(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 149665, 3756]*) (*NotebookOutlinePosition[ 150410, 3781]*) (* CellTagsIndexPosition[ 150366, 3777]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Autocorrelation of a Wavelet is the Same as the Inverse Transform of its \ Power Spectrum\ \>", "Title"], Cell["\<\ Roy A.Johnson,Department of Geosciences,University of Arizona\ \>", "Subsubtitle", FontWeight->"Bold", FontSlant->"Italic"], Cell[TextData[{ "This notebook investigates the important relation between autocorrelations \ and power spectra to demonstrate that ", StyleBox["the Autocorrelation of a Wavelet is the Same as the Inverse \ Transform of its Power Spectrum", FontWeight->"Bold"], ".", " We'll start with a sinc-like wavelet that is centered at t=0. 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{-0.746066, -6459.15, \ 0.0525126, 137.706}}] }, Open ]], Cell[TextData[{ "This shows similar values on the left, but the values diverge toward the \ right. (Also, if we used the default \"FourierParameters\" in ", StyleBox["Mathematica,", FontSlant->"Italic"], " we would have a difference in amplitude due to differences in how the \ transform values are normalized. These differences in amplitude turn out not \ to be particularly important for most practical applications). The \ fundamental problem that shows up is the fact that the discrete Fourier \ Transform (and consequently, the power spectrum) is periodic and \"wraps \ around\" such that only half of the spectrum is unique. Everything in the \ spectrum above the nyquist frequency is a mirror image of values below the \ Nyquist frequency. We can avoid this problem in our attempts at comparing \ the autocorrelation and the inverse transform of the power spectrum by \ \"pushing\" the alias lobe of the power spectrum out to the right. This \ means that we need to increase the Nyquist frequency, which we do by \ lengthening (padding) the input time series. In this case, we'll double the \ length of the wavelet." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pw\ = PadRight[w, 28, 0]\)], "Input"], Cell[BoxData[ \({100, 69, 13, \(-24\), \(-27\), \(-14\), \(-2\), 3, 5, 7, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ As before, we calculate and plot the power spectrum of this padded wavelet.\ \>", "Text"], Cell[BoxData[ \(\(fpw\ = \ Fourier[pw, \ FourierParameters \[Rule] {1, \(-1\)}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pspw\ = Abs[fpw]\^2\)], "Input"], Cell[BoxData[ \({19044.`, 16182.389919907991`, 30438.041795404548`, 48910.76680926462`, 43801.68452770204`, 29403.26177378645`, 18557.331127932342`, 10820.`, 5758.098833781474`, 4642.433321590641`, 3654.6270766631105`, 2947.947114743718`, 2798.2166385164664`, 2873.2010607065754`, 2500.`, 2873.2010607065754`, 2798.2166385164664`, 2947.947114743718`, 3654.6270766631105`, 4642.433321590641`, 5758.098833781474`, 10820.`, 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