INFINITE IMPULSE RESPONSE FILTER CHAPTER 5 INFINITE IMPULSE RESPONSE FILTER 2018/11/12

5.1 Brief Introduction to IIR CONTENTS 5.1 Brief Introduction to IIR 5.2 Impulse Invariance 5.3 Bilinear Transformation 5.4 Analog-Digital Transformation 2018/11/12

IIR--h(n) infinite length FIR--h(n) finite length Digital Filter IIR--h(n) infinite length FIR--h(n) finite length Recursively Non-recursively Different methods to design IIR filter and FIR filter ! 2018/11/12

Process of filter design Three basic steps： Requirement analysis： Type/specification (1) Lowpass / Highpass / Bandpass / 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). 2018/11/12

Delay of frequency components BASIC SPECIFICATIONS FOR DIGITAL FILTER Amplitude-frequency phase-frequency Attenuation of frequency components Delay of frequency components Transition — 3dB passband cutoff frequency Pass band Stop band 2018/11/12

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 2018/11/12

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 Geometric evaluation Pole-zero diagram 2018/11/12

（2） Iterative optimization technique Iterative optimization technique are used to minimize the error between the desired frequency response and computing generated filter. Large computation cost Computer aided 2018/11/12

（3）Application of analog filter theory Advantage: Analog approximation techniques are highly advanced. They usually yield closed form solutions. Extensive tables are available for analog filter design. Principle : DF requirement  AF requirement  AF’s Ha(s)  DF’s H (z) Basic approach: Impulse invariant; Bilinear transformation. 2018/11/12

模拟滤波器的设计方法 （教材 6.2 ） 模拟滤波器需求 Requirement Ha(s) Butterworth filter Elliptic filter Chebyshev I Chebyshev II 2018/11/12

四种模拟滤波器的比较 幅频特性： 巴特沃斯 — 整个频带内单调下降； 过渡带特性： 巴特沃斯—最差； 设计复杂性： 切比雪夫I — 通带内等纹波振动，过渡带/阻带单调下降； 切比雪夫II —阻带内等纹波振动，过渡带/通带单调下降； 椭圆—除过渡带外，通带和阻带都等纹波振动。 过渡带特性： 巴特沃斯—最差； 切比雪夫I，II—居中； 椭圆—最陡； 设计复杂性： 巴特沃斯—相同条件下，阶数最高； 椭圆—相同条件下，阶数最低； 2018/11/12

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. 两个基本的条件，一是数字滤波器的频率响应特性模仿模拟滤波器的频率响应特性。对于数字系统，单位圆上的z变换 2018/11/12

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=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] 脉冲相应不变法，冲激响应不变法，实际上它是时域采样定理在滤波器设计中的应用， 2018/11/12

Analog frequency response Digital frequency response 5.2 脉冲响应不变法 Impulse invariant method 1. 变换思想 Impulse Invariant Method Analog frequency response Analog filter Digital frequency response Digital filter 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)。 2018/11/12

？ 2 数字化设计过程 s 平面 z 平面 Inverse Laplace z-transform 模拟滤波器s-平面的极点 2018/11/12

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 映射到z平面 2018/11/12

Mapping between z-plane and s-plane 系统因果稳定 模拟频率和数字频率 2018/11/12

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: 此时DF的频响才能不失真重现AF频响。 2018/11/12

周期延拓以后，混叠严重，难以转换到数字域! 3 数字滤波器与模拟滤波器频响关系 周期延拓 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） 高通模拟滤波器 周期延拓以后，混叠严重，难以转换到数字域! 适用范围：低通，带通。 2018/11/12

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. （a） 能够保持模拟滤波器的时域瞬态信号。 （b） 线性的数字频率和模拟频率关系。 （c） Ha()必须严格带限，否则数字频域内将出现 混叠失真。 注意：Ha()不严格带限且设计要求高时，不宜采用此方法。另外此方法不能直接设计高通和带阻滤波器。 2018/11/12

5.3 Bilinear transformation Disadvantage of impulse invariance: aliasing. Maps the entire axis in s-plane to narrow band in s1 plane. Tangent transformation 2018/11/12

Mapping of the s-plane into the z-plane Image of left half plane Bilinear transformation AF DF 2018/11/12

（2）Nonlinear relationship of phase transformation drawback 2018/11/12

Frequency nonlinear change of bilinear transformation Effect of frequency warping . Prewarp critical bandedge frequency. 2018/11/12

（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 2018/11/12

（4）Procedures from AFilter to DFilter Specification Transformation：digital analog Impulse invariance ：=T Bilinear transformation Analog filter design Analog filter Digital filter 2018/11/12

例：使用双线性变换法设计一个IIR滤波器，要求： （1）通带阻带具有单调下降的特性； （2） （3） 解： 第一步：临界数字频率: 第二步：临界模拟频率: 第三步：选择巴特沃斯滤波器，根据 和 求Ha(s) 第四步：求H (z) 2018/11/12

Typical filter’s “ideal” amplitude-frequency characteristics. 5.4 Prototype transformation Prototype filter —— Analog lowpass filter Lowpass Bandpass Prototype transformation: Analog lowpass filter  Types of digital filter Highpass Bandstop Typical filter’s “ideal” amplitude-frequency characteristics. 2018/11/12

（1）High-pass, Band-pass, Band-stop a. Design method 模拟低通原型 模拟高通、带通、带阻 数字高通、带通、带阻 （1） 模拟低通原型 数字高通、带通、带阻 （2） 模拟低通原型 低通数字原型 数字高通、带通、带阻 （3） 2018/11/12

2）Direct transform HP transform Mapping 2018/11/12

Prototype transform of highpass 2018/11/12

2）Direct transform BP transform Transform Mapping 2018/11/12

2）Direct transform BP transform Transform Mapping 2018/11/12

Belongs to prototype transform. 3）z-plane transform Belongs to prototype transform. Design other kinds of digital filters using digital lowpass prototype filter. Mapping: System function H1(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 2018/11/12

Stability unchanged, from unit circle of one plane to Mapping principle: Stability unchanged, from unit circle of one plane to unit circle of another plane. Frequency response specification meets the same requirements, from unit circle of one plane to In fact, mapping is rotating on unit circle. For example: LP  HO that is rotate with  Transform equation and design equation (Next page) 2018/11/12

Determination of coefficient Transform g(z-1) Determination of coefficient Lowpass  lowpass Lowpass  highpass Lowpass  bandpass Lowpass  bandstop 2018/11/12

Digital filter transformation Lowpass  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) 2018/11/12

Review Approach to IIR design Must meets two conditions： (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 2018/11/12

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,BP,BS  DF Analog lowpass  DF LP,HP,BP,BS Analog lowpass  DF LP DF LP,HP,BP,BS 2018/11/12