# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 7 Conclusions  
This study presents the first proof of the LBCA's general applicability to any control system by implementing it in the DSKF. First, the relation between the standard STL and the DSKF is presented. Second, DARE's solution relates the DSKF's loop bandwidth, the noise covariances, and the Kalman gains in the steady-state. This relationship eases the implementation of the LBCA into the DSKF, and two approaches are presented: The LBCA-based DSKF and the LBCA-based LUT-DSKF. The LBCA-based DSKF uses the LBCA to adapt \(q\) for each loop iteration while setting \(R\) to a constant value. In contrast, the LBCA-based LUT-DSKF directly relates the LBCA's loop bandwidth update and the Kalman gains. These adaptive tracking techniques have been evaluated in a third-order Costas PLL. These techniques' complexity and static and dynamic system performance are compared with the state-of-the-art \(C/N_{0}\)-based DSKF and the LBCA-based standard STL [7]. Results show the superior performance of the LBCA-based LUT-DSKF compared to the other techniques. In terms of complexity, the LBCA-based LUT-DSKF presents less complexity compared to the other adaptive DSKFs. However, this method has greater added time complexity than the LBCA-based standard PLL. In further research, Equation (38) will be merged into the STL to achieve similar time complexity.  
An extension of the presented tracking schemes is the implementation of a LUT-DSKF that encloses the DLL, FLL, and PLL of a tracking channel. The use of the LBCA in this tracking architecture can improve the tracking performance in scenarios with different noise and signal dynamics. To cope with scenarios with multipath effects (e.g., in urban scenarios), a multi-frequency LUT-DSKF will be proposed, and the addition of the LBCA will be evaluated. Furthermore, an evaluation in real scenarios of all the LBCA-based techniques will lead to the final verification of the LBCA.  
Figure 16: System performance vs. added time complexity comparison. (**a**) Static scenario. (**b**) Dynamic scenario.  
This paper opens the door to a range of applications of the LBCA where the KF is involved. The LBCA can be implemented in the tracking stage, the interference mitigation stage (e.g., adaptive notch filtering), and the post-processing stage (e.g., adaptive loose and tightly coupling solutions) of a GNSS receiver.  
Conceptualization, I.C.; methodology, I.C. and J.R.v.d.M.; software, I.C.; validation, I.C.; formal analysis, I.C.; investigation, I.C. and J.R.v.d.M.; resources, I.C.; data curation, I.C.; writing-original draft preparation, I.C.; writing-review and editing, I.C., J.R.v.d.M., E.S.L.; visualization, I.C.; supervision, J.N., E.S.L.; project administration, W.F.; funding acquisition, W.F. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ADC & analog-to-digital converter \\ BET & backward Euler transform \\ \(C/N_{0}\) & carrier-to-noise density ratio \\ CRB & Cramer-Rao bound \\ DARE & discrete algebraic Riccati equation \\ DLL & delay locked loop \\ DSKF & direct-state Kalman filter \\ ESKF & error-state Kalman-filter \\ FAB & fast adaptive bandwidth \\ FLL & frequency locked loop \\ FPGA & field-programmable gate array \\ GNSS & global navigation satellite system \\ GPS & Global Positioning System \\ IIR & infinite impulse response \\ KF & Kalman filter \\ LBCA & loop-bandwidth control algorithm \\ LOS & line-of-sight \\ LUT & lookup table \\ MMSE & minimum mean square error \\ NCO & numerically controlled oscillator \\ PLAN & piecewise linear approximation of nonlinearities \\ PLI & phase-lock indicator \\ PLL & phase locked loop \\ PVT & position, velocity, and time \\ RFCS & radio-frequency constellation simulator \\ RFFE & radio-frequency front-end \\ SSM & state space model \\ STL & scalar tracking loop \\ SV & satellite vehicle \\ \end{tabular}