# Butterworth filter

The Butterworth filter is one of the most basic electronic filter designs. It is designed to have a frequency response which is as flat as mathematically possible in the passband.

It was first described by the British engineer S. Butterworth, (who specifically refused to publish his first name; it is thought to be Stephen) in his paper "On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental Wireless and the Radio Engineer), vol. 7, 1930, pp. 536-541.

The most basic Butterworth filter is the standard first-order 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.

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Butterworth_response.png
The frequency response of a first-order Butterworth filter

The frequency response of a first-order Butterworth low-pass filter

As mentioned, the frequency response of the Butterworth filter is maximally flat (i.e. no ripples) in the passband, and a frequency response which slopes off towards zero in the stopband. When viewed on a logarithmic Bode plot, the cut band slopes off linearly towards negative infinity. For a first-order filter, the cut line slopes off at -6 dB per octave, for second-order, -12 dB per octave, etc. All first-order filters are actually the same filter and so have the same frequency response. The Butterworth is the only filter that maintains this same shape for higher orders (just with a steeper decline in the stopband). Other varieties of filters (Bessel, Chebyshev, elliptic) have different shapes at higher orders.

The magnitude of the frequency response of an n order filter can be defined mathematically as:

[itex] \left | G(j \omega) \right | = {1 \over \sqrt{ 1 + (\omega / \omega_H) ^ {2 n}} } [itex]

where G is the gain of the filter, n is the order of the filter, ω is the frequency of the signal in radians and [itex]\omega_H[itex] is the -3dB frequency.

Normalising the expression (thus putting [itex]\omega_H = 1[itex]), the expression becomes:

[itex] \left | G(j \omega) \right | = {1 \over \sqrt{ 1 + \omega ^ {2 n}} } [itex]

Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter will require a higher order to implement, assuming all filters are designed to meet the same specifications. A Butterworth filter will also have the most linear phase response in the passband compared to the Chebyshev Type I/Type II and elliptic filters.de:Butterworthfilter it:Filtro Butterworth

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