lec9_iirdesign_01

7/13/2019 lec9_IIRdesign_01

1/64

Lecture 9

IIR Filter Design Part 1(chapter 8)


2/64

Main Contents

Lesson 1: Design Concepts

Lesson 2: Analog Filter

Lesson 3: Butterworth FilterLesson 4: Chebyshev Filter

Lesson 5: Elliptic Filter


3/64

Why IIR Filter ?

! Advantages of IIR filters:" Normally can achieve the same performance

(specification) using lower order filter (with the

price of nonlinear phase)

" Easier to design: directly borrow experience fromanalog filter design with s-to-zdomain mapping.

! IIR Filter Design by Pole-Zero Placement! IIR Filter Design byAnalog Filter Transformation


4/64

Analog Filter Transformation

! IIR filters have infinite-duration impulse responses, hencethey can be matched to analog filters, all of which generallyhave infinitely long impulse responses.

! Basic technique of IIR filter design: Transform analog filtersinto digital filters using complex-valued mappings.

! The advantage of this technique: analog filter design (AFD)tables and the mappings are available in the literature.

! However, the AFD tables are available only for low-passfilters while we also want to design other frequency-selective

filters (high-pass, band-pass, band-stop, etc.)! Therefore, need to apply frequency-band transformations

to low-pass filters. These transformations are also complex-

valued mappings, and they are also available in literature.


5/64

Design Strategies

! Two different approaches:" Matlab Toolbox" More details in digital domain.


6/64

IIR Filter Design Procedure

Steps:

! Design analoglowpass filter! Study and apply bilinear transformationsto obtain

digital lowpass filter

! Study and apply frequency-band transformationstoobtain other digital filters from digital lowpass filter

Main Problems

! We have no control over the phase characteristics of theIIR filter (not suitable for designing linear-phase filter)

! Thats why IIR filter designs will be treated as magnitude-only designs.


7/64

Main Contents






8/64

Preliminaries of Analog Filters

! A typical analog filter derived from a differential equation:

! s denotes the Laplace complex variable, anddenotes continuous-time analog frequency (rad/sec).

! Some Prototype Analog Filters: Butterworth, Chebyshev(Type I & II), and Elliptic lowpass filters.

ia(i)

d y(t)(i)

dt=

i=0

N

! ib(i)

d x(t)(i)

dti=0

M

!

" H(s) =ib i

si=0

M

!

ia i

si=0

N

!=

ib i

si=0

M

!

1+ia i

si=1

N

!=

(is+c )

i=1

M

#

(is+d)

i=1

N

#, where s =! +j$

f2!="


9/64

Frequency Response

! Let be the frequency response of an analog filter

! where is the passband ripple parameter, is the passbandcutoff frequency in rad/sec, A is the stopband attenuation parameter,

and is the stopband cutoff freq.

)( !jH

!"!"!"

!"!"!"+

,1

|)(|0

,1|)(|1

1

22

2

2

s

p

AjH

jH#

!p

!

s!


10/64

Relative Linear Scale

! Note that the power transfer function must satisfy:

! Relations between:

! Relations between:

s

p

AjH

jH

!=!=!

!=!+

=!

at,1

|)(|

at,1

1|)(|

2

2

2

2

"

2|)(| !jH

10210

10210

10

1

log10

1101

1log10

s

p

A

s

R

p

AAA

R

=!"=

"=!

+

"= #

#

2

1

1

2

1

1

2

2

1

11

1

1

2

1

1

1

1

!

!

!

!

!

!"

"!

!

+

=#=

+

$

=#

+

=

+

$

AA


11/64

Magnitude-Squared Only Specification

! Note that only the power transfer function isgiven (no phase info.), but we are interested in

evaluating s-domainsystem function H(s):

! For example:

2)( !jH

j

s

js

js

jHsHsH

sHsHjHjHjHjHjH

sHjH

=!

!=

!=

!="

"=!"!=!!=!

=!

||)(|)()(or

)()()()()()(|)(|

)()(

2

*2 |

|

))((

11||)(|)()(So

0,1

|)(|

22

2

22

2

sasasajHsHsH

aa

jH

j

s!+

=

!="=!

>

"+="

="


12/64

Pole-Zero Pattern Analysis

! The poles and zeros of the magnitude-squared functionare distributed in a mirror-image symmetry with respectto the axis

! In other words, H(s)H(-s) has the pair of poles {p, -p}and zeros {q, -q},which are across the origin of s-Plane.

! Also due to H(s) has real coefficients which lead toconjugate pole-zero pairs.

! A typical pole-zero pattern :

!j


13/64

! From this pattern we can construct H(s), which is thesystem function of our analog filter

! We want H(s)to represent a causaland stablefilter. Itmeans all poles of H(s)must lie within the left half-plane.Thus we assign all left-half polesof H(s)H(-s) to H(s).

!However, zeros of H(s) can lie anywhere in s-plane.Thus We will choose the zeros of H(s)H(-s)lying left to oron the axis as the zeros of H(s).

! The resulting filter is then called a minimum-phase filter.

Pole-Zero Pattern Analysis

!j

asH(s)

-apap

+

=

==

1

choose,


14/64

Characteristics of Analog Filters

! Briefly summarize characteristics of lowpass versionssome prototype analog filters

! Some Prototype Analog Filters:# Butterworth# Chebyshev (Type I & II)# Elliptic lowpass filters.

! Available Matlab functions to design these filters but needto understand all parameters of the filters


15/64

Main Contents






16/64

Butterworth Lowpass Filter

! A filter with flat passband and stopband responses. TheNth-order Butterworth lowpass filter has the magnitude-

squared response as:

rad/secinfreq.cutoffdB)(3theis,

1

1|)(|

2

2

cN

c

jH !

""#$%%

&'!!+

=!

2


17/64

Butterworth Lowpass Filter Properties

2


18/64

Poles of Butterworth Lowpass Filter

! To determine the system function H(s), re-write:

! The roots (poles) ofH(s)H(-s):

N

c

N

N

c

N

c

j

sjs

j

j

sjHsHsH

22

2

2

2

)(

)(

)(1

1||)(|)()(

!+

!=

!+

=!="=!

12,....,2,1,0,

)()()()1(

)12(2

22

1

22

1

!="=

"="!=

++

Nke

eeejp

NkN

j

c

c

jNkjj

c

N

k#

#

##


19/64

Some Observations


20/64

Butterworth Lowpass Filter Design! The causal and stable filter H(s)is built by selecting poles

in the left half-plane:

! Example 8.1: Design a Butterworth filter of order-3 (N=3):! "#

=

polesLHP

k

Nc

pssH

)()(

)25.05.0)(5.0(

125.0

)433.025.0)(5.0)(433.025.0(

125.0

))()(()(

5.0,

)5.0

(1

1

641

1|)(|

2

432

3

)3(26

2

+++=

+++!+=

!!!

"=

=""

+

=

"+="

sss

jssjs

pspspssH

jH

c

c


21/64

Matlab Implementation

! "#

=

polesLHP

k

N

c

pssH

)()(


22/64

Normalized Butt Analog Filter


23/64

Direct Form to Cascade Form


24/64

Example 8.2

|H(j!) |2=

1

1+ 64!6 =

1

1+ (!

0.5

)2(3)

, !c = 0.5

H(s)=!c

3

(s"p2 )(s"p3)(s"p4 )

=

0.125

(s+ 0.5)(s2 + 0.5s+ 0.25)

! Recall:


25/64

Butterworth Filter Design Procedure

! Given as specs of analog filter, find

! Solving the above two equations to obtain

! Commonly choose as the value in between.

spsp AR ,,,!!

N

c

ss

N

c

pp AR

210

2

10

)(1

1log10,

)(1

1log10

!

!+

"=

!

!+

"=

cN !and

(stopband)

110

,(passband)

110

,

log2

110

110log

2 10

c

2 1010

10

10

10

N

A

s

N

R

p

c

s

p

A

R

sp

s

p

ceilN

!

"="

!

"="

####

$

####

%

&

####

'

####

(

)

*+,-

./"

"

***

+

,

---

.

/

!

!

=

c!

cN !and


26/64

Example 8.3

! Design a low-pass Butterworth filter to satisfy:

dBA

dBR

ss

pp

16,3.0

7,2.0

==!

==!

"

"

N = ceil

log10100.7 !1

101.6 !1

"

#$

%

&'

2log100.2!

0.3!

"

#$

%

&'

(

)

**

+

**

,

-

**

.

**


27/64

! We can select the value in between as follows:


28/64

Matlab: Butterworth Lowpass Filter Design

! Note: Passband cut-off freq. is selected in Matlab


29/64

Example 8.4

! Design a low-pass Butterworth filter in Example 8.3using Matlab. We can analyze the problem as follows:

dBA

dBR

ss

pp

16,3.0

7,2.0

==!

==!

"

"

N = ceil

log10100.7 !1

101.6 !1

"

#$

%

&'

2log100.2!

0.3!

"

#$%

&'

(

)

**

+**

,

-

**

.**

= ceil{2.79} = 3, /0c =

0.2!

100.6 !16 = 0.4985 (passband);

(Check this Nand 0cto see if they satisfy the stopband specs)

The transfer function will be then constructed as follows:

)2485.04985.0)(4985.0(

1238.0)(

2+++

=

ssssH


30/64

Example 8.4 Matlab


31/64


32/64

Example 8.4 - Remark


33/64

Main Contents






34/64

Chebyshev Lowpass Filter

! Butterworth filters have monotonic response in both bands.! From equiripple FIR design (remez), we know equiripple

designs can result in lower-order filters.

! Chebyshev-I filtershave equiripple response in thepassband, while Chebyshev-II filtershave equiripple

response in the stopband.

! Magnitude-squared response of a Chebyshev-Ifilter:

!"#

$


35/64

Equiripple Response


36/64

Response Properties


37/64

Chebyshev-I Design


38/64

Chebyshev-I Design


39/64


! We design an un-normalized Chebyshev-I analogprototype filter returning Ha(s)in direct form:


40/64

Chebyshev-I Design Equations

! Given of analog filter, find

! The order N is given by:

spsp AR ,,,!! !, Nand !c


41/64

Example 8.5


42/64

Example 8.5


43/64

Example 8.5


44/64

Matlab Implementation - AFD


45/64

Example 8.6

! Design an analog Chebyshev-I lowpass filter given inExample 8.5 using Matlab

! Recall the result from Example 8.5:


46/64

Example 8.6


47/64


48/64

Chebyshev-II Lowpass Filter

! The Chebyshev-II filter hasequiripples in stopband:

! This filter has zeros on theimaginary axis.

)(1

)(

)(1

1|)(|

22

22

1

22

2

!!+

!

!

=

"#$%&

'!!+

=!(

cN

cN

cN

T

T

T

jH

)

)

)


49/64

Properties


50/64



51/64



52/64

Example 8.7


53/64

Example 8.7


54/64


55/64

Main Contents



Lesson 3: Butterworth Filter

Lesson 4: Chebyshev Filter



56/64

Elliptic Lowpass Filters

!All-pole analog filters, such as Butterworth or Chebyshev-I, contain no zeros to have quick drop in response in

stopband. Therefore the transition band is thus wider.

! These Elliptic filters exhibit equiripple behavior in thepassband as well as in the stopband

! Similar to the FIR equiripple filters. Thus they areoptimum filters in that they achieve the minimum order N

! It is therefore difficult to analyze and design.


57/64

Elliptic Lowpass Filters

! The magnitude-squared response of elliptic filters is given:

where Nis the order, is the passband ripple, andis the Nth-order Jacobian elliptic function.

)(1

1|)(|

22

2

c

NU

jH

!

!+

=!

"

!N

U


58/64

Elliptic Design Equations

! The order filter is given by:

where:


59/64



60/64



61/64

Example 8.8

! The same lowpass filter to be designed with elliptic procedure:

dBA

dBR

ss

pp

16,3.0

1,2.0

==!

==!

"

"


62/64

Example 8.7


63/64


64/64

Phase Responses of Prototype Filters

! The Elliptic filters provide optimal performance in themagnitude-squared response, but have highly nonlinear phaseresponse in the passband (not desirable).

! The Butterworth filter, while having maximally flat magnituderesponse & requiring higher-order N (more poles), exihibit a

fairly linear phase response in passband.! The Chebyshev filters having phase characteristics lie

somewhere in between (Butter & Elliptic).

! In conclusion, the choice depends on both the filter order(impacts processing speed & complexity) and the phasecharacteristics (relates phase distortion).

lec9_iirdesign_01

Documents