File:Window function (rectangular).png

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Description

rectangular window and frequency response

Date

17 December 2005

Source

Own work

Author

Bob K (original version), Olli Niemitalo

Permission
(Reusing this file)

I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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File:Window function and frequency response - Rectangular.svg is a vector version of this file. It should be used in place of this PNG file when not inferior.

File:Window function (rectangular).png → File:Window function and frequency response - Rectangular.svg

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Source code

The script below generates these .png images:

This script has not been tested in MATLAB. See the individual file histories for the simpler MATLAB scripts that were the basis of this script.

Generation of svg files by minor modification of the script displayed visual artifacts and renderer incompatibilities that could not be easily fixed. The current script fixes the visual artifacts in the png file as a post-processing step. The script generates a semi-transparent grid by taking a weighted average of two images, one with the grid and one without.

This PNG graphic was created with GNU Octave by Olli Niemitalo.

Matlab

function plotWindowLayer (w, N, gridded, wname, wspecifier)
 
  M=32;
  k=0:N-1;
  dr = 120;

  H = abs(fft([w zeros(1,(M-1)*N)]));
  H = fftshift(H);
  H = H/max(H);
  H = 20*log10(H);
  H = max(-dr,H);
 
  figure('Position',[1 1 1200 520])
  subplot(1,2,1)
  set(gca,'FontSize',28)
  area(k,w,'FaceColor', [0 1 1],'edgecolor', [1 1 0],'linewidth', 2)
  xlim([0 N-1])
  if (min(w) &gt;= -0.01)
    ylim([0 1.05])
    set(gca,'YTick', [0 : 0.1 : 1])
    ylabel('amplitude','position',[-16 0.525 0])
  else
    ylim([-1 5])
    set(gca,'YTick', [-1 : 1 : 5])
    ylabel('amplitude','position',[-16 2 0])
  endif
  set(gca,'XTick', [0 : 1/8 : 1]*(N-1))
  set(gca,'XTickLabel',[' 0'; ' '; ' '; ' '; ' '; ' '; ' '; ' '; 'N-1'])
  grid(gridded)
  set(gca,'LineWidth',2)
  set(gca,'gridlinestyle','-')
  xlabel('samples')
  if (strcmp (wspecifier, ""))
    title(cstrcat(wname,' window'))
  else
    title(cstrcat(wname,' window (', wspecifier, ')'))
  endif
  set(gca,'Position',[0.08 0.11 0.4 0.8])
  set(gca,'XColor',[1 0 1])
  set(gca,'YColor',[1 0 1])
  
  subplot(1,2,2)
  set(gca,'FontSize',28)
  h = stem(([1:M*N]-1-M*N/2)/M,H,'-');
  set(h,'BaseValue',-dr)
  ylim([-dr 6])
  set(gca,'YTick', [0 : -10 : -dr])
  set(findobj('Type','line'),'Marker','none','Color',[0 1 1])
  xlim([-M*N/2 M*N/2]/M)
  grid(gridded)
  set(findobj('Type','gridline'),'Color',[.871 .49 0])
  set(gca,'LineWidth',2)
  set(gca,'gridlinestyle','-')
  ylabel('decibels')
  xlabel('bins')
  title('Frequency response')
  set(gca,'Position',[0.59 0.11 0.4 0.8])
  set(gca,'XColor',[1 0 1])
  set(gca,'YColor',[1 0 1])

endfunction

function plotWindow (w, wname, wspecifier = "", wfilespecifier = "")

  if (strcmp (wfilespecifier, ""))
    wfilespecifier = wspecifier;
  endif

  N = size(w)(2);
  B = N*sum(w.^2)/sum(w)^2   % noise bandwidth (bins), set N = 4096 to get an accurate estimate
  
  plotWindowLayer(w, N, "on", wname, wspecifier);  % "gridded" = "on"
  print temp1.png -dpng "-S2500,1165"
  close
  plotWindowLayer(w, N, "off", wname, wspecifier);  % "gridded" = "off"
  print temp2.png -dpng "-S2500,1165"
  close
% I'm not sure what's going on here, but it looks like the author might have been able
% to save himself some time by using set(gca,"Layer","top") and set(gca,"Layer","bottom").
  I = imread ("temp1.png");
  J = imread ("temp2.png");
  info = imfinfo ("temp1.png");
  w = info.Width;
  c = 1-(double(I(:,1:w/2,1))+2*double(J(:,1:w/2,1)))/(255*3);
  m = 1-(double(I(:,1:w/2,2))+2*double(J(:,1:w/2,2)))/(255*3);
  y = 1-(double(I(:,1:w/2,3))+2*double(J(:,1:w/2,3)))/(255*3);
  c = ((c != m) | (c != y)).*(c &gt; 0).*(1-m-y);
  I(:,1:w/2,1) = 255*(1-c-m-y + 0*m + 0*y + 0*c);
  I(:,1:w/2,2) = 255*(1-c-m-y + 0*m + 0*y + 0.4*c);
  I(:,1:w/2,3) = 255*(1-c-m-y + 0*m + 0*y + 0.6*c);
  c = 1-(double(I(:,w/2+1:w,1))+2*double(J(:,w/2+1:w,1)))/(255*3);
  m = 1-(double(I(:,w/2+1:w,2))+2*double(J(:,w/2+1:w,2)))/(255*3);
  y = 1-(double(I(:,w/2+1:w,3))+2*double(J(:,w/2+1:w,3)))/(255*3);
  c = ((c != m) | (c != y)).*c;
  I(:,w/2+1:w,1) = 255*(1-c-m-y + 0*m + 0*y + 0.8710*c);
  I(:,w/2+1:w,2) = 255*(1-c-m-y + 0*m + 0*y + 0.49*c);
  I(:,w/2+1:w,3) = 255*(1-c-m-y + 0*m + 0*y + 0*c);
  if (strcmp (wfilespecifier, ""))
    imwrite (I, cstrcat('Window function and frequency response - ', wname, '.png'));
  else
    imwrite (I, cstrcat('Window function and frequency response - ', wname, ' (', wfilespecifier, ').png'));
  endif
  
endfunction

N=128;
k=0:N-1;

w = 0.42 - 0.5*cos(2*pi*k/(N-1)) + 0.08*cos(4*pi*k/(N-1));
plotWindow(w, "Blackman")

w = 0.355768 - 0.487396*cos(2*pi*k/(N-1)) + 0.144232*cos(4*pi*k/(N-1)) -0.012604*cos(6*pi*k/(N-1));
plotWindow(w, "Nuttall", "continuous first derivative")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.032*cos(8*pi*k/(N-1));
plotWindow(w, "Flat top")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.028*cos(8*pi*k/(N-1));
plotWindow(w, "SRS flat top")

w = ones(1,N);
plotWindow(w, "Rectangular")

w = (N/2 - abs([0:N-1]-(N-1)/2))/(N/2);
plotWindow(w, "Triangular")

w = 0.5 - 0.5*cos(2*pi*k/(N-1));
plotWindow(w, "Hann")

w = 0.53836 - 0.46164*cos(2*pi*k/(N-1));
plotWindow(w, "Hamming", "alpha = 0.53836")

alpha = 0.5;
w = ones(1,N);
n = -(N-1)/2 : -alpha*N/2;
L = length(n);
w(1:L) = 0.5*(1+cos(pi*(abs(n)-alpha*N/2)/((1-alpha)*N/2)));
w(N : -1 : N-L+1) = w(1:L);
plotWindow(w, "Tukey", "alpha = 0.5")

w = sin(pi*k/(N-1));
plotWindow(w, "Cosine")

w = sinc(2*k/(N-1)-1);
plotWindow(w, "Lanczos")

w = ((N-1)/2 - abs([0:N-1]-(N-1)/2))/((N-1)/2);
plotWindow(w, "Bartlett")

sigma = 0.4;
w = exp(-0.5*( (k-(N-1)/2)/(sigma*(N-1)/2) ).^2);
plotWindow(w, "Gaussian", "sigma = 0.4")

w = 0.62 -0.48*abs(k/(N-1) -0.5) +0.38*cos(2*pi*(k/(N-1) -0.5));
plotWindow(w, "Bartlett–Hann")

alpha = 2;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "alpha = 2")

alpha = 3;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "alpha = 3")

tau = N-1;
epsilon = 0.1;
t_cut = tau * (0.5 - epsilon);
T_in = abs(k - 0.5 * tau);
z_exp = ((t_cut - 0.5 * tau) ./ (T_in - t_cut) + (t_cut - 0.5 * tau) ./ (T_in - 0.5 * tau));
sigma =  (T_in &lt; 0.5 * tau) ./ (exp(z_exp) + 1);        
w = 1 * (T_in &lt;= t_cut) + sigma .* (T_in &gt; t_cut);
plotWindow(w, "Planck-taper", "epsilon = 0.1")

w = 0.35875 - 0.48829*cos(2*pi*k/(N-1)) + 0.14128*cos(4*pi*k/(N-1)) -0.01168*cos(6*pi*k/(N-1));
plotWindow(w, "Blackman-Harris")

w = 0.3635819 - 0.4891775*cos(2*pi*k/(N-1)) + 0.1365995*cos(4*pi*k/(N-1)) -0.0106411*cos(6*pi*k/(N-1));
plotWindow(w, "Blackman-Nuttall")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.032*cos(8*pi*k/(N-1));
plotWindow(w, "Flat top")

tau = (N/2);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "tau = N/2", "half window decay")

tau = (N/2)/(60/8.69);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "tau = (N/2)/(60/8.69)", "60dB decay")

alpha = 2;
w = 1/2*(1 - cos(2*pi*k/(N-1))).*exp(alpha*abs(N-2*k-1)/(1-N));
plotWindow(w, "Hann-Poisson", "alpha = 2")

Source code

Octave

Source code

function plotWindowLayer (w, N, gridded, wname, wspecifier)
 
  M=32;
  k=0:N-1;
  dr = 120;

  H = abs(fft([w zeros(1,(M-1)*N)]));
  H = fftshift(H);
  H = H/max(H);
  H = 20*log10(H);
  H = max(-dr,H);
 
  figure('Position',[1 1 1200 520])
  subplot(1,2,1)
  set(gca,'FontSize',28)
  area(k,w,'FaceColor', [0 1 1],'edgecolor', [1 1 0],'linewidth', 2)
  xlim([0 N-1])
  if (min(w) &gt;= -0.01)
    ylim([0 1.05])
    set(gca,'YTick', [0 : 0.1 : 1])
    ylabel('amplitude','position',[-16 0.525 0])
  else
    ylim([-1 5])
    set(gca,'YTick', [-1 : 1 : 5])
    ylabel('amplitude','position',[-16 2 0])
  endif
  set(gca,'XTick', [0 : 1/8 : 1]*(N-1))
  set(gca,'XTickLabel',[' 0'; ' '; ' '; ' '; ' '; ' '; ' '; ' '; 'N-1'])
  grid(gridded)
  set(gca,'LineWidth',2)
  set(gca,'gridlinestyle','-')
  xlabel('samples')
  if (strcmp (wspecifier, ""))
    title(cstrcat(wname,' window'))
  else
    title(cstrcat(wname,' window (', wspecifier, ')'))
  endif
  set(gca,'Position',[0.08 0.11 0.4 0.8])
  set(gca,'XColor',[1 0 1])
  set(gca,'YColor',[1 0 1])
  
  subplot(1,2,2)
  set(gca,'FontSize',28)
  h = stem(([1:M*N]-1-M*N/2)/M,H,'-');
  set(h,'BaseValue',-dr)
  ylim([-dr 6])
  set(gca,'YTick', [0 : -10 : -dr])
  set(findobj('Type','line'),'Marker','none','Color',[0 1 1])
  xlim([-M*N/2 M*N/2]/M)
  grid(gridded)
  set(findobj('Type','gridline'),'Color',[.871 .49 0])
  set(gca,'LineWidth',2)
  set(gca,'gridlinestyle','-')
  ylabel('decibels')
  xlabel('bins')
  title('Frequency response')
  set(gca,'Position',[0.59 0.11 0.4 0.8])
  set(gca,'XColor',[1 0 1])
  set(gca,'YColor',[1 0 1])

endfunction

function plotWindow (w, wname, wspecifier = "", wfilespecifier = "")

  if (strcmp (wfilespecifier, ""))
    wfilespecifier = wspecifier;
  endif

  N = size(w)(2);
  B = N*sum(w.^2)/sum(w)^2   % noise bandwidth (bins), set N = 4096 to get an accurate estimate
  
  plotWindowLayer(w, N, "on", wname, wspecifier);  % "gridded" = "on"
  print temp1.png -dpng "-S2500,1165"
  close
  plotWindowLayer(w, N, "off", wname, wspecifier);  % "gridded" = "off"
  print temp2.png -dpng "-S2500,1165"
  close
% I'm not sure what's going on here, but it looks like the author might have been able
% to save himself some time by using set(gca,"Layer","top") and set(gca,"Layer","bottom").
  I = imread ("temp1.png");
  J = imread ("temp2.png");
  info = imfinfo ("temp1.png");
  w = info.Width;
  c = 1-(double(I(:,1:w/2,1))+2*double(J(:,1:w/2,1)))/(255*3);
  m = 1-(double(I(:,1:w/2,2))+2*double(J(:,1:w/2,2)))/(255*3);
  y = 1-(double(I(:,1:w/2,3))+2*double(J(:,1:w/2,3)))/(255*3);
  c = ((c != m) | (c != y)).*(c &gt; 0).*(1-m-y);
  I(:,1:w/2,1) = 255*(1-c-m-y + 0*m + 0*y + 0*c);
  I(:,1:w/2,2) = 255*(1-c-m-y + 0*m + 0*y + 0.4*c);
  I(:,1:w/2,3) = 255*(1-c-m-y + 0*m + 0*y + 0.6*c);
  c = 1-(double(I(:,w/2+1:w,1))+2*double(J(:,w/2+1:w,1)))/(255*3);
  m = 1-(double(I(:,w/2+1:w,2))+2*double(J(:,w/2+1:w,2)))/(255*3);
  y = 1-(double(I(:,w/2+1:w,3))+2*double(J(:,w/2+1:w,3)))/(255*3);
  c = ((c != m) | (c != y)).*c;
  I(:,w/2+1:w,1) = 255*(1-c-m-y + 0*m + 0*y + 0.8710*c);
  I(:,w/2+1:w,2) = 255*(1-c-m-y + 0*m + 0*y + 0.49*c);
  I(:,w/2+1:w,3) = 255*(1-c-m-y + 0*m + 0*y + 0*c);
  if (strcmp (wfilespecifier, ""))
    imwrite (I, cstrcat('Window function and frequency response - ', wname, '.png'));
  else
    imwrite (I, cstrcat('Window function and frequency response - ', wname, ' (', wfilespecifier, ').png'));
  endif
  
endfunction

N=128;
k=0:N-1;

w = 0.42 - 0.5*cos(2*pi*k/(N-1)) + 0.08*cos(4*pi*k/(N-1));
plotWindow(w, "Blackman")

w = 0.355768 - 0.487396*cos(2*pi*k/(N-1)) + 0.144232*cos(4*pi*k/(N-1)) -0.012604*cos(6*pi*k/(N-1));
plotWindow(w, "Nuttall", "continuous first derivative")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.032*cos(8*pi*k/(N-1));
plotWindow(w, "Flat top")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.028*cos(8*pi*k/(N-1));
plotWindow(w, "SRS flat top")

w = ones(1,N);
plotWindow(w, "Rectangular")

w = (N/2 - abs([0:N-1]-(N-1)/2))/(N/2);
plotWindow(w, "Triangular")

w = 0.5 - 0.5*cos(2*pi*k/(N-1));
plotWindow(w, "Hann")

w = 0.53836 - 0.46164*cos(2*pi*k/(N-1));
plotWindow(w, "Hamming", "alpha = 0.53836")

alpha = 0.5;
w = ones(1,N);
n = -(N-1)/2 : -alpha*N/2;
L = length(n);
w(1:L) = 0.5*(1+cos(pi*(abs(n)-alpha*N/2)/((1-alpha)*N/2)));
w(N : -1 : N-L+1) = w(1:L);
plotWindow(w, "Tukey", "alpha = 0.5")

w = sin(pi*k/(N-1));
plotWindow(w, "Cosine")

w = sinc(2*k/(N-1)-1);
plotWindow(w, "Lanczos")

w = ((N-1)/2 - abs([0:N-1]-(N-1)/2))/((N-1)/2);
plotWindow(w, "Bartlett")

sigma = 0.4;
w = exp(-0.5*( (k-(N-1)/2)/(sigma*(N-1)/2) ).^2);
plotWindow(w, "Gaussian", "sigma = 0.4")

w = 0.62 -0.48*abs(k/(N-1) -0.5) +0.38*cos(2*pi*(k/(N-1) -0.5));
plotWindow(w, "Bartlett–Hann")

alpha = 2;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "alpha = 2")

alpha = 3;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "alpha = 3")

tau = N-1;
epsilon = 0.1;
t_cut = tau * (0.5 - epsilon);
T_in = abs(k - 0.5 * tau);
z_exp = ((t_cut - 0.5 * tau) ./ (T_in - t_cut) + (t_cut - 0.5 * tau) ./ (T_in - 0.5 * tau));
sigma =  (T_in &lt; 0.5 * tau) ./ (exp(z_exp) + 1);        
w = 1 * (T_in &lt;= t_cut) + sigma .* (T_in &gt; t_cut);
plotWindow(w, "Planck-taper", "epsilon = 0.1")

w = 0.35875 - 0.48829*cos(2*pi*k/(N-1)) + 0.14128*cos(4*pi*k/(N-1)) -0.01168*cos(6*pi*k/(N-1));
plotWindow(w, "Blackman-Harris")

w = 0.3635819 - 0.4891775*cos(2*pi*k/(N-1)) + 0.1365995*cos(4*pi*k/(N-1)) -0.0106411*cos(6*pi*k/(N-1));
plotWindow(w, "Blackman-Nuttall")

w = 1 - 1.93*cos(2*pi*k/(N-1)) + 1.29*cos(4*pi*k/(N-1)) -0.388*cos(6*pi*k/(N-1)) +0.032*cos(8*pi*k/(N-1));
plotWindow(w, "Flat top")

tau = (N/2);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "tau = N/2", "half window decay")

tau = (N/2)/(60/8.69);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "tau = (N/2)/(60/8.69)", "60dB decay")

alpha = 2;
w = 1/2*(1 - cos(2*pi*k/(N-1))).*exp(alpha*abs(N-2*k-1)/(1-N));
plotWindow(w, "Hann-Poisson", "alpha = 2")

File history

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

	Date/Time	Dimensions	User	Comment
current	16:48, 9 February 2013	2,500 × 1,123 (83 KB)	Olli Niemitalo (talk \| contribs)	Antialiasing, layout changes, larger font
	21:07, 17 December 2005	1,038 × 419 (7 KB)	Tiaguito~commonswiki (talk \| contribs)	file size. color source: http://en.wikipedia.org/wiki/Window_Function
	20:48, 17 December 2005	1,038 × 419 (8 KB)	Tiaguito~commonswiki (talk \| contribs)	source: http://en.wikipedia.org/wiki/Window_Function author: http://en.wikipedia.org/wiki/User:Bob_K

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