Fixed memory least squares filters using recursion methods

Given a set of equally spaced measurements, it is possible to curve fit a "least squares" polynomial to the
N
observed data points and obtain estimates of the past, present, or future values of the data or its derivatives by appropriate manipulations of the curve fit. This curve fitting can be accomplished by a linear weighting of the observed data over an interval
(n-1) T
. If the data is measured in real time such that a new data point is observed each
T
seconds, then the desired output (for example, the smooth or predicted value of the data) can be obtained by sliding these fixed number of weights such that the same weight always multiplies the data which is at a fixed lag with respect to the most recent data. Since these weights are zero for lags greater than
n
, they may be described as a fix-finite memory linear digital filter. In calculating the desired output for each new sample one requires a machine which can store
n
coefficients,
n
data points and performs n multiplications and
n - 1
additions in at least
T
seconds. The coefficients do not change but the multiplications and additions must be performed each
T
seconds as a new data point is measured. For large values of
n
, and small
T
, this may put a severe requirement on the real time solutions of the computer. This paper presents an alternate technique using recursion formulas to obtaining the same results as the
n
point weighting equation. The method has the advantage of requiring considerably less storage, multiplications and additions when
n \gg 1
and the degree of the curve fitting polynomial
(K)
is small.