Optimal spline digital simulators of analog filters

Abstract

A now approach is presented for obtaining a linear timevarying recursive digital filter which will optimally simulate the behavior of a linear time-varying analog filter. The property of the best approximation of linear functionals by means of generalized spline functions is invoked in the derivation of the results. In this approach, the optimal digital simulator is, viewed as a min-max estimator of the samples of the analog filter output based on the samples of its input. These samples are not required to appear at uniformly spaced instants of time. The assumption is made that the analog filter input belongs to a class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> of signals, each member of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> being generated by a known differential dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> forced by some input whose energy is less than or equal to a constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha^{2}</tex> . By interpolating the samples of the analog filter input by a generalized spline associated with the differential operator pertaining to the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> , the structure of the optimal digital simulator is derived. Error bounds are derived as functions of the maximum sampling subinterval length and the theory is illustrated by means of examples.