ON INSTANTANEOUS FREQUENCY

TLDR

Instantaneous frequency is essential for analyzing nonlinear and nonstationary processes and has traditionally been derived from the analytic signal via the Hilbert transform. This study reviews the challenges of using analytic signals for IF estimation and introduces two new methods to address them. The authors propose a direct quadrature approach and a normalized Hilbert transform, and also evaluate generalized zero‑crossing and Teager energy operator techniques. Comparisons show that the normalized Hilbert transform and direct quadrature yield the best performance, while the Teager operator is highly localized but limited to linear data and the generalized zero‑crossing is robust yet restricted to mean frequencies over a quarter wavelength, indicating that most IF determination issues are now resolved.

Abstract

Instantaneous frequency (IF) is necessary for understanding the detailed mechanisms for nonlinear and nonstationary processes. Historically, IF was computed from analytic signal (AS) through the Hilbert transform. This paper offers an overview of the difficulties involved in using AS, and two new methods to overcome the difficulties for computing IF. The first approach is to compute the quadrature (defined here as a simple 90° shift of phase angle) directly. The second approach is designated as the normalized Hilbert transform (NHT), which consists of applying the Hilbert transform to the empirically determined FM signals. Additionally, we have also introduced alternative methods to compute local frequency, the generalized zero-crossing (GZC), and the teager energy operator (TEO) methods. Through careful comparisons, we found that the NHT and direct quadrature gave the best overall performance. While the TEO method is the most localized, it is limited to data from linear processes, the GZC method is the most robust and accurate although limited to the mean frequency over a quarter wavelength of temporal resolution. With these results, we believe most of the problems associated with the IF determination are resolved, and a true time–frequency analysis is thus taking another step toward maturity.