(************** 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[ 58042, 1974]*) (*NotebookOutlinePosition[ 58784, 1999]*) (* CellTagsIndexPosition[ 58740, 1995]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Notes on Defaults in Fourier Transform and Related ", StyleBox["Mathematica", FontSlant->"Italic"], " Functions" }], "Title"], Cell[TextData[StyleBox["Roy A. Johnson, Department of Geosciences, University \ of Arizona", FontWeight->"Bold"]], "Subsubtitle"], Cell[BoxData[ \(\(SetOptions[Plot, PlotStyle \[Rule] {Thickness[0.015], Hue[0.7]}];\)\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], ", by default, defines the Fourier Transform as:" }], "Text"], Cell[BoxData[ \(\(\(\t\t\)\(\(1\/\@\(2 \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\%\ \[Infinity] f \((t)\)\ \(e\^\(i \[Omega] t\)\) \[DifferentialD]t\)\)\)\)], "Text"], Cell[TextData[{ "However, in signal processing and in several other branches of science, \ the Fourier Transform is defined somewhat differently. (The differences are \ not a problem as long as an internally consistent definition is used for \ both forward and inverse transforms; nevertheless, we will want to keep all \ of our use of the Fourier Transform definition as consistent as possible with \ the notes and textbooks). ", StyleBox["Mathematica", FontSlant->"Italic"], " makes it relatively easy to change the definition of the Fourier \ Transform to be consistent with the particular conventions used in other \ applications. This is done by providing the \"FourierParameters\" a and b in \ the pair {a,b} where these values are automatically substituted into the \ general Fourier Transform expression:" }], "Text"], Cell[BoxData[ \(\(\(\t\t\)\(\(\@\(\(\(\[VerticalSeparator]\)\(b\)\(\[VerticalSeparator]\ \)\)\/\((2 \[Pi])\)\^\(1 - a\)\)\) \(\[Integral]\_\(-\[Infinity]\)\%\ \[Infinity] f \((t)\)\ \(e\^\(i b \[Omega] t\)\) \[DifferentialD]t\)\)\)\)], "Text"], Cell[TextData[{ "In ", StyleBox["Seismic Data Processing", FontWeight->"Bold"], ", we use the signal processing definition provided by {a,b} = {0, \ -2\[Pi]}. This change of definition is done simply by using the following ", StyleBox["Mathematica", FontSlant->"Italic"], " statement, which sets the parameters for all following calculations in \ the notebook:" }], "Text"], Cell[BoxData[{ \(\(SetOptions[FourierTransform, \ FourierParameters \[Rule] {0, \(-2\) \[Pi]}];\)\), "\ \[IndentingNewLine]", \(\(SetOptions[InverseFourierTransform, \ FourierParameters \[Rule] {0, \(-2\) \[Pi]}];\)\)}], "Input"], Cell["An example:", "Text"], Cell[BoxData[ \(\(boxcar\ = \ UnitStep[t\ + 1/2]\ - \ UnitStep[t - 1/2];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(plotboxcar\ = \ Plot[boxcar, {t, \(-2\), 2}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % 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The Fourier transform of this boxcar can be found using the following \ command:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ftboxcar\ = \ FourierTransform[boxcar, t, f]\)], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ f\ \[Pi]\)\ \ \((\(-\(\[ImaginaryI]\/\(f\ \[Pi]\)\)\) + DiracDelta[f])\) + 1\/2\ \[ExponentialE]\^\(\[ImaginaryI]\ f\ \[Pi]\)\ \((\(-\(\ \[ImaginaryI]\/\(f\ \[Pi]\)\)\) + DiracDelta[f])\)\)], "Output"] }, Open ]], Cell["\<\ The result, in complex exponential form, also includes a couple of Dirac \ deltas; these are only defined at f = 0, so these deltas can basically be \ ignored. They occur here in order to make the sinc function equal to 1 at f \ = 0. We can force things into the more familiar form by using a couple of \ expansions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ftboxcar2\ = \ FunctionExpand[ ComplexExpand[FourierTransform[boxcar, t, f]]]\)], "Input"], Cell[BoxData[ \(Sin[f\ \[Pi]]\/\(f\ \[Pi]\)\)], "Output"] }, Open ]], Cell["\<\ This result is more familiar looking. 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Note that we could specify the parameter \ options within the FourierTransform function as shown below. However, this \ applies only to the specific operation and does not \"redefine\" the \ transform parameters.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ftboxcar\ = \ FunctionExpand[ ComplexExpand[ FourierTransform[boxcar, t, f, FourierParameters \[Rule] {0, \(-2\) \[Pi]}]]]\)], "Input"], Cell[BoxData[ \(Sin[f\ \[Pi]]\/\(f\ \[Pi]\)\)], "Output"] }, Open ]], Cell["\<\ If we use the default Fourier Parameters, we get a result that is \ inconsistent with our definitions:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ftboxcar\ = \ FunctionExpand[ ComplexExpand[ FourierTransform[boxcar, t, f, FourierParameters \[Rule] {0, 1}]]]\)], "Input"], Cell[BoxData[ \(\(\@\(2\/\[Pi]\)\ Sin[f\/2]\)\/f\)], "Output"] }, Open ]], Cell[TextData[{ "Thus, it's important to check (and define as necessary) these parameter \ definitions and make sure that ", StyleBox["Mathematica", FontSlant->"Italic"], " operations are doing what you expect. More information on the Fourier \ Transform and various parameters is available in the Help Browser in ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowSize->{607, 466}, WindowMargins->{{57, Automatic}, {Automatic, 7}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, StyleDefinitions -> "Classroom.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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