Laplace Autoregressive Model for Cascaded Systems

System modeling problems, such as the channel of a single-hop relay communication system in a flat-fading environment, require a cascade of two or more autoregressive (AR) processes to capture the entire system characteristics. However, for the purpose of system simulation and parameter estimations, it is more convenient if the entire system is modeled by a single AR model. In this paper, we consider a cascade system whose statistical characteristics of its subsystems is represented by two independent pth-order complex Gaussian AR processes, and model it by a single pth-order Laplace AR process. In our analysis, we first demonstrate that the marginal probability distribution functions of the real and imaginary components of a system described by a cascade of the two complex Gaussian AR processes are Laplace distributed. Then, to model the cascaded system, we develop a single complex Laplace AR process whose parameters are configured to match other statistical characteristics of the cascaded system. Specifically, we show that the autocorrelation of the developed Laplace AR process satisfies Yule-Walker type of equations and derive the steps for the design of its parameters through autocorrelation matching.