## 3 The Proposed Multiple-Signal Joint Estimation Method
### Square Root Unscented Stable Filter
For every iteration \(k\)=1,..., T (1) The prior estimate for predicting the state of the system and the square root of its covariance are as follows

\[\hat{s}_{\text{GNSS\&5G}}[t_{k}]=F\hat{s}_{\text{GNSS\&5G}}[t_{k-1}] \tag{33}\]

\[R^{-}[t_{k}]=\left(\text{qr}\left\{\left[F{R}[t_{k}]\sqrt{Q[t_{k}]}\right]^{T} \right\}\right)^{T} \tag{34}\]

(2) Generate 2L+1 sigma points

\[\chi^{-(0)}[t_{k}]=\hat{s}_{\text{GNSS\&5G}}^{-}[t_{k}] \tag{35}\] \[\chi^{-(i)}[t_{k}]=\hat{s}_{\text{GNSS\&5G}}^{-}[t_{k}]+\sqrt{ \lambda+L}\big{(}R^{-}[t_{k}]\big{)}_{i}\] \[\chi^{-(L+i)}[t_{k}]=\hat{s}_{\text{GNSS\&5G}}^{-}[t_{k}]-\sqrt{ \lambda+L}\big{(}R^{-}[t_{k}]\big{)}_{i}\]

where \(i=1\),...,\(L\). (3) Propagate the sigma points in GNSS-5G joint positioning model

\[y^{-(i)}[t_{k}]=h_{GNSS\&5G}\left(\chi^{-(i)}[t_{k}]\right)i=0,1,\ldots,2L \tag{36}\]

(4) Evaluate the predicted value of the measurement result

\[\hat{y}^{-}[t_{k}]=\sum_{i=0}^{2L}W_{i}^{(n)}\big{(}y^{-}[t_{k}]\big{)}_{i} \tag{37}\]

where \(W_{i}^{(m)}\) is the mean weights of the sigma points. (5) Compute the innovation

\[R[t_{k}]=\bigg{(}\mathrm{qr}\bigg{\{}\bigg{[}\chi^{*}[t_{k}]-y^{*}[t_{k}] \quad K[t_{k}]\sqrt{U[t_{k}]}\bigg{]}^{T}\bigg{\}}\bigg{)}^{T} \tag{31}\]0

(6) Calculate the cross covariance

\[R[t_{k}]=\bigg{(}\mathrm{qr}\bigg{\{}\bigg{[}\chi^{*}[t_{k}]-y^{*}[t_{k}] \quad K[t_{k}]\sqrt{U[t_{k}]}\bigg{]}^{T}\bigg{\}}\bigg{)}^{T} \tag{31}\]1

(7) Establish a stabilized coefficient and introduce it into the square root covariance of state prediction. \[R[t_{k}]=\bigg{(}\mathrm{qr}\bigg{\{}\bigg{[}\chi^{*}[t_{k}]-y^{*}[t_{k}] \quad K[t_{k}]\sqrt{U[t_{k}]}\bigg{]}^{T}\bigg{\}}\bigg{)}^{T} \tag{31}\]2

(8) Generate the new sigma points using the newly predicted state and execute step 2 to step 6 to repeatedlygenerate new innovation and the cross covariance. (9) Calculate the square root of the innovation covariance

\[R_{y,y}[t_{k}]=\left(\text{qr}\bigg{\{}\left[ \begin{array}{cc}y^{*}[t_{k}]& \sqrt{U[t_{k}]}\end{array} \right]^{T}\bigg{\}}\right)^{T} \tag{41}\]

(10) Calculate the Kalman gain and update the GNSS and 5G estimate state

\[K[t_{k}]=\big{(}P_{x,y}[t_{k}]/R_{y,y}[t_{k}]\big{)}/R_{y,y}^{T}[t_{k}] \tag{42}\]

\[\hat{s}[t_{k}]=\hat{s}^{-}[t_{k}]+K[t_{k}]\gamma[t_{k}] \tag{43}\]

\[R[t_{k}]=\left(\text{qr}\bigg{\{}\bigg{[}\chi^{*}[t_{k}]-K[t_{k}]y^{*}[t_{k}] \quad K[t_{k}]\sqrt{U[t_{k}]}\bigg{]}^{T}\bigg{\}}\right)^{T} \tag{44}\]

## 4 Simulation Results and Analysis

In this section, the performance of SRUSF in GNSS and the 5G joint positioning model is evaluated regarding measurements' estimation error and positioning error under different conditions. Section 4.1 describes the setup of the simulation scenario and the simulation parameters, and in Section 4.2, we validate the advantages of the proposed SRUSF algorithm. MATLAB 2022a is used to process the data, and the computer configuration is as follows: Intel i5-10400H (CPU), 24 GB (RAM), and Windows 10 (64 bit). ### Parameter Settings

Both pseudorange measurements from GNSS and TOA/AOA measurements from 5G BSs are fused for positioning. Some important simulation parameters for GNSS and 5G BSs are summarized in Table 1. The transmit signal is the 5G positioning reference signal (PRS) conforming to 3 GPP TS38.211 [31]. The time of the GNSS and 5G network can be synchronized precisely according to [32]. The simulation lasts for a total of 1000 s. The update rates of the GNSS and 5G PRS are set to 1 HZ. The GNSS-5G joint positioning receiver moves straight ahead in the x-direction at 2 m/s. The STD of the clock skew noise \(\sigma_{\varepsilon}\) is set to 1 ns. Only the line-of-sight conditions are simulated since the NLOS-suppression and observation-screening functions are not included. ### Performance Evaluation

We evaluate the horizontal localization accuracy of the proposed hybrid positioning method over different simulation scenarios and compare it with seven state-of-the-art methods, including the standard EKF, MRAKF, UKF, SRUKF, VBHUKF, and SRUSF, to verify its performance.