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Summary

Description
English: The spectral leakage of several window functions are plotted on the same coordinate system for comparison.
Date
Source Own work
Author Bob K
Permission
(Reusing this file)
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

SVG development
InfoField
 
The SVG code is valid.
V
 
This vector image was created with GNU Octave.
Gnu Octave source
InfoField
click to expand
  graphics_toolkit gnuplot
#{
The gnuplot graphics toolkit is not actively maintained and has a number
of limitations that are unlikely to be fixed.  Communication with gnuplot
uses a one-directional pipe and limited information is passed back to the
Octave interpreter so most changes made interactively in the plot window
will not be reflected in the graphics properties managed by Octave.  For
example, if the plot window is closed with a mouse click, Octave will not
be notified and will not update its internal list of open figure windows.
The qt toolkit is recommended instead.
#}
% Just a bunch of defaults copied from File:Window_function_and_frequency_response_-_Rectangular.svg
set(0, "DefaultAxesFontName", "Microsoft Sans Serif")
set(0, "DefaultTextFontName", "Microsoft Sans Serif") 
set(0, "DefaultAxesTitleFontWeight", "bold")
set(0, "DefaultAxesFontWeight",      "bold")
set(0, "DefaultAxesFontSize", 14)
set(0, "DefaultAxesLineWidth", 2)
set(0, "DefaultAxesBox", "on")
set(0, "DefaultAxesGridLineStyle", "-")
set(0, "DefaultAxesGridColor", [0 0 0])  	% black
set(0, "DefaultAxesGridAlpha", 0.1)		% opaqueness of grid
set(0, "DefaultAxesLayer", "bottom")     	% grid not visible where overlapped by graph

M = 2^12;   				% interpolation factor
% An unusually large number is needed to resolve the first null of the Tukey window

DFT_length = 2^10;
window_length = DFT_length + 1;
DTFT_length = M*DFT_length;
dr = 100; 				% dynamic range
% These are the indices of the DTFT array that will be plotted logarithmically
a = M/4 : DTFT_length/2;

% These are the bin numbers to be assigned to the indices
x = (a-1)/M;

n = 2*pi*(0:DFT_length-1)/DFT_length;

%============================================
% Rectangular window
window = ones(1,DFT_length);
window(1) = 2*window(1);	% periodic summation

H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);

figure("position", [1 1 1400 600])
semilogx(x, H, "color","black", "linewidth",1)
xlim([.25 20])		% display only 20 bins
ylim([-dr 2])
set(gca,"XTick", [.5 1:10])
set(gca,"XTickLabel",[".5"; "1"; "2"; "3"; "4"; "5"; "6"; "7"; "8"; "9"; "10"])
set(gca, "ygrid", "on");

ylabel("dB", "FontSize",14, "FontWeight","bold")  
xlabel("DFT bins", "FontSize",14, "FontWeight","bold")
title("Spectral leakage of several window functions", "FontSize",20)
hold on

%============================================
% Hann window
window = 0.5 - 0.5*cos(n);
window(1) = 2*window(1);	% periodic summation
H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
semilogx(x, H, "color","red", "linewidth",1)

%============================================
% Hamming window
window = 0.53836 - 0.46164*cos(n);
window(1) = 2*window(1);	% periodic summation
H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
semilogx(x, H, "color","blue", "linewidth",1)

%============================================
% Tukey window
alpha = 0.5;
b = alpha*DFT_length/2;
window = ones(1,window_length);
m = 0 : b;
if( max(m) == b )
  m = m(1:end-1);
endif
M = length(m);
window(1:M) = 0.5*(1-cos(pi*m/b));
window(window_length:-1:window_length-M+1) = window(1:M);

window(1) = 2*window(1);	% periodic summation
window = window(1:end-1);
H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
dark_green = [0 128 0]/256;
semilogx(x, H, "color",dark_green, "linewidth",2)

%============================================
% Blackman window
window = 0.42 - 0.5*cos(n) + 0.08*cos(2*n);
window(1) = 2*window(1);	% periodic summation
H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
gold = [251 159 3]/256;
semilogx(x, H, "color",gold, "linewidth",2)

%============================================
% Flat top window
b = [0.21557895 0.41663158 0.277263158 0.083578947 0.006947368];
window = b(1) - b(2)*cos(n) + b(3)*cos(2*n) -b(4)*cos(3*n) +b(5)*cos(4*n);
window(1) = 2*window(1);	% periodic summation
H = abs(fft(window, DTFT_length));
H = H(a);

H = H/max(H);
H = 20*log10(H);
H = max(-dr,H);
semilogx(x, H, "color","magenta", "linewidth",2)

h = legend(['rectangular'],...
           ['hann'],...
           ['hamming'],...
           ['tukey'],...
           ['blackman'],...
           ['flattop'],...
           "location","west");
legend boxoff
set(h, "fontsize",16);

Captions

The spectral leakage of several window functions are plotted on the same coordinate system for comparison.

Items portrayed in this file

depicts

1 May 2022

image/svg+xml

1845502a673ee569f1a89c513b609ff3f658bf26

3,972,356 byte

750 pixel

1,750 pixel

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current11:47, 1 May 2022Thumbnail for version as of 11:47, 1 May 20221,750 × 750 (3.79 MB)Bob Kenlarge legend and add a tick at x=.5
04:45, 1 May 2022Thumbnail for version as of 04:45, 1 May 20221,750 × 750 (3.8 MB)Bob KUploaded own work with UploadWizard

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