Summary

DescriptionWindow function (comparsion).png	English: Comparison of different window functions
Date	21 July 2006
Source	Own Work by user Marcel Müller
Author	Marcel Müller
Permission (Reusing this file)	Marcel Müller at the English-language Wikipedia, the copyright holder of this work, hereby publishes it under the following license: This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Subject to disclaimers. Attribution: Marcel Müller at the English-language 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. This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue 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. Subject to disclaimers.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

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Image details and source code

The image is created with Mathematica 3.0 and some beautifications with CorelDraw 7.0. Due to the old versions no SVG export is available.

Init

<< Graphics`Graphics`
<< Graphics`Legend`

Parameters

Len = 4096;   (* Window length *)
Oversamp = 8; (* Oversampling *)

Helper functions

CreateWindow[winfn_] := Table[winfn[p/Len], {p,0,Len}]

DoFFT[data_] := Chop@Take[
  Fourier[RotateLeft[Join[data, Table[0, {Oversamp*(Length[data]-1)-Length[data]}]], (Length[data]-1)/2]],
  (Length[data]-1)*Oversamp/2+1]

GetEnvelope[data_] := Block[
  {min = 1, max, last = data[[1]]},
  While[data[[min+1]] < last*1.001, last=data[[++min]]];
  Map[
    {#,data[[#]]}&,
    {Range[min], Select[Range[2, Length[data]-1], # > min && data[[#-1]] < data[[#]] > data[[#+1]]&]},
    {2}]
]

PlotWin[winfo_,opt___] := Block[
  {fft = Abs@DoFFT@CreateWindow@winfo[[2]]},
  ListPlot[
    20*Log[10, fft/fft[[1]]+1*^-12],
    GridLines->Automatic, PlotJoined->True, Frame->True, PlotRange->All, opt]
]

PlotWin2[winfo_,opt___] := Block[
  {fft = GetEnvelope@Abs@DoFFT@CreateWindow@winfo[[2]]},
  {LogLinearListPlot[
     {#[[1]]/Oversamp, 20*Log[10,#[[2]]/fft[[1,1,2]]+1*^-12]}& /@ fft[[1]],
     GridLines->Automatic, PlotJoined->True, Frame->True, PlotRange->All, opt],
   LogLinearListPlot[
     {#[[1]]/Oversamp, 20*Log[10,#[[2]]/fft[[1,1,2]]+1*^-12]}& /@ fft[[2]],
     GridLines->Automatic, PlotJoined->True, Frame->True, PlotRange->All, opt]}
]

Colors = Apply[
  RGBColor,
  {{0,0,0}, {1,0,0}, {0,0,1}, {0,.7,0}, {.8,0,.8}, {0,.6,.6}, {.6,.6,0}, {.5,.5,.5}},
  {1}];

Dashings = AbsoluteDashing/@(2{{} (*, {4,2}, {4,2,2,2}*)});

GetStyle[i_] :=
  {Colors[[Mod[i-1,Length[Colors]]+1]],
   Dashings[[Mod[Quotient[i-1,Length[Colors]], Length[Dashings]]+1]]}

Window Functions

GaussWin[sigma_] = Exp[-.5 * ((2#-1)/sigma)^2]&;

CosWin[a__] = Apply[Plus, {a} * Table[(-1)^p Cos[2 Pi p #], {p, 0, Length[{a}]-1}]]&;

KaiserWin[alpha_] = BesselI[0, Pi alpha Sqrt[1 - (2#-1)^2]] / BesselI[0, Pi alpha]&;

WinFN = {
  {"Recangular",        1&,                                          1.},
  {"Gauss \[Sigma]=.4", GaussWin[.4],                                1.45},
  {"Hamming",           CosWin[.53836,.46164],                       1.37},
  {"Hann",              CosWin[.5,.5],                               1.5},
  {"Bartlett",          1-Abs[2#-1]&,                                1.33},
  {"Bartlett-Hann",     .62-.48Abs[#-.5]-.38Cos[2Pi#]&,              1.46},
  {"Blackman",          CosWin[.42,.5,.08],                          1.73},
  {"Kaiser \[Alpha]=1", KaiserWin[1],                                1.2},
  {"Kaiser \[Alpha]=2", KaiserWin[2],                                1.5},
  {"Kaiser \[Alpha]=3", KaiserWin[3],                                1.8},
  {"Nutall",            CosWin[.355768,.487396,.144232,.012604],     2.01},
  {"Blackman-Harris",   CosWin[.35875,.48829,.14128,.01168],         2.01},
  {"Blackman-Nutall",   CosWin[.3635819,.4891775,.1365995,.0106411], 1.98},
  {"Flat top",          CosWin[1,1.93,1.29,.388,.032],               3.77}
};

Test Results

PlotWin[WinFN[[-1]]];

PlotWin2[WinFN[[-1]]];

ListPlot@CreateWindow[WinFN[[-1,2]]];

LogLogListPlot[DoFFT@CreateWindow[WinFN[[-1,2]]],
  PlotRange->All, PlotJoined->True, GridLines->Automatic];

DoFFT@CreateWindow[WinFN[[-1,2]]]

GetEnvelope[Abs@DoFFT[CreateWindow[WinFN[[1,2]]]]]

Results

gr = Show[MapIndexed[
  PlotWin2[#, DisplayFunction->Identity, PlotStyle->GetStyle[#2[[1]]]&, WinFN],
  DisplayFunction->$DisplayFunction, PlotRange->{All, {-145,5}}];

gr2 = ShowLegend[
  gr, {MapIndexed[{Graphics[Append[GetStyle[#2[[1]]], Line[{{0,0},{1,0}}]]],#[[1]]}&, WinFN],
  LegendShadow->{0,0}, LegendPosition->{-.8,-.5}, LegendSize->{.4,.55}, LegendTextSpace->5}];

Licensing

Marcel Müller at the English-language Wikipedia, the copyright holder of this work, hereby publishes it under the following license:

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. Subject to disclaimers.
	Attribution: Marcel Müller at the English-language 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.
	This licensing tag was added to this file as part of the GFDL licensing update.http://creativecommons.org/licenses/by-sa/3.0/CC BY-SA 3.0Creative Commons Attribution-Share Alike 3.0truetrue

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. Subject to disclaimers.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

Original upload log

• 2007-01-19 20:25 Marcel Müller 988×726×??? (82239 bytes) Frequency response of the window functions. The double logarithmic plot shows the magnitude in dB over the normalized frequency. Besides the main lobe only the envelope of the frequency response is shown. This is more clearly and allows a quick comparsion