Filter of Butterworth

A filter of Butterworth is a type of linear model of Filtre, designed to have a profit as constant as possible in its Band-width.

The filters of Butterworth were described for the first time by the British engineer Stephen Butterworth  .

Characteristics

The profit of a filter of Butterworth is most constant possible in the band-width and tends towards 0 in the band of cut. On a Diagram of Bode logarithmic curve, this answer decrease linearly towards - ∞, of -6 dB/Octave (- 20db/Décade) for a filter of first order, -12db/octave either -40dB/decade for a filter of second order, -18dB/octave or -60dB/decade for a filter of third order, etc

Transfer transfer function

As for all the linear filters, the studied prototype is the low-pass Filtre, which can be easily modified in high-pass Filtre or placed in series to form filters band pass or band suppressor.

The profit of a filter of low-pass Butterworth of order N is:

$G\_n\; (\backslash \; Omega)\; =\; \backslash \; left\; |\; H\_n\; (J\; \backslash \; Omega)\; \backslash \; right\; |\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{1\; +\; (\backslash \; Omega/\backslash \; omega\_\; \backslash \; mathrm\; \{C\})\; ^\; \{2\; N\}\}\}$

where $G\_n$ is the profit of the filter, $H\_n$ its Transfer function transfer, $j$ the unit complexes, $\backslash \; omega$ the Fréquence (angular) of the signal in rad. S ^-1 and $\backslash \; omega\_\; \backslash \; mathrm\; \{C\}$ the Frequency cut-off (angular) of the filter (to -3 dB).

By standardizing the expression (i.e. by specifying $\backslash \; omega\_\; \backslash \; mathrm\; \{C\}\; =\; 1$):

$G\_n\; (\backslash \; Omega)\; =\; \backslash \; left\; |\; H\_n\; (J\; \backslash \; Omega)\; \backslash \; right\; |\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{1\; +\; \backslash \; Omega\; ^\; \{2\; N\}\}\}$

The 2n-1 first Dérivée S from $G\_n$ are null for $\backslash \; Omega\; =\; 0$, implying a maximum constancy of the profit in the band-width.

With the high frequencies:

$=\; \{20n\}\; \{log\_\; \{10\}\; \{\backslash \; Omega\}\}$

The roll-off of the filter (the slope of the square of the profit in a diagram of Bode) is of 20n dB/décade.

Polynomials of Butterworth

The transfer transfer function standardized of a filter of Butterworth can be written in the following form:

$H\_n\; (S)\; =\; \backslash \; frac\; \{1\}\; \{B\_n\; (S)\}$

where $B\_n\; (S)$ is a Polynôme of degree N.

The following table gives the first values of these polynomials:

Comparisons

The filters of Butterworth are the only linear filters whose general form is similar for all the orders (put aside a slope different in the band from cut).

By comparison with the filters of elliptic Tchebychev or , the filters of Butterworth have a weaker roll-off which implies to use a more important order for a particular establishment. Their profit is on the other hand definitely more constant in the band-width.

Establishment

A filter of Butterworth which one knows the transfer transfer function can be implemented electronically according to the method of Cauer. K E element of such a circuit is given by:

$C\_k\; =\; 2\; sin\; \backslash \; left\; \{(2k-1)\}\{2n\}\; \backslash \; pi\; \backslash \; right$ (K odd)

$L\_k\; =\; 2\; sin\; \backslash \; left\; \{(2k-1)\}\{2n\}\; \backslash \; pi\; \backslash \; right$ (K even)

See too

Internal bonds

• linear Filtre
• Filtre of Bessel
• Filtre of Butterworth
• Filtre combs from there
• Filtre of Legendre
• Filtre of elliptic Tchebychev
• Filtre

References

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