[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/morlet-wavelet-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki19\/morlet-wavelet-wikipedia\/","headline":"Morlet wavelet – Wikipedia","name":"Morlet wavelet – Wikipedia","description":"From Wikipedia, the free encyclopedia Real-valued Morlet wavelet Complex-valued Morlet wavelet In mathematics, the Morlet wavelet (or Gabor wavelet)[1] is","datePublished":"2022-01-11","dateModified":"2022-01-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0a\/MorletWaveletMathematica.svg\/220px-MorletWaveletMathematica.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/0\/0a\/MorletWaveletMathematica.svg\/220px-MorletWaveletMathematica.svg.png","height":"136","width":"220"},"url":"https:\/\/wiki.edu.vn\/en\/wiki19\/morlet-wavelet-wikipedia\/","about":["Wiki"],"wordCount":4982,"articleBody":"From Wikipedia, the free encyclopedia  Real-valued Morlet wavelet  Complex-valued Morlet waveletIn mathematics, the Morlet wavelet (or Gabor wavelet)[1] is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope).  This wavelet is closely related to human perception, both hearing[2] and vision.[3]Table of ContentsHistory[edit]Definition[edit]Use in medicine[edit]Use in music[edit]See also[edit]References[edit]History[edit]In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution.[1]  These are used in the Gabor transform, a type of short-time Fourier transform.[2]  In 1984, Jean Morlet introduced Gabor’s work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform.[4]Definition[edit]The wavelet is defined as a constant \u03ba\u03c3{displaystyle kappa _{sigma }} subtracted from a plane wave and then localised by a Gaussian window:[5]\u03a8\u03c3(t)=c\u03c3\u03c0\u221214e\u221212t2(ei\u03c3t\u2212\u03ba\u03c3){displaystyle Psi _{sigma }(t)=c_{sigma }pi ^{-{frac {1}{4}}}e^{-{frac {1}{2}}t^{2}}(e^{isigma t}-kappa _{sigma })}where \u03ba\u03c3=e\u221212\u03c32{displaystyle kappa _{sigma }=e^{-{frac {1}{2}}sigma ^{2}}} is defined by the admissibility criterion,and the normalisation constant c\u03c3{displaystyle c_{sigma }} is:c\u03c3=(1+e\u2212\u03c32\u22122e\u221234\u03c32)\u221212{displaystyle c_{sigma }=left(1+e^{-sigma ^{2}}-2e^{-{frac {3}{4}}sigma ^{2}}right)^{-{frac {1}{2}}}}The Fourier transform of the Morlet wavelet is:\u03a8^\u03c3(\u03c9)=c\u03c3\u03c0\u221214(e\u221212(\u03c3\u2212\u03c9)2\u2212\u03ba\u03c3e\u221212\u03c92){displaystyle {hat {Psi }}_{sigma }(omega )=c_{sigma }pi ^{-{frac {1}{4}}}left(e^{-{frac {1}{2}}(sigma -omega )^{2}}-kappa _{sigma }e^{-{frac {1}{2}}omega ^{2}}right)}The “central frequency” \u03c9\u03a8{displaystyle omega _{Psi }} is the position of the global maximum of \u03a8^\u03c3(\u03c9){displaystyle {hat {Psi }}_{sigma }(omega )} which, in this case, is given by the positive solution to:\u03c9\u03a8=\u03c311\u2212e\u2212\u03c3\u03c9\u03a8{displaystyle omega _{Psi }=sigma {frac {1}{1-e^{-sigma omega _{Psi }}}}}[citation needed]which can be solved by a fixed-point iteration starting at \u03c9\u03a8=\u03c3{displaystyle omega _{Psi }=sigma } (the fixed-point iterations converge to the unique positive solution for any initial "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/morlet-wavelet-wikipedia\/#breadcrumbitem","name":"Morlet wavelet – Wikipedia"}}]}]