The Companion of Enrico's Chart for Phase Noise and Two-Sample Variances

TL;DR

This paper provides a comprehensive tutorial, review, and user guide on phase noise, frequency stability, and two-sample variances, featuring Enrico's Chart as a key reference tool for researchers and practitioners.

Contribution

It introduces Enrico's Chart, a compact reference sheet that consolidates essential concepts, formulas, and plots related to phase noise and variances, with updates and corrections.

Findings

Enrico's Chart is a practical reference for phase noise analysis.

The article clarifies the normalization of Hadamard variance.

Auxiliary materials are freely available online.

Abstract

Phase noise and frequency (in)stability both describe the fluctuation of stable periodic signals, from somewhat different standpoints. Frequency is unique compared to other domains of metrology, in that its fluctuations of interest span at least 14 orders of magnitude, from $10^{-4}$ in a mechanical watch to $10^{-18}$ in atomic clocks. The frequency span of interest is some 12-15 orders of magnitude, from $\mu$Hz to GHz Fourier frequency for phase noise, while the time span over which the fluctuations occur ranges from sub-$\mu$s to years integration time for variances. Because this domain is ubiquitous in science and technology, a common language and tools suitable to the variety mentioned are a challenge. This article is at once (1) a tutorial, (2) a review covering the most important facts about phase noise, frequency noise and two-sample (Allan and Allan-like) variances, and (3) a user guide to "Enrico's Chart of Phase Noise and Two-Sample Variances." In turn, the Chart is a reference card collecting the most useful concepts, formulas and plots in a single A4/A-size sheet, intended to be a staple on the desk of whoever works with these topics. The Chart is available under Creative Commons 4.0 CC-BY-NC-ND license from Zenodo, DOI 10.5281/zenodo.4399218. A wealth of auxiliary material is available for free on the Enrico's home page http://rubiola.org. This version includes the corrections for an unfortunate error in the normalization of the Hadamard version, and some corrections for trivial (albeit sometimes subtle) mistakes.