File:Spectral leakage caused by "windowing".svg
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Size of this PNG preview of this SVG file: 800 × 492 pixels. Other resolutions: 320 × 197 pixels | 640 × 393 pixels | 1,024 × 629 pixels | 1,280 × 787 pixels | 2,560 × 1,573 pixels | 1,152 × 708 pixels.
Original file (SVG file, nominally 1,152 × 708 pixels, file size: 146 KB)
Summary
| Description |
English: The purpose of this image is to show that windowing a sinusoid causes spectral leakage, even if the sinusoid has an integer number of cycles within a rectangular window. The leakage is evident in the 2nd row, blue trace. It is the same amount as the red trace, which represents a slightly higher frequency that does not have an integer number of cycles. When the sinusoid is sampled and windowed, its discrete-time Fourier transform also suffers from the same leakage pattern. But when the DTFT is only sampled, at a certain interval, it is possible (depending on your point of view) to: (1) avoid the leakage, or (2) create the illusion of no leakage. For the case of the blue sinusoid (3rd row of plots, right-hand side), those samples are the outputs of the discrete Fourier transform (DFT). The red sinusoid DTFT (4th row) has the same interval of zero-crossings, but the DFT samples fall in-between them, and the leakage is revealed. |
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| Date | ||||
| Source | Own work | |||
| Author | Bob K | |||
| Other versions | Spectral leakage from a sinusoid and rectangular window.png | |||
| SVG development |
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Source code
This media was created with GNU Octave (numerical computation software)
Here is a listing of the source used to create this file.
Here is a listing of the source used to create this file.
pkg load signal
% Options
frame_background_gray = true;
if frame_background_gray
graphics_toolkit("qt") % or graphics_toolkit("fltk")
frame_background = .94*[1 1 1];
d = 4; % amount to add to text sizes
ds = 8; % amount to small marker size
dl = 12; % amount to large marker size
else
graphics_toolkit("gnuplot") % background will be white regardless of value below
frame_background = .94*[1 1 1];
d=0; ds = 0;; dl = 0
endif
% (https://octave.org/doc/v4.2.1/Graphics-Object-Properties.html#Graphics-Object-Properties)
% Speed things up when using Gnuplot
set(0, "DefaultAxesFontsize",10+d)
set(0, "DefaultTextFontsize",12+d)
set(0, "DefaultAxesYlim",[-2 2])
set(0, "DefaultAxesYtick",[])
set(0, "DefaultAxesXtick",[])
set(0, "DefaultFigureColor",frame_background)
set(0, "DefaultAxesColor","white")
%=======================================================
samples_per_DFT = 64;
DFT_display_bins = samples_per_DFT/2;
samples_per_DTFT = 1024;
Hz_per_bin = 1; % Set the DFT bin spacing to 1 Hz
samples_per_sec = Hz_per_bin*samples_per_DFT; % corresponding sample rate
I = 8; % over-sample the display version of sinusoids
cycles_per_DFT = 13;
cycles_per_sample = cycles_per_DFT/(I*samples_per_DFT);
signal1I = sin(2*pi*cycles_per_sample*(-32*I:96*I)); % pad the [0,64] window with ±32
window1 = signal1I(32*I + (1:samples_per_DFT*I+1)); % extract the window
signal1 = window1(1:I:samples_per_DFT*I); % decimate the over-sample rate
cycles_per_DFT = 13.50; % repeat steps for higher frequency
cycles_per_sample = cycles_per_DFT/(I*samples_per_DFT);
signal2I = sin(2*pi*cycles_per_sample*(-32*I:96*I));
window2 = signal2I(32*I + (1:samples_per_DFT*I+1));
signal2 = window2(1:I:samples_per_DFT*I);
filter = zeros(1,length(signal1I)); % depict a rectangular window
skirt = 10;
filter(32*I + (-skirt+1:samples_per_DFT*I+skirt)) = 1.2;
%=======================================================
hfig = figure("position",[1 -89 1200 800], "color",frame_background);
x1 = .08; % left margin for annotation
x2 = .02; % right margin
ws = .05; % whitespace between plots
y1 = .08 % bottom margin
y2 = .08 % top margin
dy = .08; % vertical space between rows
height = (1-y1-y2-3*dy)/4; % space allocated for each of 4 rows
% Compute width of sinusoid plots
xwhite = x1 + ws + ws + x2; % total whitespace in row containing 3 plots
width1 = (1-xwhite)/3; % width of all sinusoid plots
y_origin = 1; % start at top of graph area
%=======================================================
% Plot the unwindowed sinusoid
x_origin = x1;
y_origin = y_origin -y2 -height; % position of top row
subplot("position",[x_origin y_origin width1 height])
plot((-32*I:96*I), signal1I, "color","black")
xlim([-32 96]*I) %ylim([-2 2])
set(gca, "xaxislocation","origin")
xlabel("unwindowed")
%=======================================================
% Plot the 13-cycle sinusoid & rectangular window
x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width1 height])
plot(0:length(window1)-1, window1, "color","blue")
xlim([-32 96]*I) %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
plot((-32*I:96*I), filter, "color","black", "linewidth",2)
xlabel("13 cycles")
%=======================================================
% Plot the 13½-cycle sinusoid & rectangular window
x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width1 height])
plot(0:length(window2)-1, window2, "color","red")
xlim([-32 96]*I) %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
plot((-32*I:96*I), filter, "color","black", "linewidth",2)
%xlabel("13½ cycles") % doesn't work
%=======================================================
% Compute and plot Fourier transforms of the 2 sinusoids
x_origin = x1;
y_origin = y_origin -dy -height;
width2 = 1 -x1 -x2;
subplot("position",[x_origin y_origin width2 height])
N = samples_per_DTFT;
Hz_per_bin = samples_per_sec/N;
S1 = abs(fft(signal1,N));
S1 = 20*log10(S1(1:N/2));
S1 = max(0,S1);
plot((0:length(S1)-1)*Hz_per_bin, S1, "color","blue", "linewidth",1);
ylim([0 max(S1)+6])
xlim([0 samples_per_sec/2])
set(gca, "xtick",0:DFT_display_bins)
% set(gca, "ytick",0:10:ylim(2)) % no, no, no. It redefines ylim.
set(gca, "ytick",0:10:(max(S1)+6))
hold on
S2 = abs(fft(signal2,N));
S2 = 20*log10(S2(1:N/2));
S2 = max(0,S2);
plot((0:length(S2)-1)*Hz_per_bin, S2, "color","red", "linewidth",1);
% Insert delta function for unwindowed transform
stem(13, 35, "^", "MarkerSize",5+ds, "linewidth",2, "color","black", "markeredgecolor","black", "markerfacecolor","black")
ylabel("decibels")
xlabel('\leftarrow frequency \rightarrow')
%=======================================================
% Replot 13-cycle sinusoid
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width1 height])
plot(0:length(window1)-1, window1, "color","blue")
xlim([0 length(window1)-1]) %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
% Overlay continuous sinusoid with discrete samples
plot(0:I:samples_per_DFT*I-1, signal1, "color","blue", ".", "MarkerSize",5+ds)
xlabel("discrete-time (sampled)")
%=======================================================
% Re-plot the Fourier transform, but truncate it to fit a smaller space
x_origin = x_origin + width1 + ws;
width3 = 1 - x_origin -x2;
subplot("position",[x_origin y_origin width3 height])
Hz_per_bin = samples_per_sec/samples_per_DTFT;
% DFT_display_bins = 32. Truncate plot to 22 bins:
N = 22.5/32*length(S1);
plot((0:N-1)*Hz_per_bin, S1(1:N), "color","blue", "linewidth",1);
ylim([0 max(S1)+6])
xlim([0 N-1]*Hz_per_bin);
set(gca, "xtick",0:22)
hold on
% Compute and overlay the discrete DFT values
N = samples_per_DFT;
Hz_per_bin = samples_per_sec/N;
S = abs(fft(signal1,N));
S = 20*log10(S(1:N/2));
S = max(0,S);
N = 23;
plot((0:N-1)*Hz_per_bin, S(1:N), "color","blue", ".", "MarkerSize",10+dl);
set(gca, "xtick",(0:N-1)*Hz_per_bin)
%=======================================================
% Replot 13½-cycle sinusoid
x_origin = x1;
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width1 height])
plot(0:length(window2)-1, window2, "color","red")
xlim([0 length(window2)-1]) %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
% Overlay continuous sinusoid with discrete samples
plot(0:I:samples_per_DFT*I-1, signal2, "color","red", ".", "MarkerSize",5+ds)
xlabel("discrete-time (sampled)")
%=======================================================
% Re-plot the Fourier transform, but truncate it to fit a smaller space
x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width3 height])
Hz_per_bin = samples_per_sec/samples_per_DTFT;
% DFT_display_bins = 32. Truncate plot to 22 bins:
N = 22.5/32*length(S2);
plot((0:N-1)*Hz_per_bin, S2(1:N), "color","red", "linewidth",1);
ylim([0 max(S2)+6])
xlim([0 (N-1)*Hz_per_bin])
set(gca, "xtick",0:22)
hold on
% Compute and overlay the discrete DFT values
N = samples_per_DFT;
Hz_per_bin = samples_per_sec/N;
S = abs(fft(signal2,N));
S = 20*log10(S(1:N/2));
S = max(0,S);
N = 23;
plot((0:N-1)*Hz_per_bin, S(1:N), "color","red", ".", "MarkerSize",10+dl);
set(gca, "xtick",(0:N-1)*Hz_per_bin)
xlabel("DFT bins")
Licensing
Bob K, the copyright holder of this work, hereby publishes it under the following license:
 
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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.
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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 | 18:17, 28 December 2025 | | 1,152 × 708 (146 KB) | wikimediacommons>Sebastian Wallroth | svgomg // Editing SVG source code using c:User:Rillke/SVGedit.js |
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