# Improved MUSIC Method against Range Dimension Deceptive Jamming Based on FDA-MIMO
## 2 The FDA-MIMO Signal Model  
Consider a one-dimensional uniform line array with a transmitting array of \(M\) elements and a receiving array of \(N\) elements. The reference carrier frequency is \(f_{0}\) and its reference array element is the first array element. The array spacings of the transmittingelements and the receiving elements are \(d_{T}\) and \(d_{R}\), respectively, and the speed of light is denoted by \(c\). Its structure is shown in Figure 1.  
The frequency offset of the \(m-\)th array element of the FDA-MIMO radar is \(\Delta f_{m}\), and its transmit frequency is  
\[f_{m}=f_{0}+m\Delta f,\ \ \ m=0,1,...,M-1 \tag{1}\]  
For the desired target point P in space, the signal to reach that desired target point can be written as  
\[s_{m}(t)=\phi_{m}(t)rect(\frac{t}{T_{p}})\ e^{2\pi f_{m}t} \tag{2}\]  
In Equation (2), \(T_{p}\) denotes the pulse duration and \(\phi_{m}(t)\) denotes the complex envelope of the \(m-\)th transmitted signal. Let us define the function \(rect(x)\) to be  
\[rect(x)=\left\{ \begin{array}{ll}1,\ 0\leq x\leq 1\\ 0,\ else\end{array} \right. \tag{3}\]  
Supposing the range and angle position parameter at a point in space is \((r_{0},\theta_{0})\), each array element signal of the FDA-MIMO satisfies the orthogonal condition, with  
\[\int\phi_{m1}{}^{*}(t)\cdot\phi_{m2}(t-\tau)\ dt=0,m1\
eq m2,\forall\tau \tag{4}\]  
Then the signal received by the \(n-\)th array element after the \(m-\)th array element transmits the signal can be written as  
\[s_{m,n}(t)=\phi_{m}(t-\tau_{m,n})rect(\frac{t-\tau_{m,n}}{T_{p}})\ e^{2\pi(f_{ 0}+\Delta f_{m})(t-\tau_{m,n})} \tag{5}\]  
In Equation (5), \(\tau_{m,n}\) represents the time delay between the \(m-\)th transmitted signal and the \(n-\)th received signal. Assuming the spatial conditions in the far field, these are  
\[\tau_{m,n}=2r_{0}/c-(md_{T}+nd_{R})\sin\theta_{0}/c \tag{6}\]  
Figure 1: The FDA-MIMO radar structure diagram. Since there is a difference in the phase between the reference array element and the signal of the \(m-\)th received array element, Equation (5) can also be approximated as  
\[s_{m,n}(t)\approx\phi_{m}(t^{\prime})rect(t^{\prime}/T_{p})\ e^{i2\pi f_{m}t^{ \prime}}e^{i\theta_{0}(md_{T}+md_{R})\sin\theta_{0}/c} \tag{7}\]  
In Equation (7), \(t^{\prime}=t-2r/c\) is the time index within the pulse, and \(\phi_{m}(t^{\prime})\) can be approximately expressed as  
\[\phi_{m}(t^{\prime})\approx 2\pi\big{[}(f_{0}+\Delta f_{m})t^{\prime}+f_{0}(md_ {T}+nd_{R})\sin\theta/c\big{]} \tag{8}\]  
After the radar receiving antenna receives the signal, it performs signal processing on the received signal, and finally, matched filtering is carried out. The signal processing flow at the receiving end is shown in Figure 2.  
After signal processing, the signal output from the output can be written as  
\[s_{m,n}(t^{\prime})\approx\xi_{s}rect(t^{\prime}/T_{p})e^{-j4\pi f_{m}t^{ \prime}/c}e^{j2\pi f_{0}(md_{T}+nd_{R})\sin\theta_{0}/c} \tag{9}\]  
In Equation (9), \(\xi_{s}\) represents the complex coefficient after processing by the signal processing equipment. After matched filtering, the output signal is only dependent on the range dimension information, but not on the time information, so Equation (9) can again be written as  
\[s^{\prime}{}_{m,n}=\xi_{s}e^{-j4\pi f_{m}t^{\prime}/c}e^{j\pi(m\sin\theta_{0}+ n\sin\theta_{0})} \tag{10}\]  
Then, the array factor of the FDA-MIMO radar is  
\[ \begin{array}{rl}AF&=\sum\limits_{n=0}^{N-1}\sum\limits_{m=0}^{M-1}s_{m,n}(t ^{\prime})\\ &=\xi_{s}rect(\frac{t^{\prime}}{T_{p}})\sum\limits_{n=0}^{N-1}\sum\limits_{m= 0}^{M-1}e^{-j4\pi f_{m}t^{\prime}/c}e^{j2\pi f_{0}(md_{T}+nd_{R})\sin\theta_{ 0}/c}\end{array}  \tag{11}\]  
From Equation (11), we know that the difference in the phase from the reference signal to the \(m-\)th transmit signal is range-dependent and independent of the time parameters. The steering vector is  
\[\mathbf{S}=\mathbf{a}(r,\theta)\otimes\mathbf{b}(\theta) \tag{12}\]  
Figure 2: Signal processing flow at the receiving end.