dsp lect 11 iir design

7/23/2019 DSP Lect 11 Iir Design

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Discrete Time Systems

Lecture # 11

IIR Filter Design



Filter Design Specifications

• The filter design process

Dr Sajjad Zaidi - DS !

"nalysis Design Implementation

$agnitude Response

hase response%ost&comple'ity

FIR&IIR

Su(type order

Transfer

function

r o ( l e m

erformance

%onstraints

latformstructure



erformance %onstraints

• )) in terms of magnitude response*

Dr Sajjad Zaidi - DS +



erformance %onstraints

• ,est Filter

– improing one usually .orsens others

• ,ut* increasing filter order /i)e) cost0 – improes all three measures

Dr Sajjad Zaidi - DS

Transition

,and

ass(andRipple

$in Stop(and

"ttenuation

2arro.est

Smallest

3reatest



ass(and Ripple

• "ssume pea4 pass(and gain 5 1

• then minimum pass(and gain 5

• 6r7 ripple

Dr Sajjad Zaidi - DS 8



Stop(and Ripple

•

ea4 pass(and gain is "9 larger than pea4 stop(and gain• :ence7 minimum stop(and attenuation

Dr Sajjad Zaidi - DS ;



Filter Type %hoice FIR s IIR

Dr Sajjad Zaidi - DS <

FIR IIR2o Feed(ac4/just =eros0

Feed(ac4/poles > =eros0

"l.ays sta(le $ay (e unsta(le%an (e linear phase Difficult to control phase

:igh order /!? @ !???0

Typ A 1&1?6rder of FIR /-!?0

Bnrelated to %T filtering Derie from analogprototype



FIR s) IIR

• If you care a(out computational cost –

Bse lo.-comple'ity IIR – %omputation no o(ject C L FIR

• If you care a(out phase response – Bse linear-phase FIR

– hase unimportant C go .ith simple IIR0




INFINITE IMPULSE RESPONSE FILTERS

Dr Sajjad Zaidi - DS E



IIR Filter Design

• IIR filters are directly related to –

analog filters /continuous time0 – ia a mapping of :/s0 /%ontinuous Time0

• to :/=0 /Discrete Time0

– It preseres many properties

• "nalog filter design is sophisticated – decades of pre-DS signal processing

• Design IIR filters ia analog prototype –

hence7 need to learn some %T filter design

Dr Sajjad Zaidi - DS 1?



"nalog Filter Design

• Decades of analysis of transistor-(ased –

filters @ sophisticated7 .ell understood• ,asic choices*

– ripples s) flatness in stop and&or pass(and

– more ripples C narro.er transition (and




%T Transfer Functions

• "nalog systems* s-transform /Laplace0

Dr Sajjad Zaidi - DS 1!



,utter.orth Filters

• $a'imally flat in (oth pass > stop (ands

Dr Sajjad Zaidi - DS 1+



,utter.orth Filters




,utter.orth Filters

• :o. to meet design specifications




,utter.orth Filters

Dr Sajjad Zaidi - DS 1;



,utter.orth G'ample

Dr Sajjad Zaidi - DS 1<

Design a ,utter.orth filter

.ith 1 d, cutoff at 14:= and

a minimum attenuation of ?

d, at 8 4:=



Steps in Filter Design

• The general filter design pro(lem can (e (riefly stated as

follo.s)

H3ien some ideal freuency response7 D/f07 find a reali=a(le IIR or

FIR digital filter .hose freuency response7 :/f07 appro'imates D/f0J

• The reali=a(le filter is found (y optimi=ing some measure

of the filterKs performance7 e)g – $inimi=ing the filter order /IIR0 or the filter length /FIR07 – $inimi=ing the .idth of the transition (ands7

– Reducing the pass(and error and&or stop(and error

• Setting up the specifications for the general filter design

pro(lem .ill define these parameters and sho. .hichtrade-offs are possi(le)




,utter.orth G'ample

• 6rder 2 5 .ill satisfy constraints

• Mhat are Nc and filter coefficients

– from a loo4up ta(le7 N-1d, 5 ?)8 .hen Nc 5 1

• So Nc 5 1???&?)8 5 1)1 4:=

– from a ta(le7 get normali=ed coefficients for

– 2 5 7 scale (y 11

• 6r7 use $atla(*

– O(7aP 5 (utter/27Qc7KsK0

Dr Sajjad Zaidi - DS 1E



"nalog Filter Types Summary

Dr Sajjad Zaidi - DS !?


http://slidepdf.com/reader/full/dsp-lect-11-iir-design 21/50Dr Sajjad Zaidi - DS !1


http://slidepdf.com/reader/full/dsp-lect-11-iir-design 22/50Dr Sajjad Zaidi - DS !!



Steps in Filter Designing

• Follo.ing are the steps to design a filter

1) %reate the Design Specifications

!) 6(tain Specs Deried from "nalog Filtering

+) Specifying an Grror $easure

) Select the Filter Type and 6rder

8) Design the Filter

;) Reali=e the Designed Filter

Dr Sajjad Zaidi - DS !+



Infinite Input Response Filter

• The impulse response h(n) has an infinite number of non-

zero samples (infinite length)

– "s an e'ample7 for a general IIR filter7 h(n)≠0 only for N o<n<∞,

where N o is a non-negatie integer

• The freuency response H(ω) is a rational funtion, i!e!, a

ratio of two finite-"egree polynomials in e #ω of the form

.here N o is an integer onstant!

• $he or"er of an %%& filter is e'ual to N, whih is the degree

of the denominator

• Bsually the degree of numerator is not greater than N





• &emember – $he or"er N also "etermines the number of preious output

samples that need to (e stored and then fed (ac4 to compute the

current output sample)

– Therefore7 IIR systems are also 4no.n as feedback systems!

– $he filter oeffiients *bn + an" *an + in the ' correspond to the

un4no.n parameters of the design)

Dr Sajjad Zaidi - DS !8




• Designing an IIR filter amounts to finding the rational

function

• $he H(ω ) shoul" best approimates the design

specifications)• In the freuency domain7 this is done (y computing the

HoptimalJ /relatie to some criteria0 coefficients bn + an" *an +

• $he filter or"er N is usually fie", but an also be

onsi"ere" as a free parameter to (e optimi=ed)

Dr Sajjad Zaidi - DS !;




• Filter transfer function7 denoted (y :/=07 is the =-transform

of h/n0 and is useful for studying the sta(ility of the system)

• For LSI filters7 sta(ility implies that a bounded input to the

filter .ill al.ays result in a bounded output)

• For IIR filters7 :/=0 is a rational function in the comple'

aria(le = and is gien (y

Dr Sajjad Zaidi - DS !<




• ,ecause the impulse response is infinitely long7

conolution can no longer (e used to implement the IIR

filters)

• Instead7 IIR filters are efficiently implemented using

fee"ba. "ifferene e'uations

• The noise characteristics of an IIR filter can (e a major

consideration .hen doing an implementation7 especially in

fi'ed-point arithmetic)

• %oefficient uanti=ation degrades the actual filter response

from that designed (y high-precision soft.are)

• $ore critical is round-off noise sensitiity .hich can (e

amplified (y the feed(ac4 loops in the filter)




Infinite Input Response Filter - ros• %ompared to FIR filters7

– IIR filters can achiee the desired design specifications .ith a relatiely lo. order /as fe.

as to ; poles0)

• Fe.er un4no.n parameters need to (e computed and stored7 .hich might

lead to a lo.er design and implementation comple'ity)

• :o.eer7 the phase response of IIR filters is neer linear7 .hich leads to the

use of all pass filters to compensate the group delay7 and – Thus raises the order of the filter and

– The comple'ity of the design process)

• IIR filters are commonly designed (y using closed-form design formulas

corresponding to classical filter types

– The order-estimating formulas for IIR filters are e'act since they are deried from

the mathematical properties of the classical prototypes)

– These formulas are ery useful to o(tain the IIR filter order needed to satisfy thedesired design specifications)

Dr Sajjad Zaidi - DS !E



Designing the Filter

• IIR filters are %ommonly designed (y – Transforming the digital design specifications in to analog design

specifications

– erforming the filter design in the analog domain)

– Transforming the analog filter into a digital filter using a suita(le

transformation)

•

opular $ethod are – ,ilinear Transformation method

1) ,utter.orth7

!) %he(yshe I

+) %he(yshe II

) Glliptic – Digital-only IIR design methods

Dr Sajjad Zaidi - DS +?



Dr Sajjad Zaidi - DS +1

G'ample @ IIR Filter Design

uestion Design a digital lo.-pass ,utter.orth filter .ith a +d, cut-off freuency

of !4:= and minimum attenuation of +?d, at )!84:= for a sampling rate of 1?4:=)



Dr Sajjad Zaidi - DS +!

R li i th D i d Filt



Reali=ing the Designed Filter

• Reali=ing the designed digital filter corresponds to

computing the output of the filter in response to any gien

input)

• For LSI filters7 this is simplified (y the fact that the input

and output signals are related through a simple conolution

operation in the time&space domain)

• If '/n0 is the input7 y/n0 the corresponding output7 and h/n0

the impulse response of the LSI filter7 then this relation is

gien (y

.here 2172! are the indices of the 1st and last non-=ero samples of h/n0)

Dr Sajjad Zaidi - DS ++

R li i th D i d Filt



Reali=ing the Designed Filter

• In the freuency /Fourier transform0 domain7 the

conolution relation /11)10 corresponds to multiplication of

the respectie Fourier transforms*

.here /Q07 :/Q07 and U/Q0 are the DTFT of '/n07 h/n07 and y/n07 respectiely)

• ,ut The aria(le is continuous and7 therefore7 it cannot (e

implemented in practice

• "n implementa(le ersion is o(tained (y using the Discrete

Fourier Transform /DFT07 .hich is a sampled ersion of the

DTFT and .hich consists of samples of the DTFTealuated at the points


20, , 1

DFT

k k N

N

π ω = = −K

2DFT is the si=e of the DFT and

corresponds to the num(er of

sample points .ithin the period !V

Si l IIR L



Simple IIR Lo.pass

• IIR C feed(ac47 =eros and poles7

– conditional sta(ility7 hOnP less useful

+8

1

1

1[ ]

1 LP

z H z K

z α

−

−

+=

−

scale to ma4egain 5 1 at Q 5 ?

pole-=erodiagram freuencyresponse FR onlog-log a'es

W 5 /1 - X0&!


Si l IIR L



Simple IIR Lo.pass

• %utoff fre) Qc from

Dr Sajjad Zaidi - DS +;

1

1

1

[ ] 1 LP

z H z K

z α

−

−

+=

−

Si l IIR :i h



Simple IIR :ighpass

Dr Sajjad Zaidi - DS +<

1

1

1[ ]

1 HP

z H z K

z α

−

−

−=

−

ass Q5V

::/-1051W5/1YX0&!

:i h d L



:ighpass and Lo.pass

• %onsider lo.pass filter*

•

then

• :o.eer

/unless :/e j.0 is pure real @ not for IIR0


Si l IIR , d



Simple IIR ,andpass

Dr Sajjad Zaidi - DS +E

Si l Filt G l



Simple Filter G'ample

Tas4

–

Design a second order IIR (andpass filter .ith Qc5?)V7+d, (and.idth of ?)1V

Dr Sajjad Zaidi - DS ?

Simple IIR ,andstop



Simple IIR ,andstop


%ascading Filter



%ascading Filter

• Repeating a filter /cascade connection0 ma4es its

characteristics more a(rupt*


Repeated roots in =-plane*

%ascading Filter



%ascading Filter

• %ascade systems are higher order

• e)g) longer /finite0 impulse response*

• In general7 cascade filters .ill not (e optimal for a gien

order


Linear phase Filters



Linear-phase Filters

• :/e jQ0 alone can hide phase distortion – differing delays for adjacent freuencies can mangle the signal

• refer filters .ith a flat phase response – e)g) [/Q0 5 ? H=ero phase filterJ

• " filter .ith constant delay \p 5 D at all freKs has

– [/Q0 5 ]DQ Hlinear phaseJ

• Linear phase can ^shiftK to =ero phase


Time reersal filtering



Time reersal filtering

• "chiees =ero-phase result

• 2ot causal_ 2eed .hole signal first

8

x[n] :/=0Time

reersal:/=0

Time

reersal

w[n

]

u[n]v[n]

y[n]


Some Filter Types



Some Filter Types

• Me hae seen the (asics of filters and a range of

simple e'amples• 2o. loo4 at a couple of other classes*

– %om( filters - multiple pass&stop (ands

– "llpass filters - only modify signal phase

Dr Sajjad Zaidi - DS ;

%om( Filter



%om( Filter

• Replace all system delays =-1 .ith longer delays =-L

• C System that (ehaes ^the sameK at a longer timescale

Dr Sajjad Zaidi - DS <

%om( Filter



%om( Filter

• ^arentK filter impulse response hOnP (ecomes com( filter

output – gOnP 5 hO?P ? ? ? ? hO1P ? ? ? ? hO!P))`

• Thus7

• Thus freuency response*

• :igh-pass response C

• pass Q 5 V&L7 +V&L7 8V&L)))

• cut Q 5 ?7 !V&L7 V&L)))


"llpass Filter



"llpass Filter

• "llpass filter has "/e jQ0! 5 W Q∀ – i)e) spectral energy is not changed

• hase response is non=ero /elsetriial0 – phase correction or special effects

• e)g)

• "llpass has special form of systemfn

• "$/=0 has poles .here D$/0 5 ?

C "$/=0 has =eros b 5 1&

Dr Sajjad Zaidi - DS E



3ood Luc4

dsp lect 11 iir design

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