## 2 System Model
### GNSS and 5G Joint Positioning Model  
Aiming to take full advantage of the measurement information offered by GNSS satellites and 5G BSs, we propose a GNSS and 5G fusion positioning model with the input of the observation data of multiple satellites and multiple 5G BSs, which are named GNSS and 5G Joint Positioning Model (GNSS and 5G JPM). The purpose of realizing the fusion positioning of GNSS and 5G is to unify the time-reference and space-reference coordinate system. In this paper, the BDS time and the Earth-Centered Earth-Fixed (ECEF) coordinate system are chosen as the unified benchmarks. Assuming that m satellites and n5G BSs can be observed in each epoch \(t\), the signal-measurement model of the GNSS and 5G Joint Positioning Receiver (JPR) can be obtained as described in Equation (5):  
\[y_{\text{GNSS \& 5G}}\left[t_{k}\right]=\left[\tau_{\text{ GNSS }}^{1}\left[t_{k}\right],\ldots,\tau_{\text{GNSS}}^{m}[t_{k}],\tau_{\text{5G}}^{1} [t_{k}],\ldots,\tau_{\text{5G}}^{n}[t_{k}],\varphi^{1}[t_{k}],\theta^{1}[t_{k} ],\ldots,\varphi^{n}[t_{k}],\theta^{n}[t_{k}]\right]^{T} \tag{7}\] \[=h_{\text{GNSS \& 5G}}\left(s_{\text{GNSS \& 5G}}\left[t_{k}\right]\right)+n_{\text{GNSS \& 5G}}[t_{k}]\]  
where \(\tau_{\text{GNSS}}^{1}[t_{k}],\ldots,\tau_{\text{GNSS}}^{m}[t_{k}]\) denotes the TOA JPR obtained from the first to the \(m\)-th satellite, \(\tau_{\text{5G}}^{1}[t_{k}],\ldots,\tau_{\text{5G}}^{n}[t_{k}],\varphi^{1}[t_{ k}],\theta^{1}[t_{k}],\ldots,\varphi^{n}[t_{k}]\), and \(\theta^{n}[t_{k}]\) represent the TOA, azimuth DOA, and elevation DOA of the first to nth 5G base stations received by the JPR, respectively. \(h_{\text{GNSS\&5G}}\) represents the measurement function of JPM \(h_{\text{GNSS\&5G}}=\left[h_{\text{GNSS}}^{1}[t_{k}],\ldots,h_{\text{GNSS}}^ {M}[t_{k}],h_{\text{5G}}^{1}[t_{k}],\ldots,h_{\text{5G}}^{N}[t_{k}]\right]^{T}\), and observation noise is composed of m satellites and n5G base stations \(n_{\text{GNSS\&5G}}[t_{k}]=\left[\omega_{\text{GNSS}}^{1}[t_{k}],\ldots, \omega_{\text{GNSS}}^{m}[t_{k}],\omega_{\text{5G}}^{1}[t_{k}],\ldots,\omega_{ \text{5G}}^{n}[t_{k}]\right]^{T}\). In the case of synchronized satellites and 5G BSs, the system state of the GNSS and 5G joint 3D positioning can be written as  
\[s_{\text{GNSS\&5G}}[t_{k}]=\left[p[t_{k}],v[t_{k}],\rho[t_{k}],\alpha[t_{k}] \right]^{T} \tag{8}\]  
The continuous white noise acceleration model in [9] is used here to characterize the transformation of the JPR's state. \[s_{\text{GNSS\&5G}}[t_{k}]=\mathbf{F}s_{\text{GNSS\&5G}}[t_{k-1}]+\mathbf{u} [t_{k}] \tag{9}\]  
where \(\mathbf{F}\) denotes the state-transition matrix, and \(\mathbf{u}[t_{k}]\) is the process noise following a Gaussian distribution. \[\mathbf{F}=\left[ \begin{array}{ccc}\mathbf{I}_{3\times 3}&\Delta t\cdot\mathbf{I}_{3 \times 3}&\mathbf{0}_{3\times 2}\\ \mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3}&\mathbf{0}_{3\times 2}\\ \mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{F}_{\text{Clock}}\end{array}  \right],\mathbf{F}_{\text{Clock}}=\left[ \begin{array}{cc}1&\Delta t\\ 0&\omega\end{array} \right] \tag{10}\]The four elements at the top-left corner of matrix \(\mathbf{F}\) represent the constant-velocity (CV) model, and \(\Delta t\) is the interval between two adjacent epochs. \(\mathbf{F}_{\text{Clock}}\) describes the JPR clock model obtained from [29]. \[\mathbf{Q}=\left[ \begin{array}{cccc}\frac{\sigma_{v}^{2}\Delta t^{3}}{3} \cdot\mathbf{I}_{3\times 3}&\frac{\sigma_{v}^{2}\Delta t^{2}}{2}\cdot\mathbf{I}_{3 \times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3\times 1}\\ \frac{\sigma_{v}^{2}\Delta t^{2}}{2}\cdot\mathbf{I}_{3\times 3}&\sigma_{v}^{2} \Delta t\cdot\mathbf{I}_{3\times 3}&\mathbf{0}_{3\times 1}&\mathbf{0}_{3 \times 1}\\ \mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&\frac{\sigma_{v}^{2}\Delta t^{2}}{ 3}&\frac{\sigma_{v}^{2}\Delta t^{2}}{2}\\ \mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&\frac{\sigma_{v}^{2}\Delta t^{2}}{ 2}&\sigma_{\eta}^{2}\Delta t\end{array}