File:Hilbert transform.svg

 

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This image is a derivative work of the following images:

• File:Hilbert_transform.png licensed with Cc-by-sa-3.0,2.5,2.0,1.0, GFDL
  • 2005-07-02T18:54:12Z Omegatron 1600x1120 (123072 Bytes) Image of the Hilbert transform of a square wave.
  • 2005-06-26T19:53:50Z Omegatron 608x433 (4690 Bytes) A graph or diagram made by [[User:Omegatron]]. (Uploaded with Wikimedia Commons.) Source: Created by [[User:Omegatron]] {{GFDL}}{{cc-by-sa-2.0}} [[Category:Diagrams]]\

Created entirely in GNUPlot using:

set term svg size 1600,1100 fsize 32
set out "Hilbert_transform.svg"

unset key
set samples 1000

set xrange [0:6]
set yrange [-1:1]
set xtics 1
set ytics 1

set grid lw 2 lc rgb "grey"

# set style of Hilbert transform
set style line 1 lw 2 lc rgb "red" 
# set style of rectangular function
set style line 2 lw 2 lc rgb "blue"

set border -1 lw 3

# Heaviside step function
u(t) = 0.5 * (1 + sgn(t))

# Rectangular function
rect(t) = u(t + 0.5) - u(t - 0.5)

# Hilbert transform of rectangular function
hrect(t) = log(abs((t + 0.5)/(t - 0.5)))/pi

# Sums from a to b of the above functions
hrectsum(t,a,b) = (a >= b) ? 0 : (hrectsum(t,a+1,b) + hrect(t - 2*a))
rectsum(t,a,b) = (a >= b) ? 0 : (rectsum(t,a+1,b) + rect(t - 2*a))

plot hsum(x,-500,500) ls 1, rectsum(x,-10,20)-0.5 ls 2

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