## 3 Multipath Interference Suppression
 \tag{15}\]

Considering the case of crossing the baseline at the moment of \(t=0\) and substituting \(x_{0}=H\tan\theta_{SV}\) into (15), we can obtain

\[\Delta\tau(t)=\frac{v^{2}t^{2}\cos\theta_{SV}\big{(}1-\cos^{2}\theta_{v}\sin^ {2}\theta_{SV}\big{)}}{2 cH}. \tag{16}\]

Based on (15), the time delay difference in the case of forward scatter can be calculated. The exact expression (11) and the approximate expression (15) are used to calculate the time delay difference, and the simulation results are shown in Figures 8 and 9. It can be seen that the curves can be described by different equations. It also shows that the above simplification is effective, which can provide theoretical support for the target sensing and target parameter estimation. From Figure 8, we can see that the influence of the speed direction angle on the delay difference is very small. Assuming that the target does not cross the baseline and the horizontal distance from the baseline when crossing the \(xOz\) plane is 50 m, the simulation is shown in Figure 9. It can be seen from Figure 9 that the delay difference in both cases are very close, and hence this paper only considers the case when the target crosses the baseline. Figure 8: Different delay when the target crosses the baseline. ### Target Sensing Statistics

The LCT of the signal after the suppression of direct wave and multipath interference is expressed as [25]

\[L(u)=\frac{1}{\sqrt{j2\pi b}}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)e^{i\frac{d}{2 }t^{2}-j\frac{k}{2}t+j\frac{dk^{2}}{2}}dt. \tag{17}\]

After discretization, this becomes

\[L(u)=\frac{1}{\sqrt{j2\pi b}}\sum_{n=0}^{N-1}f(nT_{s})e^{i\frac{d}{2}(nT_{s})^{ 2}-j\frac{k}{2}nT_{s}+j\frac{dk^{2}}{2}}, \tag{18}\]

where \(T\) stands for the signal duration, \(T_{s}\) denotes the sampling period, \(T_{s}=1/f_{s}\), \(N\) is the number of sampling points \(N{=}T_{s}\times f_{s}\). The target sensing statistic \(l\) is constructed as

\[l=|L(u)|. \tag{19}\]

### Target Detection

The detector is binary, where \(H_{1}\) indicates that the received signal, \(x(t)\) contains an echo signal, and \(H_{0}\) indicates that the received signal does not contain an echo signal, as

\[\left\{ \begin{array}{l}H_{0}:x(t)=v(t),\\ H_{1}:x(t)=A_{T}s_{r}(t)+w(t).\end{array} \right. \tag{20}\]

Based on the above binary assumption, the binary detector is

\[|l| \begin{array}{l}H_{1}\\ \stackrel{{ H_{1}}}{{\underset{H_{0}}{\overset{}{\underset{}{ \underset{}{\underset{}{\underset{}{\underset{}{\underset{}\]}}}}}}}}}T_{d}, \end{array}  \tag{21}\]

where \(T_{d}\) represents the detection threshold. The decision output \(u\) is

\[u=\left\{ \begin{array}{l}1\qquad|l|>T_{d},\\ 0\qquad else.\end{array} \right. \tag{22}\]

Figure 9: Different delay when the target does not cross the baseline. ## 5 Detection Performance Analysis

Assuming that both the real and imaginary parts of the Gaussian noise follow \(N\big{(}0,\sigma_{n}^{2}\big{)}\), one has

\[ \begin{split} L(u)&=\int_{-\frac{T}{2}}^{\frac{T}{2}}x(t )K_{A}(u,t)dt\\ &=\text{Re}\Bigg{\{}\int_{-\frac{T}{2}}^{\frac{T}{2}}x(t)K_{A}(u,t)dt\Bigg{\}}+\text{Im}\Bigg{\{}\int_{-\frac{T}{2}}^{\frac{T}{2}}x(t)\text{K}_ {A}(u,t)\text{dt}\Bigg{\}}\\ =& l_{\text{R}}+l_{\text{I}}.\end{split}  \tag{23}\]

**Lemma 1**. : _The distribution of the detection based on LCT under \(H_{1}\) is_

\[(l|H_{1})\sim CN\Bigg{(}\int_{-\frac{T}{2}}^{\frac{T}{2}}A_{T}s_{r}(t)K_{A}(u, t)dt,\sigma_{nn}^{2}\Bigg{)}, \tag{24}\]

_and the distribution of the detection based on LCT under \(H_{0}\) is_

\[(l|H_{0})\sim CN\Big{(}0,\sigma_{nn}^{2}\Big{)}, \tag{25}\]

_where \(CN(\cdot)\) represents the complex Gaussian process._

Proof. : See Appendix A. From the above analysis, the detection probability \(P_{D}\) is

\[