## 3 MMW Fuze Echo Signal Model under Rainfall Conditions  
A triangular wave frequency modulation method is used in MMW frequency-modulated fuzes. The frequency-rising segment of the frequency-modulated signal can be expressed as follows:  
\[S_{t}(t)=A_{t}\text{cos}2\pi\Big{(}f_{st}t+\beta t^{2}\Big{)},t>0, \tag{6}\]  
where \(A_{t}\) is the amplitude of the transmitted signal, \(f_{st}\) is the initial modulation frequency, \(\beta=\Delta F/T\) is twice the modulation slope, \(\Delta F\) is the modulation bandwidth, and \(T\) is the modulation period. Figure 3 illustrates the detection diagram of the MMW fuze in a rainfall environment. For this study, the effects of ground surface water and water adhering to the radome on the MMW signals are temporarily excluded, and raindrops are assumed to be independent of each other. The simulation assumes an open field for the detection environment, ignoring \begin{table}
\begin{tabular}{c c c c} \hline
**Operating Frequency** & **16 mm/h** & **50 mm/h** & **100 mm/h** \\ \hline
24 GHz & 0.09 dB & 0.29 dB & 0.57 dB \\
35 GHz & 0.18 dB & 0.51 dB & 0.93 dB \\
60 GHz & 0.35 dB & 0.84 dB & 1.38 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Rainfall attenuation at a detection distance of 20 m.  
Figure 2: Rainfall scattering-phase function. other obstacles to simplify the analysis and focus on the interaction between the target object and the detection system. Under rainy conditions, the received signals from the MMW fuze at any given moment can be categorized into three types: (1) signals reflected solely by the target surface; (2) signals after single or multiple scattering by raindrops; and (3) signals after multiple scattering by both raindrops and the target. In this case, the power of the MMW fuze echo signal can be expressed as follows:  
\[P_{r}\ =\ P_{rt}\ +\ P_{ra}\ +\ P_{rat}. \tag{7}\]  
where \(P_{rt}\), \(P_{ra}\), and \(P_{rat}\) represent the power values of the three types of echo signals mentioned above. For the first type of echo signal, assuming the target is a solid material with a rough surface, the MMW signal undergoes scattering and reflection effects on the target surface. According to the directive model [27; 28], the direction of the strongest scattered power aligns with the reflection direction, while other scattered power is concentrated around this direction. The energy distribution in different directions is given below:  
\[P_{s}=P_{z}F(\varphi)=P_{z}\bigg{(}\frac{1+\cos\varphi}{2}\bigg{)}^{\alpha}, \tag{8}\]  
where \(P_{z}\) denotes the maximum scattered power of the target surface, \(\varphi\) is the angle between the scattering direction and the reflection direction in the directive model, and \(\alpha\) is the scattering factor, describing the distribution of scattered power. Adjustments are made to the directional scattering function to account for the reflected signal. The modified expression for the directional scattering function is as follows:  
\[F(\varphi)= \begin{cases}\big{(}\frac{1+\cos\varphi}{2}\big{)}^{\alpha},\varphi \
eq 0\\ 1+P_{sz}+P_{z},\varphi=0\end{cases} , \tag{9}\]  
where \(P_{sz}\) denotes the total scattered energy. Therefore, the power of the first type of echo signal can be expressed:  
\[P_{rt}=\sum_{i=1}^{M}\frac{P_{t}G^{2}\lambda^{2}}{\left(4\pi\right)^{3}R^{4}}P _{zi}F(\varphi_{i})e^{-2\gamma_{t}R}, \tag{10}\]  
where \(M\) is the total number of ground surface target divisions. Figure 3: Detection of MMW fuze under rainfall conditions, where the dots represent raindrops with different sizes, and the arrows indicate the propagation direction of MMW signals.