Algebraic Diagnostics for Instantaneous Frequency Estimators

Algebraic Diagnostics for Instantaneous Frequency Estimators | 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 Research Article Algebraic Diagnostics for Instantaneous Frequency Estimators Takashi Matsuhisa, Kota Horizumi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9111596/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 Tracking instantaneous frequency (IF) in multi-component signals, phase-locked loops (PLLs), and reduced-order power-system models is frequently undermined by severe numerical ill-conditioning near regime boundaries. Conventional time--frequency transforms and signal-space geometric invariants can become unreliable precisely where diagnostic certainty is most needed: when denominators approach zero, frequency ridges coalesce, or a locking manifold loses robustness. We propose a unified algebraic diagnostic layer based on Stroboscopic Boundary-Ideal (BDI) constructions. Regimes (e.g.\ separable ridges, lock states, balanced harmonics) are encoded as incidence ideals in an augmented polynomial ring, and their boundaries are projected onto the design/parameter space via saturation and elimination. To quantify fragility we introduce two local indices: (i) Overlap Thickness (OT), a local intersection multiplicity measuring how ''flat'' a boundary is along a parameter drift direction, and (ii) Transversality Index (TI), the order of contact of a parameter path with the boundary. We state hypotheses under which OT and TI are finite and generator-independent, and we show an OT--TI bound for regular crossings. The framework is illustrated on a sequence of case studies: (a) Milano-type circuit diagnostics and IF paradoxes, (b) algebraic limits of multi-component separation under ridge crossing, (c) symbolic cycle-slip boundaries for PLLs, and (d) multi-delay differential equations (MDDEs), where a phase-extended stability boundary yields certificates of delay-independent stability (via infeasibility / the unit ideal) without gridding the delay space. Finally, we provide a reproducible ridge-crossing benchmark and an empirical robustness study (3,600 Monte Carlo trials across $-10$~dB to $30$~dB SNR), demonstrating that sub-bin interpolation improves the stability of a contact-order proxy and onset detection accuracy; the end-to-end runtime is on the order of tens of milliseconds per instance in our reference implementation. Electrical Engineering Applied Mathematics Instantaneous frequency boundary ideals overlap thickness phase-locked loops elimination theory delay-independent stability Full Text Additional Declarations The authors declare no competing interests. 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-9111596","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":605497839,"identity":"8a7f1e14-e712-4cbc-86f6-b805fd329248","order_by":0,"name":"Takashi Matsuhisa","email":"data:image/png;base64,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","orcid":"","institution":"IAMR KarRC RAS","correspondingAuthor":true,"prefix":"","firstName":"Takashi","middleName":"","lastName":"Matsuhisa","suffix":""},{"id":605498363,"identity":"7d3d12a0-6a99-4369-88c0-0217b5785a7d","order_by":1,"name":"Kota Horizumi","email":"","orcid":"","institution":"Ibaraki University","correspondingAuthor":false,"prefix":"","firstName":"Kota","middleName":"","lastName":"Horizumi","suffix":""}],"badges":[],"createdAt":"2026-03-13 07:26:40","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9111596/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9111596/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106093065,"identity":"ab6144e4-141b-4987-ab30-280161428a14","added_by":"auto","created_at":"2026-04-03 11:33:24","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":435708,"visible":true,"origin":"","legend":"","description":"","filename":"CSSPAlgebraicDiagonositicforIF2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9111596/v1_covered_6eeeb777-2e55-4c42-b157-da702bf2a3bd.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eAlgebraic Diagnostics for Instantaneous Frequency Estimators\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"MRI BSBH for Scientific Research","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Instantaneous frequency, boundary ideals, overlap thickness, phase-locked loops, elimination theory, delay-independent stability","lastPublishedDoi":"10.21203/rs.3.rs-9111596/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9111596/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTracking instantaneous frequency (IF) in multi-component signals, phase-locked loops (PLLs), and reduced-order power-system models is frequently undermined by severe numerical ill-conditioning near regime boundaries. Conventional time--frequency transforms and signal-space geometric invariants can become unreliable precisely where diagnostic certainty is most needed: when denominators approach zero, frequency ridges coalesce, or a locking manifold loses robustness.\u003c/p\u003e\n\u003cp\u003eWe propose a unified algebraic diagnostic layer based on Stroboscopic Boundary-Ideal (BDI) constructions. Regimes (e.g.\\ separable ridges, lock states, balanced harmonics) are encoded as incidence ideals in an augmented polynomial ring, and their boundaries are projected onto the design/parameter space via saturation and elimination. To quantify fragility we introduce two local indices: (i) Overlap Thickness (OT), a local intersection multiplicity measuring how ''flat'' a boundary is along a parameter drift direction, and (ii) Transversality Index (TI), the order of contact of a parameter path with the boundary. We state hypotheses under which OT and TI are finite and generator-independent, and we show an OT--TI bound for regular crossings.\u003c/p\u003e\n\u003cp\u003eThe framework is illustrated on a sequence of case studies: (a) Milano-type circuit diagnostics and IF paradoxes, (b) algebraic limits of multi-component separation under ridge crossing, (c) symbolic cycle-slip boundaries for PLLs, and (d) multi-delay differential equations (MDDEs), where a phase-extended stability boundary yields certificates of delay-independent stability (via infeasibility / the unit ideal) without gridding the delay space.\u003c/p\u003e\n\u003cp\u003eFinally, we provide a reproducible ridge-crossing benchmark and an empirical robustness study (3,600 Monte Carlo trials across $-10$~dB to $30$~dB SNR), demonstrating that sub-bin interpolation improves the stability of a contact-order proxy and onset detection accuracy; the end-to-end runtime is on the order of tens of milliseconds per instance in our reference implementation.\u003c/p\u003e","manuscriptTitle":"Algebraic Diagnostics for Instantaneous Frequency Estimators","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-20 17:42:40","doi":"10.21203/rs.3.rs-9111596/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6fdf5484-2e12-42b8-81ac-6639ffc5d44f","owner":[],"postedDate":"March 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":64440101,"name":"Electrical Engineering"},{"id":64440102,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2026-03-20T17:42:40+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-20 17:42:40","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9111596","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9111596","identity":"rs-9111596","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}