## 3 The Proposed Multiple-Signal Joint Estimation Method
### Square Root Unscented Stable Filter
Only 2D scenarios are considered here, but the extension to 3D is quite straightforward. (1) Positioning performance of the multiple-signal joint estimation method

The positioning error and the cumulative distribution function (CDF) of the proposed method are compared with those of the other five methods. From Figures 3 and 4, we can conclude that among the six observation fusion methods for GNSS and 5G joint positioning, the positioning algorithm based on the EKF filtering has the largest horizontal positioning error, and the horizontal positioning error is 1.98 m in 90% of cases. This is probably because the EKF in the system model uses the Jacobian matrix to calculate the observation matrix, which will bring a certain degree of precision loss. This could be due to the fact that the EKF must derive the Jacobian matrix to obtain the observation matrix, which will bring a certain degree of precision loss. Compared with EKF, the RMSE of MRAKF is reduced by more than 0.4 m on average, indicating the superiority of adaptive strategy filtering. UKF utilizes the UT transform to avoid the accuracy loss caused by nonlinear problems, and its performance is significantly better than standard EKF. In 90% of cases, the horizontal positioning error is 1.55 m. VBHUKF uses the variable Bayesian approximation to estimate the covariance of time-varying measurement noise to achieve self-adaptation. Compared with UKF, it has better filtering consistency, and the positioning accuracy is about 10% higher than that of UKF. SRUKF introduces the square root form of covariance into UKF \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline
5G carrier frequency & 3.5 GHz \\ \hline
5G bandwidth & 100 MHz \\ \hline Subcarrier spacing & 30 KHz \\ \hline Total number of satellites & 10 \\ \hline Total number of 5G BSs & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters. to ensure the semi-positive qualitative value of the state space covariance matrix, and the output numerical results are more stable. The proposed SRUSF is superior to the five other filtering methods, the positioning accuracy is slightly better than that of SRUKF and significantly better than other methods, and the positioning error of 0.97 m is achieved in 90% of cases. This is mainly due to the fact we have innovatively proposed a tight-coupled filter group. First, the observations of the GNSS and 5G joint positioning receiver are filtered independently, and then the observations of each signal source are fused to provide an optimal positioning estimate. The stabilized coefficient is introduced in the updating process of the state prediction covariance at the same time, which will undoubtedly reduce the possibility of the divergence of the filtering results due to the imperfect match between the system model and the actual situation and ultimately improve the positioning accuracy of the algorithm. (2) Positioning performance with different numbers of available signal sources

The positioning error of the GNSS-5G joint positioning system is closely related to the following factors: the modeling accuracy of the system model, the measurement error, the signal propagation error, the geometry of the signal source, the number of visible signal sources, and the positioning algorithm. The first three items are related to system design parameters and the baseband signal processing algorithm and are not within the scope of this paper. The geometry distribution of signal sources also has an important impact on positioning errors. However, under ideal conditions when the signal sources are evenly distributed, the impact on positioning errors is relatively small. This paper mainly focuses

Figure 4: CDF of positioning error for EKF, MRAKF, UKF, SRUKF, VBHUKF, and the proposed SRUSF method. Figure 3: Positioning error comparison of EKF, MRAKF, UKF, SRUKF, VBHUKF, and the proposed SRUSF method.