# Performance Analysis of Ground Target Detection Utilizing Beidou Satellite Reflected Signals
## 2 Detection Method
### Analysis of Detection Probability  
Based on the target detection mechanism, an equivalence analysis of whether the target appears or not is carried out. Additionally, a binary construction model is established (seen in Figure 4), which can be expressed as follows:  
\[H=\left\{\begin{array}{ll}\mathrm{H}_{0}:&s(t)=s_{0}(t)+n,\quad\quad\mathrm{ no\ target}\\ \mathrm{H}_{1}:&s(t)=s_{1}(t)+n,\quad\quad\mathrm{target}\end{array}\right. \tag{14}\]  
where \(s_{0}(t)\) denotes the signal received by the LHCP antenna when the target does not exist, \(s_{1}(t)\) denotes the signal received when the target exists, and the observation noise \(n\) is Gaussian noise with a mean value of 0 and a variance of \(\sigma^{2}\). When the signal \(s(t)\) is observed N times in the range of 0 to T, its two probability distributions can be expressed as follows:  
\[p(S\left|H_{0}\right.) =p(s_{1}s_{2}\cdots s_{N}/H_{0}) \tag{15}\] \[=p(s_{1}/H_{0})\cdot p(s_{2}/H_{0})\cdot\cdot\cdot p(s_{N}/H_{0})\]  
and  
\[p(S\left|H_{1}\right.) =p(s_{1}s_{2}\cdots s_{N}/H_{1}) \tag{16}\] \[=p(s_{1}/H_{1})\cdot p(s_{2}/H_{1})\cdot\cdot\cdot p(s_{N}/H_{1}).\]  
Figure 4: Binary hypothesis model of target detection using BeiDou reflected signals.  
Among them, the probability density of each observation can be expressed as  
\[p(s_{k}\left|H_{0}\right.)=\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{(s_{k}-s_{0k}) ^{2}}{2\sigma^{2}}) \tag{17}\]  
and  
\[p(s_{k}\left|H_{1}\right.)=\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{(s_{k}-s_{1k })^{2}}{2\sigma^{2}}). \tag{18}\]  
The likelihood function ratio of the binary hypothesis model can be expressed as  
\[l(s(t))=\frac{P(s(t)\left|H_{1}\right.)}{P(s(t)\left|H_{0}\right.)}\overset{H_{ 1}}{>}\frac{(C_{10}-C_{00})}{(C_{01}-C_{11})}\frac{P(H_{0})}{P(H_{1})}=l_{0} \tag{19}\]  
where \(l_{0}\) is the threshold value. \(C_{00}\), \(C_{01}\), \(C_{10}\), and \(C_{11}\) are the cost factors, which are determined by the criterion of selection. In the optimal communication system, the minimum total error probability criterion (namely Bayesian criterion) is usually used, where the cost factors \(C_{00}=C_{00}=0\) and \(C_{10}=C_{01}=1\).  
As shown in Figure 5, the results of the target detection can be divided into four situations: (1) \(P(H_{0}/H_{0})\), which means there is no target and the detection result is correct; (2) \(P(H_{0}/H_{1})\), which means there is a target and the detection result is wrong; (3) \(P(H_{1}/H_{0})\), which means there is no target and the detection result is wrong; and (4) \(P(H_{1}/H_{1})\), which means there is a target and the detection result is correct.  
Based on this, the false alarm probability \(P_{f}\), the detection probability \(P_{d}\), and the miss probability \(P_{m}\) of the target detection can be expressed as  
\[\begin{split} P_{f}&=P(H_{1}/H_{0})=\int_{l_{0}}^{ \infty}P(s(t)\left|H_{0}\right.)dt\\ &=\int_{l_{0}}^{\infty}\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{( s(t)-s_{0}(t))^{2}}{2\sigma^{2}})dt\end{split} \tag{20}\]  
and  
\[\begin{split} P_{m}&=P(H_{0}/H_{1})=\int_{-\infty}^{l_{0}}P( s(t)|H_{1})dt\\ &=\int_{-\infty}^{l_{0}}\frac{1}{\sqrt{2\pi\
u}}\exp(-\frac{(s(t)-s _{1}(t))^{2}}{2\sigma^{2}})dt\end{split} \tag{21}\]  
and  
\[\begin{split} P_{D}&=P(H_{1}/H_{1})=\int_{l_{0}}^{ \infty}P(s(t)|H_{1})dt\\ &=\int_{l_{0}}^{\infty}\frac{1}{\sqrt{2\pi\
u}}\exp(-\frac{(s(t) -s_{1}(t))^{2}}{2\sigma^{2}})dt.\end{split} \tag{22}\]