Why symmetry matters in the discrete Fourier transform : structural fidelity, phase invariance, and complementarity

Why symmetry matters in the discrete Fourier transform : structural fidelity, phase invariance, and complementarity | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Why symmetry matters in the discrete Fourier transform : structural fidelity, phase invariance, and complementarity Rui Li, Qing Zhang, Jianping Xuan, Tielin Shi, Lv Tang, Mengdi Xu, and 4 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8579188/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Although the discrete Fourier transform (DFT) is universally regarded as the digital realization of the continuous Fourier transform (FT), the structural fidelity of this correspondence has remained largely unexamined. Here we show that the ordinary DFT, through its asymmetric indexing and implicit choice of temporal origin, systematically violates fundamental symmetry relations, parity complementarity, and intrinsic phase structure inherent to the continuous transform. These violations are not matters of convention or numerical implementation, but constitute genuine structural artifacts introduced by discretization. By reformulating the DFT within a strictly centered and symmetric framework, we demonstrate that these artifacts can be eliminated, restoring the natural correspondence between even-odd signal decomposition, sampling parity, and phase behavior. The resulting symmetric DFT yields phase spectra invariant under temporal shifts and preserves the symmetry properties essential for physical interpretability. Our findings call into question the long-assumed equivalence between the continuous FT and its standard discrete implementation, and suggest that symmetry-preserving discretization is a necessary condition for faithful spectral representation across scientific and engineering applications. These results indicate that symmetry preservation is not a representational choice but a necessary structural condition for any discrete transform intended to faithfully represent continuous physical spectra. Physical sciences/Mathematics and computing/Applied mathematics Physical sciences/Mathematics and computing/Computational science Physical sciences/Mathematics and computing/Information technology Complementarity DFT FFT FT SDFT Symmetry Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8579188","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":573999690,"identity":"afb33015-2a55-47fe-8a83-54590b8a67e2","order_by":0,"name":"Rui 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Here we show that the ordinary DFT, through its asymmetric indexing and implicit choice of temporal origin, systematically violates fundamental symmetry relations, parity complementarity, and intrinsic phase structure inherent to the continuous transform. These violations are not matters of convention or numerical implementation, but constitute genuine structural artifacts introduced by discretization. By reformulating the DFT within a strictly centered and symmetric framework, we demonstrate that these artifacts can be eliminated, restoring the natural correspondence between even-odd signal decomposition, sampling parity, and phase behavior. The resulting symmetric DFT yields phase spectra invariant under temporal shifts and preserves the symmetry properties essential for physical interpretability. 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