Carrier frequencies, holomorphy and unwinding

TL;DR

This paper demonstrates that intrinsic-mode functions behave like holomorphic functions, allowing the use of complex analysis techniques such as the unwinding series to achieve stable, high-resolution, noise-robust time-frequency representations.

Contribution

It establishes the holomorphic-like behavior of intrinsic-mode functions and applies the unwinding series for improved time-frequency analysis.

Findings

Anti-holomorphic part is much smaller than holomorphic part for intrinsic-mode functions.

Unwinding series converges and stabilizes, providing high-resolution time-frequency representation.

Method is robust to noise and applicable for signal analysis.

Abstract

We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency $e^{int}$ ensures that the anti-holomorphic part is much smaller than the holomorphic part $ \| P_{-}(f)\|_{L^2} \ll \|P_{+}(f)\|_{L^2}.$ This enables us to use techniques from complex analysis, in particular the \textit{unwinding series}. We study its stability and convergence properties and show that the unwinding series can stabilize and show that the unwinding series can provide a high resolution time-frequency representation, which is robust to noise.