FIR數(shù)字濾波器的設計教學內容

1、Good is good, but better carries it.精益求精，善益求善。FIR數(shù)字濾波器的設計-第九章FIR數(shù)字濾波器的設計有限長單位脈沖響應濾波器的特點:線性相位濾波.1.線性相位FIR數(shù)字濾波器、特點1.線性相位FIRDF含義設濾波器的脈沖響應為SKIPIF10,長為N.則SKIPIF10,再表成SKIPIF10其中SKIPIF10(可正負,SKIPIF10)稱為幅度特性函數(shù),SKIPIF10稱為相位特性函數(shù).注:不是SKIPIF10如SKIPIF10的SKIPIF10它的SKIPIF10為SKIPIF10,SKIPIF10為SKIPIF10.若SKIPIF10,SKI

2、PIF10是與采樣點數(shù)N有關的常數(shù),則稱濾波器是線性相位的.系統(tǒng)的群時延定義為:SKIPIF10.對線性相位濾波器,群時延是常數(shù).2.線性相位的條件(1)SKIPIF10的特點設濾波器是線性相位的,則應有SKIPIF10即SKIPIF10從而有SKIPIF10上面二式相除且整理為SKIPIF10移項化簡為SKIPIF10求得一種情形:當SKIPIF10關于SKIPIF10奇對稱時,上式為零.SKIPIF10是偶對稱的.即滿足SKIPIF10.此時SKIPIF10.在SKIPIF10偶對稱的條件下,再分SKIPIF10和SKIPIF10(2)SKIPIF10的特點數(shù)學推導見參考文獻1,下面只給出

3、結論.當SKIPIF10是奇數(shù)時,SKIPIF10當SKIPIF10是偶數(shù)時,SKIPIF10所以在SKIPIF10偶對稱的條件下,濾波器有兩種形式(對SKIPIF10,是低通濾波器,可轉換成高通,帶通,帶阻濾波器)(對SKIPIF10也是低通濾波器,但不可轉換成高通,帶阻濾波器).(3)零點分布特點(SKIPIF10偶對稱)SKIPIF10SKIPIF10由此可得,對SKIPIF10,若SKIPIF10,則SKIPIF10.由SKIPIF10是實數(shù)列,得SKIPIF10是實系數(shù)的,所以,有三種情形的零點.例如hn=13531;zplane(hn,1);(4)極點均在SKIPIF10,且為SK

4、IPIF10階的,系統(tǒng)必穩(wěn)定.因為SKIPIF10.(5)網(wǎng)絡結構特點由SKIPIF10對SKIPIF10的對稱性,推得當SKIPIF10為偶數(shù)時,SKIPIF10當SKIPIF10為奇數(shù)時,SKIPIF10例如當SKIPIF10時,SKIPIF10SKIPIF10.可有如下網(wǎng)絡結構.直接型省了2個乘法器當SKIPIF10時,情形類似,見書P185.2用窗函數(shù)設計FIR數(shù)字濾波器線性相位的FIR時域要求是SKIPIF10對稱性.本節(jié)討論如何在幅頻特性上逼近期望濾波器.以低通為例.設SKIPIF10,則SKIPIF10SKIPIF10一般為片斷函數(shù),故SKIPIF10無限長,需處理.1.基本方法

5、(1)提出希望頻率響應函數(shù)線性相位,具有片斷特點,即SKIPIF10(2)算出SKIPIF10SKIPIF10SKIPIF10(無限長)(3)加窗SKIPIF10,長SKIPIF10,得SKIPIF10(*)要線性相位,就要SKIPIF10關于SKIPIF10偶對稱,而SKIPIF10關于SKIPIF10偶對稱,故要求SKIPIF10所以要求SKIPIF10關于SKIPIF10偶對稱.再回過來檢驗SKIPIF10是否滿足精度要求.若基本滿足,則依截取的SKIPIF10,制硬件,編軟件.2.窗函數(shù)法的性能分析由(*)式知,取點一樣時,逼近性質與窗形(值)有關.下面分析當SKIPIF10時的頻率性

6、質.由SKIPIF10,得SKIPIF10SKIPIF10SKIPIF10.其中SKIPIF10,SKIPIF10.代入卷積SKIPIF10SKIPIF10SKIPIF10SKIPIF10SKIPIF10,故SKIPIF10,SKIPIF10.相位是線性的.實際幅度=希望幅度*窗函數(shù)幅度.卷積=對每個SKIPIF10,求一積分,其值記為SKIPIF10.故有如下圖形演示.右圖為當SKIPIF10時,SKIPIF10的幅頻圖.阻帶最小衰減21dB,一般不滿足實際工程需要.過渡帶寬SKIPIF10(歸一化),這可以通過增加N來減小.這是窗函數(shù)設計的一個指標.3.典型窗函數(shù)下面給出各種窗函數(shù)的表達式

7、、時域波形、幅度特性，以及理想濾波器加窗后的波形和幅度特性.以下均設低通濾器SKIPIF10的SKIPIF10.(1)矩形窗SKIPIF10,已求得SKIPIF10,SKIPIF10矩形波形矩形波形的幅頻特性%矩形窗時域波形N=31;w=rectwin(N);n=0:30;subplot(1,2,1);stem(n,w);axis(03301.3);gridon;%矩形窗頻域特性hw,w=freqz(w,1);subplot(1,2,2);plot(w/pi,20*log10(abs(hw)/abs(hw(1);axis(01-600);gridon;pause;%理想濾波器加窗后采樣序列wc

8、=pi/2;N=31;n=0:30;t=(N-1)/2;hdn=sin(wc*(n-t)./(pi*(n-t);hdn(16)=0.5;%補點;subplot(1,2,1);stem(n,hdn);axis(033-0.20.8);gridon;%濾波器加窗后的頻域特性hw,w=freqz(hdn,1);subplot(1,2,2);plot(w/pi,20*log10(abs(hw)/abs(hw(1);axis(01-608);gridon;理想濾波器時域采樣加窗后濾波器的頻率特性過渡帶寬度SKIPIF10最小衰減SKIPIF10.當SKIPIF10時矩形窗的幅頻特為SKIPIF10與N成

9、反比,要改SKIPIF10,需另選.(2)三角窗(BartlettWindow)SKIPIF10SKIPIF10SKIPIF10各指標為:SKIPIF10.(3)升余弦窗(漢寧窗,hanningwindow)SKIPIF10,各指標為:SKIPIF10(4)改進升余弦窗(海明窗,hanningwindow)SKIPIF10,SKIPIF10(5)布萊克曼窗(blackmanwindow)SKIPIF10,各指標為:SKIPIF10.為便于選擇使用,將5種窗函數(shù)基本參數(shù)列于下表.類型窗函數(shù)的旁瓣峰值SKIPIF10過渡帶寬度SKIPIF10加窗后濾波器的阻帶最小衰減SKIPIF10rectwin

10、134/N21bartlet三角258/N25hanning318/N44hamming418/N53blackman5712/N74如阻帶最小衰減SKIPIF10,過渡帶寬度SKIPIF10.則選布萊克曼窗,且由SKIPIF10,得SKIPIF10.事實上,還有很多窗形可供選擇.見P193.4.設計步驟(1)由阻帶指標選窗型w,由過渡帶寬度選點數(shù)N,(2)構造要逼近的SKIPIF10,構造SKIPIF10(對低通)應使SKIPIF10(3)計算SKIPIF10(4)加窗SKIPIF10.例1用窗函數(shù)法設計線性相位高通FIRDF,指標為通帶截止頻率:SKIPIF10;通帶最大衰減:SKIPIF

11、10.阻帶截止頻率:SKIPIF10;阻帶最小衰減:SKIPIF10解(1)根據(jù)阻帶指標,可選漢寧和海明窗,我們選海明窗,由SKIPIF10,SKIPIF10,對高通濾波器,必須取奇數(shù)SKIPIF10.故有SKIPIF10.(2)SKIPIF10,SKIPIF10,則要逼近SKIPIF10(全通-低通)(3)求SKIPIF10SKIPIF10SKIPIF10表示全通濾波器低通濾波器(4)加窗SKIPIF10(見書,略)上述過程可用Matlab中的命令fir1來實現(xiàn).格式1:hn=fir1(N,wc,ftype,window(N+1);ftype可選high,stop;window窗名,默認ha

12、mming.格式2:hn=fir1(N,wc);階數(shù)為N,6dB截止頻率wc(01)的低通濾波器.(注h(n)的長度為N+1)當wc=wc1,wc2時,為帶通濾波器.例如上例的命令為(注設計時,對作歸一化)wc=29/66;N=32;%N=h(n)的長度-1hn=fir1(N,wc,high);subplot(1,2,1);n=0:32;stem(n,hn);axis(032-0.40.6);gridon;hw,w=freqz(hn,1);subplot(1,2,2);plot(w/pi,20*log10(abs(hw);axis(01-805);gridon;注對高通,帶阻,階數(shù)必須為偶數(shù).

13、例2用窗函數(shù)法設計一個FIR帶通濾波器,指標為阻帶下截止頻率:SKIPIF10;阻帶最小衰減SKIPIF10通帶下截止頻率:SKIPIF10;通帶最大衰減SKIPIF10通帶上截止頻率:SKIPIF10;阻帶上截止頻率:SKIPIF10;解由阻帶衰減指標,選blackman窗,由過度帶寬SKIPIF10,得SKIPIF10,通帶區(qū)間約定用SKIPIF10表示,計算如下SKIPIF10程序命令為wls=0.2*pi;wlp=0.35*pi;wup=0.65*pi;B=wlp-wls;N=ceil(12*pi/B);wp=wlp/pi-6/N,wup/pi+6/N;hn=fir1(N-1,wp,b

14、lackman(N);subplot(1,2,1);n=0:79;stem(n,hn);axis(080-0.40.4);gridon;hw,w=freqz(hn,1);subplot(1,2,2);plot(w/pi,20*log10(abs(hw);axis(01-1005);gridon;例3用窗函數(shù)法設計FIR低通濾波器,實現(xiàn)對模擬信號采樣后進行數(shù)字低通濾波,對模擬信號的指標通帶截止頻率:SKIPIF10;阻帶截止頻率:SKIPIF10;阻帶最小衰減:SKIPIF10;采樣頻率:SKIPIF10.選合適窗函數(shù),求出SKIPIF10,并畫出幅頻衰減曲線和相頻特性曲線.解(1)轉換成數(shù)字頻

15、率為通帶數(shù)字截止頻率:SKIPIF10;阻帶數(shù)字截止頻率:SKIPIF10;阻帶最小衰減:40dB;過渡帶寬度:SKIPIF10.(2)由衰減:40dB,選hamming窗,由SKIPIF10,得SKIPIF10.(3)確定SKIPIF10,命令如下:fp=2000;fs=3000;Fs=10000;wp=2*pi*fp/Fs;ws=2*pi*fs/Fs;B=ws-wp;N=ceil(8*pi/B);wc=(wp+B/2)/pi;hn=fir1(N-1,wc);n=0:N-1;subplot(1,2,1);stem(n,hn,.);gridon;hw,w=freqz(hn,1);subplot

16、(1,2,2);plot(w/pi,20*log10(abs(hw);gridon;axis(01-1004);w=-2.2:0.01:2.2;wg=sin(31*w/2)./sin(w/2);wg(221)=31;plot(w,wg);axis(-2.52.2-1032);%理想濾波器的頻域特性.ezplot(int(sin(w-x)*16)/sin(w-x)/2)/6.28,-pi/4,pi/4),-1.71.7);加窗后的幅度函數(shù)的頻域特性.附錄1對稱性數(shù)據(jù)P183n13=0:1:12;%P183h13=-0.05-0.0300.080.160.250.280.250.160.080-0.03-0.05;subplot(1,2,1);stem(n13,h13);axis(013-0.10.3)n12=0:1:11;h12=-0.05-0.0300.080.160.250.250.160.080-0.03-0.05;subplot(1,2,2);stem(n12,h12);axis(013-0.10.3)2對稱性數(shù)據(jù)P1