Summary

Related Calculations

${\displaystyle {\hat {f}}(\xi )=\int _{-{1 \over 2}}^{1 \over 2}e^{-2\pi ix\xi }dx={\frac {i}{2\pi \xi }}(e^{-\pi i\xi }-e^{\pi i\xi })={\frac {\sin \pi \xi }{\pi \xi }}.}$

${\displaystyle {\hat {g}}(\xi )=\int _{0}^{1}e^{-2\pi ix\xi }dx={\frac {i}{2\pi \xi }}(e^{-2\pi i\xi }-1)={\frac {1}{2\pi \xi }}\sin 2\pi \xi +i{\frac {1}{2\pi \xi }}(\cos 2\pi \xi -1).}$

Gnuplot Source Code

# Author: Slawekb, Ika, 2015-08-15
# Slawekb gave the analogous plot using Mathematica: https://en.wikipedia.org/wiki/File:Fourier_transform_of_rect_and_a_translation.png
# Anyone can copy and revise this code without any copyright concern.

set term svg size 800,600 enhanced font 'Times, 16'
set output 'fourier_unit_pulse.svg'

unset tics
unset border
unset colorbox
set sample 1000
set multiplot

set arrow from -3,0 to 3,0 nohead lw 2
set arrow from 0,-0.3 to 0,1.05 nohead lw 2

# Figure (a)
set size 0.5, 0.5
set origin 0.0, 0.5

unset key 
set yrange [-0.5:1.5]
plot [-3:3] -0.5 <= x && x <= 0.5 ? 1 : 0 w lines lw 2 lc rgb 'blue' t 'f'
unset yrange

# Figure (b)
set size 0.5, 0.5
set origin 0.5, 0.5

set key font "Times, 22"

plot [-3:3] (sin(pi*x))/(x*pi) w lines lw 2 t 'Re', 0 w lines t 'Im' lw 2
			

# Figure (c)
set size 0.5, 0.5
set origin 0.0, 0.0

unset key 
set yrange [-0.5:1.5]

plot [-3:3] 0.0 <= x && x <= 1.0 ? 1 : 0 w lines lw 2 lc rgb 'blue' t 'g'
unset yrange

# Figure (d)
set arrow from 0,-0.8 to 0,-0.2 nohead lw 2
set size 0.5, 0.5
set origin 0.5, 0.0

set key font "Times, 22"

plot [-3:3] (sin(2*pi*x))/(x*2*pi) w lines lw 2 t 'Re', (cos(2*pi*x)-1)/(2*pi*x) w lines t 'Im' lw 2

unset multiplot
set term windows
set output

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