# A Phase Bias Compensation Method for Anti-Interference Antenna Arrays in RTK Positioning
## 2 Materials and Methods
### Influence of Anti-Interference Algorithm on RTK Positioning  
In RTK positioning, the motion station and the reference station receive two satellites signal at the same epoch. The receiver of the motion station makes a double difference in stations and satellites to eliminate clock difference, ionospheric delay, and tropospheric delay. And the double-difference carrier phase result \(\varphi_{m,b}^{s_{1},s_{2}}\) is shown in (9). \[\varphi_{m,b}^{s_{1},s_{2}}=\lambda^{-1}(r_{m,b}^{s_{1},s_{2}})+N_{m,b}^{s_{1},s_{2}}+\varepsilon_{m,b}^{s_{1},s_{2}} \tag{9}\]  
Here, \(\lambda\) is the wavelength of the carrier. \(r_{m,b}^{s_{1},s_{2}}\) represents the double-difference geometric distance. \(N_{m,b}^{s_{1},s_{2}}\) represents the double-difference integer ambiguity. \(\varepsilon_{m,b}^{s_{1},s_{2}}\) denotes the double-difference observation noise. And the unknown parameters in (9) are \(X=[r_{m,b}^{s_{1},s_{2}}N_{m,b}^{s_{1},s_{2}}]\). Then, the Kalman filter is adopted to estimate the \(X=[r_{m,b}^{s_{1},s_{2}}N_{m,b}^{s_{1},s_{2}}]\). Its state equation and observation equation can be expressed as follows. \[\left\{ \begin{array}{l}X_{k}=\phi_{k,k-1}X_{k-1}+\Gamma_{k-1}N_{k-1}\\ L_{k}=H_{k}X_{k}+V_{k}\end{array} \right. \tag{10}\]  
Here, k is the observation epoch time. \(X_{k}\) is the state vector at k. Assuming the satellite number is 4, the \(X_{k}=[x_{k},y_{k},z_{k},\Delta N_{1},\Delta N_{2},\Delta N_{3},\Delta N_{4}]\). \((x_{k},y_{k},z_{k})\) is the motion station location. \((\Delta N_{1},\Delta N_{2},\Delta N_{3},\Delta N_{4})\) is the double-difference integer ambiguity of satellites. \(\phi_{k,k-1}\) is the state transition matrix. \(\Gamma_{k-1}\) is the control-input model that is applied to the process noise \(N_{k-1}\). \(L_{k}\) is the observation result at k. \(H_{k}\) is the observation model, which maps the true state space into the observed space. \(V_{k}\) is the observation noise, which is assumed to be zero-mean Gaussian white noise [31; 32]. However, the phase bias introduced by the anti-interference algorithm itself will directly affect the carrier phase observations, thereby influencing the positioning distance. Therefore, the carrier phase observation is shown in (11). \[\varphi_{m,b}^{s_{1},s_{2}}+\Delta\varphi_{anti}=\lambda^{-1}(r_{m,b}^{s_{1},s _{2}})+N_{m,b}^{s_{1},s_{2}}+\varepsilon_{m,b}^{s_{1},s_{2}} \tag{11}\]  
In (11), \(\Delta\varphi_{anti}\) is the phase bias that is introduced by the anti-interference algorithm. According to the prediction and update process of the Kalman filter, the bias introduced in the carrier phase observation affects the observation matrix \(L_{k}\); then, at the time of epoch k, the variable of state estimation can be shown as  
\[\hat{x}_{k,k}=\hat{x}_{k,k-1}+K_{k}[L_{k}+\Delta L_{k}-H_{k}\hat{x}_{k,k-1}] \tag{12}\]And the predicted value of state at epoch k + 1 is \(\hat{x}_{k+1,k}=\phi_{k+1,k}(\hat{x}_{k,k}+\Delta\hat{x}_{k,k})\), where \(\Delta\hat{x}_{k,k}=K_{k}\Delta L_{k}\). Therefore, the state estimation at k + 1 is as shown in (13):  
\[ \begin{array}{l}\hat{x}_{k+1,k+1}=\hat{x}_{k+1,k}+K_{k+1}[L_{k+1}+\Delta L_{k +1}-H_{k+1}\hat{x}_{k+1,k}]\\ =\phi_{k+1,k}(\hat{x}_{k,k}+\Delta\hat{x}_{k,k})+K_{k+1}[L_{k+1}+\Delta L_{k+1} -H_{k+1}\hat{\phi}_{k+1,k}(\hat{x}_{k,k}+\Delta\hat{x}_{k,k})]\end{array}  \tag{13}\]  
The state estimation error at the time of k + 1 epoch is as shown in (14):  
\[ \begin{array}{l}\Delta\hat{x}_{k+1,k+1}=\phi_{k+1,k}\Delta\hat{x}_{k,k}+K_{k +1}[\Delta L_{k+1}-H_{k+1}\phi_{k+1,k}\Delta\hat{x}_{k,k}]\\ =[I-K_{k+1}H_{k+1}]\phi_{k+1,k}\Delta\hat{x}_{k,k}+K_{k+1}\Delta L_{k+1}\end{array}  \tag{14}\]  
The observation matrix error of epoch k leads to the state estimation error of epoch k, and this error continues to be transmitted in the next epoch. And the impact on RTK positioning cannot be ignored, which will lead to a large RTK positioning bias or failure to obtain a position result. It is necessary to compensate for the bias introduced by the anti-interference algorithm to achieve the positioning of the RTK system.