A Review of Kramers-Kronig Integral Relations for Signal Integrity Applications

This paper presents an overview and applications of the Kramers-Kronig (K-K) integral equations which are used to characterize a causal system. The conventional form of the KK integral is discussed which involves a singularity in the integrand. A modified form of the K-K integral is presented which removes the singularity in theory. The causal dielectric permittivity model proposed by Djordjevic-Sarkar is used to demonstrate the evaluation of K-K integral using standard numerical integration techniques. Also, examples of passive impedance network and causal transmission line models are considered. Three different implementation methods of the K-K integral are shown, and their errors are compared. The results show a good match between the original data and K-K integral retrieved data.