Weighted Least-Squares Smoothing Filters

This paper differs from many others on least square filtering, in that no explicit note has been taken of the noise spectrum, at least no more than is taken when one fits a curve of least squares to a set of data. The concept of a least weighted error, not new in curve fitting, has, to the author's knowledge, never before been applied to a filter-- other workers have uniformly weighted their signal over a fixed interval of length
T
, with the result that the filters so derived cannot be realized with lumped constants. If this artificial constraint is removed as it is here, lumped constant filters are possible. The expression for the filter weighting function is obtained with a bare minimum of elementary mathematics; a very slight generalization leads to an expression for a time-varying filter weighting function when this is required. The nonlinear least-square filter is considered but no general solution is given. The paper is replete with examples and is directed to the average engineer. Although some original material is presented, a large part of the paper may be considered tutorial.