Designing a Butterworth filter is a trivial task. Since we know that the filter contains only poles, we know that we can write it as
buttap
command. The
real challenge of designing a Butterworth filter comes with
figuring out the optimal characteristics for the given
application.
n 










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 
Exercise 1
Design a Butterworth filter with a passband gain between 1
and 0.891 (1 dB gain) for
Solution
The first step is to determine
The next step is to find
Now, we can find the normalized transfer function. Since we know this to be a sixthorder Butterworth, we can determine from the table that
Finally, we can determine the final transfer function.