## 3 The Proposed Multiple-Signal Joint Estimation Method
### Square Root Unscented Stable Filter  
In order to enhance the numerical stability of UKF, the square root form of the state covariance is introduced. In the Square Root Unscented stable filter implementation, the square root of the state covariance is propagated directly, avoiding the requirement to refactorize the covariance at each iteration. In addition, the square root form has the additional advantages of numerical stability and positive semi-certainty of the state covariance. During the fusion of GNSS and 5G observations, the stability of the filter must be guaranteed. Therefore, a stability factor is added as a slight modification to ensure the reliability of the SRUSF. Unscented Transformation is introduced in SRUSF, utilizing the deterministic \"sampling\" method to calculate the mean and covariance terms. The relative locations of the sampling points and their corresponding weights are based on the following basic principle: they can capture the most important statistical properties of the prior random variables. \(2\text{ L}+1\) sigma points are selected according to the square root decomposition of a priori covariance. The nonlinear function is applied to each sigma point to yield a cloud of transformed points, and the estimate of the nonlinearly transformed mean and covariance can be obtained by weighting the transformed points. The initial system state \(\mathsf{s}_{\text{GNSS6\& 5G}}[t_{0}]\sim\mathcal{N}(\text{m},P)\) follows a Gaussian distribution with mean m and variance P.  
From the column of system state matrix \(s_{\text{GNSS$\&$55$}}[t_{k}]\) we can obtain \(2\,\text{L}+1\) sigma points  
\[\left\{ \begin{array}{cc}x_{0}=m\\ x_{i}=m+(\sqrt{(L+\lambda)\mathbf{P}})_{i}&i=1,\ldots,L\\ x_{i}=m-(\sqrt{(L+\lambda)\mathbf{P}})_{i-L}&i=L+1,\ldots,2L\end{array} \right. \tag{14}\]  
Calculating the corresponding weight for each sigma point,  
\[W_{0}^{(m)}=\frac{\lambda}{(L+\lambda)} \tag{15}\]  
\[W_{0}^{(c)}=\frac{\lambda}{(L+\lambda)+(1-\alpha^{2}+\beta)} \tag{16}\]  
\[W_{i}^{(m)}=W_{i}^{(c)}=\frac{1}{2(L+\lambda)},i=1,\ldots,2L \tag{17}\]  
where the scaling parameter can be expressed as \(\lambda=\alpha^{2}(L+\kappa)-L\). \(L\) is the dimension of the state space matrix, and \(\alpha\) and \(\kappa\) determine the distribution of sigma points near the mean value. The parameter \(\alpha\) is typically set to a smaller positive value \(1\times 10^{-4}\leq\alpha\leq 1\)[30]; \(\kappa\) is a secondary scaling parameter, which is often set to 0; and \(\beta\) is a scalar parameter used to incorporate prior information on probability distributions over the state space of the system. For A Gaussian distribution, \(\beta=2\) is optimal. These sigma vectors are propagated through the nonlinear function  
\[y_{i}=h(x_{i})i=0,\ldots \tag{18}\]  
The covariance of the GNSS-5G joint-positioning state-prediction error can be expressed as  
\[P^{-}[t_{k}]=FP[t_{k}]F^{T}+Q[t_{k}] \tag{19}\]  
where \(P[t_{k}]\) denotes the estimation-error covariance and its square root form \(R[t_{k}]\) is presented in (20), and \(Q[t_{k}]\) is the Gaussian process noise matrix at \(t_{k}\). \[P[t_{k}]=R[t_{k}](R[t_{k}])^{T} \tag{20}\]  
By introducing the QR decomposition, the square root of the state prediction covariance can be expressed as  
\[R^{-}[t_{k}]=\left(\text{qr}\bigg{\{}\left[FR[t_{k}]\sqrt{Q[t_{k}}\right] \bigg{]}^{T}\bigg{\}}\right)^{T} \tag{21}\]  
where the \"qr\" (orthogonal triangular decomposition) is a MATLAB function that denotes the QR decomposition. In the GNSS and 5G joint positioning SR-USF algorithm, the process noise consists of two parts, the deterministic noise and the stochastic noise. The deterministic noise can be settled using a priori knowledge, while the stochastic noise attributed to the modeling error should be compensated accurately. Because of this modeling inaccuracy, the innovation's expected mean and covariance could be divergent from their actual values in practice.