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# Butterworth Filters

Module by: Richard Baraniuk. E-mail the author

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

|Hiω|=11+ω ω c 2n H ω 1 1 ω ω c 2 n
(1)
where nn is the order of the filter and ωcωc is the cutoff frequency. The cutoff frequency is the frequency where the magnitude experiences a 3 dB dropoff (where |Hiω|=12 H ω 1 2 ).

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 nn, 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

|Hiω|=11+ω2n H ω 1 1 ω 2 n
(2)

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

1(ss1)(ss2)(ssn) 1 s s1 s s2 s sn
(3)
and a pole-zero plot of

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

Hs=1sn+ a n-1 sn1++a1s+1 H s 1 s n a n-1 s n 1 a1 s 1
(4)
From this, we may look up the ai ai from a table (like the one below) for any desired nn. 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 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

### Exercise 1

Design a Butterworth filter with a passband gain between 1 and 0.891 (-1 dB gain) for 0<ω<10 0 ω 10 and a stopband not to exceed 0.0316 (-30 dB gain) for ω20 ω 20 .

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