## 3 MMW Fuze Echo Signal Model under Rainfall Conditions
For the second type of echo signal, the power of the single scattering rain clutter can be expressed using the distributed scatterer:  
\[P_{ras}=\sum_{i=1}^{N}\frac{P_{t}G^{2}\lambda^{2}\sigma_{bi}}{(4\pi)^{3}R_{bi}^{4 }}e^{-2\gamma_{t}R_{bi}}, \tag{11}\]  
where \(N\) is the total number of raindrops in the field of view of the fuze, \(\sigma_{bi}\) is the raindrop backward scattering cross-section area, and \(\gamma_{t}\) is the rain attenuation coefficient derived from Equations (3) and (4). For the signal returning to the MMW fuze after multiple scatterings by raindrops, the received power at the first contact point between the MMW signal and the raindrop can be calculated based on the Friis transmission formula:  
\[P_{ram1}=\frac{P_{t}G\sigma_{t1}}{4\pi R_{mi}^{2}}e^{-\gamma_{t}R_{mi}}, \tag{12}\]  
where \(\sigma_{t1}\) is the raindrop attenuation cross-section area calculated with Equation (2). After the MMW signal is scattered from the \((i-1)\)-th raindrop to the \(i\)-th raindrop, the received power at the \(i\)-th raindrop is as follows:  
\[P_{rami}=\frac{\sigma_{ti}}{4\pi R_{mi}^{2}}e^{-\gamma_{t}R_{mi}}\frac{\sigma _{s(i-1)}P(\theta_{i-1})}{\sigma_{t(i-1)}}P_{ram(i-1)}, \tag{13}\]  
where \(\sigma_{s(i-1)}\) is the scattering cross-section area of the \((i-1)\)-th raindrop, \(P(\theta_{i-1})\) denotes the probability corresponding to the scattering-phase function during the scattering process from the \((i-1)\)-th raindrop to the \(i\)-th raindrop computed from Equation (5), and \(\theta_{i-1}\) is the scattering angle. The power of the MMW signal returning to the receiving antenna after undergoing \(n\) scatterings by raindrops is expressed as below:  
\[P_{ram(n+1)}=\frac{G\lambda^{2}}{(4\pi)^{3}R_{m(n+1)}^{2}}e^{-\gamma_{t}R_{m( n+1)}}\frac{\sigma_{sn}P(\theta_{n})}{\sigma_{tn}}P_{ramm}. \tag{14}\]  
Therefore, the power of the MMW signal received by the fuze after undergoing \(n\) scatterings by raindrops is as follows:  
\[P_{ram(n+1)}=\frac{P_{t}G^{2}\lambda^{2}\prod_{i=1}^{n}\sigma_{si}P(\theta_{i })}{(4\pi)^{3}\prod_{i=1}^{n+1}R_{mi}^{2}e^{\gamma_{t}R_{mi}}}. \tag{15}\]  
Then, the second type of echo signal can be expressed as follows:  
\[P_{ra}=\sum_{i=1}^{N}\frac{P_{t}G^{2}\lambda^{2}\sigma_{bi}}{(4\pi)^{3}R_{bi} ^{4}}e^{-2\gamma_{t}R_{bi}}+\sum_{n=2}^{\infty}\sum_{1}^{f(n)}\frac{P_{t}G^{2} \lambda^{2}\prod_{i=1}^{n}\sigma_{si}P(\theta_{i})}{(4\pi)^{3}\prod_{i=1}^{n+ 1}R_{mi}^{2}e^{\gamma_{t}R_{mi}}}\, \tag{16}\]  
where \(f(n)\) is the total number of scattering paths for \(n\) scatterings. Thus, the backward scattering cross-section area corresponds to the backward scattering portion of the raindrop scattering cross-section area, that is, \(\sigma_{b}=\sigma_{s}P(\theta_{1})\). The second type of echo signal can be simplified:  
\[P_{ra}=\sum_{n=1}^{\infty}\sum_{1}^{f(n)}\frac{P_{t}G^{2}\lambda^{2}\prod_{i= 1}^{n}\sigma_{si}P(\theta_{i})}{(4\pi)^{3}\prod_{i=1}^{n+1}R_{mi}^{2}e^{ \gamma_{t}R_{mi}}}. \tag{17}\]  
For the third type of echo signal, where the MMW signal is scattered by both the target and the raindrops, the power scattered by raindrops is given in the expression for the second type of echo signal. The interaction between the signal and the target surface is modeled using the directive model. Thus, the scattered power of the MMW signal when it is scattered from the transmitting antenna or the raindrop for the \(n\)-th time to the target surface can be expressed as follows:  
\[P_{tq}=\frac{1}{\left(4\pi\right)^{2}R_{q}^{2}}P_{zq}F(\varphi_{q})e^{-\gamma_{t} R_{q}}P_{iq}. \tag{18}\]  
After the MMW signal undergoes \(p\) scatterings with the target surface and \(n\) scatterings with raindrops, the received power of the MMW fuze is as below:  
Figure 3:0  
where \(g(n,p)\) represents the total number of scattering paths. Therefore, the third type of echo signal is given as follows:  
Figure 3:1  
In summary, the power of the MMW fuze echo signal can be expressed as below:  
\[