A Kalman filter algorithm based on adaptive linear predictive is proposed for the reconstruction of time-varying sparse signals in compressed sensing. The signal is observed from a sliding window. Based on the correlations between the signals of two continuous windows and the adaptive linear prediction, the state transfer equation of continuous windows signal is obtained. The equation and the modified observation equation constitute the system state-space model. In the signal reconstruction stage, a greedy algorithm is employed to determine the support set and reduced order Kalman filter iteration to get the exact solution. This paper simulates the FM, AM, WiFi RF and voice sampling signals. The simulation results show that the proposed algorithm recovering performance is improved without much increase in complexity. The reconstruction accuracy is improved about 5 percent as compared with that using the cycle spinning model, and far higher than other similar algorithms. Meanwhile, the SNR of reconstructed signal is about $1\sim 8$dB higher than that of original signal in the different noise environment, which shows that the algorithm has strong anti noise performance.