## 2 Materials and Methods
### A-VTTCES-KF Algorithm
#### 2.1.1 Preliminary Results and Analysis through Stationary Field Tests  
* Vector tracking algorithms for stationary field tests The navigator will be formed with the weighted non-linear least square (WNLS) method [42] to provide the user's position, velocity, and timing (PVT) estimations in the stationary field tests. It has been mostly accepted that the weighted matrix for the PVT estimation relies on the elevation angle of each satellite which corresponds to the priori variances for the measurements. The priori variance model is given by [69] \[\sigma^{2}=\frac{\sigma_{0}^{2}}{\sin^{2}\left(\gamma\right)}\] (1) where \(\sigma_{0}\) is assumed to be the standard deviation (STD) of the pseudo-range measurement or Doppler measurement, and it will be set to 1 in our work; \(\gamma\) stands for the elevation angle of a satellite in radians. Since the random noise on the PVT solution estimation at each epoch is independent and uncorrelated, it can be exploited to identify the characterization of the noise distribution by excluding the cross-correlation of the measurements in the VT loop through this condition. The architecture of the VT-based SDR is illustrated in Figure 1, where red lines denote the VT feedbacks, blue blocks represent the satellite vehicle (SV) ephemeris and the implementations which rely on ephemeris, DCM stands for the direction cosine matrix which includes unit vectors between the user and the line-of-sight (LOS) satellites, LR stands for the loop rate of the update. Since the VT architecture based on the WNLS navigator is not our contribution, the system model associated with this algorithm can be referred to the conference paper [70], and it will not be investigated and stated in this work. Figure 1: Vector tracking architecture based on the WNLS. LR is the loop rate of the update; DCM is short for the direction cosine matrix. * Setup for the field tests The radio frequency (RF) L1 C/A signal of the global positioning system (GPS) will be received through a single-frequency NovAtel antenna. The radio-frequency GNSS signal cannot be processed by the GNSS SDR unless it would be down-converted to the intermediate frequency (IF). In this research, a Fraunhofer GNSS front-end will be utilized to implement such procedure. A low-cost commercial U-Blox GNSS receiver was receiving the incoming signals using the same antenna as which was involved with the GNSS front-end. These two devices are connected with the antenna through a splitter and several radio-frequency cables. In addition, the high-performance multi-frequencies and multi-systems Trimble R10 receiver is set up next to the NovAtel antenna so that a high-accuracy trajectory related to the tested ground vehicle can be referred to. It should be known that, the reference receiver receives the GNSS signal with a separate antenna integrated in its body. The setup for all the field tests in this research is provided in Figure 2.  
Two open-sky scenarios are applied in the experiments to allow the implementation of the VT-based SDR in a stationary condition. In order to exclude the influence of the kinematic movement of the receiver on the clock error estimations, i.e., clock bias and clock drift errors, the preliminary experiments should be carried out in a stationary situation. Furthermore, the WNLS method instead of the KF algorithm is chosen to compute the PVT solutions to update the VT loop parameters such that the outputs of the navigator can be confirmed to be independent during the whole time spanning. The stationary experiments have been done in two situations in this work. The scenarios for Situation #1 and Situation #2 associated with the open-sky testing spot and the sky plot of the SVs are illustrated in Figures 3 and 4, respectively. The clock drift estimations of the GNSS SDR through these two situations will be subsequently analyzed with the tool of Allan variance. The time-correlated error source which is coupled in the clock drift error is expected to be specified in this way. Figure 2: Setup for the field tests.