Butterworth Filters

Summary: Describes the design of analog lowpass Butterworth filters.

The Butterworth filter is a filter that can be constructed out of passive R, L, C circuits. The magnitude of the transfer function for this filter is

Magnitude of Butterworth Filter Transfer Function

|H(ⅈω)|=

1

\1+( ω ωc )2n

(1)

where n is the order of the filter and ω_c is the cutoff frequency. The cutoff frequency is the frequency where the magnitude experiences a 3 dB dropoff (where |H(ⅈω)|=

1

\2

 ).

Figure 1: Three different orders of lowpass Butterworth analog filters: n={1,4,10}. As n increases, the filter more closely approximates an ideal brickwall lowpass response.

The important aspects of Figure 1 are that it does not ripple in the passband or stopband as other filters tend to, and that the larger n, the sharper the cutoff (the smaller the transition band).

This transfer function is often seen in its normalized form of

Magnitude of Normalized Transfer Function for Lowpass Butterworth Filter

|H(ⅈω)|=

1

\1+ω2n

(2)

Butterworth filters give transfer functions (H(ⅈω) and H(s)) that are rational functions. They also have only poles, resulting in a transfer function of the form

1

(s−s1)(s−s2)⋯(s−sn)

(3)

and a pole-zero plot of

Figure 2: Poles of a 10th-order ( n=5 ) lowpass Butterworth filter.

Note that the poles lie along a circle in the s-plane.

Designing a Butterworth Filter

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

H(s)=

1

sn+an−1sn−1+⋯+a1s+1

(4)

From this, we may look up the a_i from a table (like the one below) for any desired n. We can also find them in Matlab by using the buttap command. The real challenge of designing a Butterworth filter comes with figuring out the optimal characteristics for the given application.

TABLE 1

n	a1	a2	a3	a4	a5	a6	a7	a8	a9
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 0<ω<10 and a stopband not to exceed 0.0316 (-30 dB gain) for ω≥20.