The signal demodulation is significantly affected by the time-varying frequency offset in low earth orbit (LEO) satellite communication system. We present a method for Doppler shift and Doppler rate estimation based on 5G frame structure in satellite communication system. The integer and fractional part of the Doppler shift is estimated within a prime synchronizing signal (PSS) in two steps. Large estimation range and low complexity can be achieved in this method. Besides, a Doppler rate estimation algorithm is proposed. Precise Doppler rate is calculated by extracting the accumulated phase difference of two consecutive PSS. Furthermore, Kalman filter is utilized to improve the stability of the frequency offset estimators in low signal-to-noise ratio (SNR). Simulation results show that the estimators can perform well with large estimation range and high accuracy.