Bivariate Instantaneous Frequency and Bandwidth

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.

💡 Research Summary

The paper presents a rigorous extension of the concepts of instantaneous frequency (IF) and instantaneous bandwidth (IB) from univariate to bivariate signals. Starting from the observation that many physical systems—such as drifting oceanographic floats, polarized electromagnetic waves, or two‑axis seismic recordings—produce two coupled real‑valued time series, the authors ask how to define a unique, physically meaningful IF and IB for such data. They show that whether the bivariate signal is written as a pair of real signals ((x(t),y(t))) or as one analytic signal (z_{+}(t)=x(t)+i\mathcal{H}