A semi‐analytic approach to retrieve the unitary impulse response function

Abstract

Two kinds of errors occur in numerically transforming the transfer function (TF) to the unitary impulse response function (UIRF), the truncating error due to ignoring high frequency band contribution, and the discrete error due to numerical integration. The truncating error becomes prevailing if the TF attenuating trend is slow as the frequency approaches infinity. A semi-analytic approach is presented to alleviate this error. This approach dissects the whole frequency axis symmetrically into three bands, the central band, and two side bands extending to infinity. The contribution from the central band is calculated numerically, while the TF over the side bands is approximated as a simple function with an explicitly known inverse Laplace transform. This approach can overcome the Gibbs oscillation in computing the UIRF for a slowly attenuating TF, as is verified by the numerical examples studied here.