File:Fourier transform of rect and a translation.png

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

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Summary

[edit]
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

[edit]
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
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
You may select the license of your choice.

Original upload log

[edit]
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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Date/TimeThumbnailDimensionsUserComment
current21:47, 15 December 2018Thumbnail for version as of 21:47, 15 December 2018360 × 223 (12 KB)OgreBot (talk | contribs)(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 (talk | contribs)(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 rpgs (talk | contribs)Transferred from en.wikipedia (MTC!) (1.1.0)

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