Kramers-Kronig, Bode, and the meaning of zero

TL;DR

This paper explores the fundamental differences between Kramers-Kronig relations and Bode's relation, highlighting how causality and information delays influence their mathematical forms and physical interpretations.

Contribution

It clarifies the conceptual distinction between the equality in Kramers-Kronig relations and the inequality in Bode's relation, linking them to causality and information flow delays.

Findings

Kramers-Kronig relations are equalities derived from causality.

Bode's relation is an inequality related to gain and phase.

Delays in information flow explain the difference between the two relations.

Abstract

The implications of causality, as captured by the Kramers-Kronig relations between the real and imaginary parts of a linear response function, are familiar parts of the physics curriculum. In 1937, Bode derived a similar relation between the magnitude (response gain) and phase. Although the Kramers-Kronig relations are an equality, Bode's relation is effectively an inequality. This perhaps-surprising difference is explained using elementary examples and ultimately traces back to delays in the flow of information within the system formed by the physical object and measurement apparatus.