{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T13:41:44Z","timestamp":1765806104917,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,2]],"date-time":"2024-11-02T00:00:00Z","timestamp":1730505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Symmetry is a fundamental property of many natural systems, which is observable through signals. In most out-of-equilibrium complex dynamic systems, the observed signals are asymmetric. However, for certain operating modes, some systems have demonstrated a resurgence of symmetry in their signals. Research has naturally focused on examining time invariance to quantify this symmetry. Measures based on the statistical and harmonic properties of signals have been proposed, but most of them focused on harmonic distortion without explicitly measuring symmetry. This paper introduces a new mathematical framework based on group theory for analyzing signal symmetry beyond time invariance. It presents new indicators to evaluate different types of symmetry in non-stochastic symmetric signals. Both periodic and non-periodic symmetric signals are analyzed to formalize the problem. The study raises critical questions about the completeness of symmetry in signals and proposes a new classification for periodic and non-periodic signals that goes beyond the traditional classification based on Fourier coefficients. Furthermore, new measures such as \u201csymmetrometry\u201d and \u201cdistorsymmetry\u201d are introduced to quantify symmetry. These measures outperform traditional indicators like Total Harmonic Distortion (THD) and provide a more accurate measurement of symmetry in complex signals from applications where duty cycle plays a major role.<\/jats:p>","DOI":"10.3390\/e26110941","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T06:30:48Z","timestamp":1730788248000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetry in Signals: A New Insight"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2356-885X","authenticated-orcid":false,"given":"Jean-Marc","family":"Girault","sequence":"first","affiliation":[{"name":"Groupe ESEO, 49000 Angers, France"},{"name":"Laboratoire d\u2019Acoustique de l\u2019Universit\u00e9 du Mans (LAUM), UMR 6613, Institut d\u2019Acoustique\u2014Graduate School (IA-GS), CNRS, Le Mans Universit\u00e9, 72085 Le Mans, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1063\/1.1487570","article-title":"Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables","volume":"12","author":"Letellier","year":"2002","journal-title":"Chaos"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.chaos.2015.04.010","article-title":"Recurrence and Symmetry of time series: Application to transition detection","volume":"77","author":"Girault","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.bspc.2018.01.013","article-title":"Gait symmetry measures: A review of current and prospective methods","volume":"42","author":"Viteckova","year":"2018","journal-title":"Biomed. 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