# A Two-Stage Kalman Filter-Based Carrier Tracking Loop for Weak GNSS Signals
## 1 Introduction  
For a global navigation satellite system (GNSS) receiver, accurate navigation and timing services are readily available in open outdoor environments. However, when GNSS signals pass through challenging environments, such as dense foliage, urban canyons, and indoor environments, GNSS signals are attenuated substantially. Conventional GNSS receivers have difficulty processing such weak signals. As one of the most important parts of a GNSS receiver, the carrier tracking loop is extremely fragile under weak signal conditions, and its threshold limits the unaided receiver performance [1]. Various efforts have been devoted to improving the robustness of the carrier tracking loop under weak signal conditions. The conventional phase-locked loop (PLL) has been widely used in the carrier tracking loop. The typical strategy for improving the PLL sensitivity is to extend the integration time. Unfortunately, extending the integration time requires narrower filter bandwidths, which conflict with the dynamic performance of the PLL. The frequency-locked loop (FLL) is also commonly used in the tracking loop, and it can tolerate large, sustained frequency errors [2]. In [3], a FLL-assisted-PLL tracking loop was designed, which can resolve the conflict between the dynamic performance and the accuracy performance. However, Ref. [4] also demonstrated that the conventional tracking loopassumes that the input GNSS signal is deterministic. Consequently, the conventional loop is difficult to keep well locked in time-varying environments. The Kalman filter (KF) is a time-variant system, and it can be substituted for the conventional PLL. According to the literature, KF-based tracking shows superior performance in terms of tracking sensitivity. In [5], the KF-based tracking loop was employed to track weak GNSS signals. In addition, Refs. [6; 7] both demonstrated that KF-based tracking can track weaker signals than traditional tracking approaches. To improve tracking sensitivity, KF-based tracking also requires an increased integration time, which has been illustrated in [8]; however, the integration time should be chosen optimally to compromise between the sensitivity and the loop update rate. Hence, Ref. [9] proposed an adaptive PLL, which could adaptively adjust the integration time to track weak and dynamic signals. Moreover, Ref. [10] assumed that the data bit sign had been eliminated before employing KF-based tracking because data bit reversals could cause incorrect KF estimations. Accordingly, it is necessary to achieve bit synchronization before KF-based tracking, especially when the coherent integration time should be extended for tracking weaker signals. The conventional tracking usually achieves bit synchronization before entering the tracking stage. For instance, in [11], it was assumed that bit synchronization should be employed initially to extend the integration time in conventional tracking. Similarly, Ref. [12] proposed the control strategy in conventional tracking that achieved the bit synchronization in the first tracking phase. However, achieving bit synchronization before KF-based tracking has not yet been investigated. Additionally, the main drawback of KFs is the need of the exact prior parameters to initialize KF values. For standard KF-based tracking receivers, the KF-based tracking parameters are typically initialized according to the acquired results, which was presented in [13; 14]. However, because of the large frequency error of the acquisition results under weak signal conditions, it is difficult for the KF algorithm to converge rapidly. KF-based tracking requires an accurate initialization to keep lock under weak signal conditions. In [15], it was also demonstrated that accurate initial values can decrease the possibility of false frequency-locking in weak signal tracking. In [5], fine acquisition results were used to initialize the KF-based tracking. However, the fine acquisition stage could not achieve bit synchronization and it needed an additional Bayesian approach to determine bit signs.