \tag{16}\]  
The formula mentioned above is the one-dimensional range-Doppler compensation of multiple target reflections. In the next part, we show the Doppler and the delay two-dimensional Doppler compensation methods for Doppler delay image target peak compensation. To better express the Doppler phase changes among code element periods ranging code periods in a discrete time system, we express the Doppler phase in the form of a matrix as (17), where each row expresses a period of \(T_{P}\), and its range is from 0 to \(Q\cdot T_{P}\). \[ \begin{split} ph^{i}_{\text{B1I,r}j}&=fd^{i}_{ \text{B1I,r}j}\text{:}\\ &\begin{bmatrix}0+0,&\dots,&0+nT_{c},&\dots,&0+(N_{c}-1)T_{c}\\ \vdots&&\vdots&&\vdots\\ qT_{P}+0,&\dots,&qT_{P}+nT_{c},&\dots,&qT_{P}+(N_{c}-1)T_{c}\\ \vdots&&\vdots&&\vdots\\ (Q-1)T_{P}+0,&\dots,&(Q-1)T_{P}+nT_{c},&\dots,&(Q-1)T_{P}+(N_{c}-1)T_{c} \end{bmatrix}\end{split}  \tag{17}\]  
where we can find that the number column is incremented by \(T_{c}\) in each row and is incremented by \(T_{P}\) in each column. Therefore, we can take the FFT in the columns of the matrix to first acquire the Doppler information of targets. Next, we compensate the matrix in each \(T_{P}\) row for the duration based on the compensation phase. Finally, we can obtain the compensated DDM results by the proposed MTDC method.