## 1 Introduction  
The digital phase-locked loop (DPLL) is widely used for the carrier tracking loop of the global navigation satellite system (GNSS) receiver to synchronize the carrier phase and frequency of the locally generated replica signal to the signal received from the antenna. The DPLL measures the carrier phase difference between the two signals and controls the numerically controlled oscillator (NCO) accordingly to track the carrier phase [1]. The characteristics of the DPLL are determined by parameters such as the loop filter order, noise bandwidth, and integration time. In general, these parameters are fixed to the specific values that are suitable for the expected operational environment of the receiver in the design phase [2]. Therefore, if the operational environment characteristics, such as the signal power and dynamic stress, become different from the expected environment, the receiver's tracking performance is degraded, and the tracking loop may lose lock. Furthermore, there is a trade-off in designing the parameters of the DPLL. For example, the noise bandwidth of the DPLL should be wide enough to accommodate the dynamic stress and simultaneously be sufficiently narrow to suppress the noise effect [1]. Generally, the noise bandwidth is determined considering the expected maximum line-of-sight (LOS) dynamic stress (i.e., wide enough) of the receiver. Therefore, the loop filter cannot adequately reduce the noise effect, and the phase jitter may have degraded performance [2]. In contrast, an adaptive DPLL continually measures the receiver's current operational environment, such as the signal-to-noise ratio (SNR) and dynamic stress, and appropriately adjusts the gain of the loop filter (i.e., noise bandwidth) [3]. Consequently, the adaptive DPLL optimally tracks the incoming signal by widening the noise bandwidth during high dynamic stress while reducing the phase jitter by narrowing the noise bandwidth when the signal power is weak or interference has occurred. Performance analysis and optimization of the generalized DPLL and digital frequency-locked loop (DFLL) based on the minimum mean square error criterion are presented in [4]. They proposed an adaptive DPLL and DFLL algorithm that automatically adjusts the integration time and noise bandwidth of the loop filter for optimal performance. The simulation, which uses scenarios of high and low dynamic stress with a time-varying carrier-to-noise-density ratio (C/N\({}_{0}\)), 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. The fast adaptive bandwidth (FAB) algorithm is proposed in [6], which measures the C/N\({}_{0}\) 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]. An adaptive DPLL, where the loop gain is controlled adaptively, is presented in [7]. Mathematical models of second- and third-order DPLL with each error characteristic are derived and analyzed for various dynamics and signal strength. Experiments are performed for the proposed method with a conventional DPLL, DFLL-assisted DPLL, and FAB-based DPLL. Results show that the proposed one operates well in a highly dynamic scenario. LBCA-based adaptive carrier tracking loop for robust tracking in a sounding rocket is proposed in [8]. The LBCA is extended to the DFLL-assisted DPLL structure to improve the robustness of the tracking loop.