第八章 iir数字滤波器的设计

第八章IIR数字滤波器的设计3模拟滤波器的设计常用巴特沃斯,切比雪夫,椭圆,贝塞尔等滤波器指标相同时,滤波器阶数巴切椭阶数相同时,过渡带宽度巴切椭前3种较好逼近幅度特性,贝塞尔较好逼近相位特性三种滤波器阶数比较FP1000RP1FS1500RS60NB,WCBBUTTORDFP,FS,RP,RS,SNC1,WC1CHEB1ORDFP,FS,RP,RS,SNC2,WC2CHEB2ORDFP,FS,RP,RS,SNE,WEELLIPORDFP,FS,RP,RS,S得结果为NB19,WCB10428E003NC19,WC11000NC29,WC214535E003NE6,WE10001一般设计流程依需定指标选滤波器类型确定阶数确定参数调试2巴特沃斯滤波器的设计1幅频特性和相频特性设,221|ANCHJ则有右边曲线当时,上式写成的函数为SJS21ANCJ01230051,48NC有2N个极点12,0,21KJNKCSE取左半复平面的极点稳定,作即滤波器10ACKKHS若令,得稳定的频率响应函数SJ10NACKKJJS2设计次序N和3DB的,PSCKAH关于作归一化,得CCPSKCS或10NKKG1GPB因此设计次序可以变为N,3DB,PSCKAHS由的单调性和,可得2|AHJ|0|1AHJ20LG|LG|PAPAPJJ|SSS将代入,并取指数,得221|ANCJJ012PNPCSS消去,得,C01PSNPS令,01,PSPSPSSFK得取一个整数LGPSN再求如下C或201PN201SNC注两者不一定相等,用前式,通带刚好,阻带有余后式相反例1设低通模拟滤波器的技术指标如下通带边界频率,最大衰减40HZPF05DBP阻带截止频率,最小衰减1S4S3DB截止频率,求最小阶数N和实际5CFCF解按以上提供的公式计算或用MATLAB中命令WP2PI400WS2PI1000RP05RS40N,WCBUTTORDWP,WS,RP,RS,S得N7,WC32544手算31419FCWC/2518由此得具体设计步骤IN,3DB的,PS,PSKCII计算12,0,1JNKE或查表求得,0KKGIII将代得,即为所求滤波器/CPSPAHS例2条件同例1,求低通滤波器解I和401PS0501413PSPSKLGL367PSN214012054SCII查表求得B0B1B2B3B4B5B614491009145914591009449所以标称模拟滤波器,765432101GPBPBPBPIII代入得/CSAHS/CAPH24765324108610AS其它的系数如下A7A6A5A41146X104107X108503X1011A3A2A1A0164X1015164X1015534X1021387X1024数字都很大,工程中常用MATLAB设计设计巴特沃斯滤波器的命令和格式如下1N,WCBUTTORDWP,WS,RP,RS数字式2N,WCBUTTORDWP,WS,RP,RS,S模拟式说明格式1为数字滤波器设计的WP为通带截止频率,WS为阻带截止频率,都归一化0WP1,0WS1,其中对应关系为1FS/2RP和RS分别为通带和阻带衰减DB,返回数字滤波器的阶数N和3DB的截止频率如N,WCBUTTORD03,05,1,20配合B,ABUTTERN,WCFREQZB,A当WPWS时,自动为高通滤波器,如N,WCBUTTORD05,03,1,20这是高通配合下令B,ABUTTERN,WC,HIGH注类型FTYPE不可省FREQZB,A自动作两图当WP和WS是二维向量时,为带通或带阻滤波器如N,WCBUTTORD03,05,01,07,2,20带通B,ABUTTERN,WCFREQZB,A但注意N,WCBUTTORD01,07,03,05,2,20B,ABUTTERN,WC,STOP指为带阻FREQZB,A格式2为模拟格式,WP和WS可以是归一化的01之间数,也可以是角频率RAD/S,还有一个S,FREQZ换为FREQS,其余与格式1一样如N,WCBUTTORD005,03,1,20,S模低B,ABUTTERN,WC,SFREQSB,A也可是N,WCBUTTORD502PI,3002PI,1,20,S模低B,ABUTTERN,WC,SFREQSB,A3141025120LOGXX100025096有了阶数N和WC就可以设计滤波器的参数在MATLAB命令为1B,ABUTTERN,WC,FTYPE数字式2B,ABUTTERN,WC,FTYPE,S模拟式当FTYPE为缺省时,为低通若WCWCL,WCU,则带通滤波器当FTYPEHIGH时,为高通当FTYPESTOP时,为带阻,此时也应有WCWCL,WCU注BUTTER设计的带通和带阻都是2N阶的例如对前例的模拟滤波器的设计,可用下面程序EXAMPLEWP2PI400WS2PI1000角频率RP05RS40DBN,WCBUTTORDWP,WS,RP,RS,SB,ABUTTERN,WC,S默认低通,HIGH,STOP运行结果为N7B000000038662E024A114625E00410695E0080294E01116368E01536861E01853388E02138662E024进一步可以作出频率图H,WFREQSB,A系数从高到低SUBPLOT1,2,1PLOTW/2PI,20LOG10ABSH/ABSH1AXIS01080420GRIDONSUBPLOT1,2,2PLOTW/2PI,20LOG10ABSH/ABSH1AXIS055040GRIDON通带有余020040060080010004030201000100200300400500432103切比雪夫滤波器的设计仅介绍MATLAB命令巴特沃斯滤波器设计中为了某一性能达到标准,迫使滤波数阶数过高,过渡带过宽如果将误差均匀地分布在通带或阻带内,可使滤波器的阶数大为减低切比雪夫设计的滤波器能做到这一点切比雪夫设计的滤波器的幅频平方函数为,221|ANPHJC其中,COSAR|1HNXCX201是通带波动幅度DB见同名参考书2001版例3设低通滤波器的通带截止频率,最大衰减10HZPF1DBPP1OS|AHJ阻带过渡带通带2阻带截止频率,最小衰减20HZSF60DBS试用切比雪夫I型设计滤波器解FP100FS200RP1RS60WP2PIFPWS2PIFSN,WOCHEB1ORDWP,WS,RP,RS,S求阶数和截频B,ACHEBY1N,RP,WO,S滤波器参数,多了RPH,WFREQSB,A0100200300400806040200PLOTW/2PI,20LOGABSH/LOG10例4设计切比雪夫高通滤波器FP200FS100RP1RS40WP2PIFPWS2PIFSN,WOCHEB1ORDWP,WS,RP,RS,SB,ACHEBY1N,RP,WO,HIGH,SH,WFREQSB,APLOTW/2PI,20LOGABSH/LOG1001002003004006040200AXIS04006004椭圆滤波器的设计通带和阻带都是等波纹的,更接近理想滤波器N,WPELLIPORD,SPSB,AELLIP多了RP,RSNWFTYE例6设计低通椭圆滤波器,各指标为通带截止频率,通带波纹50HZPF1DBP阻带截止频率,阻带波纹7S40SFP500RP1FS700RS40WP2PIFPWS2PIFSN,WNELLIPORDWP,WS,RP,RS,SB,AELLIPN,RP,RS,WN,SH,WFREQSB,ASUBPLOT1,2,1PLOTW/2PI,20LOGABSH/LOG10AXIS015001000GRIDONSUBPLOT1,2,2PLOTW/2PI,ANGLEHAXIS0150044GRIDON050010001500100806040200050010001500420245高通,带通,带阻滤波器的设计1各指标如下图所示0020406081002040608100204060810020406081SPSPSLPPHSHSLPLPHSH基本设计思路是先指标转化,后滤波器转化,即将指标转化为低通滤波器指标设计低通滤波器转化为高通滤波器具体过程略设根据已获低通滤波器,如LHSP其中为3DB截止频率这里同时也作为指标PI低通高通PHHLPHSSII低通带通2LBPLPHPSIII低通带阻2LSLPPHS这些转换公式的直观意义是图形的上下翻转左右移动,或翻转下面仍用MATLAB命令来设计例7设计高通椭圆滤波器,各指标为通带截止频率,通带波纹20HZPF1DBP阻带截止频率,阻带波纹1S40S解FP200RP1FS100RS40WP2PIFPWS2PIFSN,WNELLIPORDWP,WS,RP,RS,SB,AELLIPN,RP,RS,WN,HIGH,SH,WFREQSB,AF0400W2PIFHFREQSB,A,W0200400806040200PLOTF,20LOG10ABSHAXIS0400800例8设计带通椭圆滤波器,各指标为通带下,上截止频率分别为,2HZPLFK5ZPHFK通带波纹,1DBP阻带下,上截止频率为,5SLFSHF阻带波纹40S解FP2000,5000RP1FS1500,5500RS40WP2PIFPWS2PIFSN,WNELLIPORDWP,WS,RP,RS,SB,AELLIPN,RP,RS,WN,SH,WFREQSB,AF08000W2PIFHFREQSB,A,WPLOTF,20LOG10ABSHAXIS07000800例9设计带阻椭圆滤波器,各指标为通带0,1KHZ和6KHZ,通带波纹,1DBP阻带2KHZ,5KHZ,阻带波纹40S解FS2000,5000RS400200040006000806040200FP1000,6000RP1WP2PIFPWS2PIFSN,WNELLIPORDWP,WS,RP,RS,SB,AELLIPN,RP,RS,WN,STOP,SF08000W2PIFHFREQSB,A,WSUBPLOT1,2,1PLOTF,20LOG10ABSHAXIS08000800GRIDONSUBPLOT1,2,2PLOTF,ANGLEHAXIS0800044GRIDON020004000600080008060402000200040006000800042024附录一个滤波的实例FS100T1100/FSS1SIN2PIT5S2SIN2PIT15S3SIN2PIT30时域信号XTS1S2S3SUBPLOT3,2,1PLOTT,XTTITLEXTAXIS0133频域特征XKABSFFTXT,128K0163SUBPLOT3,2,2PLOTK/6450,XK1,164TITLEXF滤波器的时域特征B,AELLIP4,01,40,10202/FSH,WFREQZB,A,512SUBPLOT3,2,4PLOTWFS/2PI,ABSH滤滤器的频域特征SUBPLOT3,2,3IMPULSEB,A滤波后的时域特征XFFILTERB,A,XTSUBPLOT3,2,5PLOTT,XFXLABELTIMESECONDSYLABELTIMEWAVEFORMAXIS0111滤波后的频域特征XK1ABSFFTXF,128K10163SUBPLOT3,2,6PLOTK1/6450,XK11,164滤波器转换的例子高通滤波器EZPLOTABSJW3/12JW2JW2JW3,05高通滤波器11111低通滤波器3阶低通作图模板EZPLOTABS1/S32S22S1,05参值提示WLP1WLS3WLC2SJW/WLCCLFSUBPLOT2,2,1EZPLOTABS1/JW32JW22JW1,03GRIDON2222低通转高通WHP1用SWLCWHP/JW1/JW替换SSUBPLOT2,2,2EZPLOTABS1/1/JW321/JW22