Filter Tunings
Overview
Filter tunings are specific tunings of filters to maximise a particular characteristic of it's response. Filter tuning directly specifies what the filters polynomials must be in it's transfer function (see What Are Transfer Functions, Poles and Zeroes for more info).
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- Butterworth Optimized for the flattest response through the pass-band, at the expense of having a low transition between the pass and stop-band.
- Chebyshev: Designed to have a steep transition between the pass and stop-band (but not the steepest, Elliptic tuning claims that award), at the expense of gain ripple in either the pass or stopband (type 1 or type 2). Also called Chevyshev, Tschebychev, Tschebyscheff or Tchevysheff, depending on exactly how you translate the original Russian name. There are two types of Chebyshev filters:
- Type 1: Type 1 Chebyshev filters (a.k.a. just a Chebyshev filter) have ripple in the passband, but no ripple in the stopband.
- Type 2: Type 2 Chebyshev filters (a.k.a. an inverse Chebyshev filter) have ripple in the stopband, but no ripple in the passband.
- Bessel: Optimized for linear phase response up to (or down to for high-pass filters) the cutoff frequency , at the expense of a slower transition to the stop-band. This is useful to minimizing the signal distortion (a linear phase response in the frequency domain is a constant time delay in the time domain).
- Elliptic: Designed to have the fastest transition from the passband to the stopband, at the expense of ripple in both of these bands (Chebyshev optimization only produces ripple in one of the bands but is not as fast in the transition). Also called Cauer filters or Rational Chebyshev filters.
These filters are explained in more detail below. If you are interested in visual comparisons, you can skip straight to the Comparisons Between Filter Tunings section.
Butterworth Tunings
Tuning a filter for a Butterworth response gives a filter which is maximally flat in the passband, and rolls off towards zero in the stopband. The price you pay for this is slower roll-off into the stop-band, compared with Chebyshev or Elliptic tunings.
It may sound dumb, but I've always remembered Butterworth as a flat passband which "slides like butter".
Butterworth tunings are defined as a filter whose magnitude is1:
where:
is the angular frequency
is the order of the filter
The normalized Butterworth polynomial of degree is given by2:
If you've never seen it before, the uppercase Pi symbol in the above equation represents the product of a series of things, such like the uppercase Sigma symbol represents the sum of a series of things. For example, .
Below is a table of the normalized factored Butterworth polynomials for order . The polynomial is useful in this form as each product forms either a first or second-order partial filter which can be directly implemented by a standard filter topology (e.g. RC filter for a first-order section, Sallen-Key for a second-order section). These polynomials were generated with the Butterworth polynomial equation above. The polynomials are normalized by setting (the characteristic frequency).
All numbers are rounded to 3 decimal places.
n | polynomial |
---|---|
1 | ((s + 1)) |
2 | ((s^2 + 1.414 s + 1)) |
3 | (\left(s + 1\right) \left(s^2 + 1.0 s + 1\right)) |
4 | (\left(s^2 + 0.765 s + 1\right) \left(s^2 + 1.848 s + 1\right)) |
5 | (\left(s + 1\right) \left(s^2 + 0.618 s + 1\right) \left(s^2 + 1.618 s + 1\right)) |
6 | (\left(s^2 + 0.518 s + 1\right) \left(s^2 + 1.414 s + 1\right) \left(s^2 + 1.932 s + 1\right)) |
7 | (\left(s + 1\right) \left(s^2 + 0.445 s + 1\right) \left(s^2 + 1.247 s + 1\right) \left(s^2 + 1.802 s + 1\right)) |
8 | (\left(s^2 + 0.39 s + 1\right) \left(s^2 + 1.111 s + 1\right) \left(s^2 + 1.663 s + 1\right) \left(s^2 + 1.962 s + 1\right)) |
Using these polynomials , we can write the transfer function of a Butterworth filter as2:
This equation is different from Butterworth magnitude equation, because this equation lacks the magnitude symbols around . This equation is the full-and-proper transfer function, which contains both the magnitude and phase information.
This transfer function gives the following bode plots (by taking the magnitude and arg of the transfer function):
![](/assets/images/butterworth-bode-plot-for-various-n-f7ebe912566b16f10cba931982b13613.png)
The Butterworth polynomial coefficients