## 3 The Proposed Multiple-Signal Joint Estimation Method
### Square Root Unscented Stable Filter
A stabilized coefficient is proposed here to solve this problem, which will undoubtedly improve the competitiveness of this method. \[R[t_{k}]=\sqrt{\varphi[t_{k}]}R^{-}[t_{k}] \tag{22}\]The estimated innovation covariance \(P_{\gamma,\gamma}[t_{k}]\) should be larger than or equal to the real one \(\mathrm{E}\{\gamma[t_{k}]\gamma^{T}[t_{k}]\}\) so as to maintain consistent estimates. \[P_{\gamma,\gamma}[t_{k}]\geq\mathrm{E}\Big{\{}\gamma[t_{k}]\gamma^{T}[t_{k}] \Big{\}} \tag{23}\]  
\[\gamma[t_{k}]=y[t_{k}]-\hat{y}[t_{k}] \tag{24}\]  
where \(\gamma[t_{k}]\) is the innovation sequence and \(\hat{y}[t_{k}]\) is the predicted measurement. It is significant to point out that the innovation covariance can be written as  
\[W_{0}^{(c)}=\frac{\lambda}{(L+\lambda)+(1-\alpha^{2}+\beta)} \tag{16}\]0  
where \(P[t_{k}]\) is the covariance of the predicted state and the cross-covariance between the measurements, and the predicted state is denoted by \(P_{x,y}[t_{k}]\):  
\[W_{0}^{(c)}=\frac{\lambda}{(L+\lambda)+(1-\alpha^{2}+\beta)} \tag{16}\]1  
Combining Equations (25) and (26) leads to Equation (27)  
\[W_{0}^{(c)}=\frac{\lambda}{(L+\lambda)+(1-\alpha^{2}+\beta)} \tag{16}\]2  
Incorporating across Equations (23) to (27) yields Equation (28), the fading factor introduced into the predicted state covariance should meet the requirements  
\[W_{0}^{(c)}=\frac{\lambda}{(L+\lambda)+(1-\alpha^{2}+\beta)} \tag{16}\]3  
where \(P^{-}[t_{k}]\) is the covariance of the predicted state \(P[t_{k}]=\varphi[t_{k}]P^{-}[t_{k}]\), and\(P_{x,y}^{-}[t_{k}]\) is the cross-covariance of the state and measurement without the fading factor \(P_{x,y}[t_{k}]=\varphi[t_{k}]P_{x,y}^{-}[t_{k}]\). Consequently, it can be verified that the lower bound of the stabilized coefficient \(\varphi[t_{k}]\) obtained in Equation (28) can be used to mitigate the modeling error by preserving the estimation consistency of the innovation. The square root of the predicted measurement covariance can be derived according to Equation (21)  
\[R_{y,y}[t_{k}]=\bigg{(}\mathrm{qr}\bigg{\{}\left[ \begin{array}{cc}y^{*}[t_{k }]&\sqrt{U[t_{k}]}\end{array} \right]\bigg{\}}^{T}\bigg{\}}\bigg{)}^{T} \tag{29}\]  
\[y^{*}[t_{k}]=\left[\left(y[t_{k}]\right)_{0}-\hat{y}[t_{k}]\cdots\left(y[t_{k} ]\right)_{i}-\hat{y}[t_{k}]\cdots\left(y[t_{k}]\right)_{2L}-\hat{y}[t_{k}] \right]\times\mathrm{diag}\Big{(}\sqrt{W^{(c)}}\Big{)} \tag{30}\]  
where L is the dimension of the receiver. \(\left(y[t_{k}]\right)_{i},i=0,1,\ldots,2L\) is the propagated sigma point, and \(y[t_{k}]\) is the predicted measurement results. \(U[t_{k}]\) is the Gaussian measurement noise matrix at \(t_{k}.\sqrt{W^{(c)}}\) denotes the weight of the sigma points. The state covariance derived from QR decomposition can be calculated using  
\[R[t_{k}]=\bigg{(}\mathrm{qr}\bigg{\{}\bigg{[}\chi^{*}[t_{k}]-y^{*}[t_{k}] \quad K[t_{k}]\sqrt{U[t_{k}]}\bigg{]}^{T}\bigg{\}}\bigg{)}^{T} \tag{31}\]  
\[\chi^{*}[t_{k}]=\Big{[}\big{(}\chi[t_{k}]\big{)}_{0}-\hat{s}[t_{k}]\cdots\big{(} \chi[t_{k}]\big{)}_{i}-\hat{s}[t_{k}]\cdots\big{(}\chi[t_{k}]\big{)}_{2t_{k}}- \hat{s}[t_{k}]\Big{]}\times\mathrm{diag}\Big{(}\sqrt{W^{(c)}}\Big{)} \tag{32}\]  
where \((\chi[t_{k}])_{i},i=0,1,\ldots,2L\) denotes the propagated sigma points and \(K[t_{k}]\) is the filtering gain. The SR-USF algorithm based on the QR decomposition for the GNSS and 5G joint positioning over the k sampling period is summed up in the Algorithm 1 in detail. ``` Initialization Denote the estimated state of GNSS and 5G joint positioning and its square root of the prediction error covariance at \(t_{k}^{i}\) by \(S_{\text{GNSS\&5G}}[t_{k}]\) and \(R[t_{k}]\).