CORRELATION FUNCTIONS FOR LINEAR ADDITIVE MARKOV CHAINS OF HIGHER ORDERS

Abstract

Subject and purpose. The task of designing various radio engineering devices, such as filters, delay lines, antennas with a given radiation pattern, requires the development of methods for generating random sequences (the values of the system parameters) with given correlation properties, since the spectral characteristics of the listed and similar to them systems are expressed in terms of the Fourier transforms of correlators. The purpose of this paper is to represent the function of the transition probability of random sequences with long-range correlations in a form convenient for numerical generation of sequences, and to study the statistical properties of the latter. Method and methodology. An adequate mathematical tool for solving such problems is the higher order Markov chains. The statistical characteristics of these objects are determined by their transition probability function, which in the general case can have a very complex form. In this paper, the transition probability function is assumed to be additive and linear with respect to the values of the random variable. It is assumed that the state space of the sequence belongs to the set of real numbers. Results. The equations that relate the correlation functions of the sequence to the weight coefficients of the memory function, determined in their turn by the transition probability function, are derived and analytically solved. Conclusion. It is shown that the correlation functions of the additive Markov chain are completely determined by the variance of the random variable and the weight coefficients of the memory function. The agreement of the obtained analytical results with the results of numerical realization of the additive Markov sequence is demonstrated. Examples of possible correlation scenarios in higher order additive linear chains are given.

Keywords

• correlation functions
• higher order linear additive markov chains
• markov sequences
• memory function