Delay Measurement Enhancement by the LSTM Embedded Kalman Filtering in Fiber-Optic Time Distribution System

To obtain precise and accurate timing signals across a wide area, the technique of fiber-optic time transfer is widely applied. As one of the main concerns in time transfer systems, transmission stability can be improved by applying filtering algorithms, and the Kalman filter (KF) usually plays a key role. The KF can be applied to enhance the precision of the transmission delay measurements within the system, and further improving the accuracy of delay compensation. Although the KF is optimal for linear Gaussian systems, it is not always effective for fiber-optic time transfer due to the nonlinear, nonstationary nature of fiber link time delay drift. The difficulty lies in the fact that the fixed state transition matrix in KF cannot track the unexpected fiber delay drift to achieve unbiased predictions. This limitation poses a potential risk of long-term stability degradation when applied to delay compensation. To address this issue, a time-varying state transition matrix is necessary. This article proposes a solution by introducing a data-driven neural network, specifically the long short-term memory (LSTM) model. The proposed method is experimentally validated over optical fiber transmission links spanning distances from 160 to 1280 km, demonstrating its ability to enhance short-term stability while preserving long-term stability, distinguishing it from the traditional KF. The short-term stability optimization is close to $1/\!\sqrt{2}$ across three different fiber lengths, which is equivalent to a 3-dB signal-to-noise ratio (SNR) improvement of the received one pulse-per-second (1PPS) signal. This result offers a promising solution for enhancing the precision and reliability of fiber-optic time transfer systems in a variety of applications.