# A Phase Bias Compensation Method for Anti-Interference Antenna Arrays in RTK Positioning
## 2 Materials and Methods
### Error Analysis  
Assume the receiving signal of antenna element k is as shown in (15).  
\[x_{k}(n)=\mathrm{a}_{k}(\theta_{s},\varphi_{s})x_{s}(n)+\mathrm{a}_{k}(\theta _{s},\varphi_{j})x_{j}(n)+x_{kn}(n) \tag{15}\]  
The satellite signal is \(x_{s}(n)=x(n)\cdot e^{i\pi n}\). \(\mathrm{a}_{k}(\theta_{s},\varphi_{s})\) is the steering vector of the satellite signal in antenna element k. The interference signal is \(x_{j}(n)\). \(\mathrm{a}_{k}(\theta_{j},\varphi_{j})\) is the steering vector of the interference signal in antenna element k. The noise is \(x_{n}(n)\).  
The antenna array steering vector \(\mathbf{a}(\theta,\varphi)\) is shown in (16):  
\[\begin{array}{l}\mathbf{a}(\theta,\varphi)=\begin{bmatrix}e^{j\frac{2\pi}{ \lambda}p_{1}\cdot r}&e^{j\frac{2\pi}{\lambda}p_{2}\cdot r}&\ldots&e^{j\frac{2 \pi}{\lambda}p_{k}\cdot r}&\ldots&e^{j\frac{2\pi}{\lambda}p_{N}\cdot r}\\ e^{j\frac{2\pi}{\lambda}p_{N}\cdot r}=e^{j\frac{2\pi}{\lambda}(\mathrm{a}_{2} \cos\theta\cos\varphi+y_{k}\cos\theta\sin\varphi+z_{k}\sin\theta)}\end{bmatrix}\end{array} \tag{16}\]  
The coordinate of antenna element k is \(p_{k}=[x_{k},y_{k},z_{k}]\). The unit vector of the incident signal direction is \(r=[\cos\theta\cos\varphi,\cos\theta\sin\varphi,\sin\theta]^{T}\). N is the element number of the antenna array. Therefore, the output signal of the antenna array is as shown in (17).  
\[\mathbf{X}(n)=\mathbf{a}(\theta_{s},\varphi_{s})\mathbf{X}_{s}(n)+\mathbf{a}( \theta_{j},\varphi_{j})\mathbf{X}_{j}(n)+\mathbf{X}_{n}(n) \tag{17}\]  
Here, \(\mathbf{X}(n)=[x_{1}(n),\cdots,x_{N}(n)]\), \(\mathbf{a}(\theta_{s},\varphi_{s})=[a_{1}(\theta_{s},\varphi_{s}),\cdots,a_{N} (\theta_{s},\varphi_{s})]\) and \(\mathbf{a}(\theta_{j},\varphi_{j})=[a_{1}(\theta_{j},\varphi_{j}),\cdots,a_{N} (\theta_{j},\varphi_{j})]\).  
For the SAP PI algorithm, the optimal weight coefficient \(\mathbf{w}_{opt}\) is a complex value that is only constrained by \(\mathbf{w}^{H}\mathbf{s}=1\). Therefore, based on (3) and (17), the output of the antenna array signal is  
\[\mathrm{Y}(n)=\mathbf{w}_{opt}^{\mathrm{H}}\mathbf{X}(n)=\mathbf{w}_{opt}^{ \mathrm{H}}\mathbf{a}(\theta_{s},\varphi_{s})\mathbf{X}_{s}(n)+\mathbf{w}_{ opt}^{\mathrm{H}}\mathbf{a}(\theta_{j},\varphi_{j})\mathbf{X}_{j}(n)+\mathbf{X}_{n}(n) \tag{18}\]  
Therefore, the phase bias of SAP PI algorithm is \(\alpha=\arctan(\mathbf{w}_{opt}^{\mathrm{H}}\mathbf{a}(\theta_{s},\varphi_{s}))\). Taking the four-element array as an example, the array distribution is shown in Figure 3.  
Assume that the reference element is at the origin point, the rest of the elements are evenly distributed on the circle, and the coordinates of the four elements are \((0,0,0)\), \((d,0,0)\), \((-\frac{1}{2}d,\frac{\sqrt{3}}{2}d,0)\), \((-\frac{1}{2}d,-\frac{\sqrt{3}}{2}d,0)\), \(d=\frac{\lambda}{2}\). And the phase bias of SAP\[\left\{\begin{array}{c}\varphi=a\tan\frac{\sin(2\pi\varepsilon)+\sin(\pi(- \varepsilon+\vartheta))}{2-\cos(2\pi\varepsilon)-\cos(\pi(\varepsilon- \vartheta))}\\ \varepsilon=\varepsilon_{s}+\varepsilon_{j}=\frac{1}{2}\cos\theta_{s}\cos \varphi_{s}+\frac{1}{2}\cos\theta_{j}\cos\varphi_{j}\\ \vartheta=\theta_{s}+\vartheta_{j}=\frac{\sqrt{3}}{2}\cos\theta_{s}\sin \varphi_{s}+\frac{\sqrt{3}}{2}\cos\theta_{j}\sin\varphi_{j}\\ \end{array}\right. \tag{19}\]  
Figure 4 shows the distribution of the phase bias.  
For the STAP PI algorithm, the optimal weight coefficient \(\mathbf{w}_{opt}\) is also a complex number, but the dimension adds the time domain parts compared with SAP. Therefore, the steering vector becomes the Kronecker product of the spatial domain steering vector and the time domain tap vector, which is the expansion of the dimension in the spatial domain, but also a complex number.  
For the SAP MVDR algorithm and STAP MVDR algorithm, the optimal weight coefficient \(\mathbf{w}_{opt}\) is a complex value constrained by \(\mathbf{w}^{H}\mathbf{a}=1\). Therefore, the algorithms do not introduce phase bias.