Filtering Technique Using Auto-Covariance Data of Signal and Observation Noise in Linear Discrete-Time Stochastic Systems

Usually, in the discrete-time estimation method of the signal, for the purpose of obtaining the state-space model for the signal, the signal process is fitted to the process model, i.e. the autoregressive model, the autoregressive moving average model, etc. Here, the signal is observed with additive white noise. This paper proposes the filtering technique, which uses the finite number of auto-covariance data of the observation process or the signal process for the positive lag time, the auto-covariance of the signal process, for the lag time 0, and the variance of the observation noise process in linear discrete-time wide-sense stationary stochastic systems. Especially, the auto-covariance data of the signal process is approximated by the Fourier cosine series expansion. The Fourier cosine series expansion leads to the expression in semi-degenerate kernel form of the auto-covariance function. Henceforth, in the current work, the filter using the covariance information calculates the filtering estimate recursively by using only the covariance information of the signal and observation noise processes.