butterworth mitres 6 007s11 lec24

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24Butterworth Filters

To illustrate some of the ideas developed in Lecture 23, we introduce in thislecture a simple and particularly useful class of filters referred to as Butter-

worthfilters.Filters in this class are specified by two parameters, the cutofffrequency and the filter order. The frequency response of these filters ismonotonic, and the sharpness of the transition from the passband to the stop-band is dictated by the filter order. For continuous-time Butterworth filters,the poles associated with the square of the magnitude of the frequency re-sponse are equally distributed in angle on a circle in the s-plane, concentric

with the origin and having a radius equal to the cutoff frequency. When thecutoff frequency and the filter order have been specified, the poles character-izing the system function are readily obtained. Once the poles are specified, itis straightforward to obtain the differential equation characterizing the filter.

In this lecture, we illustrate the design of a discrete-time filter throughthe use of the impulse-invariant design procedure applied to a Butterworthfilter. The filter specifications are given in terms of the discrete-time frequen-

cy variable and then mapped to a corresponding set of specifications for thecontinuous-time filter. A Butterworth filter meeting these specifications is de-termined. The resulting continuous-time system function is then mapped tothe desired discrete-time system function.

A limitation on the use of impulse invariance as a design procedure fo rdiscrete-time systems is the requirement that the continuous-time filter bebandlimited to avoid aliasing. An alternative procedure, called the bilineartransformation,corresponds to a mapping of the entire imaginary axis in thes-plane to once around the unit circle. Consequently, there is no aliasing intro-duced by this procedure. However, since the imaginary axis has infinite lengthand the unit circle has a finite circumference, by necessity there must be anonlinear distortion in mapping between the two frequency axes. The use ofthe bilinear transformation is therefore limited to mapping filters that are ap-

proximately piecewise constant. For filters of this type, the inherent nonlineardistortion in the frequency axis is easily accommodated by prewarping thecritical frequencies (e.g., passband edge and stopband edge) prior to carrying

out the design of the associated continuous-time system. This procedure is il-lustrated with the design of a Butterworth filter.

Suggested Reading

Section 6.5, The Class of Butterworth Frequency-Selective Filters, pages 422-428

Section 9.7.3, Butterworth Filters, pages 611-614

Section 10.8.3, The Bilinear Transformation, pages 665-667

24-1



Signals and Systems

24-2

TRANSPARENCY

24.1

Frequency response

for the class of

Butterworth filters.

BUTTERWORTH FILTERS

IB(jw)12 = 1

1 + (jw/jWc)2N

IB(w)|1.0J

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 w/wc

TRANSPARENCY

24.2The square of the

magnitude of the

frequency response

and the system

function forButterworth filters.

I B(jco) 12 + 11 + (jwIjoc)2N

B(jw) B(-jo) -+ B(s) B(-s)

B(s) B(-s)

1

1 + (s/jc)2N

poles of B(s) B(-s) at

s = (-1)1/ 2 N cC



Butterworth Filters

24-3

1B(s) B(-s) =

1 + (s/j o c)2N

In

B(s) B(-s)

x~c

\ ICRe

1B(s) B(- s) =

1 + (s/j Woe)2N

In

B(s)

x-j \

Re

TRANSPARENCY24.3Transparencies 24.3and 24.4 show a pole-zero plot associatedwith the Butterworthfilter. Show n here isthe pole-zero patternfor the square of themagnitude of thefrequency responsefor Butterworth filters.

TRANSPARENCY24.4The pole-zero plotfor the systemfunction for aButterworth filter.Since we restrict B(s)to correspond to astable, causal filter, itspoles must all be inthe left half of the

s-plane.



Signals and Systems

24-4

Sanhng\he =iokH)

9 0

Desired SeeciC.tom'

at -2.Wtk 8

Oat to - lT-.5 k W

"3(p

\ok H3 -+ 2T

1.5 Kib 40. 3 W

el \o3 c I by ")I : - L

qi a = 0. a Tr

1-,3 0 ,1 Hd ,te")\ 5 - 5

akt .nI= o 31TT

T= ? (vT)

TRANSPARENCY

24.5

Spectra illustrating

impulse invariance.

MARKERBOARD

24.1 (a)

I



Butterworth Filter

24-

TRANSPARENCY

24.6

Specifications on adiscrete-time filter to

be obtained from acontinuous-time

Butterworth filter

using impulse

invariance.

TRANSPARENCY

24.7Pole-zero plot

associated with the

Butterworth filter to

be mapped to the

desired discrete-time

filter and a summary

of the steps in the

procedure.



Signals and Systems

24-6

TRANSPARENCY24.8

Frequency response ofthe discrete-timeButterworth filterdesigned usingimpulse invariance.

TRANSPARENCY24.9The bilineartransformation formapping fromcontinuous-time to

discrete-time filters.

1.2

1.0

.8

.6

.4

.2

0

0

-10

-20

-30-40

-50

-60

-70

-80

0 .21 .4r .6r .81 w

EE--

.21 .4 .81



Butterworth Filter

24-

TRANSPARENCY24.10Mapping from thecontinuous-timefrequency axis tothe discrete-timefrequency axisresulting fromthe bilineartransformation.

ii

I

I/ = 'tan \ /

2,= an (2

Ci Ti

(A)-

TRANSPARENCY24.11Illustration of theeffect of the bilineartransformation on theapproximation to alowpass filter.

z-plane

Re

S = 2 arctan

2i0i

p

Hd(ej) I

I HC(jw) I

-- - - - - - - - - -



Signals and Systems

24-8

TRANSPARENCY24.12Determination of the

parameters fo r aButterworth filter tobe mapped to adiscrete-time filterwith the specificationsin Transparency 24.6.

2 = 2 arctan (2) T=1

wi 2 an 12)

20 log 0 G (j 2 tan ( )>-1

20 log 10 G j2 an -0.37r5

equality for N = 5.3

choose N = 6

WC= 0.76622

MARKERBOARD24.1 (b)

I--



Butterworth Filte

24

.61r .87r 7r

.2ir.6ir .8ir iT

TRANSPARENCY24.13

The pole-zero plotassociated with thesquared magnitudefunction for thedesired Butterworthfilter and the stepsinvolved in thedetermination of thediscrete-time filter.

TRANSPARENCY24.14Frequency responsefor the discrete-timefilter obtained bymapping aButterworth filterto a digital filterthrough the bilineartransformation.

1.2

1.0

.8

.6

.4

.2

0

0

-10

-20

-30

-40

-50

-60

-70

-80.67r .87r21r



Signals and Systems

24-10

TRANSPARENCY24.15Comparison of thefrequency responsesof digital filtersobtained throughimpulse invarianceand through thebilinear trans-formation.

-10

-20

-30

-40

-50

-60

-70

-80

1.0

.8

.6

.4

.2

0

.2w -41 -6w .8

0 .27r .4x .61r .81T ir



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butterworth mitres 6 007s11 lec24

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