Butterworth filter

Summary

The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".[1]

The frequency response plot from Butterworth's 1930 paper.[1]

Original paper edit

Butterworth had a reputation for solving very complex mathematical problems thought to be 'impossible'. At the time, filter design required a considerable amount of designer experience due to limitations of the theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that:

"An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies".

Such an ideal filter cannot be achieved, but Butterworth showed that successively closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated substantial ripple in the passband, and the choice of component values was highly interactive. Butterworth showed that a low-pass filter could be designed whose gain as a function of frequency (i.e. the magnitude of its frequency response) is:

 

where   is the angular frequency in radians per second and   is the number of poles in the filter—equal to the number of reactive elements in a passive filter. Its cutoff frequency (the half-power point of approximately −3 dB or a voltage gain of 1/2 ≈ 0.7071) is normalized to 𝜔 = 1 radian per second. Butterworth only dealt with filters with an even number of poles in his paper, though odd-order filters can be created with the addition of a single-pole filter applied to the output of the even-order filter. He built his higher-order filters from 2-pole filters separated by vacuum tube amplifiers. His plot of the frequency response of 2-, 4-, 6-, 8-, and 10-pole filters is shown as A, B, C, D, and E in his original graph.

Butterworth solved the equations for two-pole and four-pole filters, showing how the latter could be cascaded when separated by vacuum tube amplifiers and so enabling the construction of higher-order filters despite inductor losses. In 1930, low-loss core materials such as molypermalloy had not been discovered and air-cored audio inductors were rather lossy. Butterworth discovered that it was possible to adjust the component values of the filter to compensate for the winding resistance of the inductors.

He used coil forms of 1.25″ diameter and 3″ length with plug-in terminals. Associated capacitors and resistors were contained inside the wound coil form. The coil formed part of the plate load resistor. Two poles were used per vacuum tube and RC coupling was used to the grid of the following tube.

Butterworth also showed that the basic low-pass filter could be modified to give low-pass, high-pass, band-pass and band-stop functionality.

Overview edit

 
The Bode plot of a first-order low-pass filter

The frequency response of the Butterworth filter is maximally flat (i.e., has no ripples) in the passband and rolls off towards zero in the stopband.[2] When viewed on a logarithmic Bode plot, the response slopes off linearly towards negative infinity. A first-order filter's response rolls off at −6 dB per octave (−20 dB per decade) (all first-order lowpass filters have the same normalized frequency response). A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on. Butterworth filters have a monotonically changing magnitude function with  , unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.

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, but Butterworth filters have a more linear phase response in the passband than Chebyshev Type I/Type II and elliptic filters can achieve.

Example edit

A transfer function of a third-order low-pass Butterworth filter design shown in the figure on the right looks like this:

 
 
A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter with cutoff frequency  =1 when (for example)  =4/3 F,  =1 Ω,  =3/2 H and  =1/2 H.

A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with   = 4/3 F,   = 1 Ω,   = 3/2 H, and   = 1/2 H.[3] Taking the impedance of the capacitors   to be   and the impedance of the inductors   to be  , where   is the complex frequency, the circuit equations yield the transfer function for this device:

 

The magnitude of the frequency response (gain)   is given by

 

obtained from

 

and the phase is given by

 
 
Gain and group delay of the third-order Butterworth filter with  

The group delay is defined as the negative derivative of the phase shift with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies. The gain and the delay for this filter are plotted in the graph on the left. It can be seen that there are no ripples in the gain curve in either the passband or the stop band.

The log of the absolute value of the transfer function   is plotted in complex frequency space in the second graph on the right. The function is defined by the three poles in the left half of the complex frequency plane.

 
Log density plot of the transfer function   in complex frequency space for the third-order Butterworth filter with  =1. The three poles lie on a circle of unit radius in the left half-plane.

These are arranged on a circle of radius unity, symmetrical about the real   axis. The gain function will have three more poles on the right half-plane to complete the circle.

By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass Butterworth filter is obtained.

A band-pass Butterworth filter is obtained by placing a capacitor in series with each inductor and an inductor in parallel with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency of interest.

A band-stop Butterworth filter is obtained by placing a capacitor in parallel with each inductor and an inductor in series with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency that is to be rejected.

Transfer function edit

 
Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency  . Note that the slope is 20  dB/decade where   is the filter order.

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 gain   of an  th-order Butterworth low-pass filter is given in terms of the transfer function   as

 

where   is the order of filter,   is the cutoff frequency (approximately the −3 dB frequency), and   is the DC gain (gain at zero frequency).

It can be seen that as   approaches infinity, the gain becomes a rectangle function and frequencies below   will be passed with gain  , while frequencies above   will be suppressed. For smaller values of  , the cutoff will be less sharp.

We wish to determine the transfer function   where   (from Laplace transform). Because   and, as a general property of Laplace transforms at  ,  , if we select   such that:

 

then, with  , we have the frequency response of the Butterworth filter.

The   poles of this expression occur on a circle of radius   at equally-spaced points, and symmetric around the negative real axis. For stability, the transfer function,  , is therefore chosen such that it contains only the poles in the negative real half-plane of  . The  -th pole is specified by

 

and hence

 

The transfer (or system) function may be written in terms of these poles as

 .

where   is the product of a sequence operator. The denominator is a Butterworth polynomial in  .

Normalized Butterworth polynomials edit

The Butterworth polynomials may be written in complex form as above, but are usually written with real coefficients by multiplying pole pairs that are complex conjugates, such as   and  . The polynomials are normalized by setting  . The normalized Butterworth polynomials then have the general product form

 
 

Factors of Butterworth polynomials of order 1 through 10 are shown in the following table (to six decimal places).

n Factors of Butterworth Polynomials  
1  
2  
3  
4  
5  
6  
7  
8  
9  
10  

Factors of Butterworth polynomials of order 1 through 6 are shown in the following table (Exact).

n Factors of Butterworth Polynomials  
1  
2  
3  
4  
5  
6  

where the Greek letter phi (  or  ) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation   with a value of[4][5]

 (OEISA001622)

The  th Butterworth polynomial can also be written as a sum

 

with its coefficients   given by the recursion formula[6][7]

 

and by the product formula

 

where

 

Further,  . The rounded coefficients   for the first 10 Butterworth polynomials   are:

Butterworth Coefficients   to Four Decimal Places
                       
     
       
         
           
             
               
                 
                   
                     
                       

The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency  , as follows

  , where  

Transformation to other bandforms are also possible, see prototype filter.

Maximal flatness edit

Assuming   and  , the derivative of the gain with respect to frequency can be shown to be

 

which is monotonically decreasing for all   since the gain   is always positive. The gain function of the Butterworth filter therefore has no ripple. The series expansion of the gain is given by

 

In other words, all derivatives of the gain up to but not including the 2 -th derivative are zero at  , resulting in "maximal flatness". If the requirement to be monotonic is limited to the passband only and ripples are allowed in the stopband, then it is possible to design a filter of the same order, such as the inverse Chebyshev filter, that is flatter in the passband than the "maximally flat" Butterworth.

High-frequency roll-off edit

Again assuming  , the slope of the log of the gain for large   is

 

In decibels, the high-frequency roll-off is therefore 20  dB/decade, or 6  dB/octave (the factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule.)

Minimum order edit

To design a Butterworth filter using the minimum required number of elements, the minimum order of the Butterworth filter may be calculated as follows.[8]

 

where:

  and   are the pass band frequency and attenuation at that frequency in dB

  and   are the stop band frequency and attenuation at that frequency in dB

  is the minimum number of poles, the order of the filter.

ceil[] is a round up to next integer function.

Nonstandard cutoff attenuation edit

The cutoff attenuation for Butterworth filters is usually defined to be −3.01 dB. If it is desired to use a different attenuation at the cutoff frequency, then the following factor may be applied to each pole, whereupon the poles will continue to lie on a circle, but the radius will no longer be unity.[9] The cutoff attenuation equation may be derived through algebraic manipulation of the Butterworth defining equation stated at the top of the page.[10]

 

where:

  is the relocated pole positioned to set the desired cutoff attenuation.

  is a −3.01 dB cutoff pole that lies on the unit circle.

  is the desired attenuation at the cutoff frequency in dB (1 dB, 10 dB, etc.)

  is the number of poles, the order of the filter.

Filter implementation and design edit

There are several different filter topologies available to implement a linear analogue filter. The most often used topology for a passive realisation is the Cauer topology, and the most often used topology for an active realisation is the Sallen–Key topology.

Cauer topology edit

 
Butterworth filter using Cauer topology

The Cauer topology uses passive components (shunt capacitors and series inductors) to implement a linear analog filter. The Butterworth filter having a given transfer function can be realised using a Cauer 1-form. The k-th element is given by[11]

 
 

The filter may start with a series inductor if desired, in which case the Lk are k odd and the Ck are k even. These formulae may usefully be combined by making both Lk and Ck equal to gk. That is, gk is the immittance divided by s.

 

These formulae apply to a doubly terminated filter (that is, the source and load impedance are both equal to unity) with ωc = 1. This prototype filter can be scaled for other values of impedance and frequency. For a singly terminated filter (that is, one driven by an ideal voltage or current source) the element values are given by[3]

 

where

 

and

 
 

Voltage driven filters must start with a series element and current driven filters must start with a shunt element. These forms are useful in the design of diplexers and multiplexers.[3]

Sallen–Key topology edit

 
Sallen–Key topology

The Sallen–Key topology uses active and passive components (noninverting buffers, usually op amps, resistors, and capacitors) to implement a linear analog filter. Each Sallen–Key stage implements a conjugate pair of poles; the overall filter is implemented by cascading all stages in series. If there is a real pole (in the case where   is odd), this must be implemented separately, usually as an RC circuit, and cascaded with the active stages.

For the second-order Sallen–Key circuit shown to the right the transfer function is given by

 

We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming that  , this will mean that

 

and

 

This leaves two undefined component values that may be chosen at will.

Butterworth lowpass filters with Sallen–Key topology of third and fourth order, using only one op amp, are described by Huelsman,[12][13] and further single-amplifier Butterworth filters also of higher order are given by Jurišić et al.[14]

Digital implementation edit

Digital implementations of Butterworth and other filters are often based on the bilinear transform method or the matched Z-transform method, two different methods to discretize an analog filter design. In the case of all-pole filters such as the Butterworth, the matched Z-transform method is equivalent to the impulse invariance method. For higher orders, digital filters are sensitive to quantization errors, so they are often calculated as cascaded biquad sections, plus one first-order or third-order section for odd orders.

Comparison with other linear filters edit

Properties of the Butterworth filter are:

Here is an image showing the gain of a discrete-time Butterworth filter next to other common filter types. All of these filters are fifth-order.

 

The Butterworth filter rolls off more slowly around the cutoff frequency than the Chebyshev filter or the Elliptic filter, but without ripple.

See also edit

References edit

  1. ^ a b Butterworth, S. (1930). "On the Theory of Filter Amplifiers" (PDF). Experimental Wireless and the Wireless Engineer. 7: 536–541.
  2. ^ Bianchi, Giovanni; Sorrentino, Roberto (2007). Electronic filter simulation & design. McGraw-Hill Professional. pp. 17–20. ISBN 978-0-07-149467-0.
  3. ^ a b c Matthaei, George L.; Young, Leo; Jones, E. M. T. (1964). Microwave Filters, Impedance-Matching Networks, and Coupling Structures. McGraw-Hill. pp. 104–107, 105, and 974. LCCN 64007937.
  4. ^ Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2020-08-10.
  5. ^ OEISA001622
  6. ^ Bosse, G. (1951). "Siebketten ohne Dämpfungsschwankungen im Durchlaßbereich (Potenzketten)". Frequenz. 5 (10): 279–284. Bibcode:1951Freq....5..279B. doi:10.1515/FREQ.1951.5.10.279. S2CID 124123311.
  7. ^ Weinberg, Louis (1962). Network analysis and synthesis. Malabar, Florida: Robert E. Krieger Publishing Company, Inc. (published 1975). pp. 494–496. hdl:2027/mdp.39015000986086. ISBN 0-88275-321-5. Retrieved 2022-06-18.
  8. ^ Paarmann, Larry D. (2001). Design and Analysis of Analog Filters, A Signal Processing Perspective. Norwell, Massachusetts, US: Kluwer Academic Publishers. pp. 117, 118. ISBN 0-7923-7373-1.{{cite book}}: CS1 maint: date and year (link)
  9. ^ Paarmann, Larry D. (2001). Design and Analysis of Analog Filters, A Signal Processing Perspective. Norwell, Massachussetts, US: Kluwer Academic Publishers. pp. 118 to 120. ISBN 0-7923-7373-1.{{cite book}}: CS1 maint: date and year (link)
  10. ^ Butterworth filter#Original paper
  11. ^ US 1849656, William R. Bennett, "Transmission Network", published March 15, 1932 
  12. ^ Huelsman, L. P. (May 1971). "Equal-valued-capacitor active-RC-network realisation of a 3rd-order lowpass Butterworth characteristic". Electronics Letters. 7 (10): 271–272. Bibcode:1971ElL.....7..271H. doi:10.1049/el:19710185.
  13. ^ Huelsman, L. P. (December 1974). "An equal-valued capacitor active RC network realization of a fourth-order low-pass Butterworth characteristic". Proceedings of the IEEE. 62 (12): 1709. doi:10.1109/PROC.1974.9689.
  14. ^ Jurišić, Dražen; Moschytz, George S.; Mijat, Neven (2008). "Low-sensitivity, single-amplifier, active-RC allpole filters using tables". Automatika. 49 (3–4): 159–173.