On the mean square noise power of an optimum linear discrete filter operating on polynomial plus white noise input

In a recent article Johnson presents an asymptotic formula for the output noise power of an optimum filter designed to make a zero-lag estimate of either the input or its derivatives. It is assumed that the input function consists of a nonrandom polynomial plus stationary uncorrelated noise. It is the purpose of this paper to present an exact formula for the output noise power for the same input model. The formula presented is more general in that the estimation can be for any lag a with respect to the latest data point. Tables and graphs of the root mean square error for the zero-lag estimation of the 0th, 1st, and 2nd derivative are presented as a function of the input polynomial up to degree 5 and memory spans up to 100 sample points. A comparison is made of the relative error in root mean square using the asymptotic formula derived by Johnson.