On stationary inflection points in step responses

TL;DR

This paper proves that finite-dimensional LTI system responses cannot have inflection points with tangents parallel to the time-axis, ensuring all level crossings are transversal and not tangential.

Contribution

It establishes a fundamental property of LTI system responses, showing the impossibility of certain inflection points in their step and impulse responses.

Findings

Step and impulse responses cannot have inflection points with tangent parallel to time-axis.

Level crossings in such responses are always transversal, never tangential.

Provides a theoretical foundation for analyzing response behaviors in LTI systems.

Abstract

The step and impulse responses of a proper, rational transfer function are well-behaved analytic functions. We prove that such a response cannot have an inflection point such that the tangent at that point is parallel to the time-axis. Hence when a step or impulse response of a finite-dimensional LTI system crosses any given level, that crossing must be transversal, and can never be tangential.