# Anti-Jamming Method and Implementation for GNSS Receiver Based on Array Antenna Rotation
## 4 The Variable-Step Iteration Algorithm for Efficient Determination of Optimal Rotation Angle  
### Iteration Process to Determine Optimal Rotation Angle  
Because the mathematical expression of \(\lambda_{\min}[\mathbf{R}_{\mathbf{xx}}(\Delta\varphi)]\) is too complicated, it is hard to obtain \(\Delta\varphi_{opt}\) directly by Equation (14). Although one option is to traverse all possibilities of \(\Delta\varphi\) and find the optimal result, there is a contradiction between the computation amount and result resolution. Therefore, an iteration algorithm based on gradient descent was considered. The algorithm searches along the steepest descent direction of the residual interference-plus-noise power curve and updates \(\Delta\varphi\) according to the last iteration. The  
Figure 3: Relationship between antenna array attitude and anti-jamming performance. (**a**) ICR against rotation angle. (**b**) Residual interference power against rotation angle. (**c**) Antenna pattern at \(\Delta\varphi=19^{\circ}\). (**d**) Antenna pattern at \(\Delta\varphi=47.7^{\circ}\).  
iteration is accomplished when the minimum residual power and the corresponding \(\Delta\varphi_{opt}\) is obtained.  
The iteration based on gradient descent is shown in Formula (15). \(l\) represents the \(l\)th iteration, \(y(l)\) is the residual interference-plus-noise power, \(d(l)\) is the expected residual interference-plus-noise power, \(e(l)\) is the error, and \(\mu\) is the iteration step. Considering the interferences are expected to be completely suppressed while the noise power is close to zero, \(d(l)\) was accordingly set to zero.  
\[\left\{\begin{array}{l}y(l)=\lambda_{\min}[\mathbf{R_{xx}}(\Delta\varphi(l) )]\\ e(l)=y(l)-d(l)\\ \Delta\varphi(l+1)=\Delta\varphi(l)+\mu(l)\cdot(-\
abla(l))\end{array}\right. \tag{15}\]  
in which  
\[-\
abla(l)=-\frac{\partial e(l)}{\partial\Delta\varphi(l)} \tag{16}\]  
and the Wiener solution of optimal \(\Delta\varphi\) is  
\[\Delta\varphi_{opt}=\underset{\Delta\varphi_{opt}}{\arg}\frac{\partial e}{ \partial\Delta\varphi}=0 \tag{17}\]  
Suppose the incident direction and the power of interferences are known and the initial rotation angle is \(\Delta\varphi(0)\). A covariance matrix \(\mathbf{R_{xx}}(\Delta\varphi)\) is constructed from the information of the interference intensity, incident direction, and rotation angle. The minimum residual interference-plus-noise power was obtained by eigen-decomposition. The error \(y(l)\) equals to \(\lambda_{\min}[\mathbf{R_{xx}}(\Delta\varphi)]\). For the problem that the error expression is too complicated for direct calculation \(\
abla(l)\), the definition of a partial derivative is used to numerically approximate \(\frac{\partial e}{\partial\Delta\varphi}\), such that  
\[\frac{\partial e}{\partial\Delta\varphi}\Big{|}_{\Delta\varphi=\Delta\varphi_ {l}}=\underset{\Delta\to 0}{\lim}\frac{e(\Delta\varphi_{l}+\Delta)-e(\Delta \varphi_{l})}{\Delta} \tag{18}\]  
among which the anti-jamming algorithm in Formula (15-first equation) is not limited as above. It can be replaced by other algorithms, such as PI (power inversion) or MVDR (minimum variance distortionless response). As shown in Figure 3b, the curve of residual interference power has more than one extreme point. It is necessary to constrain the iteration step and initial value to avoid \(\Delta\varphi\) falling into a local optimum and to accelerate the convergence speed.