VARIANCE ANALYSIS OF A STOCHASTIC DELAY INTERVAL DIFFERENCE EQUATION WITH APPLICATION TO AMPLITUDE MODULATION

Abstract

Difference equations form a fundamental framework for modeling and analyzing systems that evolve in discrete time. This paper investigates a class of Stochastic Delay Interval Difference Equations (SDIDEs) and demonstrates their applicability to amplitude modulation (AM) systems. A low-frequency information signal with randomly varying amplitude over successive time intervals is modeled as a delay interval within a discrete-time Langevin formulation. The recovery of the information signal is achieved through statistical characterization of the received output, specifically by estimating its variance. The SDIDE under consideration incorporates deterministic dynamics through known nonlinear functions and stochastic perturbations represented by a scaled Brownian process. Analytical results establish that the variance of the system output exhibits a linear dependence on the square of the input delay interval. This relationship provides a theoretical basis for information extraction in AM systems using delay-based stochastic difference models. The following SDIDE is taken for analysis.
y(n)=y(n-T)+T F(y(n),y(n-T))+EG(y(n))B(n)
where F(y(n),y(n-T)) and G(y(n)) are known equations. The delay interval is T, E is a parameter which scales the noise amplitude and B(n) is a Brownian process. From the analysis, it is proved that the variance of the output is linearly proportional to the square of the delay interval of the input.