On a time-domain representation of the Kramers-Kronig dispersion relations

The development of Kramers-Kronig dispersion relations is typically carried out in the frequency domain. An alternative approach known as the time-causal theory develops dispersion relations for media with attenuation obeying a frequency power law through analysis in the time domain [T. L. Szabo, J. Acoust. Soc. Am. 96, 491-500 (1994)]. Although both approaches predict identical dispersion relations, it is perceived that these two approaches are distinct from each other. It is shown, however, that the time-causal theory is in essence a time-domain formulation of the Kramers-Kronig dispersion relations for the special case of media with attenuation obeying a frequency power law. Additionally, it is shown that time-domain representations of the Kramers-Kronig dispersion relations are available for a broader class of media than simply those with power law attenuation. The time-causal theory and the Kramers-Kronig dispersion relations can be viewed as two complementary, yet equivalent, approaches to the study of dispersion.