Understanding Harmonic Structures Through Instantaneous Frequency

Area of Science:

• Signal processing
• Non-linear dynamics
• Time-series analysis

Background:

• Analysis of harmonics and non-sinusoidal waveforms in time-series data is increasingly important.
• A precise definition of harmonic relationships in signals is currently lacking.
• Existing methods struggle with the complexities of non-sinusoidal oscillations.

Purpose of the Study:

• To propose a rigorous mathematical definition for harmonic relationships in time-series data.
• To introduce two distinct classes of harmonic structures: strong and weak.
• To validate the proposed framework with simulations and real-world data.

Main Methods:

• Defining harmonics based on integer frequency ratio, constant phase, and joint instantaneous frequency.
• Linking the definition to extrema counting and Empirical Mode Decomposition (EMD).
• Exploring the mathematical underpinnings and connections to analytic number theory.

Main Results:

• A novel, rigorous definition for harmonic relationships in signals is established.
• Two classes of harmonic structures (strong and weak) with distinct extrema behaviors are identified.
• The framework is validated across diverse datasets, including shallow water waves, neuronal models, and brain oscillations.

Conclusions:

• The proposed definition provides a robust method for identifying harmonic structures in non-sinusoidal time-series data.
• This framework enhances the understanding and analysis of complex oscillatory phenomena.
• The definition aids in addressing challenges like mode splitting in time-series decomposition methods.