Interpolation and extrapolation of sampled data

This paper is essentially an extension of the optimum filtering theory of Zadeh and Ragazzini to the case where the time function to be operated upon is available only at a sequence of sample instants. As in the latter paper the signal is taken to consist of two parts, one being a polynomial
p(t)
with unknown coefficients and known maximum degree
n
, and the other being a stationary random component
M(t)
with known autocorrelation function. It is assumed that the signal has been contaminated before filtering by the addition of stationary random noise, also of known autocorrelation function, and that the input to the filter consists of a sequence of impulse functions of constant repetition period
T
, each impulse being of area equal to the sample value of the signal plus noise at the time of occurrence of the impulse. This paper shows how to find the weighting function,
W(t)
, of a linear filter which will convert the sequence of impulse functions into a smoothed output subject to the following conditions: the weighting function
W(t)
is only nonzero over a finite range; in the absence of random components, the interpolation or extrapolation is error-free; in the presence of any random signal the approximation at the output follows a least squares law. The weighting function of the optimum filter is shown to be piece-wise continuous in the intervals
\{\gamma T \leq t \leq (\gamma + 1)T; \gamma = 0, 1, 2, \cdots N\}
and the paper concludes with a discussion of a simple example illustrating the practical application of the solution.