Signal Filtering at Jump Multiplicative Interference

The problem of constructing the approximately optimal Bayesian algorithm of filtering the output signal for a linear stochastic dynamical system is solved. Non-linear mixture of the output signal and of the multiplicative interference was measured. This interference is a continuous-valued random process with the unknown distribution law within the given limits of [c(1), c(2)], c(1) > 0, c(2) > 0, and in the frequency range of [0, ∆ω]. The filtering algorithm is synthesized by the approximately optimal signal estimation method based on the theory of systems with random jump structure and the method of two-time parametric approximation of the phase coordinates probability distribution. The method consists in approximate replacement of the unknown probability densities of phase coordinates by the known distribution laws with the unknown mathematical expectations and covariances determined as a result of solving the problem. The proposed approximate-optimal filtering algorithm is based on replacement of the continuous-valued multiplicative interference by a random jump process, i.e., Markov chain with two states c(1), c(2) and equal intensities of transitions from one state to another q′ = 2∆ω. Conditional probability densities of the output signal for fixed states of the Markov chain c(1), c(2) are approximated by gamma distributions that depend on mathematical expectations and variances of the x(t) signal at the fixed c(1), c(2) and measurements x(t) in a mixture with the Markov jump interference. An example of constructing an algorithm for evaluating the distance from an aircraft to the object based on object irradiance measurement by the infrared direction finder was considered. Irradiance was equal to the radiation strength ratio to the square of the distance. Radiation strength, i.e., multiplicative interference, was the random process with unknown continuous distribution in a limited range. The approximate-optimal filter was constructed for distance evaluation by replacing this process with a two-state Markov chain and approximating the distance probability density by the Rayleigh distribution