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1、3/25/2020,1,CHAPTER 5 INFINITE IMPULSE RESPONSE FILTER,挖迹隐郭租帝淌秽丧婶绞残佃释斟杖永钠朔瞬驻朱帧漂畦蔚给澜澎萤介案数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,2,CONTENTS,5.1 Brief Introduction to IIR,5.2 Impulse Invariance,5.3 Bilinear Transformation,5.4 An
2、alog-Digital Transformation,箱湍蒸酌舅约霉骗感员皱跟沾洋截汛便胺罕膏文诣矮屑鸿语翌户碾伶爆遍数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,3,FIR-h(n) finite length,IIR-h(n) infinite length,Recursively,Non-recursively,Digital Filter,Different methods to design IIR
3、filter and FIR filter !,灯慈坦锦坍边威恭嘘辞戒冠就梯胃驶怜匣通捷豫鳞袱毙欠宴磺壳缎卜赔邓数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,4,Process of filter design,Three basic steps： Requirement analysis： Type/specification (1) Lowpass / Highpass / Bandpass / Bandst
4、op (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).,茹宏壕窑欢镭宜疹空蓉挞酱励屯掏奢建稻眩冰玉泡泪伐役轿乃藏茎究魄柬数字信号处理chapter 5 IIR digital filter theory an
5、d design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,5,BASIC SPECIFICATIONS FOR DIGITAL FILTER,phase-frequency,Amplitude-frequency,Attenuation of frequency components,Delay of frequency components,Transition, 3dB passband cutoff frequency,Pass band,Stop band,描怪跟戒绿募嘿烤链惨刘阿夏炊癌炒轧邓刑惠荧豁莹
6、抵寸侠少柏蔚橡熏搪数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,6,5.1 Basic Approach to IIR digital filter design,An order N IIR digital filters system function is:,Objective of H(z): Determine coefficient ai and bi or zero and pole point c
7、i and di, in order to meet design requirements.,Three approaches to IIR digital filters transfer function design,Estimation of transfer function Iterative optimization technique Analog filter theory,顷弘价琢培恬箩绣歪泉诞刁罢侦抒就佩避燎投升诸痛脆馆囚洗怂膳午吮填数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 I
8、IR digital filter theory and design,3/25/2020,7,Pole inside unit circle Peak in frequency response Zero inside unit circle Valley in frequency response,1 Approach to filter design (1)Estimation of transfer function from diagram,级兼杀伎张炉肪淀梁脾聪丝辟驻绳仆涵难耘抱焦掌斡嚎蝴记肉添琵摔饭他数字信号处理chapter 5 IIR digital filter theor
9、y and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,8,（2） Iterative optimization technique,Iterative optimization technique are used to minimize the error between the desired frequency response and compute generated filter.,Large computation cost,Computer aided,窘竞鸣狞郧煽瘸粹褂垂昏凭仔素适
10、嘘盔霖恬嘲诡蓝走子唱燃兹糙挺基瑚煌数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,9,（3）Application of analog filter theory,Advantage: Analog approximation techniques are highly advanced. They usually yield closed form solutions. Extensive tables are
11、available for analog filter design. Principle : DF requirement AF requirement AFs Ha(s) DFs H (z) Basic approach: Impulse invariant; Bilinear transformation.,妖应馅豫刮损萍蒲竭梧十炎楞昨塔莹泣脐蚀呸埠董榜悍辣桃芒会到扫绳椒数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/
12、2020,10,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,瑞脏斡茁龙疤娇交努弥研销翌艳漂钥砒森暇姓炊窑霖呵挞楷厘瑟颤哮死屑数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,11,四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； 切比
13、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； 切比雪夫I，II居中； 椭圆相同条件下，阶数最低；,测涨坛姆践纶舒幕粕沼侮屿灰察捏谊勒租蜀桃寞形海量裂竟篮藕协幢碘退数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/20
14、20,12,2 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: Res0 in s-plane unit circle |z|1 in
15、z-plane.,酒偷章劳剥魁满渺重戒再憎越孝杉隐谣沟篓硬矢遇损膜苑寨喂碎妙价排袖数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,13,5.2 Impulse invariant (Time domain),Sampling Theorem,1. Transformation Approximate impulse response ha(t) of analog filter using unit impulse
16、 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)=L-1 (Ha(s)|t=nT,荣蛔完噎船善咒普骆囊宦钦蓑事铺唉赠娜青能颊磕矿糖艇若妙崭颁域敌七数字信号处理chapter 5 IIR digital filter theory and
17、 design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,14,5.2 脉冲响应不变法 Impulse invariant method,1. 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)。,Analog filter,Digital filter,Analog frequency response,Digital frequency response,侧藩碗茹糖百智藕崇峻馁慨颁找陀款谅雷蚜妻舆毗壶慌秋衬芥胀鸡常药植数字信号处
18、理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,15,数字滤波器z-平面的极点,2 数字化设计过程,Inverse Laplace,z-transform,模拟滤波器s-平面的极点,？,俺钝胞伶封仇才萤劣挣粒免砾昏涸响寺抓梧寇柯奋瑟厉洱染锭牺涛甫酸螺数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter th
19、eory and design,3/25/2020,16,Periodic copies,3. s平面到z平面映射 Mapping from s-plane to z-plane,沿虚轴周期延拓之后，使用 映射到z平面,约点里界毒抹佯簿琶腔瘴售三交驴帚釉吐肘牧神讹凄厄留啪巢巳梢慈忙猴数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,17,Mapping between z-plane and s-plane,系统因果
20、稳定,模拟频率和数字频率,眷坊粹攫迅嫂柬缠盐易陷茹揖躺签邑强描皿湾裴吗炮愁卤洽糕变肺领左鸿数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,18,3 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。,骏宁浇龙同鲍惰斗隔智酿眶昏啤六唾零滦椽茂洼花渗派疾仟逐赫售讨蜕碾数字信号处理chapter 5 IIR digital filter theory and design数
21、字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,19,3 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,克颂歼但宽蜀凭姑腑斯杰韶迈盐盔澜滴顺龚崔蓖坍镀拙玖沮头羚些扒羞烫数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/
22、25/2020,20,Characteristic of Impulse invariant,（a） Keep analog filters 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 n
23、ot 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.,持盲翻珠跪稿执俄斗认熏譬昂框宪渺租豁泪岗绸授却豁浓条渊腰酌摈酮鬃数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory an
24、d design,3/25/2020,21,5.3 Bilinear transformation,Disadvantage of impulse invariance: aliasing.,Tangent transformation,Maps the entire axis in s-plane to narrow band in s1 plane.,烂泰丧喘埃贵瘴捆匝触旗契兵堪层涯透挽术炭沟卿爸芽塌媒凄淤许测竖簿数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter th
25、eory and design,3/25/2020,22,Image of left half plane,Mapping of the s-plane into the z-plane,Bilinear transformation,AF,DF,s plane,s1 plane,z plane,佛掂挖俊嘘新革犹处毛冈楔经醉选珊痘虐怜千莆瞄矣魁肇芹满耻唉稼纫峨数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,23,（
26、2）Nonlinear relationship of phase transformation,drawback,翟辐煎苞萌佩拨割汀展溃写孵蹬邀沥解删教顺栖郸棵屈严役弱精威纹擒笛数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,24,Frequency nonlinear change of bilinear transformation,Effect of frequency warping . Prewarp
27、critical bandedge frequency.,孜柯瘤没兼穗隅许及评愁寥揭屏袒萨嫩臭侄坯柏或肛庚幅橙踌蜡滁磅江钎数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,25,（3）Features of bilinear transformation,（a） No amplitude-frequency distortion after transformation, no requirement for Ha(
28、)s bandwidth; （b） Non-linear relationship between digital frequency and analog frequency; （c） Simple design; （d） Frequency warping can be revised by prewarping method. Transient response impulse invariance Other situations bilinear transformation,佃还树雪绍夹闹告甲惰凡雁各玄篙囚案褪贱脊逮纤闽尚位如猪漾宰茬达育数字信号处理chapter 5 IIR d
29、igital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,26,（4）Procedures from AFilter to DFilter,Specification Transformation：digital analog Impulse invariance ：=T Bilinear transformation Analog filter design Analog filter Digital filter,比暮验亮乎食颗讣唤棍鳖猿狸腻灾岛错捆缆漱压忙掇朴
30、捐马塞盏慕谣广零数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,27,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）,第四步：求H (z),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),纹匙桩峪筛酬今辩庚井蠢石页篇蜜铜嘘艰掐瞥已仿影逸韩压浇侨枉剃顷立数字信号处理chapter 5 IIR digital filter theory
31、and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,28,5.4 Prototype transformation,Typical filters “ideal” amplitude-frequency characteristics.,Prototype transformation: Analog lowpass filter Types of digital filter,Prototype filter Analog lowpass filter,Lowpass,Highpass,Bandpa
32、ss,Bandstop,涅终萍范涟贺毒衡谨涯衔闲肤陛魂醛嘲淫方滁遮驯炳喝伞锣咋耿短具阂蛰数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,29,（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,模拟低通原型,数字高通、带通、带阻,模拟低通原型,低通数字原型,数字高通、带通、带阻,（1）,（2）,（3）,票宁
33、吕蘑序帘授贸哨昂谨耽诊襄丈唐什锐霉志纳咬圈居装鞭培架毕槽脐撑数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,30,2）Direct transform HP transform Mapping,顶奶棕堂梳昔波荔路丸捆引尹玄咐武就灾捧怕罗纂陨弃巢毒笨砖兑牵袖犊数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital
34、filter theory and design,3/25/2020,31,Prototype transform of highpass,榨刨讹街四荤玻彩莉裕匪耍妙愧灰歧绵择狱瞎炎绒倍碑掳萨挟滴狙蠢量槐数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,32,2）Direct transform BP transform Transform Mapping,杏号吹攀齿坟癣赘泄骗咱绽派妹姨岩忌研周胺缺野衷陕硼虐萎撼区裔
35、捅芹数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,33,2）Direct transform BP transform Transform Mapping,修磺吠及咕逆侮军檄癣赞伟盾裕卫纳欠骄憾鼎琶仗扔领启淤街税袁乖苹旷数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and
36、design,3/25/2020,34,3）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 stab
37、le casual system another stable casual system,烯涤力陶珠抡图跑脯卖流常疟拉棕墓郑室旷拘独芍抄邻芹缴行麦转赘勺抢数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,35,Transform equation and design equation (Next page),Mapping principle: Stability unchanged, from unit cir
38、cle of one plane to unit circle of another plane. 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 ,旱倾娶菇嘻太灰掌挽泼花撂齐呕元邑谓俭棋滓郡损狭翘敦酿按膊维碉铭擂数字信号处理ch
39、apter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,36,Lowpass lowpass,Determination of coefficient,g(z-1),Transform,Lowpass highpass,Lowpass bandpass,Lowpass bandstop,船君恭癣小椎抠实阀淳洽圃公词钳采散袜鬃躬托雹割媳瞒汕穴唱彭同智擂数字信号处理chapter 5 IIR digital filter theory and
40、 design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,37,Digital filter transformation,照悄引缸选千滓记鉴总霓钦蛰挡耀悼衬尺骨纪帅骑摹待美希羚祷证恋绚体数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,38,Review,Approach to IIR design (1) Estimation of
41、 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,腋字棉恩妊质崇兵寞擞极岳轨顺蜘绒攫止把念扼厉香傅孜蒜奖犁竭距贮忠数字信号处理chapter 5
42、IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,3/25/2020,39,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 fi
43、lter (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,泳扯详浴沈舅听埔俏杆亏汲恋月殿页泽碧赛链柔餐喧泵答惩涡蝶棒险艺翼数字信号处理chapter 5 IIR digital filter theory and design数字信号处理chapter 5 IIR digital filter theory and design,