# A Phase Bias Compensation Method for Anti-Interference Antenna Arrays in RTK Positioning
## 2 Materials and Methods
### Signal Model  
The antenna array is considered as comprising M ideal omnidirectional elements, ignoring the mutual coupling effects between elements [24; 25]. Moreover, the antenna unit and radio frequency (RF) channel also meet the amplitude and phase consistency. Since the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **Algorithm** & **Advantages** & **Disadvantages** & **Applicable** \\  & Space-time PI & \begin{tabular}{c} Suitable for \\ pseudo-code ranging \\ systems; avoids \\ pseudo-code \\ measurement bias \\ \end{tabular} & \begin{tabular}{c} Not applicable to \\ carrier phase \\ measurements; relies \\ on weight constraints \\ \end{tabular} & \begin{tabular}{c} Pseudo-code \\ ranging systems \\ (e.g., military \\ navigation) \\ \end{tabular} \\  & Robust real-time phase & \begin{tabular}{c} Real-time operation; no \\ processing delay; low \\ computational \\ complexity \\ \end{tabular} & \begin{tabular}{c} Accuracy limited by \\ observation window \\ size; fails if phase bias \\ exceeds discriminator \\ range \\ \end{tabular} & \begin{tabular}{c} Low-cost dynamic \\ scenarios \\ \end{tabular} \\  & Uses MPDR to & \begin{tabular}{c} Unverified \\ performance in \\ high-dynamic \\ environments \\ \end{tabular} & \begin{tabular}{c} Moderate-speed \\ dynamic scenarios \\ \end{tabular} \\  & \begin{tabular}{c} Blind adaptive \\ beamforming based on \\ Direction Lock Loop \\ (DLL) \\ \end{tabular} & \begin{tabular}{c} No prior signal \\ direction required; \\ performance \\ comparable to MVDR \\ algorithm \\ \end{tabular} & \
\begin{tabular}{c} Complex hardware \\ implementation; \\ limited practical \\ application \\ \end{tabular} &
\begin{tabular}{c} Scenarios lacking \\ prior information \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._distance from the satellite signal to the ground is far greater than the array element spacing of the antenna array, it can be assumed that both the satellite signal and the interference signal are incident with plane waves. Therefore, the signal delay is only caused by the difference in the distance between the array elements. Figure 1 is the antenna array structure. The 1, 2, 3, \(\ldots\), M is the antenna array number. And the incident signal with each antenna is shown in (1). \[x(t)=s(t)+Ka(t)+n(t) \tag{1}\]  
Here, \(s(t)\) is the GNSS signal. \(s(t)\). \(a(t)\) is the interference signal and \(K\) is its number. \(n(t)\) is the noise that cannot be ignored. The incident signal of the antenna array is as shown in (2). \[\mathbf{x}(t)=\mathbf{a}(\theta_{s},\varphi_{s})s_{0}(t)+\sum_{i=1}^{K} \mathbf{a}_{i}(\theta_{u},\varphi_{u})a_{i0}(t)+n(t) \tag{2}\]  
Here, \(\mathbf{a}(\theta_{s},\varphi_{s})=\left[1,e^{-j\
u_{0}\tau_{1}},e^{-j\
u_{0} \tau_{2}},\cdots,e^{-j\
u_{0}\tau_{M-1}}\right]^{T}\) represents the steering vector of GNSS signal in each array element. \(\mathbf{a}_{i}(\theta_{u},\varphi_{u})=\left[1,e^{-j\
u_{0}\tau_{1}},e^{-j\
u _{0}\tau_{2}},\cdots,e^{-j\
u_{0}\tau_{M-1}}\right]^{T}\) represents the steering vector of the interference signal in each array element. Figure 2a,b show the structures of SAP and STAP. The output signal of SAP is shown in (3). \[\mathbf{y}(t)=\mathbf{w}_{a}^{\mathrm{H}}\mathbf{x}_{a}(t) \tag{3}\]  
Here, \(\mathbf{w}_{a}=\left[w_{1},w_{2},\cdots,w_{M}\right]^{T}\) is the antenna element weighting coefficient. \(\mathbf{x}_{a}(t)=\left[x_{1}(t),x_{2}(t),\cdots,x_{M}(t)\right]^{T}\) is the array antenna receiving signal after RF front-end processing [26; 27]. For STAP, the anti-interference weight coefficient consists of a time domain coefficient and space domain coefficient. The output signal of STAP is shown in (4). \[\mathbf{y}(t)=\mathbf{w}_{at}^{\mathrm{H}}\mathbf{x}_{at}(t) \tag{4}\] \[\left\{