# A Phase Bias Compensation Method for Anti-Interference Antenna Arrays in RTK Positioning
## 2 Materials and Methods
### The Bias Compensation Method for Anti-Interference Algorithm
#### 2.5.1 Bias Compensation Method Based on Array Weighted-Coefficient Conjugation  
We take seven elements as an example to illustrate the bias compensation algorithm with an odd number of elements. The distribution of elements is as shown in Figure 5.  
In the antenna array, elements 2 and 5 are symmetrical, elements 3 and 6 are symmetrical, elements 4 and 7 are symmetrical, and the coordinates are \((0,0)\), \((R,0)\), \((\frac{\sqrt{2}}{2}R,\frac{\sqrt{2}}{2}R)\), \((-\frac{\sqrt{2}}{2}R,\frac{\sqrt{2}}{2}R)\), \((-R,0)\), \((-\frac{\sqrt{2}}{2}R,-\frac{\sqrt{2}}{2}R)\), and \((\frac{\sqrt{2}}{2}R,-\frac{\sqrt{2}}{2}R)\). Therefore, the antenna element delay \(\tau_{i}(i=1,\ldots,7)\) and weight coefficients \(w_{i}(i=1,\ldots,7)\) have the following relationship. \[\left\{ \begin{array}{l}\tau_{2}=-\tau_{5},\tau_{3}=-\tau_{6},\tau_{4}=-\tau_ {7}\\ w_{2}^{*}=w_{5},w_{3}^{*}=w_{6},w_{4}^{*}=w_{7}\end{array} \right. \tag{20}\]  
The phase bias of the PI algorithm in the antenna array is shown in (21). \[ \begin{array}{l}w^{H}a=[w_{1},w_{2},w_{3},w_{4},w_{5},w_{6},w_{7}]\cdot[a_{1 },a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}]^{T}\\ =[1,A_{2}e^{j\varphi_{2}},A_{3}e^{j\varphi_{3}},A_{4}e^{j\varphi_{4}},A_{5}e^{ j\varphi_{5}},A_{6}e^{j\varphi_{6}},A_{7}e^{j\varphi_{7}}]\cdot[1,e^{-jw_{ 0}\tau_{2}},e^{-jw_{0}\tau_{3}},e^{-jw_{0}\tau_{3}},e^{-jw_{0}\tau_{5}},e^{-jw_ {0}\tau_{6}},e^{-jw_{0}\tau_{7}}]\\ =1+A_{2}e^{j(\varphi_{2}-w_{0}\tau_{2})}+A_{3}e^{j(\varphi_{3}-w_{0}\tau_{3})} +A_{4}e^{j(\varphi_{4}-w_{0}\tau_{4})}+A_{5}e^{j(\varphi_{5}-w_{0}\tau_{3})} +A_{6}e^{j(\varphi_{6}-w_{0}\tau_{6})}+A_{7}e^{j(\varphi_{7}-w_{0}\tau_{7})} \\ =1+A_{2}e^{j(\varphi_{2}-w_{0}\tau_{2})}+A_{3}e^{j(\varphi_{3}-w_{0}\tau_{3})} +A_{4}e^{j(\varphi_{4}-w_{0}\tau_{4})}+A_{2}e^{-j(\varphi_{2}-w_{0}\tau_{2})} +A_{3}e^{-j(\varphi_{3}-w_{0}\tau_{3})}+A_{4}e^{-j(\varphi_{4}-w_{0}\tau_{4})} \\ =1+A_{2}\cos(\varphi_{2}-w_{0}\tau_{2})+A_{3}\cos(\varphi_{3}-w_{0}\tau_{3})+A _{4}\cos(\varphi_{4}-w_{0}\tau_{4})\end{array}  \tag{21}\]  
Therefore, the weight coefficient of the seven-element antenna array is a real value that does not introduce phase bias. Based on the analysis, we propose the seven-element zero-phase-bias PI criterion in SAP as shown in (22). \[\left\{ \begin{array}{l}\min:P=E\{|y(n)|_{2}\}=\mathbf{w}^{H}\mathbf{R}_{xx} \mathbf{w}\\ \text{\emph{subject} to:}\quad\text{w}_{2}^{*}=w_{5}\\ \text{w}_{3}^{*}=w_{6}\\ \text{w}_{4}^{*}=w_{7}\\ \text{w}_{1}=1\end{array} \right. \tag{22}\]  
Figure 5: The distribution of seven elements. The terms in (22) can be expressed as (23) with a matrix. \[\left\{\begin{array}{l}\min:\mathbf{w}^{H}\mathbf{R}_{xx}\mathbf{w}\\ \\ \text{subject to}\quad\left[ \begin{array}{cccc}1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&1&0&0\end{array} \right]\left[ \begin{array}{c}\mathbf{w}_{1}\\ \mathbf{w}_{2}\\ \mathbf{w}_{3}\\ \mathbf{w}_{4}\\ \mathbf{w}_{5}\\ \mathbf{w}_{6}\\ \mathbf{w}_{7}\end{array} \right]^{*}-\left[ \begin{array}{cccc}0&0&0&0&0&0&0 \\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\left[ \begin{array}{c}\mathbf{w}_{1}\\ \mathbf{w}_{2}\\ \mathbf{w}_{3}\\ \mathbf{w}_{4}\\ \mathbf{w}_{5}\\ \mathbf{w}_{6}\\ \mathbf{w}_{7}\end{array} \right]=\left[ \begin{array}{c}1\\ 0\\ 0\\ 0\end{array} \right]\\ \\ \text{Here, }A=\left[ \begin{array}{cccc}1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right],\text{ }\text{B}=\left[ \begin{array}{cccc}0&0&0&0&0&0 \\ 0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right],\text{ C}=\left[ \begin{array}{c}1\\ 0\\ 0\\ 0\end{array} \right],\text{ and }\mathbf{w}=\left[ \begin{array}{c}\mathbf{w}_{1}\\ \mathbf{w}_{2}\\ \mathbf{w}_{3}\\ \mathbf{w}_{4}\\ \mathbf{w}_{5}\\ \mathbf{w}_{6}\\ \mathbf{w}_{7}\end{array} \right].\text{ And}\] (23) can be simplified as (24). \[\left\{ \begin{array}{l}\mathbf{w}^{H}\mathbf{R}_{xx}\mathbf{w}\\ \mathbf{A}\mathbf{w}^{*}-\mathbf{B}\mathbf{w}=\mathbf{C}\end{array} \right.