# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 6 Discussion  
Figure 16 compares the average system performance \(\overline{\text{F}}_{\text{System}}\) and the added time complexity \(T_{\text{Added}}\) of the presented adaptive tracking techniques implemented in a third-order DSKF for carrier phase tracking. These graphs summarize the collected results combining Tables 1 and 3.  
Figure 16a shows that the LBCA-based DSKF \(T_{3}\) have the worst \(\overline{\text{F}}_{\text{System}}\) and the highest \(T_{\text{Added}}\). In contrast, the LBCA-based LUT-DSKF \(T_{4}\) achieves the best static \(\overline{\text{F}}_{\text{System}}\), although its complexity is higher than the LBCA-based standard PLL \(T_{5}\). The state-of-the-art \(C/N_{0}\)-based DSKFs \(T_{1}\), \(T_{2}\) have similar \(\overline{\text{P}}_{\text{System}}\), and \(T_{\text{Added}}\) is between \(T_{4}\) and \(T_{3}\). \(T_{5}\) presents the lowest \(T_{\text{Added}}\) with a slightly worse \(\overline{\text{P}}_{\text{System}}\) than \(T_{4}\).  
In the dynamic scenario, the superior performance of the LBCA-based techniques is clearly observed in Figure 16b. In this case, \(T_{1}\) and \(T_{2}\) present the worst \(\overline{\text{F}}_{\text{System}}\). In a highly dynamic event, the carrier tracking loop cannot follow the phase dynamics, leading to a fast shift from in-phase to quadrature component. In such a moment, the \(C/N_{0}\) estimate of [PERSON]'s method drops. Consequently, \(R\) increases, and the DSKF wrongly stops trusting the incoming carrier phase measurement. This leads to a loss of the carrier lock. In Figure 15a, a greater number of the accumulated correlation samples \(N\) and \(q\) improves the robustness to dynamics. Then, a further evaluation with higher \(q\) and \(N\) should improve  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tracking** & **Total Time Complexity** & **Iteration Time Complexity** & **Added Time Complexity** \\
**Technique** & \(T_{\text{C}}\) **[s]** & \(T_{\text{Iter}}\) **[ns]** & \(T_{\text{Added}}\) **[times]** \\ \hline Standard PLL & 6.8 & 22.7 & 1 \\ \(C/N\)-based DSKF & 41.5 & 138.3 & 6.1\(\times\) \\ LBCA-based DSKF & 61.1 & 203.7 & 8.9\(\times\) \\ LBCA-based LUT-DSKF & 25.8 & 86 & 3.8\(\times\) \\ LBCA-based PLL [7] & 16.1 & 53.7 & 2.4\(\times\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Time complexity of adaptive tracking techniques, \(3\times 10^{8}\) iterations.  
\(\overline{\mathrm{P}}_{\mathrm{System}}\) in the dynamic scenario. Among the LBCA-based techniques, \(T_{3}\) performs the worst. Once \(T_{3}\) adapts \(q\), time is needed to adjust the Kalman gains to the steady-state values. Due to this latency, \(T_{3}\) is not fast enough to adapt its coefficients in time, leading to a higher risk of losing the lock at high dynamic scenarios. \(T_{4}\) experiences no latencies since the steady-state Kalman gains are directly set based on the loop bandwidth update. This method achieves the best \(\overline{\mathrm{P}}_{\mathrm{System}}\), followed by \(T_{5}\).  
The scope of this research is to show the extension of the LBCA by applying it to the DSKF. While this has been shown, adding the LBCA into the DSKF to adapt \(q\) while fixing \(R\) does not present remarkable results, and the complexity increases significantly. Therefore, other methods have been considered to improve the performance and reduce the complexity. The LBCA-based LUT-DSKF presents superior static and dynamic \(\overline{\mathrm{P}}_{\mathrm{System}}\) among the rest of the adaptive tracking techniques. However, the LBCA-based PLL is less complex than the LBCA-based LUT DSKF, with a slight reduction of the total system performance in both static and dynamic scenarios.