# Improved MUSIC Method against Range Dimension Deceptive Jamming Based on FDA-MIMO
## 4 Improved MUSIC Method against Range Dimension Deceptive Jamming  
The existing deceptive jamming mode can utilize the range dimension degrees of freedom of the FDA-MIMO radar to create a significant deceptive interference effect on the range bin through the beamforming gain in the range dimension, and at this time, it matches the angle and distance parameter information of the desired signal at the beamforming level. When the adaptive beam is focused, the interfering beam will also be focused, but the energy entering the receiver will not be reduced, which leads to the mismatch between the position of the airspace and the position of the range bin. Through a range-angle dimension spatial spectrum search, the angle and distance parameter information of the desired signal and the jamming can be determined by utilizing the mismatch of the range dimension information against deceptive jamming. The improved MUSIC algorithm's basic idea is to first obtain the covariance matrix by maximum likelihood estimation of the received signal, and then perform the eigendecomposition of the covariance matrix to obtain two groups of mutually orthogonal signal subspaces and noise subspaces. Finally, utilizing the orthogonality of the two subspaces, the range and angle information of the received signal are determined by constructing the spatial spectrum function. The mathematical model of the radar received signal can be written as  
\[s(t)=\sum\limits_{m=1}^{M}\phi_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})rect(\frac {t-\tau_{j}/2-\tau_{j,m}}{T_{p}})e^{j2\pi f_{m}(t-\frac{\tau_{j}}{2}-\tau_{j,m})} \tag{34}\]  
where \(\tau_{j}=2r_{j}/c\) denotes the reference propagation delay. For desired targets and false targets in airspace, Equation (34) can also be written as  
\[\mathbf{s}(l)=\mathbf{s}_{d}(l)\mathbf{v}(r_{d},\theta_{d})+\sum\limits_{j=1}^ {K}\mathbf{s}_{j}(l)\mathbf{v}(r_{j},\theta_{j})+\mathbf{n}(l) \tag{35}\]After receiving the signal at the radar receiver, the covariance matrix can be found:  
\[ \begin{array}{ll}\mathbf{R}_{x}&=E\big{(}ss^{H}\big{)}\\ &=\sum\limits_{j=1}^{K+1}\alpha_{j}^{2}\mathbf{\alpha}(r_{d},\theta_{d})\mathbf{\alpha}^ {H}(r_{d},\theta_{d})+\sigma^{2}\mathbf{I}\end{array}  \tag{36}\]  
where \(\sigma^{2}\) denotes the noise signal power, and \(\mathbf{I}\) denotes the identity matrix of order \(M\times M\). Then, the eigendecomposition of \(\mathbf{R}_{x}\) is carried out. After analyzing the covariance matrix of the noise signal and the desired signal, respectively, we can get the covariance matrix of the noise signal to be  
\[\mathbf{R}_{N}=\sigma^{2}\mathbf{I} \tag{37}\]  
Decompose \(\mathbf{R}_{x}\) to get  
\[\overset{\wedge}{\mathbf{R}_{x}}=U\sum U^{H}=[S,G]\sum\left[ \begin{array}{c} S^{H}\\ G^{H}\end{array} \right] \tag{38}\]  
In Equation (38), \(\sum\) denotes the vector diagonal array, and takes the first \(D\) larger eigenvalues of \(\mathbf{R}_{x}\) relating to the eigenvector matrix to constitute the signal subspace \(S\), and takes the last \(M-D\) smaller eigenvalues of \(\mathbf{R}_{x}\) relating to the eigenvector matrix to constitute the noise subspace \(G\). To study the relationship between signal subspace \(S\) and noise subspace \(G\), multiply both sides of \(\mathbf{R}_{x}\) by \(G\), and simplify it; there are  
\[ \begin{array}{ll}\overset{\wedge}{\mathbf{R}_{x}}G&=[S,G]\sum\left[\begin{array} []{c}S^{H}\\ G^{H}\end{array}\right]G\\ &=[S,G]\sum\left[\begin{array}{c}O\\ I\end{array}\right]\\ &=\sigma^{2}G\end{array}  \tag{39}\]  
The spatial spectrum can be defined as  
\[P(\theta,r)=\frac{\mathbf{\alpha}^{H}(\theta,r)\overset{\wedge}{\mathbf{U}}\mathbf{ \alpha}(\theta,r)}{\mathbf{\alpha}^{H}(\theta,r)GG^{H}\mathbf{\alpha}(\theta,r)} \tag{40}\]  
where \(\overset{\wedge}{\mathbf{U}}=\sigma^{2}\sum\limits_{k=1}^{P}\frac{\lambda_{k }}{(\sigma^{2}-\lambda_{k})^{2}}u_{k}u_{k}^{H}\).