Diffusion approximations to output processes of nonlinear systems with wide-band inputs and applications

Many problems in communication theory involve approximations of a Markov type to outputs of nonlinear systems (with or without feedback) often so that Fokker-Planck techniques can be used. A general and powerful method is presented for getting diffusion approximations to outputs of systems with wide-band inputs. The input is parametrized by
\varepsilon
and as
\varepsilon \rightarrow 0
the bandwidth goes to infinity. It is proved that the sequence of system output processes converges weakly to a Markov diffusion process which is completely characterized. Many communication systems fit this model. The assumptions are of a type commonly used either explicitly or implicitly in either current methods of analysis of similar systems. The usefulness and relative simplicity of the method is illustrated by its application to three examples: a) a phase-locked loop (PPL), where a Markov-dfffusion approximation of the error process is developed, b) an adaptive antenna system, where an asymptotic analysis of the equations for the system is given, and c) a diffusion approximation to the output of a hard limiter followed by a bandpass filter; input-output S/N ratios are developed. Since weak convergence methods are used, the approximate "limits" yield approximations to many types of functionals of the actual systems.