# Anti-Jamming Method and Implementation for GNSS Receiver Based on Array Antenna Rotation
## 3 The Proposed Antenna Rotation Method  
The rationale for the proposed method for sup-freedom jammers follows in this section. According to the Rayleigh entropy theorem, the minimum value of residual interference-plus-noise power is the minimum eigenvalue of covariance matrix, as in  
\[p_{inres,\text{min}}=\lambda_{\text{min}}[\mathbf{R_{xx}}] \tag{9}\]  
In this case, the normalized coefficient of the spatial anti-jamming filter \(\overline{\mathbf{w}}\) is the eigenvector corresponding to the minimum eigenvalue. If the interference does not surpass the array freedom, \(\overline{\mathbf{w}}\) is orthogonal to the interference steering vector \(\mathbf{a_{jk}}\), and  
\[\overline{\mathbf{w}}^{H}\mathbf{a_{jk}}=0 \tag{10}\]  
So, the interference residual power is not affected by interference DOA. However, if the interference surpasses array freedom, the value of \(\overline{\mathbf{w}}^{H}\mathbf{a_{jk}}\) is dependent on the correlation of the interference steering vectors. Thus, the residual interference-plus-noise power is a function of incident angles and interference powers. \[p_{inres,\text{min}}=\lambda_{\text{min}}[\mathbf{R_{xx}}]=f(p_{jk},\theta_{jk},\varphi_{jk}|k=1,2,\cdots,K) \tag{11}\]  
At a certain moment, if the distances between the receiver platform and jammers are settled, the interference intensities are usually fixed. However, the incident angles can be changed by adjusting the antenna attitude. This can be utilized to reduce residual interference-plus-noise power. In order to simplify the servo structure of the antenna, let the array antenna rotate along the normal direction of the array plane, and the rotation angle is \(\Delta\varphi\). As shown in Figure 2, xOy is the coordinate system of carrier platform and x'Oy' is the coordinate system of the antenna array. The orange circles represent the array elements in coordinate xOy, and the yellow circles represent the locations of elements after antenna rotation. Suppose the interference DOA relative to the carrier platform is [\(\theta\), \(\varphi\)] and that the initial interference DOA relative to the antenna is [\(\theta\), \(\varphi\)]. After rotating the antenna array around the normal direction by angle \(\Delta\varphi\), the interference DOA relative to the platform is still [\(\theta\), \(\varphi\)], while the DOA relative to antenna becomes [\(\theta\), \(\varphi-\Delta\varphi\)]. During the rotation of the antenna array, the relative relationship between interference steering vectors in the x'Oy' coordinate have changed compared to that in the xOy coordinate. This leads to the change of the eigenstructure of the array signal covariance matrix, which further affects the residual power of the anti-jamming process. In the course of antenna array rotation, the steering vector of the \(k\)th interference is  
\[\mathbf{a_{jk}}=\mathbf{a}(\theta_{jk0},\varphi_{jk0}+\Delta\varphi) \tag{12}\]  
where \(\theta_{jk0},\varphi_{jk0}\) is the initial elevation and azimuth angle of the \(k\)th interference. When \(\theta_{jk0},\varphi_{jk0}\) and the interference intensity are known and are constant, the received signal covariance matrix is only related to the rotation angle \(\Delta\varphi\). The interference-plus-noise residual power is a function of \(\Delta\varphi\), as in  
\[p_{\textit{inres,min}}(\Delta\varphi)=\lambda_{\text{min}}[\textbf{R}_{\text{ xx}}(\Delta\varphi)] \tag{13}\]  
The rotation angle that makes the optimal residual interference-plus-noise power is  
\[\Delta\varphi_{opt}=\text{arg}\min_{\Delta\varphi}~{}~{}~{}[p_{\textit{inres, min}}(\Delta\varphi)] \tag{14}\]  
For four- or seven-element arrays, the relationships between \(p_{\textit{inres,min}}\) and incident angles are unclear, and it is hard to derive a simple expression of \(\Delta\varphi_{opt}=\text{arg}\min_{\Delta\varphi}~{}~{}~{}[p_{\textit{inres, min}}(\Delta\varphi)]\). Limited by the above reason, this section does not aim to figure out the detailed expression.