Data Fitting with Linear Transfer Functions†

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Abstract
The problem of fitting a physically realizable linear passive system (defined either by an impulse response or a frequency response function) to a given set of input data and output data is discussed. It is shown that the problem loads to an under-determined set of equations, and that additional restraint conditions defining a ‘ best ’ approximation to the data must be introduced into any solution. The various constraint conditions that have boon proposed are surveyed, and a general criterion involving a minimization of moan square error is developed. Equations defining a minimum mean square error solution are derived, and solved, for a best linear passive approximation to data collected from systems having any number n of input terminal pairs. The property of input and output data which produces a linear passive approximation, giving an exact fit, is deduced, and a criterion for the measure of success of an approximation is suggested, for situations in which an exact fit is not possible.

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