# New Nonlinear Second-Order Phase-Locked Loop with Adaptive Bandwidth Regulation
## 1 Introduction  
Phase-locked loop (PLL) is widely applied in synchronization of communication systems as a phase follow-up system [1; 2; 3]. Phase-locked loop is also extensively used in distribution grids and high-voltage DC transmission systems [4; 5; 6]. Compared with other orders of PLL, the second-order PLL is more preferred in actual engineering due to its absolute stability and easy implementation [7; 8]. Although the large loop bandwidth can lock the loop quickly, the loop noise suppression capability declines. The loop is generally designed with a small bandwidth to realize normal operation of the PLL under low signal-to-noise ratio, but this loop has poor dynamic adaptability [9; 10; 11]. In this study, both noise rejection and fast locking performance are required aspects for the PLL with regard to the area of synchronous receiving highly dynamic telemetry or GPS signals. Additionally, PLL has wide applications in cellular networks for carrier synchronization, especially the D2D communications which support the cellular networks [12; 13]. In addition, the phase-locked loop is extensively applicable in image/video and signal processing fields for accurate signal synchronization [14]. Furthermore, the PLL also has a find application in real-time motion detection for automated video surveillance systems [15]. The applicability of the phase-locked loop in real applications is very extensive, especially in communication networks. However, an irreconcilable contradiction exists between the tracking speed and accuracy of the traditional second-order PLL, and this contradiction is an important consideration in engineering applications [6; 16]. [PERSON] [17] analyzed the performance of PLL based on the Kalman filter in GPS under the influence of ionospheric scintillation. The PLL based on the Kalman filter can still maintain the locking state under serious scintillation (Scintillation index > 0.9) when the carrier-to-noise ratio decreases from 42 dB-Hz to 9 dB-Hz. The serious impacts of integral time on PLL based on the Kalman filter demand accurate processing and noise models. Nevertheless, these models are difficult to be acquired in special application environment and consumelong tracking time. [PERSON] [18] estimated the carrier phase by frequency-locked loop (FLL)-assisted PLL. Given the large acquisition range of FLL, this method can switch over three states depending on the quality of received signals. Specifically, this FLL tracks the object by using the pure PLL at slow changes in carrier frequency and by using the FLL-assisted PLL at the rapid changes in carrier frequency and phases. The loop is unlocked at the PLL tracking. The FLL-assisted PLL increases the acquisition and tracking speeds but still fails to reconcile the contradiction between the tracking speed and accuracy based on the traditional PLL. [PERSON] [19] proposed the vector tracking loop in 2008 and estimated carrier phase and Doppler frequency in an independent channel by using all available channels. On the one hand, bad channels can use other good channels for resistance. On the other hand, bad channels can contaminate healthy channels. [PERSON] [20] suggested using the PLL algorithm based on Discrete Fourier Transform (DFT) to compromise the contradiction between large acquisition bandwidth and tracking accuracy. The phase discriminator was utilized to realize the DFT algorithm. Simulation results demonstrated that the PLL based on DFT increases the acquisition speed and protects the noise robustness. However, the phase discriminator in the algorithm is achieved by DFT and increases the complexity of engineering projects. Considering the PLL with adaptive bandwidth, a set of new algorithms are proposed [21; 22; 23]. Through estimating the error signal of the discriminator, the algorithm proposed in Reference [22] adaptively adjusts the parameters of the loop filter to optimize the measurement accuracy of the carrier phase.