(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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[ 13134, 393]*) (*NotebookOutlinePosition[ 13877, 418]*) (* CellTagsIndexPosition[ 13833, 414]*) (*WindowFrame->Normal*) Notebook[{ Cell["Basic Fourier Definitions", "Title"], Cell["\<\ R. A. Johnson, Department of Geosciences, University of Arizona\ \>", "Subsubtitle", FontWeight->"Bold"], Cell["Continuous Periodic Functions", "Subtitle", FontWeight->"Bold"], Cell[TextData[{ StyleBox["Fourier Series.", FontWeight->"Bold"], " This can be used to represent any continuous periodic function, and in \ effect gives the Fourier Transform of continuous periodic functions. The \ Fourier series itself can be thought of as the inverse transform, whereas the \ coefficients ", Cell[BoxData[ \(a\_n\)]], " and ", Cell[BoxData[ \(b\_n\)]], " represent the forward transform because they give the amplitudes of the \ various frequency components." }], "Text"], Cell[BoxData[ \(f \((t)\)\ = \ a\_0\/2\ + \ \[Sum]\+\(n = 1\)\%\[Infinity]\((\(a\_n\) Cos[2 \[Pi]\ f\_n\ t]\ + \ \(b\_n\) Sin[2 \[Pi]\ f\_n\ t])\)\)], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["where", "Text"], Cell[BoxData[ \(a\_n\ = \ \(2\/T\) \(\[Integral]\_\(-\(T\/2\)\)\%\(T\/2\)f \((t)\)\ \ Cos[2 \[Pi]\ f\_n\ t] \[DifferentialD]t\ \ \ \ \ \ \ \ \ \ {n\ = \ 0, 1, 2, ... }\ \ \ \ \ and\)\)], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(b\_n\ = \ \(2\/T\) \(\[Integral]\_\(-\(T\/2\)\)\%\(T\/2\)f \((t)\)\ \ Sin[2 \[Pi]\ f\_n\ t] \[DifferentialD]t\ \ \ \ \ \ \ \ \ \ {n\ = \ 1, 2, ... }\)\)], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "and ", Cell[BoxData[ \(f\_n\)]], " = n ", Cell[BoxData[ \(f\_0\)]], " = ", Cell[BoxData[ FormBox[ StyleBox[\(n/T\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], " (T = period; ", Cell[BoxData[ \(f\_0\)]], " = fundamental frequency)." }], "Text"], Cell[TextData[{ "Another representation of the Fourier Series (essentially the same thing \ as above, ", StyleBox["but admitting negative frequencies", FontWeight->"Bold"], ") is:" }], "Text", CellFrame->{{0, 0}, {0, 0.5}}], Cell[BoxData[ \(f \((t)\)\ = \ \[Sum]\+\(n = \(-\[Infinity]\)\)\%\[Infinity] C\_n\ \ \[ExponentialE]\^\(\(+\[ImaginaryI]\)\ 2 \[Pi]\ f\_n\ t\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((Inverse)\)\ \ \ \ \ {n\ = \ \ \(\(..\) \(-1\)\), 0, 1, ... }\)], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(C\_n\ = \ \(1\/T\) \(\[Integral]\_\(-\(T\/2\)\)\%\(T\/2\)\ f \((t)\)\ \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ 2 \[Pi]\ f\_n\ \ t\)\) \[DifferentialD]t\ \ \ \ \ \ \((Forward)\)\ \ \ \ \ \ {n\ = \ \(\(..\) \ \(-1\)\), 0, 1, ... }\)\)], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Note that the amplitudes of the ", Cell[BoxData[ FormBox[ StyleBox[\(C\_n\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], " terms are half those calculated using the first definition for ", Cell[BoxData[ FormBox[ StyleBox[\(a\_n\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]], FontVariations->{"CompatibilityType"->0}], " and ", Cell[BoxData[ FormBox[ StyleBox[\(b\_n\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]], FontVariations->{"CompatibilityType"->0}], " because the ", Cell[BoxData[ FormBox[ StyleBox[\(C\_n\), FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}], TraditionalForm]]], " terms exist for both positive and negative n. These last two expressions \ are in a form that more closely resembles the inverse and forward Fourier \ Transforms below. Because the function f(t) is periodic, the frequencies are \ discrete (i.e., just points, not continuous functions). ", StyleBox["Continuous periodic functions transform into discrete \ non-periodic functions", CellFrame->True, Background->GrayLevel[0.849989], FontVariations->{"Underline"->True}], "." }], "Text", CellFrame->{{0, 0}, {0.5, 0}}], Cell["Discrete Periodic Functions (Number Series)", "Subtitle"], Cell[TextData[{ StyleBox["Discrete Fourier Transform.", FontWeight->"Bold"], " This can be used to give the forward Fourier Transform of the discrete \ periodic time series ", Cell[BoxData[ FormBox[ SubscriptBox["g", StyleBox["k", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], TraditionalForm]]], " or the inverse Fourier Transform of the discrete periodic frequency \ series ", Cell[BoxData[ FormBox[ SubscriptBox["G", StyleBox["n", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], TraditionalForm]]], ". The coefficients ", Cell[BoxData[ \(G\_n\)]], " represent the forward transform of the time series ", Cell[BoxData[ FormBox[ SubscriptBox["g", StyleBox["k", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], TraditionalForm]]], " and give the amplitudes of the discrete frequencies contained in ", Cell[BoxData[ FormBox[ SubscriptBox["g", StyleBox["k", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], TraditionalForm]]], ". The forward transform of a time series is a frequency series that \ repeats with a period of ", Cell[BoxData[ \(TraditionalForm\`1\/\[CapitalDelta]t\)]], " where \[CapitalDelta]t is the sample interval in time. Note that N \ samples represent the function in both the time and frequency domains, and \ that ", StyleBox["discrete periodic functions transform to discrete periodic \ functions", CellFrame->True, Background->GrayLevel[0.849989], FontVariations->{"Underline"->True}], "." }], "Text"], Cell[BoxData[ RowBox[{\(G\_n\), " ", "=", " ", RowBox[{\(1\/N\), RowBox[{\(\[Sum]\+\(k = 0\)\%\(N - 1\)\), " ", RowBox[{ SubscriptBox["g", StyleBox["k", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], " ", \(\[ExponentialE]\^\(\(\(-\[ImaginaryI]\)\ 2 \[Pi]\ n\ \ k\)\/N\)\), " ", \((Forward\ )\), " ", \((Period\ = \ \(1\/\[CapitalDelta]t\ = \ N\))\)}]}]}]}]], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ RowBox[{" ", RowBox[{ FormBox[ SubscriptBox["g", StyleBox["k", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]], "TraditionalForm"], " ", "=", " ", \(\[Sum]\+\(n = 0\)\%\(N - 1\)G\_n\ \[ExponentialE]\^\(\(\(+\ \[ImaginaryI]\)\ 2 \[Pi]\ n\ k\)\/N\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((Inverse)\)\ \ \ \ \ \ \ \ \ \ \((Period\ = \ \(1\/\[CapitalDelta]f\ = \ N\))\)\)}]}]], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["Continuous, Non-Periodic Functions", "Subtitle"], Cell[TextData[{ "General Fourier Transform.", StyleBox[" This is used to give the Fourier Transform of continuous \ functions that are non-periodic (never repeat). These expressions constitute \ a Fourier Transform pair, and give the forward and inverse transformations to \ represent the continuous frequency or time functions in the other domain. 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