Image:Butterworth response.png

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The frequency response of a Butterworth filter with logarithmic axes (Bode plot) and various labels. Cutoff frequency is normalized to 1 rad/s. Gain is normalized to 0 dB in the passband

Many orders on one plot: Image:Butterworth orders.png

Version with no text available at Image:Butterworth plain.png, though you should probably just modify the source code and regenerate it in your own language:

[edit] Instructions

See Wikipedia graph-making tips.

Generated in gnuplot with the following script (save as butterworth.plt and then open in gnuplot):

set samples 2001
set terminal postscript enhanced landscape color lw 2 "Times-Roman" 20
set output "butterworth.ps"

# Butterworth amplitude response and decibel calculation.  n is the order, which is just 1 in this image.
G(w,n) = 1 / (sqrt(1 + w**(2*n)))
dB(x) = 20 * log10(abs(x))

# Gridlines
set grid

# Set x axis to logarithmic scale
set logscale x 10

# Set range of x and y axes
set xrange [0.001:1000]
set yrange [-60:10]

# Create x-axis tic marks once per decade (every multiple of 10)
set xtics 10

# Use 10 x-axis minor divisions per major division
set mxtics 10

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

# No need for a key
set nokey #0.1,-25

# Frequency response's line plotting style
set style line 1 lt 1 lw 2

# Draw a separator between passband and stopband and label them
set style line 2 lt 2 lw 1 
set style arrow 2 nohead ls 2
set arrow 3 from 1,-60 to 1,10 as 2

# Label coordinates are relative to the graph window, not to the function, centered at the 1/4 and 3/4 width points
set label 1 "Passband" at graph 0.25, graph 0.1 c
set label 2 "Stopband" at graph 0.75, graph 0.1 c

# Asymptote lines and slope lines are the same "arrow" style
set style line 3 lt 3 lw 1
set style arrow 3 nohead ls 3

# Draw asymptote lines
set arrow 1 from 1,0 to 1000,-60 as 3
set arrow 2 from .001,0 to 1,0 as 3

# -3 dB arrow style and arrow
set style line 4 lt 4 lw 1
set style arrow 4 head filled size screen 0.02,15,45 ls 4
set arrow 4 from 2,3 to 1,0 as 4

# "Cutoff frequency" label uses same coordinates as the function
set label 3 "Cutoff frequency" at 2,4 l

# "-3 dB" label
set arrow 5 from 0.5,-6 to 1,-3 as 4   
set label 4 "-3.01 dB" at 0.5,-7 r

# Draw slope lines and label
set arrow 6 from 100,-20 to 12,-20 as 3
set arrow 7 from 100,-20 to 100,-39 as 3
set label 5 "Slope: -20 dB/decade" at 100,-18 c

# Plot the filter response
plot \
     dB(G(x,1)) ls 1  title "1st-order response"

Then I opened the butterworth.ps file in a text editor to edit the line colors and linestyles, as per this description. This avoids needing to open in proprietary software, and really isn't that difficult (especially if you don't know the commands in the proprietary software either). ;-) Identify the lines easily by their color (the arrow is currently magenta and I want it to be black. Ah, there is the entry with 1 0 1, red + blue = magenta) or by using the gnuplot linestyle−1. (For instance, gnuplot's linestyle 3 corresponds to the ps file's /LT2.) Then you can edit the colors and dashes by hand. I changed the original:

/LT0 { PL [] 1 0 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def
/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def

into this:

/LT0 { PL [] 0 0 1 DL } def
/LT1 { PL [4 dl 2 dl] 0.5 0.5 0.5 DL } def
/LT2 { PL [6 dl 3 dl] 0.3 0.3 1 DL } def
/LT3 { PL [] 0 0 0 DL } def

/LT4–/LT8 I left unchanged. (I don't know what they're used for anyway.) /LTw, /LTb, and /LTa are for the grid lines and such.

To convert the PostScript file to PNG:

1. 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
2. Crop off extra whitespace (Shift+C if you can't find it in the toolbox)
3. Possibly need to rotate it: Click Image → Transform → Rotate 90 degrees clockwise
4. 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.)
5. Image → Scale Image...
   • Width and Height at 25%
   • Cubic interpolation
6. You can view at normal size if you want by pressing 1, Ctrl+E
7. Save as Butterworth response.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". العربية | Česky | Deutsch | English | Español | Français | Italiano | 日本語 | 한국어 | Nederlands | Polski | Português | Slovenčina | Svenska | עברית +/- This file is licensed under the Creative Commons Attribution ShareAlike license versions 2.5, 2.0, and 1.0 You may select the license of your choice.