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)
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