## 3 Proposed STPAP Beamforming Algorithm  
### Novel MVDR-Based Criterion  
Assume that \(L\) desired GNSS signals are incident from angular direction \((\theta_{l},\varphi_{l})\) with polarized parameter \((\gamma_{l},\eta_{l})(l=1,2,\cdots,L)\) and \(Q\) interferences are incident from angular direction \((\theta_{q},\varphi_{q})\) with polarized parameter \((\gamma_{q},\eta_{q})(q=1,2,\cdots,Q)\). Using the DPSA for the STPAP, the received signal model for the \(n\)-th block can be denoted as  
\[\mathbf{X}(n)=\sum\limits_{l=1}^{L}\mathbf{A}_{l}(\theta_{l},\varphi_{l},\gamma_{l}, \eta_{l})\mathbf{s}_{l}(n)+\sum\limits_{q=1}^{Q}\mathbf{A}_{q}(\theta_{q},\varphi_{q},\gamma_{q},\eta_{q})\mathbf{s}_{q}(n)+\mathbf{G}(n), \tag{8}\]  
where \(\mathbf{s}_{l}(n)=[s_{l}(t),\cdots,s_{l}(t-(K-1)T_{0})]^{\rm H}\) and \(\mathbf{s}_{q}(n)=[s_{q}(t),\cdots,s_{q}(t-(K-1)T_{0})]^{\rm H}\) are respectively the complex envelope of the \(l\)-th desired GNSS signal and the \(q\)-th interference for the \(n\)-th block, \(\mathbf{A}_{l}(\theta_{l},\varphi_{l},\gamma_{l},\eta_{l})\) and \(\mathbf{A}_{q}(\theta_{q},\varphi_{q},\gamma_{q},\eta_{q})\) respectively denote the joint space-time-polarization steering vector of the \(l\)-th desired GNSS signal and \(q\)-th interference, and \(\mathbf{G}(n)\in\mathbb{C}^{2 MK\times 1}\) represents the additive white Gaussian noise vector for the \(n\)-th block. Figure 2: The STPAP architecture using the DPSA. Due to the aforementioned STPAP architecture, the joint space-time-polarization steering vector \(\mathbf{A}(\theta,\varphi,\gamma,\eta)\in\mathbb{C}^{2 MK\times 1}\) is given by  
\[\mathbf{A}(\theta,\varphi,\gamma,\eta)=\mathbf{I}_{K\times K}\otimes\mathbf{a}_{\text{s}}( \theta,\varphi)\otimes\mathbf{a}_{\text{p}}(\theta,\varphi,\gamma,\eta) \tag{9}\]  
with  
\[ \begin{array}{cccc}\mathbf{a}_{\text{s}}(\theta,\varphi)=\left[\begin{array}{ ccccc}1&e^{-j\Delta(\theta,\varphi)}&e^{-j\Delta\varphi(\theta,\varphi)}&\cdots&e^{-j(M-1) \Delta\phi(\theta,\varphi)}\end{array}\right]^{\text{H}},\\ \mathbf{a}_{\text{p}}(\theta,\varphi,\gamma,\eta)=\left[\begin{array}{c}-\sin \varphi\cos\gamma+\cos\theta\cos\varphi\sin\gamma e^{j\eta}\\ \cos\varphi\cos\gamma+\cos\theta\sin\varphi\sin\gamma e^{j\eta}\end{array} \right],\end{array}  \tag{10}\]  
where \(\otimes\) denotes the Kronecker product and \(\mathbf{I}_{K\times K}\) is the \(K\)-th order identity matrix. In addition, \(\mathbf{a}_{\text{s}}(\theta,\varphi)\) and \(\mathbf{a}_{\text{p}}(\theta,\varphi,\gamma,\eta)\) respectively represent the spatial steering vector and polarized steering vector, \(\Delta\phi(\theta,\varphi)\) denotes the phase difference between the adjacent elements, and \(j=\sqrt{-1}\). The traditional MVDR criterion is designed to minimize the array output power subject to the constraint that keeping the array gain towards the desired GNSS satellites undistorted and its expression is given by  
\[ \begin{array}{c}\underset{\mathbf{W}(n)}{\text{Minimize }}\mathbf{W}^{\text{H}}(n)\mathbf{R}(n)\mathbf{W}(n)\\ \text{s.t. }\mathbf{W}^{\text{H}}(n)\mathbf{C}=1\quad n=1,2,\cdots N,\end{array}  \tag{11}\]  
where \(\mathbf{C}\in\mathbb{C}^{2 MK\times 1}\) represents the constraint vector and \(\mathbf{R}(n)\in\mathbb{C}^{2 MK\times 2 MK}\) is the covariance matrix. In practice, it is infeasible to acquire the theoretical covariance matrix and thus it can be replaced with the sample covariance matrix \(\widetilde{\mathbf{R}}(n)=\frac{1}{N}\sum\limits_{n=1}^{N}\mathbf{X}(n)\mathbf{X}^{\text{ H}}(n)\), in which \(N\) is the number of blocks. In this paper, a novel constraint vector \(\overline{\mathbf{C}}\in\mathbb{C}^{2 MK\times J}\) for the STPAP, which keeps the array gain towards \(J\) desired GNSS satellites in the FOV undistorted at once while minimizing the array output power, is proposed. We can give the expression of \(\overline{\mathbf{C}}\) as  
\[\overline{\mathbf{C}}=\left[