## 3 Signal Level Compatibility Analysis
### Analysis of the Effect of Interference on the Capture Performance of BeiDou  
The presence of interfering signals may make it difficult for the receiving equipment to accurately capture the target signal. In a low signal-to-noise ratio environment, the interfering signal may mask the target signal, making it difficult for the receiving equipment to correctly identify and capture the target signal, thus affecting communication quality and reliability. This section focuses on the analysis of the impact of interference on the performance of BeiDou capture, according to the literature [17] to establish a binary signal detection model. \(H_{0}\) expresses the condition that only noise exists; \(H_{1}\) expresses the condition that interference and noise exist at the same time, then the false alarm probability of the signal can be expressed as:  
\[P_{f}=\int_{V_{T}}^{\infty}p(z\mid H_{0})dz \tag{7}\]  
The signal detection probability can be expressed as:  
\[P_{d}=\int_{V_{T}}^{\infty}p(z\mid H_{1})dz \tag{8}\]  
Figure 7: Code tracking error varies with carrier-to-noise ratio. where \(V_{t}\) is a detection threshold; \(p(z\mid H_{1})\) is the probability density of detection of the input signal powerz in the presence of both interference and noise; and \(p(z\mid H_{0})\) is the probability density of detection of the input signal power \(z\) in the presence of only noise. When the condition \(H_{1}\) holds, \(z\) follows a non-central Cartesian distribution with 2 M degrees of freedom, and when the condition holds \(H_{0}\), \(z\) follows a central Cartesian distribution with 2 M degrees of freedom. The conditional probability density of \(z\) is given by the following expression [17]:  
\[P(z\mid H_{1})=\frac{1}{2}(\frac{z}{\lambda})^{\frac{1}{2}(M-1)}exp(-\frac{1} {2}(z+\lambda))I_{M-1}(\sqrt{z\lambda}) \tag{9}\]  
\[P(z\mid H_{0})=\frac{1}{2^{M}\Gamma(M)}z^{M-1}exp(-\frac{z}{2}) \tag{10}\]  
\[\lambda=2M(C/N_{0})T_{L} \tag{11}\]  
where \(\lambda\) denotes the non-central parameter of the non-central Cartesian distribution, \(T_{L}\) is the pre-detection equivalent carrier-to-noise ratio, \(C/N_{0}\) is the pre-detection integration time, \(\Gamma(M)\) is the gamma function, defined as \(\Gamma(M)=(M-1)!\), and \(I_{M-1}(\sqrt{z\lambda})\) is the first class \(M-1\)-order Bessel function. Firstly, the false-alarm probability \(P_{f}\) of signal detection is determined; then, subsequently, the judgment threshold \(V_{t}\) can be obtained according to Equation (7), and finally the detection probability is obtained. The false alarm probability of signal detection should be minimized as much as possible during the capture process. In this paper, the capture performance is simulated in the following simulation by taking the pre-detection integration time of 20 ms and 10 ms for the case of false alarm probability as \(1\times 10^{-6}\). The signal detection probability is usually used to measure the accuracy of the receiver in detecting signals and is one of the important indicators for evaluating the performance of a communication system. From Figure 8, it can be seen that there is a positive correlation between the signal detection probability and the equivalent carrier-to-noise ratio, and the signal detection probability increases as the equivalent carrier-to-noise ratio increases. In order to obtain the effect of 5G transmit power on the BeiDou signal, 5G is introduced as an interfering signal into the interference-to-signal ratio ISR, and the relationship between the equivalent carrier-to-noise ratio and the interference-to-signal ratio is:  
\[ISR=G_{SVi}-G_{J}+10 lg\bigg{[}QR_{c}(10^{-\frac{(CN_{0})_{0}dLN}{10}}-10^{- \frac{(CN_{0})_{0}}{N}})\bigg{]} \tag{12}\]  
where \(G_{SVi}\) is the antenna gain pointing to the first satellite, \(G_{J}\) is the antenna gain, and \(CN_{0}\) is the receiver carrier-to-noise ratio in the case of no interference.