The purpose of this paper is to investigate the use of asymptotic methods in the design of optimal state estimators for nonlinear systems and to use these methods to estimate the complexity of such filters. More specifically, we develop expansions for the autocorrelation function of the solution of a stochastic differential equation which is close to linear in a suitable sense. We then construct a realization of the corresponding autocorrelation function using a linear system driven by white noise. Finally, we explore the significance of the structure of this linear filter in the original nonlinear context.