Sampling and Parameter Estimation for a Second Order Linear System with a Fractional Brownian Motion

A scalar input-scalar output linear second order system has a fractional Gaussian noise input. The fractional Gaussian noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval (1/2,1). A family of estimators for a linear system with a fractional Gaussian noise and some unknown parameters that is obtained from the continuous time least square equations for the corresponding linear system with a white Gaussian noise is known to be strongly consistent. In this paper, it is shown that the asymptotic behaviour of this family of estimators obtained from the discretized least squares equations with only samples of the system output do not converge to the asymptotic behavior of the family of estimators of the continuous time least squares equations. This phenomenon is important for the numerical computation of parameter estimates.