# Improved MUSIC Method against Range Dimension Deceptive Jamming Based on FDA-MIMO
## 3 Model and Method of Range Dimension Deceptive Jamming
### Range Dimension Deceptive Jamming Method  
The existing deceptive jamming can create a significant deceptive interference effect for the FDA-MIMO radar at the range bin estimation level through the range dimensional beam gain over the airspace. The spatial location of the jammer is \((r_{j},\theta_{j})\), and its intercepted FDA-MIMO radar signal is  
\[s(t)=\sum_{m=1}^{M}\phi_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})rect(\frac{t- \tau_{j}/2-\tau_{j,m}}{T_{p}})e^{j2\pi\tau_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m })} \tag{22}\]  
In Equation (22), \(\tau_{j}=2r_{j}/c\) denotes the propagation delay, and \(\tau_{j,m}\) denotes the propagation delay difference among the reference array to the \(m-\)th transmitting array. After the interferer receives the signal transmitted by the radar, it first passes through the power amplifier for power amplification, then after the local oscillator \(f_{FTG}\) for down-conversion, and finally performs filtering and other processing. Figure 4 is the signal processing flow at the receiving end of the spoofing interference model. After filtering the radar-received signal at the receiver end through a filter bank with center frequency \(p\Delta f\) and bandwidth \(\Delta f\), the output signal of the \(p-\)th filter can be written as  
\[s_{1}(t)=\varphi_{\tau}\sum_{m=1}^{M}\phi_{m}(t-\tau_{j}/2-\tau_{j,m})rect( \frac{t-\tau_{j}/2-\tau_{j,m}}{T_{p}})\times e^{j2\pi[\Delta f_{m}(t-\frac{ \tau_{j}}{2}-\tau_{j,m})-f_{0}(\frac{\tau_  
In Equation (23), there is \(p=m\leq M-1\), \(\varphi_{\tau}\) denotes the public phase change items. The filter is first matched-filtered for the intercepted signal, and then modulated with a delayed forwarding such as its phase \(\varphi_{j}\) and time delay \(\tau_{j}\), where \(\varphi_{j}=2\pi\Delta f_{m}\tau_{j}\) and \(\tau_{j}=2\Delta r/c\). The final false target is  
\[ \begin{array}{ll}s_{2}(t)&=A\varphi\sum\limits_{m=1}^{M}\phi_{m}(t-\tau_{j}/ 2-\tau_{j,m}-\tau_{jam})rect(\frac{t-\tau_{j}/2-\tau_{j,m}}{\Gamma_{p}})\times \\ &e^{j2\pi[\Delta f_{m}(-\frac{\tau_{j}}{2}-\tau_{j,m}-\tau_{j,m})-f_{0}(\frac{ \tau_{j}}{2}+\tau_{j,m})+\Delta f_{m}\tau_{j,m}]}\end{array}  \tag{24}\]  
where \(A\) is the amplitude modulation, and \(\varphi=\varphi_{\tau}\exp\{-j2\pi f_{0}\tau_{jam}\}\) denotes the public phase change items. By mixing with the local oscillator signal and then up-conversion processing, the processed signal is  
\[s_{3}(t)=A\varphi^{\prime}{}_{\tau}\sum\limits_{m=1}^{M}\phi_{m}(t-\tau_{j}/2 -\tau_{j,m}-\tau_{jam})\times rect(\frac{t-\tau_{j}/2-\tau_{j,m}}{T_{p}})e^{j2 \pi[f_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})]} \tag{25}\]  
where \(\varphi^{\prime}{}_{\tau}\) denotes the public phase change items after up-conversion. Thus, the signal of the \(n-\)th received array element at the receiver is  
\[ \begin{array}{ll}y_{j,n}=A\varphi^{\prime}{}_{\tau}\sum\limits_{m=1}^{M} \phi_{m}(t-\tau_{j}-\tau_{j,m}-\tau_{j,n}-\tau_{jam})\times rect(\frac{t-\tau _{j}-\tau_{j,m}-\tau_{j,n}-\tau_{jam}}{T_{p}})e^{j2\pi f_{m}(t-\tau_{j}-\tau_{ j,m}-\tau_{j,n})}\end{array}  \tag{26}\]  
where \(\tau_{j,n}\) denotes the transmission time delay difference among the reference array to the \(n-\)th receiving array, and \(\tau_{j,n}=-d_{R}n\sin\theta/c\). In the far-field spatial conditions, the time delay between the \(n-\)th received array and the \(m-\)th transmitted array is  
\[\tau_{n,m}=2\tau_{j}/c-nd_{T}\sin\theta/c-md_{R}\sin\theta/c \tag{27}\]  
Let \(\tau_{0,0}=2r/c\), and \(rect[(t-\tau_{n,m}-\tau_{jam})/T_{p}]\) is approximated as \(rect[(t-\tau_{0,0}-\tau_{jam})/T_{p}]\), then Equation (26) be approximated as  
\[y_{j,n}\approx A\varphi^{\prime}\sum\limits_{m=1}^{M}\phi_{m}(t^{\prime})rect (\frac{t^{\prime}}{T_{p}})e^{j2\pi f_{m}(t-\frac{\tau_{j}}{c})}e^{j\pi(m\sin \theta_{j}+n\sin\theta_{j})} \tag{28}\]  
In Equation (28), \(t^{\prime}=t-2r/c-2\Delta r/c\) denotes the pulse time index.