An extension of the minimum mean square prediction theory for sampled input signals

A method is developed for finding the ordinates of a digital filter which will produce a general linear operator of the signal
S(t)
such that the mean square error of prediction will be a minimum. The input to the filter is sampled at intervals
\Delta t
. The samples contain stationary noise
N(j\Delta t)
, a stationary signal component,
M(j\Delta t)
, and a nonrandom signal component, \begin{equation} P(j\Delta t) = \sum_{k=0}^n a_k P_k (j\Delta t) \end{equation} where the subset of nonrandom functions
P_k (t)
are known a priori, but the parameter vector
a = (a_o, a_l, \cdots, a_n)
need not be. The solution is obtained as a matrix equation which relates the ordinates of the digital filter to the autocorrelation properties of
M(t)
and
N(t)
and the nature of the prediction operation.