Complex Frequency as Generalized Eigenvalue

TL;DR

This paper introduces complex frequency as a geometric generalization of eigenvalues for LTI systems, providing a unified interpretation linking linear system theory with differential geometry.

Contribution

It demonstrates that complex frequency extends eigenvalues through a geometric lens, applicable to linear systems and offering insights into system flow and dynamics.

Findings

Complex frequency coincides with eigenvalues for diagonalizable LTI systems.

Provides a geometric interpretation of eigenvalues using differential geometry.

Highlights limitations of this equivalence for nonlinear systems.

Abstract

This paper shows that the concept of complex frequency, originally introduced to characterize the dynamics of signals with complex values, constitutes a generalization of eigenvalues when applied to the states of linear time-invariant (LTI) systems. Starting from the definition of geometric frequency, which provides a geometrical interpretation of frequency in electric circuits that admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion, respectively, we show that complex frequency arises as its restriction to the two-dimensional Euclidean plane. For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order. The paper provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves. The paper also highlights that this equivalence does not generally hold for nonlinear systems. On the other hand, the geometric frequency of the system can always be defined, providing a geometrical interpretation of the system flow. A variety of examples based on linear and nonlinear circuits illustrate the proposed framework.