# Table-Based Adaptive Digital Phase-Locked Loop for GNSS Receivers Operating in Moon Exploration Missions

^{*}

## Abstract

**:**

## 1. Introduction

_{0}), showed that the proposed adaptive DPLL is superior to the traditional fixed-gain DPLL. However, the proposed algorithm uses only the C/N

_{0}value as an input, so the dynamic performance of the adaptive loop filter is fixed. An adaptive DPLL algorithm for high dynamic applications is suggested in [5] and is verified to have a better performance compared to the traditional DPLL using simulation.

_{0}and dynamic stress to calculate the optimal noise bandwidth for the current environment by estimating the pole location of the loop transfer function that minimizes the thermal noise. Adaptive DPLL algorithms, including FAB, fuzzy logic (FL), and loop-bandwidth control algorithm (LBCA), are evaluated and compared in terms of the performance and computational complexity for the various scenarios in [3].

_{0}and the signal dynamics. The tuning method for the signal tracking KF using the relationship between the C/N

_{0}and the equivalent noise bandwidth is introduced in [10]. An adaptive KF structure that adjusts the measurement and process noise covariance (which are fixed in the case of the standard KF) is presented in [11].

## 2. Moon Exploration Mission Trajectory

_{0}and jerk value for each visible satellite, is calculated for the global positioning system (GPS) using the GPS orbit information and presented in Figure 3 and Figure 4, respectively. For the calculation of the LOS information, the side-lobe signal of the transmitter antenna pattern is utilized. Each color in the figures indicates a different GPS satellite. As expected, the C/N

_{0}information in Figure 3 has a reversed arc shape because the free space loss is increased when the distance between the spacecraft and the GPS satellites increases. The C/N

_{0}has a range from 1.48 to 56.58 dB-Hz. The calculated overall LOS jerk information and the enlarged version of the maximum jerk region are presented in Figure 4. The jerks occur periodically due to the propulsion power of the spacecraft, and the magnitude of each jerk gradually decreases. The maximum jerk occurs at the beginning of the trajectory, and its magnitude is approximately 411 g/s, remaining for only 1–2 s.

## 3. Optimal Bandwidth Table

_{0}and dynamic stress. The optimal noise bandwidth is calculated using the theoretical measurement error modeling of the DPLL that incorporates the thermal noise, Allan deviation, vibration-induced oscillator phase noise, and dynamic stress error. This paper assumes that the order of the DPLL is third-order to effectively track the dynamics components leading up to acceleration. In addition, the receiver is assumed to track the pure pilot channel (i.e., phase transition due to the navigation data bit or secondary code does not exist) to track very weak signals such that the C/N

_{0}value is as low as a few dB-Hz. However, the other orders, including the data channel case, can be applied by an analogous method.

^{2}] are the clock parameters related to the Allan deviation. In this work, the oven-controlled crystal oscillator (OCXO) is assumed to be used for the spacecraft. The corresponding clock parameters used for the paper are presented in Table 1.

^{2}/Hz], and ${\omega}_{1}$ and ${\omega}_{2}$ are the lower and upper limits [rad/s] of the constant vibration PSD, respectively. Additionally, the parameter values for the vibration-induced oscillator phase noise calculation are selected based on the assumption that the receiver uses a high-quality oscillator and suffers a moderate amount of vibration. These parameter values are presented in Table 2.

^{3}]. Since the third-order DPLL can track up to the acceleration, the dynamic stress error of the third-order DPLL is induced by only the jerk value. Notably, the dynamic stress error is inversely proportional to the natural frequency of the loop filter, so the noise bandwidth (which is proportional to the natural frequency) should be widened to reduce the dynamic stress error.

_{0}and the jerk are selected to be similar to the analysis result. However, the minimum value of the C/N

_{0}is selected as 5.4 dB-Hz (not 1.48 dB-Hz, as can be observed in Figure 3) because it is the smallest C/N

_{0}that the receiver can track in a theoretical sense (i.e., the smallest C/N

_{0}where the theoretical measurement error is less than 30 degrees). The figure shows that the measurement error varies with the noise bandwidth value, and one optimal point at which the measurement error is minimized exists for each condition. As expected, the optimal bandwidth for the low SNR condition is very narrow (${B}_{n}$ = 0.7 Hz) to suppress the large noise effect due to the very low C/N

_{0}of 5.4 dB-Hz. Furthermore, the optimal bandwidth is very wide (${B}_{n}$ = 213.3 Hz) for the case of the high dynamics to track fast variations in the signal component.

_{0}and jerk value. If the minimum RMSE is less than 30 degrees, the optimal bandwidth is saved to the optimal bandwidth table; otherwise, the calculated value is discarded. A two-dimensional table is outputted by the algorithm and is used for the table-based adaptive DPLL algorithm.

Algorithm 1 Optimal bandwidth table generation algorithm. | |

1: | Initialization: |

2: | Set target range of $C/{N}_{0}$ and jerk ${R}^{\prime \prime \prime}$ |

3: | Set carrier frequency ${f}_{c}$ |

4: | Set clock parameters ${h}_{0}$, ${h}_{-1}$, ${h}_{-2}$ |

5: | Set parameters of vibration-induced oscillator phase noise ${k}_{g}$, ${G}_{g}$, ${\omega}_{1}$, ${\omega}_{2}$ |

6: | Optimal bandwidth calculation: |

7: | for each $C/{N}_{0}$ do |

8: | for each ${R}^{\prime \prime \prime}$ do |

9: | Calculate ${B}_{opt}$ and ${\sigma}_{min}$ for current $C/{N}_{0}$ and ${R}^{\prime \prime \prime}$ |

10: | if ${\sigma}_{min}<30$ then |

11: | Save ${B}_{opt}$ to optimal bandwidth table |

12: | else |

13: | Do not save B_{opt} |

14: | end if |

15: | end for |

16: | end for |

_{0}and the jerk, respectively. The target generation ranges for each bin were selected as the maximum expected values (i.e., 57 dB-Hz for C/N

_{0}and 411 g/s for jerk) for the operating environment. Figure 6a shows the overall shape of the calculated optimal bandwidth table in a three-dimensional view, as the optimal bandwidth for each bin is represented as a height in the figure. The same shape is presented in Figure 6b,c with respect to the C/N

_{0}and jerk, respectively. A top view of the optimal bandwidth table is illustrated in Figure 6d. Since the optimal bandwidth is not saved to the optimal bandwidth table if the minimum RMSE exceeds 30 degrees, empty spaces can be observed in Figure 6d. Therefore, the empty space means the receiver cannot operate in that condition in a theoretical sense.

## 4. Table-Based Adaptive DPLL

_{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.

_{0}and jerk values from each estimator. The matching index of the optimal bandwidth table is determined, and the corresponding optimal bandwidth value is extracted from the table. Since the optimal bandwidth value is saved only if the minimum RMSE does not exceed the threshold, the existence of the optimal bandwidth value means the receiver can theoretically operate in the current condition. Therefore, the optimal bandwidth is applied to the noise bandwidth through (8); otherwise, the recent value of the noise bandwidth is maintained. Finally, the new noise bandwidth is handed over to the loop filter, and the integration time for the next integration is calculated using (9).

Algorithm 2 Table-based adaptive DPLL algorithm. | |

1: | Get estimated $C/{N}_{0}$ and ${R}^{\prime \prime \prime}$ |

2: | Noise bandwidth transition: |

3: | Find index for $C/{N}_{0}$ and $R\prime \prime \prime $ |

4: | Get ${B}_{opt}$ from optimal bandwidth table |

5: | if ${B}_{opt}$ is exist then |

6: | ${B}_{n}\left[k+1\right]=\alpha {B}_{opt}+\left(1-\alpha \right){B}_{n}\left[k\right]$ |

7: | else |

8: | ${B}_{n}\left[k+1\right]={B}_{n}\left[k\right]$ |

9: | end if |

10: | Return ${B}_{n}\left[k+1\right]$ to the loop filter |

11: | Integration time calculation: |

12: | $T=\mathsf{\Delta}T\times B{T}_{norm}/\left(\mathsf{\Delta}T\times {B}_{n}\right)$ |

13: | Set next integration time to $T$ |

_{0}and jerk are wrong, the resulting noise bandwidth is not optimal anymore. Therefore, the algorithm fits the environment that has slow variations of C/N

_{0}and/or jerks where the estimators can estimate them without much difficulty. Additionally, scenarios in which the trajectory is fixed so that the receiver can know its approximate location and signal reception characteristics are also appropriate conditions for the algorithm. The Moon exploration mission corresponds to both conditions. Dedicated estimators would be required to apply the algorithm to other conditions, such as having fast variations, etc.

## 5. Simulation

_{0}starts from 57 dB-Hz and decreases to 5.4 dB-Hz for 300 s. The value of 5.4 dB-Hz is selected as a minimum C/N

_{0}because it is the minimum C/N

_{0}that allows the receiver to operate, which is obtained from the optimal bandwidth table generation procedure in Section 3. The C/N

_{0}decreases with a repeated uniform pattern that maintains its value for 30 s and decreases by 10 dB-Hz for 30 s. Furthermore, the C/N

_{0}maintains 5.4 dB-Hz starting from 300 s for 150 s and then quickly recovers to 57 dB-Hz at 450 s. After the SNR has recovered, the high dynamics region begins; the jerk dynamic stresses occur four times, starting from 510 s with an interval of 10 s. Each of these occurrences is set to a magnitude of 411 g/s and a length of 1 s because the observed jerk dynamic stress in the Moon exploration mission lasted only for 1–2 s.

_{0}region, so the low signal power and high dynamic stress do not coexist in our situation. Therefore, the low SNR region and high jerk region were intentionally separated in the simulation to imitate such mission conditions. From this point, it is thought that the configured simulation scenario well reflects the characteristics of the mission, and the harshness of both situations is similar.

_{0}decreases. From 300 s to 450 s, the noise bandwidth is set to 0.7 Hz, which is very narrow to suppress the noise effect. When the jerk dynamic stress exists after 500 s, the noise bandwidth is widened to 213.3 Hz to track the high dynamics of signal components.

_{0}is low. The errors in the Doppler frequency and Doppler rate increase temporally for the high dynamics region and converge to zero after the jerk dynamic stress disappears.

_{0}= 17 dB-Hz), while the proposed algorithm maintains its lock. The carrier tracking errors for the high dynamics region (500–550 s) are presented in Figure 15. The proposed algorithm continues the tracking during the high dynamics region with the temporally increased tracking error in the existence of the jerk dynamic stress. However, the fixed-bandwidth loop loses lock immediately after the first jerk dynamic stress occurs and cannot be recovered. Figure 14 and Figure 15 show that the proposed algorithm operates stably in harsh environments where the conventional fixed-bandwidth tracking loop loses lock.

## 6. Evaluation

_{0}and jerk, respectively. Each point in the figures is obtained with 100 iterations of the 150 s simulation. As expected, Figure 16 shows that the fixed-bandwidth loop has the largest jitter value, and the proposed algorithm has a similar performance to the FAB. The FL and LBCA have slightly better performances than the proposed algorithm. Figure 17 shows that all the adaptive DPLL algorithms have similar performances for the jerk dynamic stress, while the fixed-bandwidth loop cannot track high dynamic signals.

^{9}iterations for each algorithm. Each execution time result of the adaptive DPLL algorithm contains the execution time of the standard loop filter. The comparison results show that the proposed algorithm has a 2.4–5.4 times faster execution time compared to the other algorithms.

_{0}and jerk dynamic stresses, while also having a faster execution time.

## 7. Conclusions

_{0}and dynamic stress and extracting the noise bandwidth value from the optimal bandwidth table. The structure of the algorithm, as well as the method which calculates the proper integration time to maintain the normalized bandwidth to constant, was proposed.

_{0}has a range of 5.4–57 dB-Hz, and the maximum jerk dynamic stress is 411 g/s. The proposed algorithm operated stably throughout the simulation, while a conventional fixed-bandwidth loop lost its lock. For the low SNR region, the noise bandwidth was narrowed to 0.7 Hz, and the integration time was increased to 420 ms to track the noisy signal. The noise bandwidth was widened to 213.3 Hz at the high dynamics region to track the fast variation in the signal components. Most of the time, the normalized bandwidth, which is a product of the noise bandwidth and the integration time, was maintained below the target normalized bandwidth value, so the stability of the loop filter was maintained.

_{0}and dynamic stress. Moreover, the trajectory during the mission is fixed, so the receiver can know its approximate C/N

_{0}and/or dynamic stress. Nevertheless, the proposed algorithm can be applied to any condition using a regenerated optimal bandwidth table with modified parameters. However, in the case of a situation where the signal power and dynamic stress change rapidly, sophisticated algorithms estimating C/N

_{0}and dynamic stress would be needed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Overall Moon exploration mission trajectory in the earth-centered, earth-fixed (ECEF) coordinate.

**Figure 3.**Line-of-sight (LOS) carrier-to-noise-density ratio (C/N

_{0}) information of the Moon exploration spacecraft during the mission trajectory. Each color in the figure indicates a different global positioning system (GPS) satellite.

**Figure 4.**LOS jerk information of the Moon exploration spacecraft during the mission trajectory. Each color in the figure indicates a different GPS satellite.

**Figure 5.**Example of the measurement error calculation results with respect to the noise bandwidth variation for low signal-to-noise ratio (SNR) and high dynamics conditions.

**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.

**Figure 8.**Structure of the proposed table-based adaptive digital phase-locked loop (DPLL) algorithm.

**Figure 9.**Configured simulation scenario information. The C/N

_{0}has a range of 5.4–57 dB-Hz and the maximum jerk is 411 g/s.

**Figure 10.**Optimal bandwidth variation during the simulation. The noise bandwidth is narrowed to 0.7 Hz as the C/N

_{0}is lowered to 5.4 dB-Hz and widened to 213.3 Hz as the jerk dynamic stress increases to 411 g/s.

**Figure 11.**Integration time variation during the simulation varied with a step of $\mathsf{\Delta}T$ (20 ms for this study). It increases to 420 ms for the C/N

_{0}of 5.4 dB-Hz and reduces to the lower bound value in the high dynamics region.

**Figure 12.**Normalized bandwidth variation during the simulation. The normalized bandwidth is maintained below the target normalized bandwidth value (i.e., 0.3) most of the time, except for the high dynamics region due to the unavoidable limitation of the integration time by the lower bound condition.

**Figure 13.**Carrier tracking results of the proposed table-based adaptive DPLL algorithm: (

**a**) Carrier phase; (

**b**) Doppler frequency; (

**c**) Doppler rate. The proposed algorithm stably tracks the signal components for the simulation scenario.

**Figure 14.**Carrier tracking error at the low SNR region (210–270 s): (

**a**) Proposed algorithm; (

**b**) Fixed bandwidth (B

_{n}= 15 Hz, T = 20 ms). The fixed-bandwidth loop loses lock at approximately 253 s (C/N

_{0}= 17 dB-Hz) while the proposed algorithm maintains its lock.

**Figure 15.**Carrier tracking error at the high dynamics region (500–550 s): (

**a**) Proposed algorithm; (

**b**) Fixed bandwidth (B

_{n}= 15 Hz, T = 20 ms). The fixed-bandwidth loop loses lock at approximately 510 s immediately after the jerk dynamic stress occurs. The error for the proposed algorithm increases temporally while the jerk dynamic stress exists and converges to zero immediately after disappearing.

**Figure 16.**Numerical jitter calculation result with respect to the C/N

_{0}. The proposed algorithm has a similar performance to the fast adaptive bandwidth (FAB). Fuzzy logic (FL) and loop-bandwidth control algorithm (LBCA) have slightly better performances.

**Figure 17.**Numerical jitter calculation result with respect to the jerk dynamic stress. All adaptive DPLL algorithms have similar performances.

**Figure 18.**Execution time measurement results of each adaptive DPLL algorithm. The proposed algorithm has approximately 2.4–5.4 times faster execution time compared to the other adaptive DPLL algorithms.

**Table 1.**Clock parameters for oven-controlled crystal oscillator [26].

Parameter | Value |
---|---|

${h}_{0}$ | 2.51 × 10^{−26} [s] |

${h}_{-1}$ | 2.51 × 10^{−23} [s/s] |

${h}_{-2}$ | 2.51 × 10^{−22} [s/s^{2}] |

**Table 2.**Parameter values for the vibration-induced oscillator phase noise calculation [26].

Parameter | Value |
---|---|

${k}_{g}$ | 2 × 10^{−10} [parts/g] |

${G}_{g}$ | 0.05 [g^{2}/Hz] |

${\omega}_{1}$ | 25 × 2π [rad/s] |

${\omega}_{2}$ | 2500 × 2π [rad/s] |

Algorithm | Addition (Subtraction) | Multiplication | Division | Remarks |
---|---|---|---|---|

Table-based (proposed) | 5 | 4 | 1 | Floor(·) |

FAB | 1 | 16 | 5 | Seventh-root |

FL | 15 | 20 | 3 | - |

LBCA | 8 | 5 | 2 | Exp(·) |

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**MDPI and ACS Style**

Song, Y.-J.; Won, J.-H.
Table-Based Adaptive Digital Phase-Locked Loop for GNSS Receivers Operating in Moon Exploration Missions. *Sensors* **2022**, *22*, 10001.
https://doi.org/10.3390/s222410001

**AMA Style**

Song Y-J, Won J-H.
Table-Based Adaptive Digital Phase-Locked Loop for GNSS Receivers Operating in Moon Exploration Missions. *Sensors*. 2022; 22(24):10001.
https://doi.org/10.3390/s222410001

**Chicago/Turabian Style**

Song, Young-Jin, and Jong-Hoon Won.
2022. "Table-Based Adaptive Digital Phase-Locked Loop for GNSS Receivers Operating in Moon Exploration Missions" *Sensors* 22, no. 24: 10001.
https://doi.org/10.3390/s222410001