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Fourier_transform_of_rect_and_a_translation.png (360 × 223 pixels, file size: 12 KB, MIME type: image/png)

Summary

Description
English: A rectangular pulse function f(t), shown beside its Fourier transform in the first row. In the second row, a shifted rectangular pulse g(t), with its Fourier transform, exhibiting a phase shift in each frequency.
Date
Source Created in Mathematica 9.0
Author Slawekb 
f[t_] = HeavisideTheta[1 - t]*HeavisideTheta[1 + t]
fhat[s_] = FourierTransform[f[t], t, s]
p1 = Plot[f[t], {t, -4, 4}, Filling -> Axis, AxesLabel -> {"\!\(\*
StyleBox[\"t\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)", "\!\(\*
StyleBox[\"f\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"(\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"t\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\")\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)"}, Ticks -> False, 
  PlotStyle -> {Thickness[0.008]}]
p2 = Plot[fhat[s], {s, -10, 10}, AxesLabel -> {"\!\(\*
StyleBox[\"\[Omega]\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)", "\!\(\*
StyleBox[OverscriptBox[\"f\", \"^\"],\nFontFamily->\"Sans Serif\",\n\
FontSize->14,\nFontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"(\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"\[Omega]\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\")\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)"}, Ticks -> False, 
  PlotStyle -> {Thickness[0.005]}]
g[t_] = f[t - 1]
ghat[s_] = FourierTransform[g[t], t, s]/.s->-s
p3 = Plot[g[t], {t, -4, 4}, Filling -> Axis, AxesLabel -> {"\!\(\*
StyleBox[\"t\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)", "\!\(\*
StyleBox[\"g\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"(\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"t\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\")\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)"}, Ticks -> False, 
  PlotStyle -> {Thickness[0.008]}]
p4 = Plot[{Re[ghat[s]], Im[ghat[s]]}, {s, -10, 10}, PlotRange -> All, 
  AxesLabel -> {"\!\(\*
StyleBox[\"\[Omega]\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)", "\!\(\*
StyleBox[\"Re\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\" \",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[OverscriptBox[
StyleBox[
RowBox[{\" \", \"g\"}],\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"], \"^\"],\nFontFamily\
->\"Sans Serif\",\nFontSize->14,\nFontWeight->\"Plain\",\nFontSlant\
->\"Italic\"]\)\!\(\*
StyleBox[\",\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\" \",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"Im\",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\" \",\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[OverscriptBox[
StyleBox[
RowBox[{\" \", \"g\"}],\nFontFamily->\"Sans Serif\",\nFontSize->14,\n\
FontWeight->\"Plain\",\nFontSlant->\"Italic\"], \"^\"],\nFontFamily\
->\"Sans Serif\",\nFontSize->14,\nFontWeight->\"Plain\",\nFontSlant\
->\"Italic\"]\)"}, Ticks -> False, PlotStyle -> {Thickness[0.005]}]
GraphicsGrid[{{p1, p2}, {p3, p4}}]

Licensing

Slawekb at English Wikipedia, the copyright holder of this work, hereby publishes it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Attribution: Slawekb at English Wikipedia
You are free:
  • to share – to copy, distribute and transmit the work
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Under the following conditions:
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Original upload log

The original description page was here. All following user names refer to en.wikipedia.
Date/Time Dimensions User Comment
2015-08-13 11:47:20 360 × 223 Slawekb Reverted to version as of 11:46, 13 August 2015
2015-08-13 11:47:03 360 × 223 Slawekb Reverted to version as of 11:42, 13 August 2015
2015-08-13 11:46:38 360 × 223 Slawekb
2015-08-13 11:42:27 360 × 223 Slawekb Changed from Mathematica's wacky Fourier convention to the usual one.
2014-12-13 15:20:56 736 × 455 Slawekb Uploading a self-made file using [[Wikipedia:File_Upload_Wizard|File Upload Wizard]]

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13 December 2014

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Date/TimeThumbnailDimensionsUserComment
current21:47, 15 December 2018Thumbnail for version as of 21:47, 15 December 2018360 × 223 (12 KB)OgreBot(BOT): Reverting to most recent version before archival
21:47, 15 December 2018Thumbnail for version as of 21:47, 15 December 2018736 × 455 (28 KB)OgreBot(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2014-12-13 15:20:56 by Slawekb
14:33, 15 December 2018Thumbnail for version as of 14:33, 15 December 2018360 × 223 (12 KB)I love rpgsTransferred from en.wikipedia (MTC!) (1.1.0)

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