Image:Butterworth orders.png

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Butterworth filter frequency response of several orders. Cutoff frequency is normalized to 1 rad/s. Gain is normalized to 0 dB in the passband

[edit] Instructions

See Wikipedia graph-making tips.

Generated in gnuplot with the following script:

 set samples 1001
 set terminal postscript enhanced landscape color solid lw 2 "Times-Roman" 20
 set output "butterworth_orders.ps"

 G(w,n) = 1 / (sqrt(1 + w**(2*n)))

 dB(x) = 20 * log10(abs(x))

 set grid

 # Set x axis to logarithmic scale
 set logscale x 10

 # Set range of x and y axes
 set xrange [0.01:100]
 set yrange [-100:10]

 # Create 10 x tick marks per decade
 set xtics 10

 set xlabel "Angular frequency (rad/s)"
 set ylabel "Gain (dB)"

 # Put legend in bottom left corner
 set key 0.1,-45

 # Plot five orders of the filter
 plot \
     dB(G(x,1)) title "1st order", \
     dB(G(x,2)) title "2nd order", \
     dB(G(x,3)) title "3rd order", \
     dB(G(x,4)) title "4th order", \
     dB(G(x,5)) title "5th order"

To convert the PostScript file to PNG:

Open it in the GIMP (make sure you have ghostscript installed! - Windows instructions)
- Enter 500 under Resolution (it doesn't say "DPI" but I think that's what it means)
- Uncheck Try bounding box (since the bounding box cuts off the edge, unfortunately. You can try with the bounding box first.)
- Enter large values for Width and Height
- Check Color
- Check Strong anti-aliasing for both graphics and text
Crop off extra whitespace (Shift+C if you can't find it in the toolbox)
Possibly need to rotate it: Click Image → Transform → Rotate 90 degrees clockwise
Filters → Blur → Gaussian blur at 2.0 px (No need to blur if you use strong anti-aliasing during conversion. I see no significant difference between end results.)
Image → Scale Image...
- Width and Height at 25%
- Cubic interpolation
You can view at normal size if you want by pressing 1, Ctrl+E
Save as Butterworth orders.png

Source: Created by User:Omegatron using gnuplot, possibly with post-processing in the GIMP (PNG) or Inkscape (SVG)

I, the author of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".