Butterworth Filter - Transfer Function

Transfer Function

Like all filters, the typical prototype is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form band-pass and band-stop filters, and higher order versions of these.

The gain of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as

where

  • n = order of filter
  • ωc = cutoff frequency (approximately the -3dB frequency)
  • is the DC gain (gain at zero frequency)

It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below ωc will be passed with gain, while frequencies above ωc will be suppressed. For smaller values of n, the cutoff will be less sharp.

We wish to determine the transfer function H(s) where (from Laplace transform). Since H(s)H(-s) evaluated at s = jω is simply equal to |H(jω)|2, it follows that

The poles of this expression occur on a circle of radius ωc at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by

-\frac{s_k^2}{\omega_c^2} = (-1)^{\frac{1}{n}} = e^{\frac{j(2k-1)\pi}{n}} \qquad\mathrm{k = 1,2,3, \ldots, n}

and hence;

The transfer function may be written in terms of these poles as

The denominator is a Butterworth polynomial in s.

Read more about this topic:  Butterworth Filter

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