## 2 Multistatic Radar System Model
### Measurement Model  
On the basis of the signal model established in the above subsection, the measurement models are introduced for physical targets and active false targets. In the multistatic radar system T/R-R\({}^{N}\), as shown in Figure 1, the location coordinates of the receiver stations are \(\mathbf{Z}_{Rn}=[x_{n},y_{n},z_{n}]^{T}\) (\(n=0,1,\ldots,N\)). The location and velocity vectors of the physical target are assumed as \(\mathbf{Z}_{PT}=[x,y,z]^{T}\) and \(\mathbf{v}_{PT}=[v_{x},v_{y},v_{z}]^{T}\), respectively. The measurement information of a target in the \(n\)-th receiver \(\mathbf{R}_{n}\) includes the bistatic radial distance sum \(\rho^{\prime}_{n}\), azimuth angle \(\alpha^{\prime}_{n}\), pitch angle \(\beta^{\prime}_{n}\) and bistatic radial velocity \(v^{\prime}_{n}\). The measurement information can be expressed as  
\[\mathbf{\Omega}=[\rho^{\prime}_{n},\alpha^{\prime}_{n},\beta^{\prime}_{n},v^{ \prime}_{n}]^{T}=[\rho_{n},\alpha_{n},\beta_{n},v_{n}]^{T}+\mathbf{\omega}_{n} \tag{6}\]  
where \(\rho_{n}\), \(\alpha_{n}\), \(\beta_{n}\) and \(v_{n}\) are the corresponding true values. \(\mathbf{\omega}_{n}\) is the measurement error vector whose elements are an independently distributed Gaussian distribution with zero mean, \(\mathbf{\omega}_{n}\)\(\sim\)\(N(\mathbf{0}_{4\times 1},\mathbf{R}_{e})\)[21; 34]. \(\mathbf{0}_{n\times 1}\) is an \(n\)\(\times\)\(1\) all-zero vector. \(\mathbf{R}_{e}\) is the covariance matrix, \(\mathbf{R}_{e}=diag\left(\sigma^{2}_{p,n},\sigma^{2}_{a,n},\sigma^{2}_{p,n}, \sigma^{2}_{v,n}\right)\). \(\sigma_{p,n},\sigma_{a,n},\sigma_{p,n}\), and \(\sigma_{v,n}\) are the measurement error standard deviation values of the range, azimuth angle, pitch angle and radial speed in the receiver R\({}_{n}\), respectively. For the physical target, the geometrical relationship between the physical radar and the bistatic T/R-R\({}_{n}\) can be represented as  
\[\left\{ \begin{aligned} \rho_{n}&=R_{T}+R_{n}=R_{0}+R_{n }=\|\mathbf{Z}_{R0}-\mathbf{Z}_{PT}\|+\|\mathbf{Z}_{Rn}-\mathbf{Z}_{PT}\|\\ \alpha_{n}&=\arctan\frac{y-y_{Rn}}{x-x_{Rn}}\\ \beta_{n}&=\arctan\frac{z-z_{Rn}}{\sqrt{(x-x_{Rn})^{ 2}+(y-y_{Rn})^{2}}}\\ v_{n}&=\frac{v_{T}+v_{nr}}{2}=\frac{v_{0r}+v_{nr}}{2 }=\frac{\langle\mathbf{v}_{PT},\mathbf{e}_{R0}\rangle+\langle\mathbf{v}_{PT}, \mathbf{e}_{Rn}\rangle}{2}\end{aligned} \right. \tag{7}\]  
where arctan is the inverse function of the tangent. \(R_{n}\) and \(v_{nr}\) are the range and the radial velocity of the PT with regard to the receiver R\({}_{n}\), respectively. \(\|\mathbf{u}\|=\sqrt{\langle\mathbf{u},\mathbf{u}\rangle}\) is the norm of the vector \(\mathbf{u}\). \(\langle\mathbf{u}_{1},\mathbf{u}_{2}\rangle=\mathbf{u}_{1}^{T}\mathbf{u}_{2}\) is the inner product between two vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). \(\mathbf{e}_{Rn}\) the unit vector from the target pointing receiver Rn  
\[\mathbf{e}_{Rn}=-[\cos\alpha_{n}\cos\beta_{n},\sin\alpha_{n}\cos\beta_{n},\sin \beta_{n}]^{T} \tag{8}\]  
For the joint range-velocity deception false target generated by the self-defense repeater jammer of the physical target, the geometrical relationship between the physical radar and the bistatic T/R-R\({}_{n}\) can be represented as  
\[\left\{ \begin{aligned} \rho_{n}&=R_{T}+R_{n}+R_{D}=\| \mathbf{Z}_{R0}-\mathbf{Z}_{PT}\|+\|\mathbf{Z}_{Rn}-\mathbf{Z}_{PT}\|+R_{D} \\ \alpha_{n}&=\arctan\frac{y-y_{Rn}}{x-x_{Rn}}\\ \beta_{n}&=\arctan\frac{z-z_{Rn}}{\sqrt{(x-x_{Rn})^{ 2}+(y-y_{Rn})^{2}}}\\ v_{n}&=\frac{v_{T}+v_{nr}+v_{D}}{2}=\frac{\langle \mathbf{v}_{PT},\mathbf{e}_{R0}\rangle+\langle\mathbf{v}_{PT},\mathbf{e}_{Rn} \rangle+v_{D}}{2}\end{aligned} \right. \tag{9}\]  
where \(R_{D}\) and \(v_{D}\) are the corresponding deception range and velocity caused by the jammer's time delay modulation \(\triangle\tau\) and doppler frequency modulation \(\triangle f_{d}\) for the intercepted radar signals to generate the FT.