File:Fourier transform – Rectangular.svg

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Description
English: Fourier transform: Rectangular window
Italiano: Trasformata di Fourier: Finestra rettangolare
Date
Source Own work
Author Bob K (original version), Olli Niemitalo, BobQQ, Luca Ghio
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Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
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File:Window function and frequency response - Rectangular.svg

File:Window function – Rectangular.svg
SVG development
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[edit]

The script below generates these SVG images:

The script is not MATLAB-compatible. 

pkg load signal
 
graphics_toolkit gnuplot
set (0, "defaultaxesfontname", "sans-serif")
set (0, "defaultaxesfontsize", 12) 
set (0, "defaultaxeslinewidth", 1)
 
function plotWindow (w, wname, wfilename = "", wspecifier = "", wfilespecifier = "")
 
  M = 32; % Fourier transform size as multiple of window length
  Q = 512; % Number of samples in time domain plot
  P = 40; % Maximum bin index drawn
  dr = 130; % Maximum attenuation (dB) drawn in frequency domain plot
 
  N = length(w);
  B = N*sum(w.^2)/sum(w)^2 % noise bandwidth (bins)
 
  k = [0 : 1/Q : 1];
  w2 = interp1 ([0 : 1/(N-1) : 1], w, k);
 
  if (M/N < Q)
    Q = M/N;
  endif
 
  figure('position', [1 1 1200 600])
  subplot(1,2,1)
  area(k,w2,'FaceColor', [0 0.4 0.6], 'edgecolor', [0 0 0], 'linewidth', 1)
  if (min(w) >= -0.01)
    ylim([0 1.05])
    set(gca,'YTick', [0 : 0.1 : 1])
  else
    ylim([-1 5])
    set(gca,'YTick', [-1 : 1 : 5])
  endif
  ylabel('amplitude')
  set(gca,'XTick', [0 : 1/8 : 1])
  set(gca,'XTickLabel',[' 0'; ' '; ' '; ' '; ' '; ' '; ' '; ' '; 'N-1'])
  grid('on')
  set(gca,'gridlinestyle','-')
  xlabel('samples')
  if (strcmp (wspecifier, ""))
    title(cstrcat(wname,' window'), 'interpreter', 'none')
  else
    title(cstrcat(wname,' window (', wspecifier, ')'), 'interpreter', 'none')
  endif
  set(gca,'Position',[0.094 0.17 0.38 0.71])
 
  H = abs(fft([w zeros(1,(M-1)*N)]));
  H = fftshift(H);
  H = H/max(H);
  H = 20*log10(H);
  H = max(-dr,H);
  k = ([1:M*N]-1-M*N/2)/M;
  k2 = [-P : 1/M : P];
  H2 = interp1 (k, H, k2);
 
  subplot(1,2,2)
  set(gca,'FontSize',28)
  h = stem(k2,H2,'-');
  set(h,'BaseValue',-dr)
  xlim([-P P])
  ylim([-dr 6])
  set(gca,'YTick', [0 : -10 : -dr])
  set(findobj('Type','line'),'Marker','none','Color',[0.8710 0.49 0])
  grid('on')
  set(findobj('Type','gridline'),'Color',[.871 .49 0])
  set(gca,'gridlinestyle','-')
  ylabel('decibels')
  xlabel('bins')
  title('Fourier transform')
  set(gca,'Position',[0.595 0.17 0.385 0.71])
 
  if (strcmp (wfilename, ""))
    wfilename = wname;
  endif
  if (strcmp (wfilespecifier, ""))
    wfilespecifier = wspecifier;
  endif
  if (strcmp (wfilespecifier, ""))
    savetoname = cstrcat('Window function and frequency response - ', wfilename, '.svg');
  else
    savetoname = cstrcat('Window function and frequency response - ', wfilename, ' (', wfilespecifier, ').svg');
  endif
  print(savetoname, '-dsvg', '-S1200,600')
  close
 
endfunction
 
N=2^17; % Window length, B is equal for Triangular and Bartlett from 2^17
k=0:N-1;

s = 0.1*N;
w = exp(-(k - (N - 1)/2).^2./(4*s.^2)) - exp(-(-1/2 - (N - 1)/2).^2./(4*s.^2)).*(exp(-(k+N - (N - 1)/2).^2./(4*s.^2)) + exp(-(k-N - (N - 1)/2).^2./(4*s.^2)))./(exp(-(-1/2+N - (N - 1)/2).^2./(4*s.^2)) + exp(-(-1/2 - N - (N - 1)/2).^2./(4*s.^2)));
plotWindow(w, "App. conf. Gaussian", "Approximate confined Gaussian", '&#963;&#8348; = 0.1N', "sigma_t = 0.1N");

w = parzenwin(N).';
plotWindow(w, "Parzen");

w = 1-((k-(N-1)/2)/((N+1)/2)).^2;
plotWindow(w, "Welch");

alpha = 5; % Attenuation in 20 dB units
w = chebwin(N, alpha * 20).';
plotWindow(w, "Dolph&#8211;Chebyshev", "Dolph-Chebyshev", '&#945; = 5', "alpha = 5")
 
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&#8211;Harris", "Blackman-Harris")
 
w = 0.62 -0.48*abs(k/(N-1) -0.5) +0.38*cos(2*pi*(k/(N-1) -0.5));
plotWindow(w, "Bartlett&#8211;Hann", "Bartlett-Hann")
 
w = 0.53836 - 0.46164*cos(2*pi*k/(N-1));
plotWindow(w, "Hamming", "Hamming", '&#945; = 0.53836', "alpha = 0.53836")
 
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")
 
alpha = 2;
w = 1/2*(1 - cos(2*pi*k/(N-1))).*exp(alpha*abs(N-2*k-1)/(1-N));
plotWindow(w, "Hann&#8211;Poisson", "Hann-Poisson", '&#945; = 2', "alpha = 2")
 
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", "Nuttall", "continuous first derivative")
 
w = ones(1,N);
plotWindow(w, "Rectangular")
 
w = (N/2 - abs(k-(N-1)/2))/(N/2);
plotWindow(w, "Triangular")
 
w = 0.5 - 0.5*cos(2*pi*k/(N-1));
plotWindow(w, "Hann")
 
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", "Tukey", '&#945; = 0.5', "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", "Gaussian", '&#963; = 0.4', "sigma = 0.4")
 
alpha = 2;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "Kaiser", '&#945; = 2', "alpha = 2")
 
alpha = 3;
w = besseli(0,pi*alpha*sqrt(1-(2*k/(N-1) -1).^2))/besseli(0,pi*alpha);
plotWindow(w, "Kaiser", "Kaiser", '&#945; = 3', "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 < 0.5 * tau) ./ (exp(z_exp) + 1);        
w = 1 * (T_in <= t_cut) + sigma .* (T_in > t_cut);
plotWindow(w, "Planck-taper", "Planck-taper", '&#949; = 0.1', "epsilon = 0.1")
 
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&#8211;Nuttall", "Blackman-Nuttall")
 
tau = (N/2);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "Exponential", '&#964; = N/2', "half window decay")
 
tau = (N/2)/(60/8.69);
w = exp(-abs(k-(N-1)/2)/tau);
plotWindow(w, "Exponential", "Exponential", '&#964; = (N/2)/(60/8.69)', "60dB decay")

tau = N-1;
alpha = 4.45;
epsilon = 0.1;
t_cut = tau * (0.5 - epsilon);
t_in = k - 0.5 * tau;
T_in = abs(t_in);
z_exp = ((t_cut - 0.5 * tau) ./ (T_in - t_cut) + (t_cut - 0.5 * tau) ./ (T_in - 0.5 * tau));
sigma =  (T_in < 0.5 * tau) ./ (exp(z_exp) + 1);
w = (1 * (T_in <= t_cut) + sigma .* (T_in > t_cut)) .* besseli(0, pi*alpha * sqrt(1 - (2 * t_in / tau).^2)) / besseli(0, pi*alpha);
plotWindow(w, "Planck&#8211;Bessel", "Planck-Bessel", "&#949; = 0.1, &#945; = 4.45", "epsilon = 0.1, alpha = 4.45")

N = 4097;
k=0:N-1;
alpha = 2;
s = sin(alpha*2*pi/N*[1:N-1])./[1:N-1];
c0 = [alpha*2*pi/N,s];
A = toeplitz(c0);
[V,evals] = eigs(A, 1);
[emax,imax] = max(abs(diag(evals)));
w = abs(V(:,imax));
w = w.';
w = w / max(w);
plotWindow(w, "DPSS", "DPSS", '&#945; = 2', "alpha = 2")

N = 4097;
k=0:N-1;
alpha = 3;
s = sin(alpha*2*pi/N*[1:N-1])./[1:N-1];
c0 = [alpha*2*pi/N,s];
A = toeplitz(c0);
[V,evals] = eigs(A, 1);
[emax,imax] = max(abs(diag(evals)));
w = abs(V(:,imax));
w = w.';
w = w / max(w);
plotWindow(w, "DPSS", "DPSS", '&#945; = 3', "alpha = 3")

global N T P abar target_stnorm;
N = 2049; % This is will take a lot of time with N = 2049
k = 0:N-1;
target_stnorm = 0.1;
function [g,sigma_w,sigma_t] = CGWn(alpha, n)
  % determine eigenvectors of M(alpha)
  global N P T
  opts.maxit = 10000;
  if(n ~= N)
    [g,lambda] = eigs(P + alpha*T, n, 'sa', opts);
  else
    [g,lambda] = eig(P + alpha*T);
  end
  sigma_t = sqrt(diag((g'*T*g) / (g'*g)));
  sigma_w = sqrt(diag((g'*P*g) / (g'*g)));
end
function [h1] = helperCGW(anorm)
  global N abar target_stnorm
  [~,~,sigma_t] = CGWn(anorm*abar,1);
  h1 = sigma_t - target_stnorm * N;
end
% define alphabar, and matrices T and P
T = zeros(N,N);
P = zeros(N,N);
for k=1:N
  T(k,k) = (k - (N+1)/2)^2;
  for l=1:N
    if k ~= l
      P(k,l) = 2*(-1)^(k-l)/(k-l)^2;
    else
      P(k,l) = pi^2/3;
    end
  end
end
abar = (10/N)^4/4
[anorm, aval] = fzero(@helperCGW, 0.1/target_stnorm);
[CGWg, CGWsigma_w, CGWsigma_t] = CGWn(anorm*abar,1);
w = CGWg * sign(mean(CGWg));
w = w'/max(w);
plotWindow(w, "Confined Gaussian", "Confined Gaussian", '&#963;&#8348; = 0.1N', "sigma_t = 0.1N");

The generated SVG files should be post-processed by the following Perl script, to change the SVG metadata title from "Gnuplot" to the body of the file name, and to unobfuscate Unicode character codes (for Greek symbols and en dash). The script will scan the current directory for .svg files and makes the necessary changes if they have not already been made. 

#!/usr/bin/perl

opendir (DIR, '.') or die $!;
while ($file = readdir(DIR)) {
  $ext = substr($file, length($file)-4);
  if ($ext eq '.svg') {
    print("$file\n");
    ($pre, $name) = split(" - ", substr($file, 0, length($file)-4));
    @lines = ();
    open (INPUTFILE, "<", $file) or die $!;
    while ($line = <INPUTFILE>) {
      $line =~ s/&amp;/&/g; 
      if ($line eq "<title>Gnuplot</title>\n") {
        $line = '<title>Window function and its Fourier transform &#8211; '.$name."</title>"."\n";
      }
      @lines[0+@lines] = $line;
    }
    close(INPUTFILE);
    open (OUTPUTFILE, ">", $file) or die $!;  
    for ($t = 0; $t < @lines; $t++) {
      print(OUTPUTFILE $lines[$t]);
    }
    close(OUTPUTFILE);
  }
}
closedir (DIR);

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current21:50, 11 June 2016Thumbnail for version as of 21:50, 11 June 2016282 × 270 (79 KB)Luca Ghio (talk | contribs){{Information |Description={{en|Fourier transform: Rectangular window}} {{it|Trasformata di Fourier: Finestra rettangolare}} |Source={{own}} |Date={{subst:CURRENTYEAR}}-{{subst:CURRENTMONTH}}-{{subst:CURRENTDAY2}} |Author=Bob K (original...

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