Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Olhede S. and Walden A.T. 2003Polarization phase relationships via multiple Morse wavelets. I. FundamentalsProc. R. Soc. Lond. A.459413–444http://doi.org/10.1098/rspa.2002.1049SectionRestricted accessPolarization phase relationships via multiple Morse wavelets. I. Fundamentals S. Olhede S. Olhede Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author and A.T. Walden A.T. Walden Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author S. Olhede S. Olhede Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author and A.T. Walden A.T. Walden Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 February 2003https://doi.org/10.1098/rspa.2002.1049AbstractThe purpose of this paper is to provide a detailed analysis of the use of multiple generalized Morse wavelets for polarization analysis of multi–component recordings where phase relationships between components of the signal can be transient. Special attention will be given to the case of three components. The use of complex and analytic wavelets enables detection of coherent motion with elliptical polarization because of sensitivity to phase shifts among components. We adopt a singular–valuedecomposition approach. The principal polarization is given by the first eigenvector of the multiple–wavelet multivariate scalogram at scale a and time b. Having described in detail a deterministic signal plus random noise model for the recorded data, we show that for (a,b) in a domain Ω, we can approximate the phase between components of the original multi–component signal via the estimated phase of the components of the first right–singular vector; detailed results on the accuracy of the approximation are given. The exact form of the domain Ω, which depends on the transient nature of the phase relationship, is described. 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Olhede S and Walden A (2003) Polarization phase relationships via multiple Morse wavelets. II. Data analysis, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 10.1098/rspa.2002.1050, 459:2031, (641-657), Online publication date: 8-Mar-2003. This Issue08 February 2003Volume 459Issue 2030 Article InformationDOI:https://doi.org/10.1098/rspa.2002.1049Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/02/2003Published in print08/02/2003 License: Citations and impact Keywordsphasenon–stationary time–seriespolarizationmultiple waveletsstatistical analysismulti–component signals