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  • Very close to the ideal near ω = 0 and ω = ,
  • Very smooth at all frequencies with a monotonic decrease from ω = 0 to , and
  • Largest difference between the ideal and actual responses near thetransition at ω = 1 where | F ( j 1 ) | 2 = 1 / 2 .

Although not part of the approximation addressed, the phase curve is also very smooth.

An important feature of the Butterworth filter is the closed- form formula for the solution, F ( s ) . The expression for F F ( s ) may be determined as

F ( s ) F ( - s ) = 1 1 + ( - s 2 ) N

This function has 2 N poles evenly spaced around a unit radius circle and 2 N zeros at infinity. The determination of F ( s ) is very simple. In order to have a stable filter, F ( s ) is selected to have the N left-hand plane poles and N zeros at infinity; F ( - s ) will necessarily have the right-hand plane poles and the other N zeros at infinity. Thelocation of these poles on the complex s plane for N = 1 , 2 , 3 , and 4 is shown in [link] .

Figure two consists of four graphs. Each has a horizontal axis labeled, real part of s, with values ranging from -2 to 2 in increments of 1. Each has a vertical axis labeled, imaginary part of s, with values ranging from -1.5 to 1.5 in increments of 0.5. Each graph has a circle of radius one centered at the origin. The first graph is titled First Order BW Filter Poles, and there is a mark on the leftmost potion of the circle, at (-1, 0). The second graph is titled Second Order BW Filter Poles, and there are two marks on the left side of the circle, one in the middle of the portion of the circle in the second quadrant, and one in the middle of the portion of the circle in the third quadrant. The third graph is titled Third order BW Filter Poles, and there are three evenly-spaced marks on the outside of the left half of the circle. The fourth graph is titled Fourth Order BW Filter Poles, and there are four evenly-spaced marks on the left half of the circle.
Pole Locations for Analog Butterworth Filter Transfer Function on the Complex s Plane

Because of the geometry of the pole positions, simple formulas are easy to derive for the pole locations. If the realand imaginary parts of the pole location are denoted as

s = u + j w

the locations of the N poles are given by

u k = - cos ( k π / 2 N )
ω k = sin ( k π / 2 N )

for N values of k where

k = ± 1 , ± 3 , ± 5 , . . . , ± ( N - 1 ) for N even
k = 0 , ± 2 , ± 4 , . . . , ± ( N - 1 ) for N odd

Because the coefficients of the numerator and denominator polynomials of F ( s ) are real, the roots occur in complex conjugate pairs. The conjugate pairs in [link] , [link] can be combined to be the roots of second-order polynomials so that for N even, F ( s ) has the partially factored form of

F ( s ) = k 1 s 2 + 2 cos ( k π / 2 N ) s + 1

for k = 1 , 3 , 5 , . . . , N - 1 . For N odd, F ( s ) has a single real pole and, therefore, the form

F ( s ) = 1 s + 1 k 1 s 2 + 2 cos ( k π / 2 N ) s + 1

for k = 2 , 4 , 6 , , N - 1

This is a convenient form for the cascade and parallel realizations discussed in elsewhere.

A single formula for the pole locations for both even and odd N is

u k = - sin ( ( 2 k + 1 ) π / 2 N )
ω k = cos ( ( 2 k + 1 ) π / 2 N )

for N values of k where k = 0 , 1 , 2 , . . . , N - 1

One of the important features of the Butterworth filter design formulas is that the pole locations are found by independentcalculations which do not depend on each other or on factoring a polynomial. A FORTRAN program which calculates these values isgiven in the appendix as Program 8. Mathworks has a powerful command for designing analog and digital Butterworth filters.

The classical form of the Butterworth filter given in [link] is discussed in many books [link] , [link] , [link] , [link] , [link] . The less well-known form given in [link] also has many useful applications [link] . If the frequency location of unwanted signals is known, the zeros of the transfer function given by thenumerator can be set to best reject them. It is then possible to choose the pole locations so as to have a passband as flat as theclassical Butterworth filter by using [link] . Unfortunately, there are no formulas for the pole locations; therefore, thedenominator polynomial must be factored.

Summary

This section has derived design procedures and formulas for a class of filter transfer functions that approximate the idealdesired frequency response by a Taylor's series. If the approximation is made at ω = 0 and ω = , the resulting filter is called a Butterworth filter and the response iscalled maximally-flat at zero and infinity. This filter has a very smooth frequency response and, although not explicitly designed for,has a smooth phase response. Simple formulas for the pole locations were derived and are implemented in the design program in theappendix of this book.

Butterworth filter design procedures

This section considers the process of going from given specifications to use of the approximation results derived in theprevious section. The Butterworth filter is the simplest of the four classical filters in that all the approximation effort is placed attwo frequencies: ω = 0 and ω = . The transition from passband to stopband occurs at a normalized frequency of ω = 1 . Assuming that this transition frequency or bandedge can later be scaled to any desired frequency, the only parameter tobe chosen in the design process is the order N .

The filter specifications that are consistent with what is optimized in the Butterworth filter are the degree of “flatness" at ω = 0 (DC) and at ω = . The higher the order, the flatter the frequency response at these two points. Because ofthe analytic nature of rational functions, the flatter the response is at ω = 0 and ω = , the closer it stays to the desired response throughout the whole passband and stopband. Anindirect consequence of the filter order is the slope of the response at the transition between pass and stopband. The slope ofthe squared-magnitude frequency response at ω = 1 is

s = F F ' ( j 1 ) = - N / 2

The effects of the increased flatness and increased transition slope of the frequency response as N increases are illustrated in Figure 1 from Design of Infinite Impulse Response (IIR) Filters by Frequency Transformations .

In some cases specifications state the response must stay above or below a certainvalue over a given frequency band. Although this type of specification is more compatible with a Chebyshev erroroptimization, it is possible to design a Butterworth filter to meet the requirements. If the magnitude of the frequency response of thefilter over the passband of 0 < ω < ω P must remain between unity and G , where ω p < 1 and G < 1 , the required order is found by the smallest integer N satisfying

N log ( ( 1 / G ) 2 - 1 ) 1 log ( ω p )

This is illustrated in [link] where | F | must remain above 0.9 for ω up to 0.9, i.e., G = 0.9 and ω p = 0.9. These requirements require an order of at least N = 7 .

Figure three is a graph titled Analog Butterworth Filter Frequency Response. The horizontal axis is labeled Normalized Frequency ω, and ranges in value from 0 to 3 in increments of 0.5. The vertical axis is labeled Magnitude Response, and ranges in value from 0 to 1 in increments of 0.2. There is one curve on the graph, and it starts at (0, 1), moving horizontally for 0.5 units, and then proceeds to decrease at a increasing rate. At approximately (1.1, 0.5), the curve then begins decreasing at a decreasing rate, until it terminates still with a slightly negative slope at (3, 0.1). at a horizontal value of approximately 0.8, an arrow points at the horizontal axis, labeled ω_P. At this horizontal point, there is a vertical line up to the point where it intersects with the aforementioned curve.
Passband Specifications for Designing a Butterworth Filter

If stopband performance is stated in the form of requiring that the response stay below a certain value for frequency above acertain value, i.e., | F | < G for ω > ω s , the order is determined by the same formula [link] with ω p replaced by ω s .

Note | F ( j 1 ) | = 1 / ( 2 ) which is called the “half power" frequency because | F ( j 1 ) | 2 = 1 / 2 . This frequency is normalized to one for the theory but can be scaled to any value for applications.

Design of a butterworth lowpass iir filter

To illustrate the calculations, a lowpass Butterworth filter is designed. It is desired that the frequency response stay above0.8 for frequencies up to 0.9. The formula [link] for determining the order gives a value of 2.73; therefore, the order isthree. The analytic function corresponding to the squared-magnitude frequency response in [link] is

| F ( j ω ) | 2 = 1 1 + ω 6

The transfer function corresponding to the left-half-plane poles of F'(s) are calculated from [link] to give

F ( s ) = 1 ( s + 1 ) ( s + 0 . 5 + j 0 . 866 ) ( s + 0 . 5 - j 0 . 866 )
F ( s ) = 1 ( s + 1 ) ( s 2 + s + 1 )
F ( s ) = 1 s 3 + 2 s 2 + 2 s + 1

The frequency response is obtained by setting s = j ω which has a plot illustrated in [link] for N = 3 . The pole locations are the same as shown in [link] c.

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Source:  OpenStax, Filter design - sidney burrus style. OpenStax CNX. May 07, 2009 Download for free at http://cnx.org/content/col10701/1.1
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