Improving the Approximation to a Prescribed Time Response

The topic considered in this paper is the problem of obtaining the Laplace transform of a prescribed impulse response, under the constraint that this transform must be a realizable rational function. In general, the solution of this problem requires that approximations be made, either in the time domain or in the frequency domain. A procedure is developed which provides a systematic method for improving the approximation by making small changes in the poles and residues of the transfer function. The effects of such changes on the impulse response are evaluated by means of a Taylor series expansion of the impulse response. It is shown that only the first two terms of this expansion provide a reasonably accurate estimate of these effects. A set of normalized curves are prepared which allow the designer to determine how a given pole or residue should be changed in order to improve the approximation in the time domain. The procedure is demonstrated by applying it to a numerical example.