## 4 Table-Based Adaptive DPLL  
A simplified structure of a conventional carrier tracking loop of the GNSS receiver is shown in Figure 7. Note that the code tracking components are omitted in the figure for the simplicity of discussion. The carrier tracking process starts at the correlator. The correlator correlates the received signal transferred from the radio frequency front-end and the replica signal produced in the carrier generator. Then, the discriminator measures the difference (\(e_{\phi}\)) of the carrier phase between the two signals using the in-phase and quadrature prompt correlator outputs (\(I_{p}\), \(Q_{p}\)), which are inputted to the discriminator. The discriminator output is filtered in the loop filter, with the loop filter also estimating the differential components that depend on the loop filter order: the carrier phase (\(\phi\)), Doppler frequency (\(f_{D}\)), and Doppler rate (\(\dot{f_{D}}\)) in the case of the third-order DPLL. Finally, the carrier generator and the NCO (which is omitted in the figure) generate a new replica signal to close the loop. Figure 6: Result example of the generated optimal bandwidth table: (**a**) Overall shape; (**b**) Optimal bandwidth with respect to the C/N\({}_{0}\); (**c**) Optimal bandwidth with respect to the jerk; (**d**) Top view. The structure of the proposed table-based adaptive DPLL is presented in Figure 8. The optimal bandwidth table, the C/N\({}_{0}\) estimator, and the jerk estimator are newly added to the conventional structure. The C/N\({}_{0}\) estimator estimates the C/N\({}_{0}\) of a current channel using a C/N\({}_{0}\) estimation algorithm (such as the variance-summing method or power-ratio method explained in [27]). As presented in [5], the LOS jerk is estimated in the jerk estimator by differencing the adjacent Doppler rate, which is estimated in the loop filter with a time interval. The optimal bandwidth table returns the optimal bandwidth of the current environment efficiently using the pre-generated table with a slight calculation for finding the index of the table. Since the immediate and rapid change in the noise bandwidth can incur the instability of the tracking loop, the optimal bandwidth is reflected smoothly as follows:  
\[B_{n}[k+1]=\alpha B_{opt}+(1-\alpha)B_{n}[k] \tag{8}\]  
where \(\alpha\) is a coefficient for a smooth transition and is heuristically set to 0.1. The DPLL is stable if the normalized bandwidth, which is a product between the noise bandwidth and the integration time, is less than one (especially when the normalized bandwidth is close to zero) because the corresponding analog counterpart can be correctly represented [28]. The proper value of the integration time is calculated in consideration of this condition as follows:  
\[T=\Delta T\times\left\lfloor\frac{BT_{norm}}{\Delta T\times B_{n}}\right\rfloor \text{[s]} \tag{9}\]  
where \(\lfloor\cdot\rfloor\) is a floor function, \(\Delta T\) is a minimum unit value [s] for the variation in the integration time and \(BT_{norm}\) is the pre-defined target normalized bandwidth [unitless]. Here, \(\Delta T\) is used to prevent excessively frequent changes in the integration time value, which can be a reason for the loop filter's instability. Considering this, \(\Delta T\) is set to 20 ms,  
Figure 8: Structure of the proposed table-based adaptive digital phase-locked loop (DPLL) algorithm. Figure 7: Simplified structure of conventional carrier tracking loop. so the integration time is changed for the step of 20 ms. \(BT_{norm}\) is set to 0.3, which is identical to the typical value that can be obtained from the noise bandwidth of 15 Hz and the integration time of 20 ms for the GPS L1 C/A receiver. Consequently, the normalized bandwidth of the table-based adaptive DPLL is maintained at values below the \(BT_{norm}\), if possible. Sometimes, the integration time value of zero can be obtained from (9) if the noise bandwidth is relatively large.