# PBR Clutter Suppression Algorithm Based on Dilation Morphology of Non-Uniform Grid
## 2 Non-Uniform Polar Grid Construction for PBR  
According to the bi-static radar position principle, Figure 2 demonstrates the geometrical relationship between target, transmitter and receiver. \(\mathrm{R_{r}}\) stands for the range of the target, that is the distance between target and receiver. \(\mathrm{R_{t}}\) is the distance between target and transmitter. L stands for the baseline range between transmitter and receiver. \(\mathrm{\beta}\) stands for the bi-static angle, that is the intersection angle between the line from receiver to target and the line from transmitter to target. \(\mathrm{\theta_{r}}\) stands for the azimuth angle, that is the supplementary angle between the line from receiver to transmitter and the line from receiver to target. To locate the target, we should find its azimuth and range. The azimuth can be directly measured through the passive bi-static radar system. However, the range can only be calculated through the bi-static range sum. Suppose \(\mathrm{R_{S}}\) is the bi-static range sum. Thus,  
\[\mathrm{R_{s}}=R_{t}+R_{r} \tag{1}\]  
Referring to the cosine law, the relationships between \(\mathrm{R_{r}}\) and \(\mathrm{R_{s}}\) can be derived into Equations (2) and (3). \[\mathrm{R_{r}}=(R_{\mathrm{s}}^{2}-L^{2})/(2(R_{s}+Lcos\mathrm{\theta_{r}})) \tag{2}\]  
\[\mathrm{R_{s}}=R_{r}+\sqrt{R_{r}^{2}+L^{2}+2R_{r}Lcos(\mathrm{\theta_{r}})} \tag{3}\]  
Figure 2: The geometrical relationship of PBR. From Equation (2), we can find that the mono-static range \(\rm R_{r}\) is a non-linear function of the bi-static range sum \(\rm R_{S}\) and the azimuth angle \(\rm\theta_{r}\). In passive bi-static radar system, firstly, achieve the time synchronization of the echo channel with the help of the reference channel. Then, measure the difference between \(\rm R_{S}\) and \(\rm L\) through the time delay of echo. Since \(\rm L\) is fixed, \(\rm R_{S}\) can be directly measured from echo channel. As \(\rm R_{S}\) and \(\rm\theta_{r}\) are direct measurements, the measuring and detecting error transfer to \(\rm R_{r}\) by Equation (2) at last. Due to non-linear relationship between \(\rm R_{S}\) and \(\rm R_{r}\), it is necessary for us to construct a non-uniform grid for the latter clutter suppression processing. At first, we derive the first order Taylor series expansion of Equation (2) at an arbitrary position \((\rm R_{s0},\theta_{0})\), shown in Equation (4). \[ \begin{array}{ll}\rm R_{r}(R_{s0}+\Delta r,\theta_{0}+\Delta\theta)\\ \qquad\qquad\qquad\qquad\qquad=\rm R_{r}(R_{s0},\theta_{0})+\Delta r\frac{ \partial\rm R_{r}(R_{s0},\theta_{0})}{\partial R_{s}}|_{R_{s}=R_{s0}}+\Delta \theta\frac{\partial\rm R_{r}(R_{s0},\theta)}{\partial\theta}|_{\theta=\theta _{0}}+\rm o(\Delta r,\Delta\theta)\end{array}  \tag{4}\]  
Thus, the transfer error of the \(\rm R_{r}\) caused by \(\rm R_{S}\) and \(\theta\) at the position \(\rm M(R_{s0},\theta_{0})\) is derived below. \[ \begin{array}{ll}\Delta R_{r}&=R_{r}(R_{s0}+\Delta r,\theta_{0}+\Delta\theta )-R_{r}(R_{s0},\theta_{0})\\ &=\Delta r\frac{\partial\rm R_{r}(R_{s0},\theta)}{\partial R_{s}}|_{R_{s}=R_{ s0},\theta=\theta_{0}}+\Delta\theta\frac{\partial\rm R_{r}(R_{s0},\theta)}{ \partial\theta}|_{R_{s}=R_{s0},\theta=\theta_{0}}+\rm o(\Delta r,\Delta\theta) \\ &=\Delta r\frac{R_{s0}^{2}+L^{2}+2R_{s0}L\rm cos(\theta_{0})}{2(R_{s0}+L\rm cos (\theta_{0}))^{2}}+\Delta\theta\frac{(R_{s0}^{2}-L^{2})L\rm sin(\theta_{0})}{2( R_{s0}+L\rm cos(\theta_{0}))^{2}}+\rm o(\Delta r,\Delta\theta)\end{array}  \tag{5}\]  
Assume \(\rho_{1}(R_{s0},\theta_{0})=\frac{R_{s0}^{2}+L^{2}+2R_{s0}L\rm cos(\theta_{0}) }{2(R_{s0}+L\rm cos(\theta_{0}))^{2}}\), \(\rho_{2}(R_{s0},\theta_{0})=\frac{(R_{s0}^{2}-L^{2})L\rm sin(\theta_{0})}{2(R_ {s0}+L\rm cos(\theta_{0}))^{2}}\). Then, omit the Peano remainder term \(\rm o(\Delta r,\Delta\theta)\).