## 4 Simulation of MMW Fuze Echo Signals Using the Monte Carlo Method
### Particle Collision Settings  
Here, the scattering direction and energy after particle collision are determined. Based on the position of the \(m\)th collision of the particle, whether the particle collides with the target is determined. If the particle reaches the target surface, it is considered to have collided with the target surface. If it does not reach the target surface, it is considered to have collided with a raindrop. When a particle collides with the target, the scattering direction after the collision is determined by random sampling from the directive model. The scattering direction is illustrated in Figure 4b, where \(XO^{\prime}Y\) denotes the target plane, \(\overset{\rightarrow}{DO^{\prime}}\) is the incident direction of the particle, \(\overset{\rightarrow}{O^{\prime}A}\) is the actual scattering direction, \(\overset{\rightarrow}{O^{\prime}A^{\prime}}\) is the specular reflection direction, and \(\varphi\) is the angle between \(\overset{\rightarrow}{O^{\prime}A}\) and \(\overset{\rightarrow}{O^{\prime}A^{\prime}}\). Additionally, \(\overset{\rightarrow}{O^{\prime}B}\), \(\overset{\rightarrow}{O^{\prime}C}\), \(\overset{\rightarrow}{O^{\prime}B^{\prime}}\), and \(\overset{\rightarrow}{O^{\prime}C^{\prime}}\) are the projections of \(\overset{\rightarrow}{O^{\prime}A}\) and \(\overset{\rightarrow}{O^{\prime}A^{\prime}}\) onto the \(XO^{\prime}Z\) and \(YO^{\prime}Z\) planes, respectively; the angles between these four projection vectors and the \(Z\) axis are \(\alpha\), \(\beta\), \(\alpha^{\prime}\), and \(\beta^{\prime}\), respectively. Based on uniform random sampling from the probability distribution function of the directive model, two projections of angle \(\varphi\) are determined, represented by \(\varphi_{n1}\) and \(\varphi_{n2}\). Then, angles \(\alpha\) and \(\beta\) are calculated as follows:  
\[\left\{ \begin{aligned} &\alpha=\varphi_{n1}+\alpha^{\prime}\\ &\beta=\varphi_{n2}+\beta^{\prime}\\ \end{aligned} \right.. \tag{30}\]  
Therefore, the scattering direction of the particle after colliding with the target is as below:  
\[\left\{ \begin{aligned} & u_{x,m}=\tan\alpha/\sqrt{\tan^{2}\alpha+\tan^{2} \beta+1}\\ & u_{y,m}=\tan\beta/\sqrt{\tan^{2}\alpha+\tan^{2}\beta+1}\\ & u_{z,m}=-1/\sqrt{\tan^{2}\alpha+\tan^{2}\beta+1}\\ \end{aligned} \right.. \tag{31}\]  
After the collision, the energy of the particle is \(W_{m}=\eta_{t}W_{m-1}\), where \(\eta_{t}\) is the target reflection coefficient, which is in the range \([0,\ 1]\). For simplicity, \(\eta_{t}\) is set to \(1\), assuming the target acts as a strong reflector. When a particle collides with a raindrop, the scattering direction is shown in Figure 4c, where \(\overset{\rightarrow}{O^{\prime\prime}A}\) is the scattering direction, \(\theta\) is the elevation angle between \(\overset{\rightarrow}{O^{\prime\prime}A}\) and the \(Z\) axis, and \(\phi\) is the azimuth angle between the projection of \(\overset{\rightarrow}{O^{\prime\prime}A}\) within the \(XO^{\prime\prime}Y\) plane and the \(X\) axis. The elevation angle \(\theta_{n1}\) is determined by random sampling based on the probability distribution function of the Mie scattering phase function. Meanwhile, the azimuth angle is determined by applying uniform random sampling. The expressions for the scattered elevation angle and azimuth angle are as follows:  
\[\left\{ \begin{aligned} &\theta=\theta_{n1}\\ &\phi=2\pi\xi_{2}\\ \end{aligned} \right.. \tag{32}\]Then, the scattering direction of the particle after colliding with the raindrop is presented below:  
\[\left\{ \begin{aligned} \mu_{x,m}&=\frac{\sin\theta}{\sqrt{1- \mu_{z,m-1}^{2}}}(\mu_{x,m-1}\mu_{z,m-1}\cos\phi-\mu_{y,m-1}\sin\phi)+\mu_{x,m- 1}\cos\theta\\ \mu_{y,m}&=\frac{\sin\theta}{\sqrt{1-\mu_{z,m-1}^{2} }}(\mu_{y,m-1}\mu_{z,m-1}\cos\phi-\mu_{x,m-1}\sin\phi)+\mu_{y,m-1}\cos\theta\\ &\mu_{z,m}=-\sin\theta\cos\phi\sqrt{1-\mu_{z,m-1}^{2}}+\mu_{z,m- 1}\cos\theta\end{aligned}