(************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 715277, 17251]*) (*NotebookOutlinePosition[ 716018, 17276]*) (* CellTagsIndexPosition[ 715974, 17272]*) (*WindowFrame->Normal*) Notebook[{ Cell["Shift of Delta Function and Phase Relations", "Title"], Cell["\<\ Roy A. Johnson, Department of Geosciences, University of Arizona\ \>", "Subsubtitle", FontWeight->"Bold", FontSlant->"Italic"], Cell[TextData[{ "This notebook demonstrates that ", StyleBox["a shift of a function in one domain results in a linear phase \ shift in the other domain", FontVariations->{"Underline"->True}], ". This is demonstrated with the Dirac delta function. A Dirac delta \ function convolved with any function returns that same function. If the \ Dirac delta is shifted away from zero, the convolution operation results in \ the function being shifted to the position of the delta function. The \ amplitude spectrum of the shifted function will be the product of the \ amplitude spectrum of the Dirac delta (a constant = 1) and the spectrum of \ the function. The phase spectra will add. Thus, the act of shifting the \ function adds a linear phase shift to its phase spectrum. This notebook \ shows the linear phase shift caused by shifting the Dirac delta." }], "Text", FontWeight->"Bold"], Cell["\<\ Start by clearing any old definitions and then define a delta function with \ an arbitrary shift of \"a.\"\ \>", "Text", FontWeight->"Bold"], Cell[BoxData[ \(Clear[a, t, f, ft, ftazero, ff, amp, ftft]\)], "Input"], Cell[BoxData[ \(\(ft\ = \ DiracDelta[t - a];\)\)], "Input"], Cell["\<\ The transform of a unit delta function is a \"wall,\" which effectively is a \ constant value of 1 in the other dimension. 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Expanding this result into Cosine and Sine \ values gives:" }], "Text", FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(ff\ = \ ComplexExpand[ftft]\)], "Input"], Cell[BoxData[ \(Cos[2\ a\ f\ \[Pi]] - \[ImaginaryI]\ Sin[2\ a\ f\ \[Pi]]\)], "Output"] }, Open ]], Cell[TextData[{ "The amplitude spectrum of the shifted spike should remain unchanged (it is \ still a constant \"wall\" of height 1). This can be checking by squaring the \ Real and Imaginary parts, summing and taking the square root. The result, as \ expected, is 1 (because ", Cell[BoxData[ \(TraditionalForm\`Cos\^2\)]], " + ", Cell[BoxData[ \(TraditionalForm\`Sin\^2\)]], " = 1)." }], "Text", FontWeight->"Bold"], Cell[TextData[{ "The phase spectrum will depend on the shift value \"a.\" Below is a \ sequence of plots (that can be animated by double clicking) that shows the \ variation in phase with changing \"a\" from 0.1 to 1.0 in steps of 0.05. 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