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Original file (SVG file, nominally 531 × 540 pixels, file size: 63 KB)
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Summary
| Description |
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. |
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| 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:
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| Other versions |
This file was derived from: Processing losses for 3 window functions.gif |
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| SVG development |  This W3C-invalid vector image was created with GNU Octave. |
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| Octave/gnuplot source | click to expand
This graphic was created by the following Octave script: pkg load signal
% Options
frame_background_gray = true;
if frame_background_gray
graphics_toolkit("qt") % has "insert text" option
% graphics_toolkit("fltk") % has cursor coordinate readout
frame_background = .94*[1 1 1];
d = 4; % amount to add to text sizes
else
graphics_toolkit("gnuplot") % background will be white regardless of value below
frame_background = .94*[1 1 1];
d=0;
endif
% (https://octave.org/doc/v4.2.1/Graphics-Object-Properties.html#Graphics-Object-Properties)
% Speed things up when using Gnuplot
set(0, "DefaultFigureColor",frame_background)
set(0, "DefaultAxesFontsize",10+d) % size of numeric tick labels
set(0, "DefaultLineLinewidth",1)
set(0, "DefaultTextFontsize",10+d)
hfig= figure("position",[100 50 574 609]);
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")
xlim([-40 40])
ylim([-20 2])
ylabel("decibels")
text(-40, 3.5, "Rectangular window")
text(-12.0, -8.1, "X dB")
text(3.6, -8.1, "X - 3.92 dB")
text(-28.7, -22.7, "No scallop loss")
text(9.8, -22.7, "Maximum scallop loss")
title({"Processing losses for sinusoids in additive noise"; ""; ""}, "fontsize",12+d);
%
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")
text(-12.4, -8.1, "X - 1.76 dB")
text(3.6, -8.1, "X - 3.18 dB")
text(-28.7, -22.7, "No scallop loss")
text(20, -22.7, 'frequency \rightarrow')
%
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")
text(-12.0, -8.1, "X - 3.01 dB")
text(3.6, -8.1, "X - 3.85 dB")
text(-28.7, -22.7, "No scallop loss")
text(9.8, -22.7, "Maximum scallop loss")
% After this call, the cursor units change to a normalized ([0,1]) coordinate system, spanning all plots
annotation("line", [.334 .5], [.906 .906])
annotation("line", [.52 .7], [.87 .87])
annotation("doublearrow", [.388 .388], [.906 .727],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
annotation("doublearrow", [.545 .545], [.87 .727],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
annotation("line", [.334 .5], [.602 .602])
annotation("line", [.52 .7], [.589 .589])
annotation("doublearrow", [.388 .388], [.602 .438],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
annotation("doublearrow", [.545 .545], [.589 .438],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
annotation("line", [.334 .5], [.3 .3])
annotation("line", [.52 .7], [.292 .292])
annotation("doublearrow", [.388 .388], [.3 .147],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
annotation("doublearrow", [.545 .545], [.292 .147],...
"head1style","vback1", "head2style","vback1",...
"head1length",5, "head1width",5, "head2length",5, "head2width",5)
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File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 21:07, 28 January 2020 | | 531 × 540 (63 KB) | Bob K | change frame background from white to gray |
| 02:34, 26 March 2019 |  | 718 × 761 (126 KB) | Bob K | version created by export from figure (instead of print function) is more true to programmed dimensions | |
| 20:08, 25 March 2019 |  | 512 × 384 (66 KB) | Bob K | move an x-axis label to the right hand side | |
| 19:55, 25 March 2019 |  | 512 × 384 (66 KB) | Bob K | User created page with UploadWizard |
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