Hybrid kalman filtering algorithm with wavelet packet data processing for linear dynamical systems

The paper develops a hybrid algorithm for predicting a linear dynamic system based on a combination of an adaptive Kalman filter with preprocessing using a wavelet packet analysis of the initial data of the background of the system under study. Being based on Fourier analysis, wavelet analysis and wavelet packet analysis are quite acceptable for time-frequency analysis of a signal, but they cannot be performed recursively and in real time and, therefore, cannot be used for dynamic analysis of random processes. In combination with the Kalman filter, a combination of the characteristics of the multiple-resolution wavelet transform and the recurrent formulas of the Kalman filter in real time is obtained. Since the original signal is usually given in the form of discrete measurements, to implement their convolution used in the Kalman filter, it is necessary to use cyclic convolutions with periodic continuation of the signal for any time interval. In the case of different values of the original signal at the ends of the considered time interval [0,T], the periodized signal can have large values and sharp different amplitude at the ends of the periodization interval. To smooth out the values of the periodized signal at the ends of the periodization interval, a cascade decomposition and recovery algorithm was used using Dobshy boundary wavelets with a finite number of moments. Signal recovery is performed in a series of operations comparable to the duration of the time interval under consideration. The smoothed signal obtained in this way is used as a Kalman filter platform for predicting the dynamic system under study. Taking into account that the correlation functions of the noise in the observation equation and the phase state of the system are usually unknown, the adaptation of the Kalman filter to these noises (interference) is carried out on the basis of a zeroing sequence. The manuscript does not contain related data