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ORIGIN OF NAME


The Butterworth type filter was first described by the British Engineer Stephen Butterworth (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.
of a first-order Butterworth low-pass filter]]

example]]


OVERVIEW


The frequency response of the Butterworth filter is maximally flat (has no ripples) in the passband, and rolls off towards zero in the stopband. When viewed on a logarithmic Bode Plot , the response slopes off linearly towards negative infinity. For a first-order filter, the response rolls off at −6 DB per Octave (−20 dB per Decade ). (All first-order filters, regardless of name, are actually identical and so have the same frequency response.) For a second-order Butterworth filter, the response decreases at −12 dB per octave, a third-order at −18 dB, and so on. Butterworth filters have a monotonically decreasing magnitude function with ω.
The Butterworth is the only filter that maintains this same shape for higher orders (but with a steeper decline in the stopband) whereas other varieties of filters ( Bessel , Chebyshev , Elliptic ) have different shapes at higher orders.

Compared with a Chebyshev Type I/Type II filter or an Elliptic Filter , the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular Stopband specification. However, Butterworth filter will have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters.


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 magnitude squared frequency response:


  1/(1 + &epsilon<sup>2</sup>) band edge value of H(ω)<sup>2</sup><br>
  Since ''H(s)H(-s)'' Evaluated At ''s jω'' is simply equal to H(&omega)<sup>2</sup>, it follows that<br><center>


and hence,
s_k = \omega_ce^{ rac{j\pi}{2}}e^{ rac{j(2k+1)\pi}{2n}}
''k = 0, 1, 2, ...., n-1''


The magnitude of the Frequency Response of an ''n''th order lowpass filter can be defined mathematically as:







At maximum or minimum,  
{dg \over d \omega} = 0

\left ( -1 ight ) \left ({1+{\omega}^{2n}} ight ) ^{-2} \left ( {2n{\omega}^{2n-1}} ight )^{-2} = 0

{\omega}^{2n-1} \left ({1+{\omega}^{2n}} ight )^{-2} = 0

  ω &nbsp &nbsp 0 &nbsp&nbsp&nbsp &nbsp&nbsp&nbsp&nbsp&nbspω = -1 &nbsp
(not possible as ω is Real )</center></big>


Hence, the high frequency roll off = 20n dB/decade


FILTER REALIZATION


The Butterworth filter having a given transfer function can be realised using Cauer - 1 form:
kth element is given by:
C_k = 2 \sin \left [ rac {(2k-1)}{2n} \pi ight ]; k = odd

L_k = 2 \sin \left [ rac {(2k-1)}{2n} \pi ight ]; k = even


NORMALIZED BUTTERWORTH POLYNOMIALS

  { Class "wikitable" style="text-align: center"



COMPARISON WITH OTHER LINEAR FILTERS


Here is an image showing the Butterworth filter next to other common kind of filters obtained with the same number of coefficients:

As is clear from the image, the Butterworth filter rolls off more slowly than all the others but it shows no ripples.


SEE ALSO