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Summary

Description

English: Comparison of three window functions in terms of their effects on equal-strength sinusoids with additive noise. The noise "floor" is smoothed by averaging many DFTs to reveal the substantial difference in levels, caused by the different window functions. In each case, the sinusoid on the left suffers no scalloping and the one on the right exhibits worst-case scalloping. The rectangular window produces the most scalloping but lowest noise-floor. The Hann window has a higher noise floor but much less potential scalloping, which results in the lowest "worst case processing loss" of these 3 functions.

Date

25 February 2013, 10:16:32

Source

Own work

Author

Bob K

Permission
(Reusing this file)

I, the copyright holder of this work, hereby publish it under the following license:

	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. CC0falsefalse

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Category:Self-published work

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GIF development

This GIF graphic was created with LibreOffice.

Category:PNG created with LibreOffice#Processing%20losses%20for%203%20window%20functions.gif

Octave/gnuplot source

click to expand

This graphic was created with the help of the following Octave script:

pkg load signal
graphics_toolkit gnuplot
clear all; close all; clc

  hfig= figure("position",[100 0 574 609]);
 
  N = 256;                % sams_per_fft;
  window1 = rectwin(N)'/sum(rectwin(N));
  window2 = hann(N)'/sum(hann(N));
  window3 = blackmanharris(N)'/sum(blackmanharris(N));
%
  Fs = N;                 % sams_per_sec;
  HzPerBin = Fs/N;
  F1 = -20.0*HzPerBin;    % bin -20
  F2 = +20.5*HzPerBin;    % bin 20.5
%
  L = 100000;
  n = 1:L;
  x = exp(j*2*pi*F1/Fs*n) + exp(j*2*pi*F2/Fs*n);
  x = x + (randn(1,L) +j*randn(1,L))*1.4;
%
  sams_per_offset = 0.75*N;    % overlap = 25%
%
% number of samples available beyond just one FFT
  excess_sams = length(x) - N;
%
  j1 = floor( excess_sams / sams_per_offset );
  sams_per_offset = floor( excess_sams / j1 );
  num_ffts = 1 + j1;
%
% define the first block of data
  samples = 1:N;
%
  amplitude1 = zeros(1,N);
  amplitude2 = zeros(1,N);
  amplitude3 = zeros(1,N);
%
% Loop over all the available blocks
  for j1 = 1 : num_ffts
    amplitude1 = amplitude1 + abs(fft( x(samples) .* window1 ));
    amplitude2 = amplitude2 + abs(fft( x(samples) .* window2 ));
    amplitude3 = amplitude3 + abs(fft( x(samples) .* window3 ));
    samples = samples + sams_per_offset;
  end
%
  amplitude1  = 20*log10(fftshift(amplitude1/num_ffts));
  amplitude2  = 20*log10(fftshift(amplitude2/num_ffts));
  amplitude3  = 20*log10(fftshift(amplitude3/num_ffts));
% 
  abscissa = -40:40;
  set(gca,'FontSize',8)
  subplot(3,1,1);
  h = area(abscissa, amplitude1(abscissa +N/2+1), ...
  'FaceColor', [.871 .49 0], 'edgecolor', [.871 .49 0]);
  set(h,'BaseValue',-20)
  set(gca, 'XTick', [-20 0 20.5], 'XTickLabel',[' '; ' '; ' '], 'YGrid','on', 'color', 'white')
  title('Processing losses for sinusoids in additive noise','fontsize', 12);
  xlim([-40 40])
  ylim([-20 2])
  ylabel('decibels')
  text(-40, 3.5, 'Rectangular window')
%
  subplot(3,1,2);
  h = area(abscissa, amplitude2(abscissa +N/2+1), ...
  'FaceColor', [.871 .49 0], 'edgecolor', [.871 .49 0]);
  set(h,'BaseValue',-20)
  set(gca, 'XTick', [-20 0 20.5], 'XTickLabel',[' '; ' '; ' '], 'YGrid','on', 'color', 'white')
  xlim([-40 40])
  ylim([-20 2])
  ylabel('decibels')
  text(-40, 3.5, 'Hann window')
%
  subplot(3,1,3);
  h = area(abscissa, amplitude3(abscissa +N/2+1), ...
  'FaceColor', [.871 .49 0], 'edgecolor', [.871 .49 0]);
  set(h,'BaseValue',-20)
  set(gca, 'XTick', [-20 0 20.5], 'XTickLabel',[' '; ' '; ' '], 'YGrid','on', 'color', 'white')
  xlim([-40 40])
  ylim([-20 2])
  ylabel('decibels')
  text(-40, 3.5, 'Blackman-Harris')

Category:Window function Category:Created with GNU Octave Category:Images with Octave source code Category:Images with Gnuplot source code