# Research on the Compound Jamming Method against FDA-MIMO Beamforming
## 2 FDA-MIMO Basic Principle and Jamming Model  
### Basic Principle of FDA-MIMO  
Consider an FDA-MIMO scheme whose structure is shown in Figure 1. The transmitting carrier frequency is \(f_{0}\), the interval between adjacent array elements of the transmitting array is \(d_{T}\), and the interval between adjacent array elements of the receiving array is \(d_{R}\), both of which are half-wavelength \(\lambda/2\). The number of elements of the transmitting array and the number of elements of the receiving array are \(M\) and \(N\), respectively. The transmitting frequency of the \(m\)-th array element is  
\[f_{m}=f_{0}+\Delta f_{m},\;m=0,1,\,\ldots\,,M-1 \tag{1}\]  
where \(\Delta f_{m}\) is the frequency offset increment of the \(m\)-th array element. Then the signal emitted by the \(m\)-th array element can be expressed as  
\[s_{m}(t)=\phi_{m}(t)rect(\frac{t}{T_{p}})\;e^{j2\pi f_{m}t},\;m=0,1,\ldots,M-1 \tag{2}\]  
where \(\phi_{m}(t)\) is the complex envelope of the \(m\)-th signal and \(T_{p}\) is the pulse duration. The function \(rect(x)\) can be defined as  
\[rect(x)=\left\{ \begin{array}{ll}1,\;0\leq x\leq 1\\ 0,\;else\end{array} \right. \tag{3}\]  
Figure 1: The structure diagram of FDA-MIMO. After the \(m\)-th transmitting array element transmits the signal, the signal received by the \(n\)-th receiving array element can be expressed as  
\[s_{m,n}(t)=\phi_{m}(t-\tau_{m,n})rect(\frac{t-\tau_{m,n}}{T_{p}})\,e^{j2\pi(f_{0} +\Delta f_{m})(t-\tau_{m,n})} \tag{4}\]  
where \(\tau_{m,n}\) is the propagation delay from the \(m\)-th transmitting array element to the \(n\)-th receiving array element, which can be expressed as  
\[\tau_{m,n}=2r/c-(md_{T}+nd_{R})\sin\theta/c \tag{5}\]  
It should be noted that the phase difference between the \(n\)-th receiving array element and the reference array element cannot be neglected here, so Equation (4) can be expressed as  
\[s_{m,n}(t)\approx\phi_{m}(t^{\prime})rect(t^{\prime}/T_{p})\,e^{j2\pi\tau_{m} t^{\prime}}e^{j6(md_{T}+nd_{R})\sin\theta/c} \tag{6}\]  
where \(t^{\prime}=t-2r/c\) is the time index within the pulse. Figure 2 shows the signal processing flow at the receiver side, where the signal is received by the receiving antenna, it is first mixed with the \(e^{j2\pi f_{0}t}\) signal and the \(e^{j2\pi\Delta f_{m}t}\) signal in the signal processor, and then matched filtering with the \(\phi_{m}(t^{\prime})\) signal in the digital signal processing equipment. Then the output signal of the \(n\)-th receiving array element related to the \(m\)-th transmitting array element can be expressed as  
\[s_{m,n}(t^{\prime})\approx\xi_{s}rect(t^{\prime}/T_{p})e^{-j4\pi f_{m}t^{ \prime}/c}e^{j2\pi f_{0}(md_{T}+nd_{R})\sin\theta/c} \tag{7}\]  
where \(\xi_{s}\) is the complex coefficient after matched filtering. After the matched filtering process, the signal at the output end is independent of the time parameter and is only related to the range parameter, so Equation (7) can also be expressed as  
\[s^{\prime}_{m,n}=\xi_{s}e^{-j4\pi f_{m}t^{\prime}/c}e^{j\pi(m\sin\theta+n\sin \theta)} \tag{8}\]  
Thus, its array factor can be expressed as  
\[ \begin{array}{ll}AF&=\sum\limits_{n=0}^{N-1}\sum\limits_{m=0}^{M-1}s_{m,n} (t^{\prime})\\ &=\xi_{s}rect(\frac{t^{\prime}}{T_{p}})\sum\limits_{n=0}^{N-1}\sum\limits_{m=0 }^{M-1}e^{-j4\pi f_{m}t^{\prime}}c\ e^{j2\pi f_{0}(md_{T}+nd_{R})\sin\theta/c} \end{array}  \tag{9}\]  
Figure 2: Signal processing flow at the receiver side. Its steering vector can be expressed as  
\[\mathbf{S}=\mathbf{a}(r,\theta)\otimes\mathbf{b}(\theta) \tag{10}\]  
where \(\otimes\) is the Kronecker operator, \(\mathbf{a}(r,\theta)\) and \(\mathbf{b}(\theta)\) represents the transmit steering vector and the receive steering vector, which can be expressed as  
\[\mathbf{a}(r,\theta)= \begin{bmatrix}1\\ e^{2\pi\left[-\Delta f_{1}2r/c+f_{0}d_{T}\sin\theta/c\right]}\\ \vdots\\ e^{2\pi\left[-\Delta f_{M-1}2r/c+f_{0}(M-1)d_{T}\sin\theta/c\right]}\end{bmatrix}  \tag{11}\]  
\[\mathbf{b}(\theta)=