# Research on the Compound Jamming Method against FDA-MIMO Beamforming
## 4 Jamming Methods and Models  
The existing false target jamming methods cannot produce an effective jamming effect on FDA-MIMO radar. Therefore, this paper studies a suppression and range deceptive compound jamming method against FDA-MIMO. Firstly, the range dimension beamforming gain is analyzed in the spatial domain, and then an effective compound jamming effect is formed at the range bin estimation level for FDA-MIMO. The spatial position of the jammer is \((r_{j},\theta_{j})\), and the intercepted signal can be expressed as  
\[s(t)=\sum_{m=1}^{M}\phi_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})rect(\frac{t-\tau _{j}/2-\tau_{j,m}}{T_{p}})e^{j2\pi f_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})} \tag{33}\]  
where \(\tau_{j,m}\) represents the propagation delay difference of the \(m\)-th transmitting array element relative to the reference array element, and \(\tau_{j}=2r_{j}/c\) represents the reference propagation delay. After receiving the radar signal, the receiver of the jammer first amplifies its power by a low-noise amplifier, then down-converts it, and finally passes it through a band-pass filter bank (the bandwidth of the filter bank is \(\Delta f\), and the center frequency is \(p\Delta f\)), and its signal processing flow is shown in Figure 4. Then the output signal of the \(p\)-th filter output can be expressed as  
\[s_{1}(t)=\phi_{\tau}\sum_{m=1}^{M}\phi_{m}(t-\tau_{j}/2-\tau_{j,m})rect(\frac {t-\tau_{j}/2-\tau_{j,m}}{T})\times e^{j2\pi[\Delta f_{m}(t-\frac{\tau_{j}}{2} -\tau_{j,m})-f_{0}(\frac{\tau_{j}}{2}+\tau_{j,m})]} \tag{34}\]  
where \(p=n\leq N-1\), \(\varphi_{\tau}\) represents the common phase change term related to \(\tau_{FTG}\). The signal is filtered by the band-pass filter bank, and then modulated by the time delay \(\tau_{jam}=2\Delta r/c\) and the phase \(\varphi_{j}=2\pi\Delta f_{m}\tau_{j}\), the obtained false target signal can be expressed as  
\[ \begin{array}{ll}s_{2}(t)&=A\varphi\sum\limits_{m=1}^{M}\phi_{m}(t-\tau_{j}/ 2-\tau_{j,m}-\tau_{jam})rect(\frac{t-\tau_{j}/2-\tau_{j,m}}{T_{p}})\\ &\times e^{j2\pi[\Delta f_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m}-\tau_{jam})-f_{0} (\frac{\tau_{j}}{2}+\tau_{j,m})+\Delta f_{m}\tau_{jam}]}\end{array}  \tag{35}\]  
where \(A\) is the amplitude, \(\varphi\) represents the common phase change term, and \(\varphi=\varphi_{\tau}\exp\{-j2\pi f_{0}\tau_{jam}\}\). The false target signal is mixed with the local oscillator signal \(f_{FTG}\), and then upconverting it, the final signal can be expressed as  
\[s_{3}(t)=A\varphi^{\prime}\sum\limits_{m=1}^{M}\phi_{m}(t-\tau_{j}/2-\tau_{j, m}-\tau_{jam})rect(\frac{t-\tau_{j}/2-\tau_{j,m}}{T_{p}})e^{j2\pi[\tau_{j}(t- \frac{\tau_{j}}{2}-\tau_{j,m})]} \tag{36}\]  
Figure 4: The signal processing flow of the jamming model. where \({q^{\prime}}_{\tau}\) is the common term after up-conversion. The signal at the receiver side of the \(n\)-th receiver array can be expressed as  
\[ \begin{array}{ll}y_{j,n}&=A{\varphi^{\prime}}_{\tau}rect\big{(}\frac{t-\gamma _{j}-\tau_{j,n}-\tau_{j,n}}{\Gamma_{p}}\big{)}e^{j2\pi f_{m}(t-\tau_{j}-\tau_{ j,n}-\tau_{j,n})}\\ &\times\sum\limits_{m=1}^{M}{\phi_{m}(t-\tau_{j}-\tau_{j,m}-\tau_{j,n}-\tau_{ j,m})}\end{array}  \tag{37}\]  
where \(\tau_{j,n}=-d_{R}n\sin\theta/c\) represents the propagation time delay difference between the \(n\)-th received array element and the reference array element. The Equation (37) can be approximately expressed as  
\[y_{j,n}\approx A{\varphi^{\prime}}rect\big{(}\frac{t^{\prime}}{T_{p}}\big{)} \times\sum\limits_{m=1}^{M}{\phi_{m}(t^{\prime})e^{j2\pi f_{m}(t-\frac{\gamma _{j}}{c})}e^{j\pi(m\sin\theta_{j}+n\sin\theta_{j})}} \tag{38}\]  
where \(t^{\prime}=t-2r/c-2\Delta r/c\) represents the intra-pulse time index in this case. In the signal processing process at the receiving side, the received signal is mixed with the signal \(e^{j2\pi f_{0}t}\) and the signal \(e^{j2\pi n\Delta ft}\), and then matched filtering with the signal \(\phi_{m}(t^{\prime})\). The final signal can be expressed as  
\[