## 3 Multipath Interference Suppression
Figure 6: Linear canonical transform of the signal before multipath interference suppression. ## 4 Aerial Target Passive Sensing Based on Linear Canonical Transform

Based on the above analysis, the amplitude and phase of the received signal are mainly reflected in the shadow occlusion effect and the approximate chirp Doppler phenomenon. Using these characteristics, and considering the outstanding performance of LCT in chirp signal processing, this paper proposes the aerial target sensing based on LCT. ### Signal Delay Analysis

One sees that the change in the time delay change of the signal makes the echo signal equivalent to a frequency-modulated signal. Since the satellite is far away from the ground, it is assumed that the target echo signal is parallel to the direct incident wave arriving at the receiver, or assumed a plane wave. Assuming that the time is zero when the target traverses the \(xOz\) plane, the flying height of the target is \(H\), and the distance between the projection on the ground and the receiver is \(x_{0}\), the position of the target trajectory can be expressed as

\[\left\{ \begin{array}{l}x_{SV}(t)=L\sin\theta_{SV},\\ y_{SV}(t)=0,\\ z_{SV}(t)=L\cos\theta_{SV},\end{array} \right. \tag{8}\]

and the trajectory of the satellite is

\[\left\{ \begin{array}{l}x(t)=vt\cos\theta_{v}+x_{0},\\ y(t)=vt\sin\theta_{v},\\ z(t)=H,\end{array} \right. \tag{9}\]

where \(L\) denotes the baseline length at \(t=0\). Under the assumption of plane waves, \(R_{D}=R_{T}+R_{R}\cos\theta\); therefore, the delay difference can be written as

\[\Delta\tau(t)=\frac{R_{T}+R_{R}-R_{D}}{c}=\frac{R_{R}-R_{R}\cos\theta}{c}. \tag{10}\]

Figure 7: Linear canonical transform of the signal after multipath interference suppression. Applying the Pythagorean theorem and substituting the expressions of target trajectory and satellite trajectory into (10), we can obtain

\[ \begin{split}\Delta\tau(t)=&\frac{R_{R}}{c}\frac{\left[ \underline{x}(t)\underline{y}(t)\underline{z}(t)\right]\left[x_{SV}(t)y_{SV}( t)z_{SV}(t)\right]^{T}}{c\sqrt{x_{SV}^{2}(t)+y_{SV}^{2}(t)+z_{SV}^{2}(t)}}\\ =&\frac{R_{R}}{c}\frac{vt\cos\theta_{v}\sin\theta_{ SV}+x_{0}\sin\theta_{SV}+H\cos\theta_{SV}}{c},\end{split}  \tag{11}\]

where \(R_{R}\) can be expressed as

\[ \begin{split} R_{R}&=\sqrt{\left(vt\cos\theta_{v}+x _{0}\right)^{2}+\left(vt\sin\theta_{v}\right)^{2}+H^{2}}\\ &=\sqrt{v^{2}t^{2}+2 vtx_{0}\cos\theta_{v}+x_{0}{}^{2}+H^{2}}. \end{split}  \tag{12}\]

Taking the third-order Maclaurin of \({x_{0}}^{2}+H^{2}\) about \(v^{2}t^{2}+2 vtx_{0}\cos\theta_{v}\), we can obtain

\[ \begin{split} R_{R}&\approx\sqrt{{x_{0}}^{2}+H^{2} }+\frac{v^{2}t^{2}+2 vtx_{0}\cos\theta_{v}}{2\sqrt{{x_{0}}^{2}+H^{2}}}-\frac{ \left(v^{2}t^{2}+2 vtx_{0}\cos\theta_{v}\right)^{2}}{8({x_{0}}^{2}+H^{2})\sqrt{ {x_{0}}^{2}+H^{2}}}\\ &=\sqrt{{x_{0}}^{2}+H^{2}}-\frac{v^{4}t^{4}+4v^{3}t^{3}x_{0}\cos \theta_{v}}{8({x_{0}}^{2}+H^{2})\sqrt{{x_{0}}^{2}+H^{2}}}\\ &+\frac{v^{2}t^{2}\bigg{(}1-\frac{x_{0}^{2}\cos^{2}\theta_{v}}{x_ {0}^{2}+H^{2}}\bigg{)}+2 vtx_{0}\cos\theta_{v}}{2\sqrt{{x_{0}}^{2}+H^{2}}}. \end{split}  \tag{13}\]

By omitting the third term and the fourth term of \(t\), (13) can be rewritten as

\[R_{R}\approx\sqrt{{x_{0}}^{2}+H^{2}}+\frac{v^{2}t^{2}\bigg{(}1-\frac{x_{0}^{2 }\cos^{2}\theta_{v}}{x_{0}^{2}+H^{2}}\bigg{)}+2 vtx_{0}\cos\theta_{v}}{2\sqrt{ {x_{0}}^{2}+H^{2}}}. \tag{14}\]

Substituting (14) into (11), the available delay difference is expressed as

\[ \begin{split}\Delta\tau(t)&=\frac{\sqrt{{x_{0}}^{2 }+H^{2}}-x_{0}\sin\theta_{SV}-H\cos\theta_{SV}}{c}+\frac{v^{2}t^{2}\bigg{(}1- \frac{x_{0}^{2}\cos^{2}\theta_{v}}{x_{0}^{2}+H^{2}}\bigg{)}}{2c\sqrt{{x_{0}}^{ 2}+H^{2}}}\\ &+\frac{vt}{c}\Bigg{(}\frac{x_{0}\cos\theta_{v}}{\sqrt{{x_{0}}^{ 2}+H^{2}}}-\cos\theta_{v}\sin\theta_{SV}\Bigg{)}.\end{split}