CHAPTER 5 INFINITE IMPULSE RESPONSE FILTER 2021228 1

CHAPTER 5 INFINITE IMPULSE RESPONSE FILTER 2021/2/28 1

CONTENTS 5. 1 Brief Introduction to IIR 5. 2 Impulse Invariance 5. 3 Bilinear Transformation 5. 4 Analog-Digital Transformation 2021/2/28 2

Digital Filter IIR--h(n) infinite length FIR--h(n) finite length Recursively Non-recursively Different methods to design IIR filter and FIR filter ! 2021/2/28 3

Process of filter design Three basic steps： • Requirement analysis： Type/specification (1) Lowpass / Highpass / Bandstop (2) Bandedge frequency (3) Passband ripple / Stopband attenuation • Objective of digital filter is to develop a casual and stable transfer function H(z) meeting the frequency response specification. • Implementation of H(z) 2021/2/28 4

BASIC SPECIFICATIONS FOR DIGITAL FILTER Amplitude-frequency phase-frequency Attenuation of frequency components Delay of frequency components Pass band Transition 2021/2/28 — 3 d. B passband cutoff frequency Stop band 5

5. 1 Basic Approach to IIR digital filter design An order N IIR digital filter’s system function is: Objective of H(z): Determine coefficient ai and bi or zero and pole point ci and di, in order to meet design requirements. Three approaches to IIR digital filter’s transfer function design Estimation of transfer function Iterative optimization technique Analog filter theory 2021/2/28 6

1 Approach to filter design (1)Estimation of transfer function from diagram Pole inside unit circle Peak in frequency response Zero inside unit circle Valley in frequency response Pole-zero diagram 2021/2/28 Geometric evaluation 7

（2） Iterative optimization technique are used to minimize the error between the desired frequency response and computing generated filter. Large computation cost Computer aided 2021/2/28 8

（3）Application of analog filter theory Advantage: (1) Analog approximation techniques are highly advanced. (2) They usually yield closed form solutions. (3) Extensive tables are available for analog filter design. Principle : DF requirement AF’s Ha(s) DF’s H (z) Basic approach: Impulse invariant; invariant Bilinear transformation. 2021/2/28 9

模拟滤波器的设计方法 （教材 6. 2 ） 模拟滤波器需求 Requirement Butterworth filter Chebyshev I 2021/2/28 Ha(s) Elliptic filter Chebyshev II 10

2 Two basic rules from analog to digital filter （1）Imaginary axis j in s-plane be mapped onto unit circle ejw of z-plane. (Ha(s) H(z)) （2）A stable and casual analog transfer function be transformed into a stable and casual digital transfer function. That is: Re[s]<0 in s-plane unit circle |z|<1 in z-plane. 2021/2/28 12

5. 2 Impulse invariant (Time domain) Sampling 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=n. T 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=n. T] 2021/2/28 13

5. 2 脉冲响应不变法 Impulse invariant method 1. 变换思想 Impulse Invariant Method Analog filter Analog frequency response Digital filter Digital frequency response 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)。 2021/2/28 14

3. s平面到z平面映射 Mapping from s-plane to z-plane Periodic copies Sampling 沿虚轴周期延拓之后，使用 2021/2/28 映射到z平面 16

Mapping between z-plane and s-plane 系统因果稳定 2021/2/28 模拟频率和数字频率 17

Characteristic of Impulse invariant （a） Keep analog filter’s transient signal in time domain. （b） Linear relationship between digital frequency and analog frequency. （c） Ha( ) must be band limited on , otherwise distortion will happen in digital frequency domain. Attention: This method is not suitable in following situation: (1) Ha( ) is not band limited or ha(t) change unstably, and the design requirement is very high (2) Highpass filter and bandstop filter. 2021/2/28 20

5. 3 Bilinear transformation Disadvantage of impulse invariance: aliasing. Maps the entire axis in s-plane to narrow band in s 1 plane. Tangent transformation 2021/2/28 21

Mapping of the s-plane into the z-plane s 1 plane z plane Image of left half plane Bilinear transformation 2021/2/28 AF DF 22

（2）Nonlinear relationship of phase transformation 2021/2/28 23

Frequency nonlinear change of bilinear transformation Effect of frequency warping. Prewarp critical bandedge frequency. 2021/2/28 24

（3）Features of bilinear transformation （a） No amplitude-frequency distortion after transformation, no requirement for Ha( )’s bandwidth; （b） Simple design; （c）Non-linear relationship between digital frequency and analog frequency; （d） Frequency warping can be revised by prewarping method. Transient response impulse invariance Other situations bilinear transformation 2021/2/28 25

（4）Procedures from AFilter to DFilter (1) Specification (2) Transformation：digital analog Impulse invariance ： = T Bilinear transformation (3) Analog filter design (4) Analog filter Digital filter 2021/2/28 26

5. 4 Prototype transformation Prototype filter —— Analog lowpass filter Lowpass Bandpass Prototype transformation: Analog lowpass filter Highpass 2021/2/28 Types of digital filter Bandstop Typical filter’s “ideal” amplitude-frequency characteristics. 28

2）Direct transform HP transform Mapping 2021/2/28 30

Prototype transform of highpass 2021/2/28 31

2）Direct transform BP transform Transform Mapping 2021/2/28 32

2）Direct transform BP transform Transform Mapping 2021/2/28 33

3）z-plane transform Belongs to prototype transform. Design other kinds of digital filters using digital lowpass prototype filter. Mapping: System function H 1(z) of digital lowpass prototype filter Required system function Hd(Z) of other filter One z-plane another z-plane One stable casual system another stable casual system 2021/2/28 34

Mapping principle: n Stability unchanged, from unit circle of one plane to unit circle of another plane. n Frequency response specification meets the same requirements, from unit circle of one plane to unit circle of another plane. In fact, mapping is rotating on unit circle. For example: LP HO that is rotate with Transform equation and design equation (Next page) 2021/2/28 35

Transform g(z-1) Determination of coefficient Lowpass lowpass Lowpass highpass Lowpass bandstop 2021/2/28 36

Digital filter transformation Digital filter specification Analog filter specification (1) Analog frequency transform Analog/digital transform Analog lowpass prototype (2) Digital lowpass prototype Digital/digital frequency transform Digital filter’s system function Lowpass Highpass, bandpass and bandstop 2021/2/28 37

Review 1. Approach to IIR design (1) Estimation of transfer function (2) Iterative 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 2021/2/28 38

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 transformation s plane z plane (frequency domain) Transform pair: Non-linear frequency mapping Design Simply. (3) Prototype transform Prototype: Lowpass analog filter Other types of filter Three types: Analog lowpass AF LP, HP, BS DF Analog lowpass DF LP, HP, BS 2021/2/28 39