Weighted least-squares smoothing filters

IN THIS PAPER the concept of a minimum weighted squared error that has often been used in curve fitting1 is applied to a filter operating upon an input signal. The filter output is required to be a weighted least-squared-fit to its input. The steady-state error of the filter in reproducing its input is then zero when the input signal is contained in the ensemble of the chosen. fitting functions. The statistical aspects of the problem2 are subordinated to the requirement of zero steady state error to a proper input. The separate provinces of time-independent and time-varying filters in this sort of problem are defined; a slight generalization leads to a solution in the time-varying case. The proper domain of nonlinear least-squares filters is pointed out, but no general solution is given.