Generalization of Kramers-Krönig relations for evaluation of causality in power-law media

Abstract

Classical Kramers-Krönig (K–K) relations connect real and imaginary parts of the frequency-domain response of a system. The K–K relations also hold between the logarithm of modulus and the argument of the response, e.g. between the attenuation and the phase shift of a solution to a wave-propagation problem. For square-integrable functions of frequency, the satisfaction of classical K–K relations implies causality in the time domain. On the other hand, when the K–K relations are checked for the logarithm of the system response, the function is not a square integrable one. Then one can employ classical K–K relations with subtractions, but their satisfaction for the logarithm of the system response does not imply causality of the original function. In this paper, the K–K relations are generalized towards functions which are not square-integrable, also allowing for causality evaluation when the logarithm of the system response is considered. That is, we propose generalization of the K–K relations with subtractions, whose validity for the logarithm of the system response and the satisfaction of additional assumptions imply causality of the originally considered function. The derived theory is then applied to electromagnetic media characterized by power-law frequency dispersion, i.e. the media which are described by fractional-order models (FOMs). In this case, the subtraction procedure generates functions which may be not square integrable, or even not locally integrable. However, we can rigorously analyse causality of the media described by FOM using the derived theory, as well as the parameter ranges for which such models are causal.