Filtros IIR - Ejemplos de Diseño

8/17/2019 Filtros IIR - Ejemplos de Diseño

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This chapter discusses various IIR flter design methods. The our standard

types o flters are used here:

low-pass flter;

high-pass flter;

band-pass flter; and

band-stop flter.

The design method used here is known as bilinear transormation.

The IIR flter design process can be split into several steps described in

hapter !.".# $esigning IIR flters by bilinear transormation. These are:

$efning flter specifcations;

%peciying the type o analog prototype flter;

omputing the flter order according to the flter specifcations and specifedanalog prototype flter;

omputing the transer unction o reerence analog prototype flter;

Transormation into analog flter by range scaling;

Transormation into digital flter by bilinear transormation; and

I the resulting flter doesn&t satisy the given specifcations or i it is possible

to decrease the flter order' then it is necessary to do it. The specifed flter

order is increased or decreased according to needs' and steps (' # and ) are

repeated ater that as many times as needed.

%ome steps are skipped in some cases. I the flter order is known' step ! is

skipped. I the type o reerence analog prototype flter is predetermined'

step " is skipped.

In every given e*ample' the IIR flter design process will be described

through these steps in order to make it easier or you to observe similarities

and di+erencies between various design methodes' analog prototype flters

and design o various types o flters as well.

,igure !-)- illustrates the design steps along with input and output data or

each o them.

,igure !-)-. %teps in designing digital IIR flter

The frst block reers to design o reerence analog prototype flter o

appropriate order. The output data is a reerence analog prototype flter

transer unction a/s0. Regardless o the type o reerence analog prototypeflter in use' the transer unction is given by:


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where:

1 is a constant;

2k is the k-th 2ero o the reerence analog prototype flter transer unction;

3 is a number o 2eros o the reerence analog prototype flter transer

unction;

pk is the k-th pole o the reerence analog prototype flter transer unction;

and

4 is a number o poles o the reerence analog prototype flter transer

unction and flter order as well.

Reerence analog prototype flter is always a low-pass flter. The ne*t step is

the transormation into an analog flter o appropriate type. The e*pression

used to transorm the reerence analog prototype flter transer unction

depends on the type o flter that needs to be obtained. The fnal result is

the transer unction /s0 given by:

where:

1 is a constant;

2k is the k-th 2ero o the reerence analog prototype flter transer unction;

3 is a number o 2eros o the reerence analog prototype flter transer

unction;

pk is the k-th pole o the reerence analog prototype flter transer unction;

and

4 is a number o poles o the reerence analog prototype flter transer

unction and flter order as well.5s seen' the transer unctions o reerence analog prototype flter and

analog flter are very similar. They di+er only in the value o constant 1'

the values o the transer unction poles and 2eros 2k and pk as well as in

the number o transer unction 2eros 3. The flter order is the same i the

analog flter is a low-pass or high-pass flter' whereas it is di+erent i the

analog flter is a band-pass or band-stop flter. In the later case' the analog

flter order /40 is twice that o the reerence analog prototype flter.

The ne*t step is the transormation into appropriate digital IIR flter using

bilinear transormation given by e*pression:


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,ilter $esigner Tool is used or testing and analysing the resulting IIR flters

in this chapter. 5ll data are calculated with the accuracy o ( decimal digits'which is su6cient or most e*amples.

!.). ,ilter design using 7utterworth flter

!.).. 8*ample

%tep :

Type o flter 9 low-pass flter

,ilter specifcations:

,ilter order 9 4";

%ampling reuency 9 s"1


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,irst it is necessary to compute the analog prototype flter cut-o+ reuency

>c.

The analog flter transer unction is obtained rom the reerence analog

prototype flter transer unction using e*pression:

5s the 7utterworth reerence prototype flter has no 2eros' the e*pression

or transer unction is simpler:

%tep ):

The transormation into a digital flter through bilinear transormation:

?enerally' substituting the comple* variable s into the e*pression or analog

flter transer unction' the ollowing is obtained:


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where:

2k are the 2eros o analog flter transer unction; and

pk are the poles o analog flter transer unction.

This general e*pression can be written in a simpler way in this e*ample:

5 more condensed orm o the previous e*pression is:

The result is the IIR flter transer unction.

%tep @:

The flter order is predetermined.

There is no need to additionally change it.

,ilter reali2ation:

,igure !-)-" illustrates the direct reali2ation o designed IIR flter' whereas

,igure !-)-! illustrates the reuency response o the flter obtained using

,ilter $esigner Tool.


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,igure !-)-". $igital IIR flter direct reali2ation in this e*ample

,igure !-)-!. $igital IIR flter reuency characteristic in this e*ample

!.).." 8*ample "

%tep :

Type o flter 9 high-pass flter


,ilter order 9 4!;



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The 7utterworth reerence prototype flter transer unction has no 2eros'

only poles. These can be computed via e*pression:

5s 4 !' the values o poles are:

The reerence analog prototype flter transer unction is:

%tep #:


>c.






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%tep ):




where:






%tep @:



,ilter reali2ation:

,igure !-)-( illustrates the direct reali2ation o designed IIR flter.


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,igure !-)-(. $igital IIR flter direct reali2ation in this e*ample

,igure !-)-#. $igital IIR flter reuency characteristic in this e*ample


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!.)..! 8*ample !

%tep :

Type o flter 9 band-pass flter


,ilter order 9 4(;

%ampling reuency 9 s"1c.


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%tep ):


?enerally' by substituting the comple* variable s into the e*pression or

analog flter transer unction' the ollowing is obtained:


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where:



In this e*ample' the general e*pression can be written in a simpler way:



%tep @:

,ilter order is predetermined.


,ilter reali2ation:

,igure !-)-) illustrates the direct reali2ation o designed IIR flter.

,igure !-)-). $igital IIR flter direct reali2ation in this e*ample


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,igure !-)-@. $igital IIR flter reuency characteristic in this e*ample

!.)..( 8*ample (

%tep :

Type o flter 9 band-stop flter


,ilter order 9 4(;



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Ahen designing an IIR band-stop flter' the reerence prototype flter order ishal that o the reuired IIR flter order. In this e*ample or 4 (' the order

o reerence prototype flter is "' so the values o poles are:


%tep #:


>c.




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%tep ):




where:






%tep @:



,ilter reali2ation:


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,igure !-)-B illustrates the direct reali2ation o designed IIR flter.

,igure !-)-B. $igital IIR flter direct reali2ation in this e*ample


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,igure !-)-C. $igital IIR flter reuency characteristic in this e*ample

!.)." ,ilter design using hebyshev flter

!.).". 8*ample

%tep :

Type o flter 9 low-pass flter


%ampling reuency 9 s((112;

=assband cut-o+ reuency9 c#


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progressively until some pre-determined reuirements are satisfed. Det&s

assume that the initial flter order is (.

%tep (:

The hebyshev reerence prototype flter transer unction has no 2eros'


5s 4 (' the values o poles are:

The hebyshev flter transer unction is e*pressed as:

In this case' the value o constant 51 is:

51 1."(#@so that the hebyshev reerence analog prototype flter transer unction is:


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%tep #:

,irst it is necessary to compute the analog prototype flter cut-o+ reuency>c.



5s the hebyshev reerence prototype flter has no 2eros' the e*pression or

transer unction is simpler:

%tep ):




where:





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%tep @:

7y analy2ing the resulting flter using ,ilter $esigner Tool' it is obvious that

the attenuation amounting to !."d7 appro*imately at the reuency o

B


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It is necessary to additionally redefne the flter order until the predefned

reuirements are satisfed. The flter order is incremented by and is #

thereore. 5ll steps starting with step ! are iterated.

%tep!:

The flter order is incremented in the second iteration. 5 new flter order is #.

%tep (:



5s 4 #' the values o poles are:

The hebyhsev flter transer unction is e*pressed as:


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51 9 1.""B

so the hebyshev reerence analog prototype flter transer unction is:

%tep #:


>c.

The analog flter transer unction is obtained rom the reerence analogprototype flter transer unction using e*pression:



%tep ):





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where:






%tep @:


the attenuation amounting to (.)d7 appro*imately at the reuency o

B


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,igure !-)-. ,reuency characteristic o the resulting IIR flter

It is not necessary to urther increase the flter order as this one is

appropriate. %ometimes more iterations are needed to determine the flter

order. The whole procedure is the same' only it takes more time.

,ilter reali2ation:

,igure !-)-" illustrates the direct reali2ation o designed IIR flter' whereas

,igure !-)-! illustrates the reuency characteristic o the flter obtained

using ,ilter $esigner Tool.


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,igure !-)-". $igital IIR flter direct reali2ation


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In this case' the value o constant 51 is:51 9 1.(C!


%tep #:


>c.


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5ter substitution o poles and >c into e*pression:

%tep ):




where:





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%tep @:



,ilter reali2ation:

,igure !-)-( illustrates the direct reali2ation o designed IIR flter.

,igure !-)-(. $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-#. ,reuency characteristic o digital IIR flter in this e*ample

!.)..! 8*ample !

%tep :



,ilter order 9 4 (;

%ampling reuency 9 s "1


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Ahen designing an IIR band-pass flter' the reerence prototype flter orderis hal that o the reuired IIR flter order. In this e*ample or 4 (' the

order o reerence prototype flter is "' so the values o poles are:


In this e*ample' the value o constant 51 is:

51 1.CB")


%tep #:

,irst it is necessary to compute the analog prototype flter cut-o+ reuency>c.


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%tep ):





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where:



In this e*pression' the general e*pression can be written in a simpler way:



%tep @:



,ilter reali2ation:

,igure !-)-) illustrates direct reali2ation o designed IIR flter.


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,ilter !-)-). $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-@. ,reuency characteristic o digital IIR flter in this e*ample

!.)..( 8*ample (

%tep :

Type o flter 9 band-stop flter,ilter specifcations:

,ilter order 9 4 (;



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51 1.CB")


%tep #:


>c.




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%tep ):




where:



In this e*pression' the general e*pression can be written in a simpler way:




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%tep @:



,ilter reali2ation

,igure !-)-B illustrates the direct reali2ation o designed IIR flter.

,igure !-)-B. $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-C. ,reuency characteristic o digital IIR flter in this e*ample

!.).! ,ilter design using inverse hebyshev flter

!.).!. 8*ample

%tep :

Type o flter 9low-pass flter


%ampling reuency 9 s ((112;

=assband cut-o+ reuency 9 c #


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,ilter order is not pre-determined' so it is necessary to choose an initial

solution rom which iterative method starts. The solution is redefned

progressively until some pre-determined reuirements are satisfed. Det&s

assume that the initial flter order is (.

%tep (:



5s 4 (' the values o poles are:

The inverse hebyhsev flter transer unction is e*pressed as:


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1 1.1

so that the hebyshev reerence analog prototype flter transer unction is:

%tep #:


>c.



In this e*ample' the analog flter transer unction becomes:


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%tep ):




where:






%tep @:


the attenuation amounting to #."d7 appro*imately at the reuency o

#


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,igure !-)-"1. ,reuency charactersitic o designed IIR flter

The attenuation at #


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5s 4 #' the values o poles are:

The hebyhsev flter transer unction is e*pressed as:


1 9 1.1#


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so the inverse hebyshev reerence analog prototype flter transer unction

is:

%tep #:


>c.



In this e*ample' the analog flter transer unction becomes:

%tep ):





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where:






%tep @:

7y analysing the resulting flter using ,ilter $esigner Tool' it is obvious that

the attenuation at B


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,igure !-)-". ,reuency characteristic o IIR flter

It is not necessary to urther increase the flter order as this one is

approriate. %ometimes more iterations are needed to determine the flter

order. The whole procedure is the same' only it takes more time.

,ilter reali2ation:

,igure !-)-"" illustrates the direct reali2ation o designed IIR flter' whereas

,igure !-)-"! illustrates the reuency characteristic o the flter obtained

using ,ilter $esigner Tool.

,igure !-)-"". $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-"!. ,reuency characteristic o digital IIR flter in this e*ample

!.).." 8*ample "

%tep :

Type o flter 9 high-pass flter


,ilter order 9 4 !;



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The inverse hebyshev flter transer unction is e*pressed as:


1 -1.1!


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%tep #:


>c.



In this e*ample' the transer unction is:

5ter substituting the poles and >c into e*pression:

%tep ):





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where:






%tep @:

The flter order is predetermined. There is no need to additionally change it.

,ilter reali2ation:

,igure !-)-"( illustrates the direct reali2ation o designed IIR flter.


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,igure !-)-"(. $irect reali2ation o digital IIR flter in this e*ample

,igure !-)-"#. ,reuency characteristic o digital IIR flter in this e*ample


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!.)..! 8*ample !

%tep :


,ilter specifcation:

,ilter order 9 4 (;



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1 1.1


%tep #:


>c.


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,or this e*ample' the analog flter transer unction is:

%tep ):




where:





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%tep @:



,ilter reali2ation:

,igure !-)-") illustrates direct reali2ation o designed IIR flter.

,igure !-)-"). $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-"@. ,reuency characteristic o digital IIR flter in this e*ample

!.)..( 8*ample (

%tep :

Type o flter 9 band-stop flter

,ilter specifcation:

,ilter order 9 4(;



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The inverse hebyshev reerence prototype flter transer unction has no

2eros' only poles. These can be computed via e*pression:

Ahen designing an IIR band-stop flter' the reerence prototype flter order is

hal that o the reuired IIR flter order. In this e*ample or 4 (' the order

o reerence prototype flter is "' so the values o poles are:

The inverse hebyshev flter transer unction is e*pressed as:


1 1.1


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%tep #:


>c.

The analog flter transer unction is obtained rom the reerence analogprototype flter transer unction using e*pression:

In this e*ample' the analog flter transer unction is:

%tep ):



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where:


pk are the poles o analog flter transer unction.In this e*pression' the general e*pression can be written in a simpler way:



%tep @:



,ilter reali2ation:

,igure !-)-"B illustrates the direct reali2ation o designed IIR flter.


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,igure !-)-"B. $irect reali2ation o digital IIR flter in this e*ample


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,igure !-)-"C. ,reuency characteristic o digital IIR flter in this e*ample

%hare via 8mail

Table o ontents

lose

Introduction

. Introduction to $igital ,ilter $esign

. 7asic concepts o digital fltering

.".Types o digital flters

". ,inite Impulse Response /,IR0 ,ilter

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". Introduction

"." ,inite impulse response /,IR0 flter design methods

".! Aindow unctions

".( 8*amples ".# ,inite word-length e+ects

!. Infnite Impulse Response /IIR0 ,ilters

!. Introduction

!." Infnite impulse response /IIR0 flter design

!.! Reerence 5nalog =rototype ,ilter

!.( 5nalog prototype flter to analog flter transormation

!.# 7ilinear transormation

!.) 8*amples

!.) 8*amples by 3ikro8lektronika is licensed under a reative ommons

5ttribution (.1 International Dicense' e*cept where otherwise noted.

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