## 2 Attenuation and Scattering Characteristics of MMW Signals under Rainfall Conditions  
Attenuation refers to the loss of energy when MMW signals are absorbed by raindrops as they pass through a rain field. In contrast, scattering involves the deflection of MMW signal propagation paths by raindrops, leading to the dispersion and spread of energy in various directions. The effects of rainfall on MMW scattering and attenuation depend on factors such as rainfall intensity, raindrop shape, raindrop size, and the operating frequency of the fuze. Raindrops are typically ellipsoidal or oblate spheroidal in shape but are often modeled as spherical in engineering applications. This simplification is used because the spherical model is less complex, and its simulation results are relatively close to those of ellipsoidal raindrops [22]. The raindrop size distribution, which describes the number of raindrops of various sizes, is commonly represented by the Marshall-Palmer distribution [23]. The expression for this distribution is as follows:  
\[N(I,D)=8000\text{exp}(-4.1I^{-0.21}D), \tag{1}\]  
where \(N(I,D)\) is the number of raindrops per cubic meter, \(I\) is the rainfall intensity, and \(D\) is the raindrop diameter. In rainy environments, MMW signals undergo refraction, reflection, and absorption by raindrops, leading to signal attenuation and scattering. When the signal frequency exceeds 10 GHz, Mie scattering theory should be used to study the attenuation and backscatter characteristics of raindrop particles [24]. The calculation methods for the attenuation cross-section and scattering cross-section of raindrops based on Mie scattering theory [25] are as follows:  
\[\sigma_{t}(\lambda,D)=\frac{2\pi}{k^{2}}\text{$\sum_{n=1}^{\infty}$}(2n+1) \text{Re}(a_{n}+b_{n}); \tag{2}\]  
\[\sigma_{b}(\lambda,D)=\frac{\pi}{k^{2}}\text{$\left|\text{$\sum_{n=1}^{\infty }$}(2n+1)(-1)^{n}(a_{n}-b_{n})\right|^{2}$}, \tag{3}\]  
where \(k=2\pi/\lambda\) is the wave number, \(\text{Re}(\cdot)\) represents the real part of a number, and \(a_{n}\) and \(b_{n}\) are Mie coefficients, with the empirical upper value of n referenced in [25]. The MMW attenuation coefficient, which reflects the attenuation capacity of signals under rainy conditions, is given by [26]:  
\[\gamma_{t}=4.343\cdot 10^{-6}\int_{0}^{\infty}N(I,D)\sigma_{t}(\lambda,D)dD. \tag{4}\]  
Figure 1 shows the variation of the attenuation coefficient with rainfall intensity, indicating that the MMW attenuation coefficient increases with both frequency and rainfall intensity. The detection range of MMW proximity fuzes is typically within 20 m, therefore, the calculated rainfall attenuation at a detection distance of 20 m is presented in Table 1. The three commonly used operating frequencies for MMW fuzes are 24, 35, and 60 GHz, while 16, 50, and 100 mm/h correspond to heavy rain, torrential rain, and torrential rainstorm, respectively. Figure 1: Attenuation coefficient. In addition, raindrop particles cause scattering effects on MMW signal transmission, leading to the uneven distribution of energy in various directions after scattering. The scattering-phase function describes the spatial distribution of energy following MMW signal scattering by raindrops [25]. Its expression is as follows:  
\[P(\theta)=\frac{\lambda^{2}}{2\pi\mathcal{Q}_{s}}\Big{[}\big{|}S_{1}(\theta) \big{|}^{2}+\big{|}S_{2}(\theta)\big{|}^{2}\Big{]}, \tag{5}\]  
where \(S_{1}(\theta)\) and \(S_{2}(\theta)\) are the vertical and parallel components of the complex amplitude function, respectively, and \(Q_{s}\) is the raindrop scattering coefficient. Figure 2 shows the scattering-phase function values for three different raindrop sizes at an operating frequency of 35 GHz. When the raindrop diameter is small, the scattering-phase function exhibits two peaks at 0 rad and \(\pi\) rad, with both peaks having approximately the same magnitude. This indicates that the scattering energy is concentrated in forward and backward directions.