## 3 Proposed STPAP Beamforming Algorithm
### Novel MVDR-Based Criterion
\begin{array}{cccc}c_{\text{h},1,1}&c_{\text{h},1, 2}&\cdots&c_{\text{h},1,J}\\ c_{\text{v},1,1}&c_{\text{v},1,2}&\cdots&c_{\text{v},1,J}\\ \vdots&\vdots&\vdots&\vdots\\ c_{\text{h},M,1}&c_{\text{h},M,2}&\cdots&c_{\text{h},M,J}\\ c_{\text{v},M,1}&c_{\text{v},M,2}&\cdots&c_{\text{v},M,J}\\ 0&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0\end{array} \right] \tag{12}\]  
with  
\[ \begin{array}{ll}c_{\text{h},m,l}&=a_{\text{s},m}(\theta_{l},\varphi_{l})a_ {\text{p},\text{h}}(\theta_{l},\varphi_{l},\gamma=\pi/4,\eta=\pi/2)\\ &=(-\sin\varphi_{l}\cos\frac{\pi}{4}+\cos\theta_{l}\cos\varphi_{l}\sin\frac{ \pi}{4}e^{j\frac{\pi}{2}})e^{-j(m-1)\Delta\phi(\theta_{l},\varphi_{l})},\\ c_{\text{v},m,l}&=a_{\text{s},m}(\theta_{l},\varphi_{l})a_{\text{p},\text{v}}( \theta_{l},\varphi_{l},\gamma=\pi/4,\eta=\pi/2)\\ &=(\cos\varphi_{l}\cos\frac{\pi}{4}+\cos\theta_{l}\sin\varphi_{l}\sin\frac{ \pi}{4}e^{j\frac{\pi}{2}})e^{-j(m-1)\Delta\phi(\theta_{l},\varphi_{l})},\end{array}  \tag{13}\]  
where \(a_{\text{s},m}(\theta_{l},\varphi_{l})\) denotes the \(m\)-th component of the spatial steering vector corresponding to the \(l\)-th GNSS signal. In addition, \(a_{\text{p},\text{h}}(\theta_{l},\varphi_{l},\gamma=\pi/4,\eta=\pi/2)\) and \(a_{\text{p},\text{v}}(\theta_{l},\varphi_{l},\gamma=\pi/4,\eta=\pi/2)\) represents the horizontal and vertical component of the polarized steering vector corresponding to the \(l\)-th GNSS signal, respectively. Note that the DOAs of the desired GNSS signals, \((\theta_{l},\varphi_{l})\), are assumed to be known as a priori through the existing GNSS DOA estimation methods proposed in [32; 33; 34; 35], such as the inertial navigation device assisting method. Moreover, the polarized parameters for all desired GNSS signals are \((\gamma,\eta)=(\pi/4,\pi/2)\) as the polarization modes of all GNSS signals are RHCP. Furthermore, the parameter \(J\) is worth discussing because it determines the number of adaptive filters that used in the proposed STPAP beamforming algorithm when multiple GNSS satellites exist in the FOV. When \(J=L\), only one adaptive filter is required for all the desired GNSS satellites in the FOV; When \(2\leq J\leq L-1\), at least two but no more than \(L-1\) adaptive filters is needed to deal with all the desired GNSS satellites in the FOV; When \(J=1\), the proposed MVDR-CMS criterion degenerates into the MVDR-CSS criterion in [22]. Therefore, the parameter \(J\) is defined as \(2\leq J\leq L\) in the proposed MVDR-CMS criterion. Obviously, the computational complexity of the proposed criterion is lower than that of the MVDR-CSS criterion in [22]. With the novel constraint vector in (12), the expression in (11) can be rewritten as  
\[ \begin{array}{ll}\underset{\mathbf{W}_{\text{all}}(n)}{\text{Minimize}}&\mathbf{W}_ {\text{all}}^{\text{H}}(n)\widetilde{\mathbf{R}}(n)\mathbf{W}_{\text{all}}(n)\\ \text{s.t.}&\mathbf{W}_{\text{all}}^{\text{H}}(n)\widetilde{\mathbf{C}}=\overset{ \rightharpoonup}{1}_{J}&n=1,2,\cdots N,\end{array}  \tag{15}\]  
where \(\mathbf{W}_{\text{all}}(n)\in\mathbb{C}^{2 MK\times 1}\) denotes the weight vector for the \(J\) GNSS satellites in the FOV and \(\overset{\rightharpoonup}{1}_{J}\) a \(1\times J\) vector containing all ones. Moreover, the weight vector \(\mathbf{W}_{\text{all}}(n)\) are found from the solution to (14) and it can be presented as \(\mathbf{W}_{\text{all}}(n)=\widetilde{\mathbf{R}}^{-1}(n)\overline{\mathbf{C}}[\overline{ \mathbf{C}}^{\text{H}}\widetilde{\mathbf{R}}^{-1}(n)\overline{\mathbf{C}}]\overset{ \rightharpoonup}{1}_{J}\). It is noticed that the weight vector \(\mathbf{W}_{\text{all}}(n)\) can be applied to the \(J\) desired GNSS satellites in the FOV at once, which can greatly reduce the computational complexity, especially as the parameter \(J\) increases. For comparison, the aforementioned MVDR-CSS criterion in [22] for the STPAP when using the DPSA is presented as  
\[