Fil:Window function (rectangular).png

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Beskrivelse

rectangular window and frequency response

Dato

17. december 2005

Kilde

Eget arbejde

Forfatter

Bob K (original version), Olli Niemitalo

Tilladelse
(Genbrug af denne fil)

Jeg, indehaveren af ophavsretten til dette værk, udgiver dette værk som offentlig ejendom. Dette gælder i hele verden.
I nogle lande er dette ikke juridisk muligt. I så fald:
Jeg giver enhver ret til at anvende dette værk til ethvert formål, uden nogen restriktioner, medmindre sådanne restriktioner er påkrævede ved lov.

Andre versioner

En vektorversion af dette billede (SVG) er tilgængelig. Det bør bruges i stedet for punktgrafikbilledet når det er fordelagtigt.

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

For mere information om vektorgrafik: Læs om overgangen til SVG på Commons.
Se også information om MediaWikis understøttelse af SVG-billeder.

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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.

N

PNG Rastergrafik blev lavet med GNU Octave af 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")

Kildekode

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")

Filhistorik

Klik på en dato/tid for at se filen som den så ud på det tidspunkt.

	Dato/tid	Dimensioner	Bruger	Kommentar
nuværende	9. feb. 2013, 17:48	2.500 × 1.123 (83 KB)	Olli Niemitalo	Antialiasing, layout changes, larger font
	17. dec. 2005, 22:07	1.038 × 419 (7 KB)	Tiaguito~commonswiki	file size. color source: http://en.wikipedia.org/wiki/Window_Function
	17. dec. 2005, 21:48	1.038 × 419 (8 KB)	Tiaguito~commonswiki	source: http://en.wikipedia.org/wiki/Window_Function author: http://en.wikipedia.org/wiki/User:Bob_K

Filanvendelse

Den følgende side bruger denne fil:

Bruger:Peh-herlev/sandkasse-Vinduesfunktion

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Anvendelser på et.wikipedia.org
- Kasutaja:U.schults/Aknafunktsioon

Fil:Window function (rectangular).png

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copyrighted, dedicated to the public domain by copyright holder ^engelsk

licens

released into the public domain by the copyright holder ^engelsk

etablerings- / skabelsesdato

17. december 2005

kilde til fil

eget værk af uploader

medietype

image/png

Filhistorik

Filanvendelse

Global filanvendelse

Beskrivelse

Matlab

Octave

Captions

Elementer som er med i denne fil

Denne egenskab har en værdi, men den er ukendt

copyrighted, dedicated to the public domain by copyright holder engelsk

released into the public domain by the copyright holder engelsk

17. december 2005

image/png

Filhistorik

Filanvendelse

Global filanvendelse

copyrighted, dedicated to the public domain by copyright holder ^engelsk

released into the public domain by the copyright holder ^engelsk