## 3 Materials and Methods
### Proposed Deception Velocity-Based (DVB) Discrimination Method
\begin{array}{cc}\frac{\partial v_{D}}{\partial a_{0}}& \frac{\partial v_{D}}{\partial\beta_{0}}&\frac{\partial v_{D}}{\partial v_{0 }}&\frac{\partial v_{D}}{\partial\alpha_{1}}&\frac{\partial v_{D}}{\partial \beta_{1}}&\frac{\partial v_{D}}{\partial v_{1}}&\frac{\partial v_{D}}{ \partial\alpha_{2}}&\frac{\partial v_{D}}{\partial\beta_{2}}&\frac{\partial v _{D}}{\partial\alpha_{2}}\end{array} \right] \tag{27}\]  
and \(\boldsymbol{\Lambda}\) is the  
\[\boldsymbol{\Lambda}=\text{diag}\Big{(}\sigma_{a_{0}}^{2},\sigma_{\bar{b}_{0 }}^{2},\sigma_{v_{0}}^{2},\sigma_{\alpha_{1}}^{2},\sigma_{\bar{b}_{1}}^{2}, \sigma_{v_{1}}^{2},\sigma_{\bar{a}_{2}}^{2},\sigma_{\bar{b}_{2}}^{2},\sigma_{ \bar{b}_{2}}^{2}\Big{)} \tag{28}\]  
Under hypothesis H\({}_{1}\), although the true deception velocity \(v_{D,FT}\) of the active false target is unknown, its estimation \(\hat{v}_{D,FT}\) approximately follows a Gaussian distribution with mean \(v_{D,FT}\) and variance \(\sigma_{v_{D}}^{2}\), namely \(\hat{v}_{D,FT}\sim N(v_{D,FT},\sigma_{v_{D}}^{2})\). Meanwhile, under hypothesis H\({}_{0}\), the virtual deception velocity \(\hat{v}_{D,PT}\) of the physical target approximately follows a Gaussian distribution with zero mean and variance \(\sigma_{v_{D}}^{2}\), namely \(\hat{v}_{D,PT}\sim N(0,\sigma_{v_{D}}^{2})\). Therefore, the discrimination factor can be constructed as follows  
\[\mu=\frac{v_{d}^{2}}{\sigma_{v_{D}}^{2}} \tag{29}\]  
For physical targets, \(\mu\) approximately follows the Chi-square distribution with 1 degree of freedom based on the analysis that the virtual deception velocity \(\hat{v}_{D,PT}\) of the physical target follows a Gaussian distribution \(\hat{v}_{D,PT}\sim N(0,\sigma_{v_{D}}^{2})\), namely \(H_{0}:\mu\sim\chi_{1}^{2}\)[35]. Accordingly, the following hypothesis test is established  
\[\mu\overset{H_{0}}{\underset{H_{12}}{\underset{H_{12}}{\overset{<}{\sim}}}}\epsilon \tag{30}\]  
where \(\epsilon\) is the discrimination threshold, which is determined by the significance level \(\kappa\) of the hypothesis test  
\[\epsilon=\Phi_{\chi_{1}^{2}}^{-1}(1-\kappa) \tag{31}\]  
where \(\Phi_{\chi_{1}^{2}}^{-1}\) is the inverse cumulative distribution function of the Chi-square distribution \(\chi_{1}^{2}\) with 1 degree of freedom. Actually, the significance level \(\kappa\) is namely the expectedmisjudment probability of the physical target \(P_{PT}^{\prime}\), which is the expected probability that a physical target is judged as a false target. Therefore, the expected discrimination probability of the physical target \(P_{PT}\) is  
\[P_{PT}=1-P_{PT}^{\prime}=1-\kappa \tag{32}\]  
As will be shown in the simulation section, the discrimination performance is affected by the deception speed, SNR/JNR, measurement accuracy and location. Therefore, in practice, the set of the expected misjudment probability of the physical target \(P_{PT}^{\prime}\) should take the prior information of the above factors into consideration. If the expected misjudment probability of the physical target \(P_{PT}^{\prime}\) is set too little, the false target cannot be discriminated effectively. The discrimination criterion is set as  
* I. When all the combinations composed of one target in \(\mathrm{T/R_{0}}\) and all targets in \(\mathrm{R_{1}}\) and \(\mathrm{R_{2}}\) are judged to be \(\mathrm{H_{12}}\), the target is discriminated to be an active false target;
* II. When more than one target combination composed of one target is judged to be \(\mathrm{H_{0}}\), the target combination with the minimum \(\mu\) is reserved, and for one specific target, no more than one combination can pass the hypothesis testing to avoid ambiguity. Compared with the VVA discriminator, the proposed DVB method directly makes use of the measurements to construct the discrimination statistics, reducing the complexity of the process. Moreover, the demand for the receiver number decreases by transforming the coordinate system to reduce the data dimension. As shown in Figure 3, the processing flow of the proposed DVB method is summarized as follows:  
1.