## 2 Theoretical Analysis
### Transmission from the Target to the Receiver Plane
\tag{18}\]  
From Equation (18), we can obtain the following equations  
\[\frac{1}{\sigma_{ijout}^{2}}=-\frac{k}{i}(\mathbf{m}_{21}+m_{24})-\frac{2k}{i} m_{22}. \tag{19}\]  
By substituting Equations (6), (9), (13), (16), (18) and (19), we obtain the following equations  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]  
\[a_{1 ij}=8R_{1 ij}^{2}z^{2}\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{4}, \tag{21}\]  
\[b_{1 ij}=16R_{1 ij}^{2}z^{2}\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{4}, \tag{22}\]  
\[c_{1 ij}=R_{1 ij}^{2}z^{2}\sigma_{1 ij}^{2}\Big{[}16\rho_{0}^{2}\sigma_{1 ij}^{2}+ \delta_{1 ij}^{2}\Big{(}8\rho_{0}^{2}+32\sigma_{1 ij}^{2}\Big{)}\Big{]}, \tag{23}\]  
\[d_{1 ij}= 8k^{2}R_{1 ij}z\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{4}+4k^{2}z^{2 }\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{4} \tag{24}\] \[+R_{1 ij}^{2}\Big{[}4k^{2}\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij }^{4}+z^{2}\Big{(}4\rho_{0}^{2}\sigma_{1 ij}^{2}+\delta_{1 ij}^{2}\Big{(}\rho_{0 }^{2}+8\sigma_{1 ij}^{2}\Big{)}\Big{)}\Big{]}\;,\] \[e_{2 ij}=16R_{1 ij}^{2}k^{2}\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{4}\;, \tag{25}\]  
\[f_{2 ij}=4R_{1 ij}^{2}k^{2}\rho_{0}^{2}\delta_{1 ij}^{2}\sigma_{1 ij}^{2}, \tag{26}\]  
\[\delta_{1 ij}^{2}=\frac{4k^{2}\rho_{0}^{4}\delta_{ij}^{2}\sigma^{4}+z^{2}\rho_ {0}^{2}\Big{(}4\rho_{0}^{2}\sigma^{2}+\delta_{ij}^{2}\Big{(}\rho_{0}^{2}+8 \sigma^{2}\Big{)}\Big{)}}{k^{2}\rho_{0}^{2}\Big{(}24\delta_{ij}^{2}+4\rho_{0}^ {2}\Big{)}\sigma^{4}+z^{2}\Big{(}8\rho_{0}^{2}\sigma^{2}+\delta_{ij}^{2}\Big{(}2 \rho_{0}^{2}+12\sigma^{2}\Big{)}\Big{)}}, \tag{27}\]  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]0  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]1  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]2  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]3  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]4  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]5  
By applying Equations (35) and (36), we obtain the following solutions for \(l_{c}^{2}\):  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]6  
\[\frac{1}{\sigma_{ijout}^{2}}=\frac{e_{2 ij}l_{c}^{2}w_{R}^{2}+f_{2 ij}l_{c}^{4}w_ {R}^{4}}{a_{1 ij}w_{R}^{4}+2b_{1 ij}w_{R}^{2}+b_{1 ij}l_{c}^{2}+c_{1 ij}l_{c}^{2}w_ {R}^{2}+d_{1 ij}l_{c}^{4}w_{R}^{4}}\;, \tag{20}\]7  
Because of the properties of quadratic equations, we get two sets of solutions for \(w_{R}^{2},l_{c}^{2}\) by solving Equations (37), (38), (43) and (44). In fact, there is only one set of solutions that corresponds to the target. Therefore, it is necessary to find the correct solutions.