# Phase space decomposition for phase noise and synchronization analysis of planar nonlinear oscillators

## TL;DR

This paper introduces a phase space decomposition method for analyzing phase noise and synchronization in planar nonlinear oscillators, providing analytical tools to improve understanding and control of oscillator performance.

## Contribution

It presents a novel decomposition approach for perturbations in oscillators, with analytical formulas and insights into control implications.

## Key findings

- Derived analytical formulas for perturbation vectors.
- Clarified the validity of orthogonal projection in analysis.
- Enhanced understanding of phase noise and synchronization effects.

## Abstract

Synchronization phenomena, frequency shift and phase noise are often limiting key factors in the performances of oscillators. The perturbation projection method allows to characterize how the oscillator's output is modified by these disturbances. In this brief we discuss the appropriate decomposition of perturbations for synchronization and phase noise analysis of planar nonlinear oscillators. We derive analytical formulas for the vectors spanning the directions along which the perturbations have to be projected. We also discuss the implications of this decomposition in control theory and to what extent a simple orthogonal projection is correct.