Recursive Filter With Exponential Kernel for Nonstationary Systems

Nonstationary stochastic systems in the Wiener-Kolmogorov sense have properties defined by their moments of probability, entropy, and distribution function. Filtering Theory, in general, describes indirectly, a stochastic system through the processes of parameter estimation and state identification. The objective of this article is to develop a Recursive Filter with an Exponential Kernel (RFEK) to reconstruct the response of a nonstationary stochastic system. To achieve this, first, a system viewed as a Black Box (BB) is analyzed. These systems are those whose internal dynamics are unknown, only their input-output is known from a set of responses measured with respect to a particular excitation. From these measurements and applying the proposed filter, a set of estimated parameters and identified states are obtained as a characterization of the system. Subsequently, a comparison is made between the filter output signal and the reference signal over time; that is, measuring their point-to-point convergence. The convergence of the stochastic reference makes it possible to indirectly observe its stability from a bounded estimation region. As a case study, bioelectric signals of the electroencephalographic (EEG) type are analyzed giving an improved approximation with respect to the Kalman filter results.