Modeling of random processes based on Karhunen-Loeve decomposition

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.