# Performance Analysis of Ground Target Detection Utilizing Beidou Satellite Reflected Signals
## 2 Detection Method  
### Phase Difference Analysis of Reflected Signal  
When aiming to extract the navigation message from the GNSS satellite, the carrier of the signal should be removed. To accomplish this goal, the receiver needs to adjust the code phase and the carrier frequency of locally generated replicas to align with the corresponding satellite signal. In the process, the carrier phase can be estimated by the phase locked loop (PLL) in the tracking loop [29]. PLL uses a feedback function to control the behavior of the numerically controlled oscillator (NCO), which can guarantee that the replica signal is accurate. The core role of the feedback loop is to measure the differences in frequency and phase among the received signals and the locally generated replicas [30]. Based on these differences, the controller commands the NCO to increase or decrease its frequency. In the ground target detection application, information about the target was carried in the reflected signal. The phase difference information in the PLL can be used to detect it effectively. In the BeiDou software receiver used in this paper, the second-order Costas loop was used for the tracking operation, which is shown in Figure 1.  
In the absence of a ground target, the reflected signal received by the left-handed circularly polarized (LHCP) antenna came from the ground, whose geometric relationship is shown in Figure 2. In this case, the input signal of PLL \(u_{i_{1}}\) can be expressed as  
\[u_{i_{1}}=u_{ref-road}(t). \tag{1}\]  
Figure 1: Block diagrams of the phase lock loop in the tracking of BeiDou reflected signals. After mixing the input signals and the locally generated replica signals, the low pass filter was used to filter out the high frequency components contained in the mixing results. Its in-phase (\(I\)) branch and quadrature (\(Q\)) branch are expressed as  
\[I_{P}(t)=aD(t)\cos(w_{e}t+\theta_{e}) \tag{2}\]  
and  
\[Q_{P}(t)=aD(t)\sin(w_{e}t+\theta_{e}), \tag{3}\]  
where \(w_{e}\) and \(\theta_{e}\) are the frequency and initial phase differences between the input signals and the locally generated replicas, respectively. According to the above two formulas, two signals, the \(I\) branch and the \(Q\) branch, in the phase discriminator can also be expressed as  
\[\gamma_{P}(t)=I_{P}(t)+jQ_{P}(t)=aD(t)e^{j(w_{e}t+\theta_{e})}. \tag{4}\]  
The phase difference, \(\Delta\varphi_{e}\), between the two signals is calculated by using the arc tangent function:  
\[\Delta\varphi_{e}=\arctan(\frac{Q_{P}(t)}{I_{P}(t)}). \tag{5}\]  
The variance of the phase difference can be expressed as  
\[\mathrm{var}(\Delta\varphi_{e})=\frac{B}{C/N_{0}}(1+\frac{1}{2T_{coh}C/N_{0}} )rad^{2}, \tag{6}\]  
where \(B\) is the noise bandwidth, \(T_{coh}\) is the integration time, and \(C/N_{0}\) is the carrier-to-noise ratio information of BeiDou signals [31], which can be expressed as  
\[C/N_{0}=\frac{1}{T_{coh}}(\frac{M}{M+2}Z-1). \tag{7}\]  
Figure 2.0 Ground target detection using BeiDou satellite reflected signals when there is no target appearing. \(s_{1}\), \(s_{2}\), and \(s_{3}\) are the direct signals of the BeiDou satellite; \(r_{ground}\) is the signal reflected by the ground. In this paper, when calculating the carrier-to-noise ratio of BeiDou reflected signals, the \(M\) value is 20, and the \(Z\) value can be expressed as  
Figure 2.1  
where \(I_{p}\) and \(Q_{p}\) represent the results of coherent integration, and \(I_{N}\) represents the results of the coherent integration of noise. When there is a ground target in the footprint of the antenna, the geometrical relationship is as shown in Figure 2.2 The received reflected signal can be expressed as  
Figure 2.3  
where \({u_{{}_{ref-ground}}^{*}(t)}\) is the reflected signal from the ground, which does not include the reflected signal from the target itself, which is represented by \({u_{{}_{ref-target}}^{*}(t)}\).