(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 493232, 13670]*) (*NotebookOutlinePosition[ 518607, 14561]*) (* CellTagsIndexPosition[ 518531, 14555]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Appendix C*", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["\n\n", FontSize->16, FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox["A BRIEF (AND NECESSARILY INCOMPLETE) EXCURSION INTO FOURIER \ TRANSFORMS", FontWeight->"Bold", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth, TextAlignment->Center], Cell[TextData[{ StyleBox["*The ", FontSize->9, FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSize->9, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" Notebooks in this dissertation benefited from the substantial \ assistance of Marlyn L. Hicks", FontSize->9, FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "This is the first of four ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks included in the Appendices of this dissertation. The first \ two, Appendices C and D, deal with Fourier and Inverse Fourier Transforms, \ the third, contained in Appendix E, describes a method to model optimal \ holding period, and the fourth, Appendix G, contains a random number \ generator that simulates \"efficient\" frontiers using stable random \ variates.\n\nAll of the notebooks provided were produced with ", StyleBox["Mathematica", FontSlant->"Italic"], " version 4.0 and should be executable in ", StyleBox["Mathematica", FontSlant->"Italic"], " version 3.0 with few or no modifications. They all may be viewed with ", StyleBox["MathReader", FontSlant->"Italic"], ", a free software produced by Wolfram Research and downloadable from \ http://www.Wolfram.com/", StyleBox["MathReader.", FontSlant->"Italic"], " The electronic version of this dissertation on CD-Rom contains ", StyleBox["MathReader", FontSlant->"Italic"], ". Opening the notebook in ", StyleBox["MathReader", FontSlant->"Italic"], " is recommended.\n\nViewing or printing in color considerably enhances the \ graphics created in these notebooks.\n\nThis is the first of two Appendices \ intended to supplement the main text in the area of Fourier Transforms. \ Here, Appendix C expands on the general nature of Inverse Fourier Transforms \ (IFT) briefly discussed in Chapter 2. Appendix D is intended to supplement \ Chapter 5, covering the more complex application of the IFT to stable \ distributions and showing how many of the illustrations in this dissertation \ are produced.\n\nTo provide context, I begin by replicating the \ illustration, also shown in Chapter 2, from Farlow(1993), depicting the \ reason Fourier Transforms are so helpful" }], "Text", PageWidth->PaperWidth], Cell[TextData[{ StyleBox[Cell[GraphicsData["Metafile", "\<\ CF5dJ6E]HGAYHf4PEfU^I6mgLb15CDHPAVmbKF5d0@00020020 4P0000`000010000DP0004`100010000c_ooo`0000000000000009010000000014004U@0J@1]06D0 L`0P04h0I@1g0200DP1_06d0H@1^0000000000000000000000000000000000000000000000000901 00020PH31@@50P<4QgX080000808000000000?l100000000E01Y06d0I@1c0200CP1U07L0801B06l0 K@1Q06h0002ZTb]0000000000000P03`nGcjolcS4P000000003HV0M000000000083n?`4000010000 g=hB00000034XPP`a=hB0040003LgQ80000000P4Q`0L05P2/>DB06PhE`;Oai@`oooooooooooooal0 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Farlow, \ ", FontSize->9, FontColor->GrayLevel[0]], StyleBox["Partial Differential Equations for Scientists and Engineers", FontSize->9, FontColor->GrayLevel[0], FontVariations->{"Underline"->True}], StyleBox[", 1993, Dover Publications, New York, p. 73.", FontSize->9, FontColor->GrayLevel[0]], StyleBox["\n\n", FontColor->GrayLevel[0]], StyleBox[ "In our case, we wish to illustrate how two parameters of the stable \ distribution, its shape parameter (\[Alpha]) and its skewness parameter (\ \[Beta]) relate to one another. Doing so requires graphing the utility of \ different risk choices as modeled by different probability distributions. The \ \"obstacle\" overcome by the Fourier Transform is the fact that the stable \ distribution does not have a probability distribution function (pdf). \ Chapters 10-12 of Farlow constitute a good resource for Fourier review. The \ ch.f. and the pdf of the stable distribution constitute a Fourier transform \ pair as described in Farlow.", FontFamily->"Times New Roman", FontSize->12, FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell[BoxData[ \(CleanSlate[]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(CleanSlate[]\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[BoxData[{ \(<< Calculus`\), "\[IndentingNewLine]", \(<< Calculus`FourierTransform`\), "\n", \(<< Statistics`\), "\n", \(<< Graphics`\)}], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(Off[General::spell1];\)\n \(Off[General::spell];\)\), FontFamily->"Courier New"]], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(Null\^2\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "Below is an example of a Fourier Series, an expansion series in the same \ general family as a Taylor series. 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In \ ", FontFamily->"Times New Roman", FontSize->12], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic"], StyleBox[" Version 3.0 the statement ?FourierTransform yielded:\n\n\ FourierTransform[expr, t, w] gives a function of w, which is the Fourier \ transform of expr, a function of t. It is defined by FourierTransform[expr, \ t, w] = FourierOverallConstant * Integrate[Exp[FourierFrequencyConstant I w \ t] expr, {t, -Infinity, Infinity}].", FontFamily->"Times New Roman", FontSize->12] }], "Text", PageWidth->PaperWidth], Cell[TextData[{ "Using the variables common to the characteristic function for Stable \ distributions, the above ", StyleBox["Mathematica", FontSlant->"Italic"], " description becomes:\n\nFourierTransform[fx1, x, u] gives a function of\n \ u, which is the Fourier transform of fx1, a function of x. It is defined \ by FourierTransform[fx1, x, u] = \ FourierOverallConstant*Integrate[Exp[FourierFrequencyConstant I u x] fx1, \ {x, -Infinity, Infinity}].\n \nThe original function, ", StyleBox["f(x1)", FontSlant->"Italic"], ", is a function of x, the FT of this function, ", StyleBox["f(u1)", FontSlant->"Italic"], " is a function of u (making this whole operation somewhat like a \ transformation of a variable). \n\n(A note about notation in this monograph: \ Generally, for our notation, the x is the original function and the u is the \ transformed function)\n\nThe analogy when we apply this methodology to the \ Stable distribution problem is: The notation for the pdf will be f(x", Cell[BoxData[ \(TraditionalForm\`*\)]], ") and the notation for the characteristic function (ch.f.) will be f(u", Cell[BoxData[ \(TraditionalForm\`*\)]], "), where the symbol \" ", Cell[BoxData[ \(TraditionalForm\`*\)]], " \" has the usual wildcard meaning and normally will be represented by the \ sequential numbering of functions. 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It is called an \ \"improper function\" because it fails to satisfy a requirement of a \ function, namely to map each value of x to a unique value of y. DiracDelta \ is a function which is 0 for anything other than DiracDelta[0] and \ \[Integral]DiracDelta[0]=1. ", StyleBox["Mathematica", FontSlant->"Italic"], "'s description from version 3.0 is:\n\n", StyleBox["DiracDelta[x] is a distribution that is 0 for x != 0 and \ satisfies Integrate[DiracDelta[x], {x, -Infinity, Infinity}] = 1. \ DiracDelta[x1, x2,...] is a distribution that is 0 for x1 != 0 || x2 != 0 || \ ... and satisfies Integrate[DiracDelta[x1, x2, ...], {x1, -Infinity, \ Infinity}, {x2, -Infinity, Infinity}, ...] = 1.", FontColor->GrayLevel[0]] }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell["Hence:", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%3 DiracDelta[ x - 2.9999999999] \[DifferentialD]x\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(1.`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]\_1\%3 DiracDelta[ x - 3.000000000001] \[DifferentialD]x\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(0\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ A critical property of the Fourier transform that makes it useful is the \ existence of its inverse. Together, the FT and the IFT are known as a \ \"Fourier transform pair\" (Farlow p. 90). The IFT is defined below. Following that is the translation of the IFT into \ the variables we are working with, where \[Omega] = w and t = x. A and B are \ user defined values with defaults of 1.\ \>", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[BoxData[ \(TextForm\`f \((t)\) = \(B\/\(2 A \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\( F \((\[Omega])\)\) e\^\(\(-B\) i \[Omega] t\)\ d \[Omega]\)\)], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell["becomes:", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[BoxData[ \(TextForm\`f \((x)\) = \(B\/\(2 A \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity] F \((w)\) e\^\(-Biwx\)\ \[DifferentialD]w\ \n\n \(\(Note\)\(\ \)\(a\)\(\ \)\(change\)\(\ \)\(of\)\(\ \)\(variable\)\(\ \ \)\(here\)\(\ \)\(to\)\(\ \)\(make\)\(\ \)\(this\)\(\ \)\(equation\)\(\ \ \)\(consistent\)\(\ \)\(with\)\(\ \)\(the\)\(\ \)\(use\)\(\ \ \)\(of\)\(\ \ \)\(w\)\(\ \ \)\(as\)\(\ \)\(the\)\(\ \)\(inverse\)\(\ \ \)\(transform\)\(\ \)\(variable\)\(\ \)\(\(below\)\(.\)\)\(\t\t\t\)\(\n\)\)\)\ \)], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, TextAlignment->Left, FontFamily->"Times New Roman", FontSize->12, Background->None], Cell[TextData[{ "The next function returns a table which is a function of [u,w]. Note that \ values are 0 except where [w-u] = 0. The restriction of the range of u to \ {-3,3} and the sampling interval of 1 (resulting in integers only) results in \ seven terms.\n\nThe improvement is that this function is integrable term by \ term, hence transformable. Note that u is evaluated at points where [w-u] = \ 0, while a is unevaluated.\n\nThis procedure is valid only if a is \ independent of u and w, ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\(u, w\)\ a\)]], "=0.\n \nNote that the result of the IFT is as predicted by\n\n", Cell[BoxData[ \(TraditionalForm\`f(x) = B/\((2 A \[Pi])\) \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\( F( w)\) e\^\(-Biwx\)\ \[DifferentialD]w\)\)]], " where F[w] = DiracDelta[w-u]." }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[BoxData[ \(\(\(fu2a\)\(:=\)\(Table[ a\ DiracDelta[w - u] u, {u, \(-3\), 3, 1}]\)\(\ \)\)\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[CellGroupData[{ Cell[BoxData[ \(fu2a\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({\(-3\)\ a\ DiracDelta[3 + w], \(-2\)\ a\ DiracDelta[ 2 + w], \(-a\)\ DiracDelta[1 + w], 0, a\ DiracDelta[\(-1\) + w], 2\ a\ DiracDelta[\(-2\) + w], 3\ a\ DiracDelta[\(-3\) + w]}\)], "Output",\ PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ StyleBox["f(u2a) ", FontSlant->"Italic", Background->None], StyleBox["is a function of w with u evaluated. IFT is performed by \ integrating each term, producing the same seven terms as \ InverseFourierTransform[fu2,w,x].", Background->None] }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx2a = \((1/\((2\ \[Pi])\))\) \(\[Integral]\_\(-\[Infinity]\)\%\ \[Infinity] fu2a\ \(E\^\(\(-I\)\ w\ x\)\) \[DifferentialD]w\)\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({\(-\(\(3\ a\ \[ExponentialE]\^\(3\ \[ImaginaryI]\ x\)\)\/\(2\ \ \[Pi]\)\)\), \(-\(\(a\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ x\)\)\/\[Pi]\)\), \ \(-\(\(a\ \[ExponentialE]\^\(\[ImaginaryI]\ x\)\)\/\(2\ \[Pi]\)\)\), 0, \(a\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ x\)\)\/\(2\ \[Pi]\), \(a\ \ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ x\)\)\/\[Pi], \(3\ a\ \ \[ExponentialE]\^\(\(-3\)\ \[ImaginaryI]\ x\)\)\/\(2\ \[Pi]\)}\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["One may integrate a single term to check...", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx2b = \((1/\((2\ \[Pi])\))\)\ Integrate[\(-3\)\ a\ DiracDelta[ 3 + w]\ E\^\(\(-I\)\ w\ x\), \ {w, \(-\[Infinity]\), \ \[Infinity]}]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(\(-\(\(3\ a\ \[ExponentialE]\^\(3\ \[ImaginaryI]\ x\)\)\/\(2\ \[Pi]\)\)\ \)\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "...or perform ", StyleBox["Mathematica", FontSlant->"Italic"], "'s IFT on it" }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx2c = InverseFourierTransform[fu2a, w, x]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({\(-\(\(3\ a\ \[ExponentialE]\^\(3\ \[ImaginaryI]\ x\)\)\/\@\(2\ \ \[Pi]\)\)\), \(-a\)\ \[ExponentialE]\^\(2\ \[ImaginaryI]\ x\)\ \ \@\(2\/\[Pi]\), \(-\(\(a\ \[ExponentialE]\^\(\[ImaginaryI]\ x\)\)\/\@\(2\ \ \[Pi]\)\)\), 0, \(a\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ x\)\)\/\@\(2\ \[Pi]\), a\ \[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ x\)\ \@\(2\/\[Pi]\), \(3\ \ a\ \[ExponentialE]\^\(\(-3\)\ \[ImaginaryI]\ x\)\)\/\@\(2\ \[Pi]\)}\)], \ "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "We can convert ", StyleBox["f(x2a)", FontSlant->"Italic"], " to seven complex trigonometric functions." }], "Text", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ExpToTrig\)\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \("ExpToTrig[expr] converts exponentials in expr to trigonometric \ functions."\)], "Print", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpToTrig[fx2a]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({\(-\(\(3\ a\ Cos[ 3\ x]\)\/\(2\ \[Pi]\)\)\) - \(3\ \[ImaginaryI]\ a\ Sin[3\ \ x]\)\/\(2\ \[Pi]\), \(-\(\(a\ Cos[ 2\ x]\)\/\[Pi]\)\) - \(\[ImaginaryI]\ a\ Sin[2\ \ x]\)\/\[Pi], \(-\(\(a\ Cos[ x]\)\/\(2\ \[Pi]\)\)\) - \(\[ImaginaryI]\ a\ Sin[x]\)\/\(2\ \ \[Pi]\), 0, \(a\ Cos[x]\)\/\(2\ \[Pi]\) - \(\[ImaginaryI]\ a\ Sin[x]\)\/\(2\ \ \[Pi]\), \(a\ Cos[2\ x]\)\/\[Pi] - \(\[ImaginaryI]\ a\ Sin[2\ x]\)\/\[Pi], \ \(3\ a\ Cos[3\ x]\)\/\(2\ \[Pi]\) - \(3\ \[ImaginaryI]\ a\ Sin[3\ x]\)\/\(2\ \ \[Pi]\)}\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ When we specify a value for w, as below where w = 2, we get zero for all \ values except where DiracDelta[w-u]=DiracDelta[0]\ \>", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fu2a /. w -> 2\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({0, 0, 0, 0, 0, 2\ a\ DiracDelta[0], 0}\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "Setting a = 1, carrying out the IFT on ", StyleBox["f(u2a)", FontSlant->"Italic"], " table and summing the terms produces a pdf (plotted next) but not a very \ useful one. Recall that in this case the sampling interval is 1." }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx3 = \[Sum]\+\(n = 1\)\%7 fx2c[\([n]\)] /. a -> 1\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(\[ExponentialE]\^\(\(-2\)\ \[ImaginaryI]\ x\)\ \@\(2\/\[Pi]\) - \ \[ExponentialE]\^\(2\ \[ImaginaryI]\ x\)\ \@\(2\/\[Pi]\) + \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ x\)\/\@\(2\ \[Pi]\) - \ \[ExponentialE]\^\(\[ImaginaryI]\ x\)\/\@\(2\ \[Pi]\) + \(3\ \ \[ExponentialE]\^\(\(-3\)\ \[ImaginaryI]\ x\)\)\/\@\(2\ \[Pi]\) - \(3\ \ \[ExponentialE]\^\(3\ \[ImaginaryI]\ x\)\)\/\@\(2\ \[Pi]\)\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[Abs[fx3], {x, \(-5\), 5}]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 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The sampling interval below is .1, determining the unambiguous \ range of x after IFT. The range of x is 1/sampling interval. The ambiguity, \ in sampling theory, is known as aliasing. In the above example we have \ deliberately undersampled to show the effect. The effect is that the function \ changes direction within the range of x, resulting in the multi-modal shape. \ Our goal is to eliminate the direction change within the range of interest, \ or to move the ambiguity outside the range. We do this by reducing the size \ of the sampling interval." }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[TextData[StyleBox["[Note for later: When using this methodology on \ stable distributions, the variable \"t\" in Nolan's ChF becomes u which is \ defined as u = 2\[Pi]f (as shown below), where f is the frequency (an \ engineering convention). Alternatively, 2\[Pi] could have been the constant \ \"B\" in the IFT (See Shannon for derivation and proof). The 2 arises from \ the fact that samples in the DF domain are complex, therefore each sample \ counts as two, the real value and the imaginary value.]\n\nRecalling the \ original function and its FT:", FontColor->GrayLevel[0], Background->None]], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell[BoxData[{ \(fx1\), "\n", \(fu1\)}], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(\[ExponentialE]\^\(-\(x\^2\/10\)\)\)], "Output", PageWidth->PaperWidth], Cell[BoxData[ \(\@5\ \[ExponentialE]\^\(-\(\(5\ u\^2\)\/2\)\)\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["\<\ Defining the variable \"u\" as we will do in the Stable distribution case, we \ multiply the DiracDelta function times the FT of the original function\ \>", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[{ \(\(u = 2\ \[Pi]\ f;\)\), "\n", \(fu2 = \(\(DiracDelta[w - u]\)\(\ \)\(fu1\)\(\ \)\)\)}], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(\@5\ \[ExponentialE]\^\(\(-10\)\ f\^2\ \[Pi]\^2\)\ DiracDelta[ 2\ f\ \[Pi] - w]\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[StyleBox["The problem with our first attempt to simulate a pdf \ (resulting in the multi-modal plot), sampled from the table created where\n\n\ fu2a:=Table[a DiracDelta[w-u]u,{u,-3,3,1}],\n\nwas that we undersampled. The \ pdf is ambiguous at the points x = +/- 1/SI, where SI is the sampling \ interval - in this case 1. The unambiguous range must include all significant \ values of the pdf. As the transforms are not known to exist for the type of \ functions we are working in and as we know of no analytical method of \ determining the sampling interval that produces all significant values prior \ to the transform, the method we use is to plot the transform to determine if \ the pdf is very near 0 well within the ambiguous range. If it is not, we \ shorten the sampling interval.\n\nThe following reduces the sampling interval \ from 1 to .1. Note the last character in the command line is \".1\".", FontColor->GrayLevel[0]]], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, TextAlignment->Left, FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[1, 0, 1]], Cell[CellGroupData[{ Cell[BoxData[ \(fu3 = Table[fu2, {f, \(-1\)/\[Pi], 1/\[Pi], .1}]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({0.00010151732912263184`\ DiracDelta[\(-2.`\) - w], 0.020261025251823245`\ DiracDelta[\(-1.3716814692820414`\) - w], 0.5617197663372443`\ DiracDelta[\(-0.7433629385640828`\) - w], 2.1632915267311175`\ DiracDelta[\(-0.1150444078461239`\) - w], 1.1573039780178833`\ DiracDelta[\(\(0.5132741228718346`\)\(\ \[InvisibleSpace]\)\) - w], 0.0860036526799066`\ DiracDelta[\(\(1.1415926535897931`\)\(\ \[InvisibleSpace]\)\) - w], 0.0008878168869486618`\ DiracDelta[\(\(1.7699111843077522`\)\(\ \[InvisibleSpace]\)\) - w]}\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "With a revised sample, ", StyleBox["f(u3)", FontSlant->"Italic"], ", we again perform the IFT on it." }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx4 = InverseFourierTransform[fu3, w, x]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \({0.00004049955478044552`\ \[ExponentialE]\^\(2.`\ \[ImaginaryI]\ x\), 0.008082979617233378`\ \[ExponentialE]\^\(1.3716814692820414`\ \ \[ImaginaryI]\ x\), 0.22409376452914018`\ \[ExponentialE]\^\(0.7433629385640828`\ \ \[ImaginaryI]\ x\), 0.8630284548472089`\ \[ExponentialE]\^\(0.1150444078461239`\ \ \[ImaginaryI]\ x\), 0.4616974881081039`\ \[ExponentialE]\^\(\(-0.5132741228718346`\)\ \ \[ImaginaryI]\ x\), 0.03431049332297473`\ \[ExponentialE]\^\(\(-1.1415926535897931`\)\ \ \[ImaginaryI]\ x\), 0.0003541876934582001`\ \[ExponentialE]\^\(\(-1.7699111843077522`\)\ \ \[ImaginaryI]\ x\)}\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell["We then sum them.", "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(fx5 = \[Sum]\+\(n = 1\)\%\(Length[fx4]\)fx4[\([n]\)]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(0.8630284548472089`\ \[ExponentialE]\^\(0.1150444078461239`\ \ \[ImaginaryI]\ x\) + 0.4616974881081039`\ \[ExponentialE]\^\(\(-0.5132741228718346`\)\ \ \[ImaginaryI]\ x\) + 0.22409376452914018`\ \[ExponentialE]\^\(0.7433629385640828`\ \ \[ImaginaryI]\ x\) + 0.03431049332297473`\ \[ExponentialE]\^\(\(-1.1415926535897931`\)\ \ \[ImaginaryI]\ x\) + 0.008082979617233378`\ \[ExponentialE]\^\(1.3716814692820414`\ \ \[ImaginaryI]\ x\) + 0.0003541876934582001`\ \[ExponentialE]\^\(\(-1.7699111843077522`\)\ \ \[ImaginaryI]\ x\) + 0.00004049955478044552`\ \[ExponentialE]\^\(2.`\ \[ImaginaryI]\ x\)\)], \ "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "Calibration is done by noting that the value of the original function, ", StyleBox["f(x1)", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`e\^\(\(-x\^2\)/10\)\)]], ", should = 1 at x = 0. We need a divisor, mx1 below, that will produce \ that result in sum of the IFT. The result is a plot which, while closer to \ the shape we are interested in, still falls short. " }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(mx1 = Abs[fx5 /. x -> 0]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], Cell[BoxData[ \(1.5916078676729`\)], "Output", PageWidth->PaperWidth] }, Open ]], Cell[TextData[{ "While it is possible to calculate the constant \"A\" for the IFT for a \ specific situation, it is easier to supply the calibration constant after the \ transform. Here we relied on our knowledge of the function value at x = 0. \ When we work with the Stable ch.f. we will apply a more general rule, relying \ on the knowledge that ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-\[Infinity]\)\%\[Infinity] PDF \(\(\[DifferentialD]\)\(X\)\(\ \)\)\)]], StyleBox["should", FontSlant->"Italic"], " = 1 and adjust the function accordingly." }], "Text", PageWidth->PaperWidth, ShowSpecialCharacters->True, FontFamily->"Times New Roman", FontSize->12, FontColor->GrayLevel[0], Background->None], Cell[CellGroupData[{ Cell[BoxData[ \(px3 = Plot[Abs[fx5/mx1], {x, \(-10\), 10}]\)], "Input", PageWidth->PaperWidth, ShowSpecialCharacters->True], 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 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