Correlation Properties of Additive Linear High-Order Markov Chains

The aim of this paper is to study the statistical properties of the Nth order correlated Markov chains, which have a large application area in the UWB signal generation, transmission and processing. The state space of the chain under consideration is supposed to be a finite set of real numbers. The transition probability function is assumed to be homogeneous, additive and linear with respect to the values of random variable. The equations relating the correlation functions of the sequence to the memory function, which determined by the transition probability function, are derived and solved. The obtained analytical solutions for the correlation functions are compared with the results of numerical simulations. Examples of possible correlation scenarios in additive linear chains of higher orders are given. The obtained results can be useful both in analytical and simulative context, for example, for the modeling of the UWB transmission channels, for the refinement of transmitter and receiver design, for the enhancement of coding methods suitable for the UWB communication systems.