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CHAPTER 5 CHAPTER 5

INFINITE IMPULSE RESPONSE FILTERINFINITE IMPULSE RESPONSE FILTER

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CONTENTSCONTENTS

5.1 Brief Introduction to IIR5.1 Brief Introduction to IIR

5.2 5.2 Impulse InvarianceImpulse Invariance

5.3 5.3 Bilinear TransformationBilinear Transformation

5.4 5.4 Analog-Digital TransformationAnalog-Digital Transformation

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( )x n ( )h n ( )y n

FIR--h(n) finite length

IIR--h(n) infinite length

Recursively Non-recursively

0 1

( ) ( ) ( )N N

i ii i

y n a x n i b y n i

1

0

N

i

)in(x)i(h)n(y

N

n

n

H z h n z

1

0

N

k

ki

M

k

ki

zb

za)z(H

1

0

1

Digital Filter

Different methods to design IIR filter and FIR filter !

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Process of filter designProcess of filter design

Three basic stepsThree basic steps ：：

• Requirement analysisRequirement analysis ：：

Type/specificationType/specification

(1) Lowpass / Highpass (1) Lowpass / Highpass / / Bandpass / BandstopBandpass / Bandstop

((22) ) Bandedge frequencyBandedge frequency

((33) ) Passband ripple Passband ripple / / Stopband attenuationStopband attenuation

• Objective of digital filter is to develop a Objective of digital filter is to develop a

casual casual

and stable transfer function and stable transfer function H(z)H(z) meeting meeting

the the

frequency response specification. frequency response specification.

• Implementation of H(z)Implementation of H(z).

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BASIC SPECIFICATIONS FOR DIGITAL FILTERBASIC SPECIFICATIONS FOR DIGITAL FILTER

)(jjj e|)e(H|)e(H phase-frequencyAmplitude-frequency

Attenuation of frequency components

Delay of frequency components

Transition

|)e(H|lg|)e(H|lg

s

p

js

j

p

2020

dBp 3

c — 3dB passband cutoff frequency

Pass bandStop band

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N

ii

M

ii

N

i

ii

M

i

ii

)zd(

)zc(A

zb

za)z(H

1

1

1

1

0

0

1

1

1

5.1 5.1 Basic Approach to IIRBasic Approach to IIR digital filter digital filter designdesignAn order N IIR digital filter’s system function is:An order N IIR digital filter’s system function is:

Objective of H(z):Objective of H(z): Determine coefficient Determine coefficient aai i andand bbi i or zero or zero

and pole point and pole point ci ci and and didi, in order to meet design requirements. , in order to meet design requirements.

Three approaches to IIR digital filter’s transfer function designThree approaches to IIR digital filter’s transfer function design

Estimation of transfer functionEstimation of transfer function

Iterative optimization techniqueIterative optimization technique

Analog filter theoryAnalog filter theory

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Pole inside unit circle Pole inside unit circle PeakPeak in frequency response in frequency response

Zero inside unit circle Zero inside unit circle ValleyValley in frequency response in frequency response

1 1 Approach to filter designApproach to filter design

(1)(1)Estimation of transfer function from diagramEstimation of transfer function from diagram

Geometric evaluationGeometric evaluationPole-zero diagramPole-zero diagram

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（（ 22 ） ） Iterative optimization techniqueIterative optimization technique

Iterative optimization technique are used to minimize the Iterative optimization technique are used to minimize the

error between the desired frequency response and computing error between the desired frequency response and computing

generated filter.generated filter.

N

ii

M

ii

N

i

ii

M

i

ii

zd

zcA

zb

zazH

1

1

1

1

0

0

)1(

)1(

1)(

)e(H jw

M

i

jwd

jw |)e(H||)e(H|1

2

Large computation Large computation

costcostComputer Computer

aidedaided

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（（ 33 ）） Application of analog filter Application of analog filter

theorytheoryAdvantage: Advantage:

(1)(1) Analog approximation techniques are highly advanced.Analog approximation techniques are highly advanced.

(2)(2) They usually yield closed form solutions.They usually yield closed form solutions.

(3)(3) Extensive tables are available for analog filter design.Extensive tables are available for analog filter design.

PrinciplePrinciple :: DF requirementDF requirement AF requirement AF requirement AF’s HAF’s Haa(s) (s) DF’s H (z) DF’s H (z)

Basic approach: Basic approach: Impulse invariantImpulse invariant;

Bilinear transformation.Bilinear transformation.

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模拟滤波器的设计方法模拟滤波器的设计方法 （（教材 教材 6.26.2

））模拟滤波器需求模拟滤波器需求 Requirement

Ha(s)

Butterworth filterButterworth filter Elliptic filterElliptic filter

Chebyshev Chebyshev II

Chebyshev IIChebyshev II

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四种模拟滤波器的比较四种模拟滤波器的比较 幅频特性：幅频特性：

巴特沃斯 — 整个频带内单调下降；巴特沃斯 — 整个频带内单调下降； 切比雪夫切比雪夫 I — I — 通带内等纹波振动，过渡带通带内等纹波振动，过渡带 // 阻带单调下降；阻带单调下降； 切比雪夫切比雪夫 II —II — 阻带内等纹波振动，过渡带阻带内等纹波振动，过渡带 // 通带单调下降；通带单调下降； 椭圆—除过渡带外，通带和阻带都等纹波振动。椭圆—除过渡带外，通带和阻带都等纹波振动。

过渡带特性：过渡带特性： 巴特沃斯—最差；巴特沃斯—最差； 切比雪夫切比雪夫 II ，， II—II— 居中；居中； 椭圆—最陡；椭圆—最陡；

设计复杂性：设计复杂性： 巴特沃斯—相同条件下，阶数最高；巴特沃斯—相同条件下，阶数最高； 切比雪夫切比雪夫 II ，， IIII—— 居中；居中； 椭圆—相同条件下，阶数最低；椭圆—相同条件下，阶数最低；

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2 2 Two basic rules from analog to digital Two basic rules from analog to digital

filterfilter

（（ 11 ）） Imaginary axis Imaginary axis jj in s-plane be mapped onto in s-plane be mapped onto

unit unit

circle ecircle ejw jw of z-plane. (Hof z-plane. (Haa(s) (s) H(z)) H(z))

（（ 22 ）） A stable and casual analog transfer function be A stable and casual analog transfer function be

transformed into a stable and casual digital transfer transformed into a stable and casual digital transfer

function. That is:function. That is:

Re[s]<0 in s-plane Re[s]<0 in s-plane unit circle unit circle ||z|<1 in z-plane.z|<1 in z-plane.

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5.2 5.2 Impulse invariant (Time domain)

Sampling TheoremSampling Theorem

1. Transformation

Approximate impulse response ha(t) of analog

filter using unit impulse response h(n) of digital

filter. Let h(n) equal to ha(t)’s sampling value,

that is:

h(n)=ha(t)|t=nT

If transfer function of analog filter is Ha(s),

then transfer function of required digital filter is:

H(z)=ZT[L-1 [(Ha(s)]|t=nT]

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5.2 5.2 脉冲响应不变法 脉冲响应不变法 Impulse invariant Impulse invariant

methodmethod1. 变换思想 Impulse Invariant Method

使数字滤波器的单位脉冲响应序列 h(n) 模仿模拟滤波器的冲激响应

ha(t) 。

ah t

ah n h nT

aH s ax t ay t

H z x n y n

Analog filter

Analog filter

Digital filter

Digital filter

aH j

Analog frequency response

Analog frequency response

jH e

Digital frequency response

Digital frequency response

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( )aH s

( )H z

is Te数字滤波器数字滤波器 z-z- 平面的极点平面的极点

2 数字化设计过程

1

( ) ( )i

Ns t

a ii

h t A e u t

1

( ) ( ) ( )i

Ns nT

a ii

h n h nT A e u nT

1

Ni

i i

A

s s

Inverse Laplace

z-transform

模拟滤波器模拟滤波器 s-s- 平面的极点平面的极点 is

s s 平面平面 z z 平面平面

11 1 i

Ni

s Ti

A

e z T

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aH s

ah t ah nT h n

aH s H z

ˆ ( ) ( ) snTa

n

H s h nT e 1

( ) ( )N

n

i

H z h n z

1 2ˆ ( ) ( )a am

H s H s j mT T

sTz ePeriodic copies

3. s 平面到 z 平面映射 Mapping from s-plane to z-

plane

Sampling

aH s 沿虚轴周期延拓之后，使用 映射到 z 平面沿虚轴周期延拓之后，使用 映射到 z 平面sTz e H z

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( )sT j j T T j Tz e re e e e = =

Mapping between z-plane and s-Mapping between z-plane and s-

planeplane

Tr e 系统因果稳定系统因果稳定 T 模拟频率和数字频率模拟频率和数字频率

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3 数字滤波器与模拟滤波器频响关系

若 AF 的频响是带限的，且 :T

||,)j(Ha

0

此时 DF 的频响才能不失真重现 AF 频响。

T ah t

ah nT

h n

( )

1 2( )

j

am

H e

H j j mT T T

aH jΩ

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3 数字滤波器与模拟滤波器频响关系

实际模拟滤波器的频响是非带限的 频谱混叠（该方法严重的缺点）

适用范围：低通，带通。

周期延拓

周期延拓以后，混叠严重，难以转换到数字域 !

高通模拟滤波器

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Characteristic of Impulse invariantCharacteristic of Impulse invariant

（（ aa ） ） Keep analog filter’s Keep analog filter’s transient signal in time transient signal in time

domain. domain.

（（ bb ） ） Linear relationship between digital frequency and Linear relationship between digital frequency and

analog frequency.analog frequency.

（（ cc ） ） HHaa(()) must be band limited on must be band limited on , otherwise , otherwise

distortion will happen in digital frequency domain. distortion will happen in digital frequency domain.

Attention: This method is not suitable in following Attention: This method is not suitable in following

situation:situation:

(1) (1) HHaa(() is not band limited or h) is not band limited or haa(t) change unstably, (t) change unstably,

and the design requirement is very highand the design requirement is very high

(2) Highpass filter and bandstop filter.(2) Highpass filter and bandstop filter.

TT

,

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5.3 5.3 Bilinear transformationBilinear transformation

Disadvantage of impulse invariance: aliasing.Disadvantage of impulse invariance: aliasing.

js)s(Ha ，

111 js)s(Ha ，

Ttan

T 12

12

001 从时，从TT

Tangent transformation

Maps the entire axis in s-plane Maps the entire axis in s-plane

to narrow band in sto narrow band in s11 plane. plane.T

j

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Image of left half plane

Mapping of the s-plane into the z-plane

1

1

2 1

1

s T

s T

es

T e

1

1

1

12

z

z

Ts

Tsez 1

s

Ts

Tz

22Bilinear transformation

AFAF1

12)()(

z

z

Ts

a sHzH DFDF

s plane s1 plane z plane

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（（ 22 ）） Nonlinear relationship of phase Nonlinear relationship of phase transformationtransformation

2

12

2

121 tan

TTtan

Tzss 1

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Frequency nonlinear change of bilinear transformationEffect of frequency warping .Effect of frequency warping .

Prewarp critical bandedge frequency.Prewarp critical bandedge frequency.

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（（ 33 ）） Features of bilinear Features of bilinear

transformationtransformation（（ aa ） ） No No amplitude-frequencyamplitude-frequency distortion after transformation, no distortion after transformation, no

requirement for Hrequirement for Haa(()’s bandwidth;)’s bandwidth;

（（ bb ） ） Simple design; Simple design;

（（ cc ）） Non-linear relationship between digital frequency and analog Non-linear relationship between digital frequency and analog

frequency;frequency;

（（ dd ） ） Frequency warpingFrequency warping can be revised by prewarping method.

Transient response Transient response impulse invarianceimpulse invariance

Other situations Other situations bilinear transformationbilinear transformation

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（（ 44 ）） Procedures from AFilter to Procedures from AFilter to

DFilterDFilter(1) Specification

(2) Transformation ： digital analog

Impulse invariance ： =T

Bilinear transformation

(3)Analog filter design

(4)Analog filter Digital filter

2

12tan

T

)z(H)s(Ha

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例：使用双线性变换法设计一个例：使用双线性变换法设计一个 IIRIIR 滤波器，要求：滤波器，要求： （（ 11 ）通带阻带具有单调下降的特性；）通带阻带具有单调下降的特性；

（（ 22 ））

（（ 33 ））

1 2

2 1 1 21

0.06745 0.1349 0.06745( ) ( )

1 1.143 0.4128s

a zs

T z

z zH z H s

z z

第四步：求第四步：求 HH (z)(z)

: 100Hz, 300Hz, 1000 Hzp s sDF f f F

3dB, 20dBp s

解：解： 第一步：临界数字频率第一步：临界数字频率 :: 0.2 , 0.6 ,p s

第二步：临界模拟频率第二步：临界模拟频率 ::2

tan( / 2) 685.8 2 109(Hz)

2tan( / 2) 2452.76 2 438(Hz)

p ps

s ss

T

T

第三步：选择巴特沃斯滤波器，根据 和 求第三步：选择巴特沃斯滤波器，根据 和 求 HHaa(s)(s), , p s p s

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5.4 5.4 Prototype Prototype transformationtransformation

Typical filter’s “ideal” amplitude-frequency characteristics. Typical filter’s “ideal” amplitude-frequency characteristics.

PrototypePrototype transformation:transformation:

Analog lowpass filterAnalog lowpass filter

Types of digital filterTypes of digital filter

Prototype filterPrototype filter

—— —— Analog lowpass filterAnalog lowpass filter

Lowpass

Highpass

Bandpass

Bandstop

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（（ 11 ）） High-pass, Band-pass, Band-stopHigh-pass, Band-pass, Band-stop

a. Design methoda. Design method

模拟低通原型模拟低通原型 模拟高通、带模拟高通、带通、带阻通、带阻

数字高通、带数字高通、带通、带阻通、带阻

模拟低通原型模拟低通原型 数字高通、带数字高通、带通、带阻通、带阻

模拟低通原型模拟低通原型 低通数字原型低通数字原型 数字高通、带数字高通、带通、带阻通、带阻

（（ 11 ））

（（ 22 ））

（（ 33 ））

aH s H z z f s

aH s 1H z

z f s

2H Z

)Z(gz 11

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22 ）） Direct transformDirect transform

HP transformHP transform

MappingMapping

1

1

1

12

z

z

Ts

0 1

0 1

s plane origin z

s j z

，，

2

ctg

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Prototype transform of highpassPrototype transform of highpass

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22 ）） Direct transformDirect transform

BP transformBP transform

TransformTransform

MappingMapping0

0 00

0 1

js plane origin z e

s j z

，，

sin

coscosz

coszz

zz

ezezs

jj

0

20

2

1

12

11

00

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22 ）） Direct transformDirect transform

BP transformBP transform

TransformTransform

MappingMapping

0

0

000

，，

coscos

sincoszz

zs

0

02

2

12

1

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33 ）） z-plane transformz-plane transform

Belongs to prototype transform.Belongs to prototype transform.

Design other kinds of digital filters using digital lowpass prototype filter. Design other kinds of digital filters using digital lowpass prototype filter.

Mapping:Mapping:

System function HSystem function H11(z) of (z) of digital lowpass prototype filterdigital lowpass prototype filter

Required system function HRequired system function Hdd(Z) of other filter(Z) of other filter

One z-plane One z-plane another z-planeanother z-plane

One stable casual systemOne stable casual system another stable casual systemanother stable casual system

)Z(gz 11

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Transform equation and design equationTransform equation and design equation (Next page)(Next page)

Mapping principle: Mapping principle:

Stability unchanged, from unit circle of one plane to Stability unchanged, from unit circle of one plane to

unit circle of another plane.unit circle of another plane.

Frequency response specification meets the same Frequency response specification meets the same

requirements, from unit circle of one plane to requirements, from unit circle of one plane to

unit circle of another plane.unit circle of another plane.In fact, mapping is rotating on unit circle.In fact, mapping is rotating on unit circle.

For example: LP For example: LP HO that is rotate with HO that is rotate with

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Lowpass Lowpass

lowpasslowpass

Determination of coefficientDetermination of coefficientg(zg(z-1-1))TransformTransform

Lowpass Lowpass

highpasshighpass

Lowpass Lowpass

bandpassbandpass

Lowpass Lowpass

bandstopbandstop

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Digital filter transformationDigital filter transformation

Lowpass Lowpass Highpass, bandpass and bandstop Highpass, bandpass and bandstop

Digital filter specification

Analog filter specification

Analog low- pass prototype

Analog frequency transform

Digital lowpass

prototype

Analog/digital transform

Digital/digital frequency transform

Digital filter’s system

function

(1)

(2)

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Review

1. Approach to IIR design

(1) Estimation of transfer functionEstimation of transfer function

(2) Iterative optimization techniqueIterative optimization technique

(3) Design digital filter with analog filter (mapping)

Must meets two conditions ：

Frequency response simulation: j on unit circle

Casual and stability invariable:

left half on s-plane unit circle

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2. Approach to IIR filter design (1) Impulse invariance

Application of sampling theorem (time-domain)Transform: Linear frequency mapping Frequency aliasing Applicable to lowpass filter and bandpass filter

(2) Bilinear transformations plane z plane (frequency domain)Transform pair: Non-linear frequency mappingDesign Simply.

(3) Prototype transformPrototype: Lowpass analog filter Other types of filter

Three types: Analog lowpass AF LP,HP,BP,BS DF

Analog lowpass DF LP,HP,BP,BS

Analog lowpass DF LP DF LP,HP,BP,BS