File:Aliasing between a positive and a negative frequency.png

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Aliasing_between_a_positive_and_a_negative_frequency.png ‎(694 × 452 pixels, file size: 44 KB, MIME type: image/png)

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Summary

Description
English: This figure depicts two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points. They are thus aliases of each other when sampled at the rate (fs) indicated by the grid lines. The gold-colored function depicts a positive frequency, because its real part (the cos function) leads its imaginary part by 1/4 of one cycle. The cyan function depicts a negative frequency, because its real part lags the imaginary part.
Date
Source Own work
Author Bob K
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Creative Commons CC-Zero 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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This graphic was created with the help of the following Octave script: 

graphics_toolkit gnuplot
gold = [251 159 3]/256;         % arbitrary color choice
sam_per_sec = 1;
T = 1/sam_per_sec;              % sample interval
dt = T/20;                      % time-resolution of continuous functions
cycle_per_sec = sam_per_sec/8;  % sam_per_sec = 8 * cycle_per_sec (satisfies Nyquist)

figure
subplot(3,1,1)
xlim([-2 10])
ylim([-1.3 1.3])

% Plot cosine function
start_time_sec = -2;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = cos(2*pi*cycle_per_sec*x);
plot(x, y, "color", gold, "linewidth", 4)
box off           % no border around plot please
hold on           % same axes for next 3 plots

% Plot sine function
start_time_sec = 0;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = sin(2*pi*cycle_per_sec*x);
plot(x, y, "color", gold, "linewidth", 4)

% Sample cosine function at sample-rate (1/T)
start_time_sec = 0;
stop_time_sec = 8;
x = start_time_sec : T : stop_time_sec;
y = cos(2*pi*cycle_per_sec*x);
plot(x, y, "color", "black", ".")

% Sample sine function
y = sin(2*pi*cycle_per_sec*x);
plot(x, y, "color", "black", ".")

set(gca, "xaxislocation", "origin")
set(gca, "yaxislocation", "origin")
set(gca, "xgrid", "on");
set(gca, "ygrid", "off");
set(gca, "ytick", [0]);
set(gca, "xtick", [0:8]);

subplot(3,1,2)
xlim([-2 10])
ylim([-1.3 1.3])
cycle_per_sec2 = cycle_per_sec - sam_per_sec;   % negative frequency

% Re-plot same cosine function on new axes
start_time_sec = -2;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = cos(2*pi*cycle_per_sec*x);
plot(x, y, "color", gold, "linewidth", 4)
box off
hold on

% Plot other cosine function
start_time_sec = -2;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = cos(2*pi*cycle_per_sec2*x);
plot(x, y, "color", "cyan", "linewidth", 4)

% Sample cosine functions at sample-rate (1/T)
start_time_sec = 0;
stop_time_sec = 8;
x = start_time_sec : T : stop_time_sec;
y = cos(2*pi*cycle_per_sec*x);
plot(x, y, "color", "black", ".")

set(gca, "xaxislocation", "origin")
set(gca, "yaxislocation", "origin")
set(gca, "xgrid", "on");
set(gca, "ygrid", "off");
set(gca, "ytick", [0]);
set(gca, "xtick", [0:8]);

subplot(3,1,3)
xlim([-2 10])
ylim([-1.3 1.3])

% Re-plot original sine function on new axes
start_time_sec = 0;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = sin(2*pi*cycle_per_sec*x);
plot(x, y, "color", gold, "linewidth", 4)
box off
hold on

% Plot other sine function
start_time_sec = -2;
stop_time_sec = 10;
x = start_time_sec : dt : stop_time_sec;
y = sin(2*pi*cycle_per_sec2*x);
plot(x, y, "color", "cyan", "linewidth", 4)

% Sample sine functions at sample-rate (1/T)
start_time_sec = 0;
stop_time_sec = 8;
x = start_time_sec : T : stop_time_sec;
y = sin(2*pi*cycle_per_sec*x);
plot(x, y, "color", "black", ".")

set(gca, "xaxislocation", "origin")
set(gca, "yaxislocation", "origin")
set(gca, "xgrid", "on");
set(gca, "ygrid", "off");
set(gca, "ytick", [0]);
set(gca, "xtick", [0:8]);

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inception

27 March 2013

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image/png

checksum

176af466b89d1b29d7b8c6750686d08053d33e6f

determination method: SHA-1

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height

452 pixel

width

694 pixel

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current07:26, 27 March 2013Thumbnail for version as of 07:26, 27 March 2013694 × 452 (44 KB)wikimediacommons>Bob KUser created page with UploadWizard

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