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Butterworth Filter Coefficients

Module by: Matthew Moravec. E-mail the author

Based on: Butterworth Filters by Richard Baraniuk

Summary: A table of Butterworth filter coefficients.

Butterworth Filter Coefficients

The transfer function of a Butterworth filter is:

H⁢s=1sn+ a n-1 ⁢sn−1+ ⋯ +a1⁢s+1 H s 1 s n a n-1 s n 1 ⋯ a1 s 1 (1)
From this, we may look up the ai ai from a table (like the one below) for any desired filter order nn. We can also find them in Matlab by using the buttap command.

Table 1
n a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 a7a7 a8a8 a9a9
2 1.414214
3 2.000000 2.000000
4 2.613126 3.414214 2.613126
5 3.236068 5.236068 5.236068 3.236068
6 3.863703 7.464102 9.141620 7.464102 3.863703
7 4.493959 10.097835 14.591794 14.591794 10.097835 4.493959
8 5.125831 13.137071 21.846151 25.688356 21.846151 13.137071 5.125831
9 5.758770 16.581719 31.163437 41.986386 41.986386 31.163437 16.581719 5.758770
10 6.392453 20.431729 42.802061 64.882396 74.233429 64.882396 42.802061 20.431729 6.392453

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