Butterworth filter

Butterworth filters are continuous frequency filters that are designed so that the frequency response below the cutoff frequency ω _g runs horizontally as long as possible. Only shortly before this cut-off frequency should the transfer function decrease and transition into the transmission loss of n · 20 dB per frequency decade ( n is the order of the Butterworth filter). The simplest form of the 1st order Butterworth filter is the RC element . A modern practical application of the filter is common in computer animation ; it serves to reduce curve points without changing the general shape of the curve.

The Bode diagram of a Butterworth first order low pass filter

The Butterworth filter simplifies the point density of a curve without changing the basic curve shape.

A signal with the cutoff frequency is attenuated to -fold the original signal, i.e. H. the attenuation at the cutoff frequency is approx. 3 dB . Butterworth filters have a uniform (smooth) course of the transfer function in both the passband and the stopband. ${\displaystyle {\frac {1}{\sqrt {2}}}\approx 0{,}7071}$

The Butterworth filter was named after the British physicist Stephen Butterworth , who first described this type of filter.

Orders 1 to 5 Butterworth low-pass filters

Example: 2nd order Butterworth filter, low-pass , implemented as a Sallen-Key filter .

Transfer function

This results in the requirement for the transfer function:

${\displaystyle \left|{\underline {A}}\right|^{2}={\frac {A_{0}^{2}}{1+k_{2n}\Omega ^{2n}}}}$

With

${\displaystyle A_{0}}$ DC voltage gain

${\displaystyle \Omega ={\frac {f}{f_{g}}}}$Frequency normalized to the cutoff frequency

${\displaystyle n}$ Order of the filter

The coefficients of the Butterworth filter are obtained by comparing the coefficients with the general transfer function.

Coefficients

If one brings the transfer function into the normalized form ( ): ${\displaystyle P={\frac {p}{\omega _{0}}}}$

${\displaystyle A[P]={\frac {A_{0}}{\prod _{i}(1+a_{i}P+b_{i}P^{2})}}}$

result for the coefficients and the following relationships: ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$

Order n of the filter even:

${\displaystyle i=1\ldots {\frac {n}{2}}}$

${\displaystyle a_{i}=2\cos {\frac {(2i-1)\pi }{2n}}}$

${\displaystyle b_{i}=1\ }$

Order n of the filter is odd:

${\displaystyle i=2\ldots {\frac {(n+1)}{2}}}$

${\displaystyle a_{1}=1\ }$

${\displaystyle b_{1}=0\ }$

${\displaystyle a_{i}=2\cos {\frac {(i-1)\pi }{n}}}$

${\displaystyle b_{i}=1\ }$

properties

The Butterworth filter has the following properties:

• monotonous amplitude response both in the passband and in the stopband
• rapid kinking at the cutoff frequency, improves with order
• considerable overshoot in step response , deteriorates with order
• the phase course has a small non-linearity
• relatively frequency-dependent group delay
• great implementation effort with high order

Filter realization

The Butterworth filter with a given transfer function can be implemented in the following form:

The k th element is given by:

${\displaystyle C_{k}=2\sin \left[{\frac {(2k-1)}{2n}}\pi \right]}$for k odd

${\displaystyle L_{k}=2\sin \left[{\frac {(2k-1)}{2n}}\pi \right]}$for k straight

In digital signal processing , Butterworth filters can be implemented by selecting appropriate filter coefficients in IIR filters (recursive filter structure). The cascading of two Butterworth filters of the nth order results in a Linkwitz-Riley filter of the 2nd order.

Normalized Butterworth Polynomials

The Butterworth polynomials are usually written as complex conjugate poles s ₁ and s _n . The polynomials are also normalized by the factor ω _c = 1. The normalized Butterworth polynomials thus have the following form:

${\displaystyle B_{n}(s)=\prod _{k=1}^{\frac {n}{2}}\left[s^{2}-2s\cos \left({\frac {2k+n-1}{2n}}\,\pi \right)+1\right]}$for n straight

${\displaystyle B_{n}(s)=(s+1)\prod _{k=1}^{\frac {n-1}{2}}\left[s^{2}-2s\cos \left({\frac {2k+n-1}{2n}}\,\pi \right)+1\right]}$for n odd

Exactly to 4 decimal digits they are:

n Polynomial factors ${\displaystyle B_{n}(s)}$ 1 ${\displaystyle s+1}$ 2 ${\displaystyle s^{2}+{\sqrt {2}}s+1}$ 3 ${\displaystyle \left(s+1\right)\left(s^{2}+s+1\right)}$ 4th ${\displaystyle \left(s^{2}+{\sqrt {2-{\sqrt {2}}}}s+1\right)\left(s^{2}+{\sqrt {2+{\sqrt {2}}}}s+1\right)}$ 5 ${\displaystyle \left(s+1\right)\left(s^{2}+{\sqrt {\frac {3-{\sqrt {5}}}{2}}}s+1\right)\left(s^{2}+{\sqrt {\frac {3+{\sqrt {5}}}{2}}}s+1\right)}$ 6th ${\displaystyle \left(s^{2}+{\sqrt {2-{\sqrt {3}}}}s+1\right)\left(s^{2}+{\sqrt {2}}s+1\right)\left(s^{2}+{\sqrt {2+{\sqrt {3}}}}s+1\right)}$ 7th ${\displaystyle \left(s+1\right)\left(s^{2}+0{,}4450s+1\right)\left(s^{2}+1{,}2470s+1\right)\left(s^{2}+1{,}8019s+1\right)}$ 8th ${\displaystyle \left(s^{2}+{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}s+1\right)\left(s^{2}+{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}s+1\right)\left(s^{2}+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}s+1\right)\left(s^{2}+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}s+1\right)}$

Individual evidence

1. ↑ In this example, the Butterworth filter is used as a low-pass filter, which detects noise in the high point density of the upper curve (points that are distributed around the smooth curve instead of lying on it) and, with a preset sample rate, basically one creates a similar but much simpler curve. The application comes from computer animation.
2. ↑ Stephen Butterworth: On the Theory of Filter Amplifiers In: Wireless Engineer , Volume 7, 1930, pp. 536-541

See also

• Bessel filter
• Cauer filter
• Chebyshev filter
• Linkwitz-Riley filter

Web links

• Analog filter , Othmar Marti and Alfred Plettl, Ulm University
• Online Butterworth Low Pass Filter Calculator