## 3 Methods  
### Multiple Targets Detection Doppler Compensation  
In order to remove the Doppler phase influence during multiple targets detection, we further studied echo processing. The B3I signal is set as an example to express the formula, and the B1I signal has the same form. Take Formulas (1) and (3) into (5) as  
\[ \begin{split} QR^{i,j}_{\text{B3I,r}}&=\sum\limits_ {q=1}^{Q}u_{T_{P}}(t-q*T_{P})\cdot R^{i,j}_{\text{B3I,r}}\\ &=a^{i,j}_{\text{B3I,r}}A^{i}_{\text{B3I}}C^{i}_{\text{B3I}}\cdot \sum\limits_{q=1}^{Q}\left\{u\left(t-\tau^{i}_{rj}-q\cdot T_{P}\right)\right.\\ &\left.\cdot\cos(2\pi fd^{i,j}_{\text{B3I,r}}(t-\tau^{i,j}_{\text {B1I,r}}-q\cdot T_{P})+\varphi^{i}_{\text{B3I}})\right\}\end{split}  \tag{6}\]  
where \(\mathcal{Q}R^{i,j}_{\text{B3I,r}}\) is the received reflected B3I signals transmitted by satellite \(i\) and reflected by target \(j\) during \(Q\) ranging code periods. According to the ranging code \(C^{i}_{\text{B3I}}\), we find the duty ratio of the signal to be 1 in a ranging code period. Therefore, the Doppler phase changes during a signal ranging code period cannot be ignored. The echoes model (6) is improved as  
\[ \begin{split} QR^{i,j}_{\text{B3I,r}}&=a^{i,j}_{ \text{B3I,r}}A^{i}_{\text{B3I}}C^{i}_{\text{B3I}}\cdot\sum\limits_{q=1}^{Q} \sum\limits_{n=1}^{N_{C}}\left\{u_{T_{c}}\left(t-\tau^{i,j}_{r}-qT_{P}-nT_{c} \right)\right.\\ &\left.\cdot\cos(2\pi fd^{i,j}_{\text{B3I,r}}(t-\tau^{i,j}_{ \text{B3I,r}}-qT_{P}-nT_{c})+\varphi^{i}_{\text{B3I}})\right\}\end{split}  \tag{7}\]  
where \(N_{c}\) is the number of range code elements, and \(T_{c}\) is the range code element period. In (7), the Doppler phase of echoes changes with each ranging code element, and it influences the target range peak result level of the matched filter. The normalized autocorrelation peak level changing with Doppler frequency is shown in Figure 2, where the peak level reduces slowly first and then drops quickly to lower than 0.1 when the Doppler frequency is more than 1000 Hz. Moreover, the relevant parameters are described in the simulation section. Focusing on the Doppler phase influence, we express (7) as  
\[ \begin{array}{l}\mathcal{Q}Ph_{\text{B3I,r}}^{i,j}=\mathcal{Q}R_{\text{B3I,r}}^ {i,j}=\mathcal{Q}R_{\text{B3I,r}}^{i,j}(Ph_{\text{B3I,r}}^{i,j})\\ \mathcal{Q}Ph_{\text{B3I,r}}^{i,j}=\sum\limits_{q=1}^{Q}\sum\limits_{n=1}^{N_{ C}}u_{T_{c}}(t-qT_{p}-nT_{c})\cdot 2\pi fd_{\text{B3I,r}}^{i,j}(t-qT_{p}-nT_{c})\\ =\mathcal{Q}Ph_{\text{B3I,r}}^{i,j}(fd_{\text{B3I,r}}^{i,j})\end{array}  \tag{8}\]  
where we can find that \(\mathcal{Q}Ph_{\text{B3I,r}}^{i,j}\) is the sequence with a length of \(Q\cdot N_{c}\) and is up to \(fd_{\text{B3I,r}}^{i,j}\). Moreover, each target reflects echo carriers with different Doppler frequencies, and we needed to reduce multiple Doppler phase influences for better detection of the target peak. Before the Doppler phase compensation, the compensatory Doppler frequency of target echoes needed to be acquired. We propose two methods: method 1, without estimation and method 2, with estimation. In method 1, we acquired the Doppler frequency range of targets by satellite navigation code data and receiver data as  
\[ \begin{array}{l}fd_{\text{B1I,min}}^{i}=\\ \min(fd_{\text{B1I,sat\_rec}}^{i}+fd_{\text{B1I,rec\_tar\_max}}^{i},fd_{\text{ B1I,sat\_rec}}^{i}-fd_{\text{B1I,rec\_tar\_max}}^{i})\\ fd_{\text{B1I,max}}^{i}=\\ \max(fd_{\text{B1I,sat\_rec}}^{i}+fd_{\text{B1I,rec\_tar\_max}}^{i},fd_{\text{ B1I,sat\_rec}}^{i}-fd_{\text{B1I,rec\_tar\_max}}^{i})\end{array}  \tag{9}\]  
and the compensated Doppler frequencies are acquired by sampling in the range with an interval of 125 Hz as  
\[ \begin{array}{l}^{1}fd_{\text{B1I,r}}^{i,j}=\prescript{1}fd_{\text{B1I,min}}^ {i}+250*(j^{\prime}-1)\\ \prescript{1}fd_{\text{B1I,min}}^{i}\leq\prescript{1}fd_{\text{B1I,r}}^{i,j^{ \prime}}\leq\prescript{1}fd_{\text{B1I,max}}^{i}\end{array}  \tag{10}\]  
Figure 2: Target peak curves with the Doppler channel in the detection of BDS signals.