# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 1 Introduction
The difficulty of finding the correct \(\mathbf{Q}\) and \(\mathbf{R}\) values increases even more in time-varying scenarios since these parameters are continuously changing [20]. Different methods to estimate the noise covariances of the KF have been summarized in a review study [22]. One solution can be to implement a moving average filter to estimate \(\mathbf{Q}\) and \(\mathbf{R}\)[23]. Moreover, it is possible to implement a carrier-to-noise density ratio (\(C/N_{0}\))-based DSKF, in which \(\mathbf{R}\) depends on the variance of the STL discriminator's output [24]. \(\mathbf{Q}\) can also be adapted according to the dynamic stress error [25]. Another solution is the implementation of an ESKF combining long non-coherent integrations to improve tracking sensitivity [26]. Moreover, a weighted adaptive ESKF can be implemented for scenarios with unknown \(C/N_{0}\)[27]. Figure 1 presents a roadmap of the studies. Previous research [7] evaluated three adaptive tracking techniques in the standard STL: the loop-bandwidth control algorithm (LBCA), the fast adaptive bandwidth (FAB), and the Fuzzy logic. The LBCA was superior to the other techniques regarding tracking performance, system performance, and complexity. This technique adjusts the STL's loop bandwidth based on the statistics of the discriminator's output [28]. The current research extends the LBCA by implementing it in the DSKF. The relationship between the STL and the DSKF, referred to as the dashed red line, facilitates the LBCA's implementation in the DSKF. The LBCA's estimated loop bandwidth can be related to \(\mathbf{Q}\) and \(\mathbf{R}\) in the steady-state. Suppose one of the covariances is  
set to a constant value. In that case, the remainder covariance can be updated based on the LBCA's bandwidth update. In addition, the DSKF can be simplified considering the convergence of the Kalman gains in the steady-state, referred to as the dashed blue line. This low-complexity tracking scheme is the so-called lookup table (LUT)-DSKF. The LBCA can also be implemented in the LUT-DSKF and directly update the Kalman gains based on the estimated loop bandwidth. This paper evaluates the performance of the LBCA-based DSKF and the LBCA-based LUT-DSKF. These adaptive techniques are compared with the \(C/N_{0}\)-based DSKF, a well-known state-of-the-art method that adapts \(\mathbf{R}\) based on the estimated \(C/N_{0}\)[24]. The DSKF, the LUT-DSKF, and the mentioned adaptive techniques are implemented in the carrier phase synchronization tracking stage of the GOOSE(r) receiver [29]. Each adaptive technique's tracking performance and system performance are evaluated under simulated scenarios with different dynamics and noise levels. These methods are also compared with the LBCA-based standard STL [7]. This research expands a conference paper [20] by describing in detail the relationship between STL and the DSKF, explaining the DSKF's steady-state convergence that leads to the LUT-DSKF, adapting the LUT-DSKF using the LBCA, and improving the scope of the results. The rest of the paper is organized as follows: Section 2 compares the standard STL with the DSKF, analyzes the steady-state convergence and derives the tracking scheme of the LUT-DSKF. Section 3 shows the architecture of the adaptive techniques used in the DSKF and the LUT-DSKF. The experimental setup and implementation in an open software interface GNSS hardware receiver are described in Section 4. Section 5 presents the adaptive tracking techniques' complexity, tracking performance, and system performance results. These results are discussed in Section 6. Finally, Section 7 concludes and indicates future work.