Bivariate Instantaneous Frequency and Bandwidth

TL;DR

This paper extends the concepts of instantaneous frequency and bandwidth to bivariate signals, providing a geometric interpretation through evolving elliptical representations and analyzing their stability and modulation.

Contribution

It introduces a unique, comprehensive framework for defining and interpreting bivariate instantaneous frequency and bandwidth, including their geometric and physical significance.

Findings

Bivariate bandwidth decomposes into three modulation components.

Ellipse stability relates to amplitude, eccentricity, and orientation changes.

Application demonstrates practical utility in oceanographic data analysis.

Abstract

The generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived. These are uniquely defined whether the signal is represented as a pair of real-valued signals, or as one analytic and one anti-analytic signal. A nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time, and this representation provides a simple geometric interpretation for the bivariate instantaneous moments. The bivariate bandwidth is shown to consist of three terms measuring the degree of instability of the time-varying ellipse: amplitude modulation with fixed eccentricity, eccentricity modulation, and orientation modulation or precession. An application to the analysis of data from a free-drifting oceanographic float is presented and discussed.