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(shè)計(jì)-第九章FIR數(shù)字濾波器的設(shè)計(jì)有限長(zhǎng)單位脈沖響應(yīng)濾波器的特點(diǎn):線性相位濾波.1.線性相位FIR數(shù)字濾波器、特點(diǎn)1.線性相位FIRDF含義設(shè)濾波器的脈沖響應(yīng)為SKIPIF10,長(zhǎng)為N.則SKIPIF10,再表成SKIPIF10其中SKIPIF10(可正負(fù),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(diǎn)數(shù)N有關(guān)的常數(shù),則稱濾波器是線性相位的.系統(tǒng)的群時(shí)延定義為:SKIPIF10.對(duì)線性相位濾波器,群時(shí)延是常數(shù).2.線性相位的條件(1)SKIPIF10的特點(diǎn)設(shè)濾波器是線性相位的,則應(yīng)有SKIPIF10即SKIPIF10從而有SKIPIF10上面二式相除且整理為SKIPIF10移項(xiàng)化簡(jiǎn)為SKIPIF10求得一種情形:當(dāng)SKIPIF10關(guān)于SKIPIF10奇對(duì)稱時(shí),上式為零.SKIPIF10是偶對(duì)稱的.即滿足SKIPIF10.此時(shí)SKIPIF10.在SKIPIF10偶對(duì)稱的條件下,再分SKIPIF10和SKIPIF10(2)SKIPIF10的特點(diǎn)數(shù)學(xué)推導(dǎo)見(jiàn)參考文獻(xiàn)1,下面只給出

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

4、IPIF10階的,系統(tǒng)必穩(wěn)定.因?yàn)镾KIPIF10.(5)網(wǎng)絡(luò)結(jié)構(gòu)特點(diǎn)由SKIPIF10對(duì)SKIPIF10的對(duì)稱性,推得當(dāng)SKIPIF10為偶數(shù)時(shí),SKIPIF10當(dāng)SKIPIF10為奇數(shù)時(shí),SKIPIF10例如當(dāng)SKIPIF10時(shí),SKIPIF10SKIPIF10.可有如下網(wǎng)絡(luò)結(jié)構(gòu).直接型省了2個(gè)乘法器當(dāng)SKIPIF10時(shí),情形類似,見(jiàn)書(shū)P185.2用窗函數(shù)設(shè)計(jì)FIR數(shù)字濾波器線性相位的FIR時(shí)域要求是SKIPIF10對(duì)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(shè)SKIPIF10,則SKIPIF10SKIPIF10一般為片斷函數(shù),故SKIPIF10無(wú)限長(zhǎng),需處理.1.基本方法

5、(1)提出希望頻率響應(yīng)函數(shù)線性相位,具有片斷特點(diǎn),即SKIPIF10(2)算出SKIPIF10SKIPIF10SKIPIF10(無(wú)限長(zhǎng))(3)加窗SKIPIF10,長(zhǎng)SKIPIF10,得SKIPIF10(*)要線性相位,就要SKIPIF10關(guān)于SKIPIF10偶對(duì)稱,而SKIPIF10關(guān)于SKIPIF10偶對(duì)稱,故要求SKIPIF10所以要求SKIPIF10關(guān)于SKIPIF10偶對(duì)稱.再回過(guò)來(lái)檢驗(yàn)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(diǎn)一樣時(shí),逼近性質(zhì)與窗形(值)有關(guān).下面分析當(dāng)SKIPIF10時(shí)的頻率性

6、質(zhì).由SKIPIF10,得SKIPIF10SKIPIF10SKIPIF10.其中SKIPIF10,SKIPIF10.代入卷積SKIPIF10SKIPIF10SKIPIF10SKIPIF10SKIPIF10,故SKIPIF10,SKIPIF10.相位是線性的.實(shí)際幅度=希望幅度*窗函數(shù)幅度.卷積=對(duì)每個(gè)SKIPIF10,求一積分,其值記為SKIPIF10.故有如下圖形演示.右圖為當(dāng)SKIPIF10時(shí),SKIPIF10的幅頻圖.阻帶最小衰減21dB,一般不滿足實(shí)際工程需要.過(guò)渡帶寬SKIPIF10(歸一化),這可以通過(guò)增加N來(lái)減小.這是窗函數(shù)設(shè)計(jì)的一個(gè)指標(biāo).3.典型窗函數(shù)下面給出各種窗函數(shù)的表達(dá)式

7、、時(shí)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(shè)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(shí)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(bǔ)點(diǎn);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(shí)域采樣加窗后濾波器的頻率特性過(guò)渡帶寬度SKIPIF10最小衰減SKIPIF10.當(dāng)SKIPIF10時(shí)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(biāo)為:SKIPIF10.(3)升余弦窗(漢寧窗,hanningwindow)SKIPIF10,各指標(biāo)為:SKIPIF10(4)改進(jìn)升余弦窗(海明窗,hanningwindow)SKIPIF10,SKIPIF10(5)布萊克曼窗(blackmanwindow)SKIPIF10,各指標(biāo)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(guò)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(guò)渡帶寬度SKIPIF10.則選布萊克曼窗,且由SKIPIF10,得SKIPIF10.事實(shí)上,還有很多窗形可供選擇.見(jiàn)P193.4.設(shè)計(jì)步驟(1)由阻帶指標(biāo)選窗型w,由過(guò)渡帶寬度選點(diǎn)數(shù)N,(2)構(gòu)造要逼近的SKIPIF10,構(gòu)造SKIPIF10(對(duì)低通)應(yīng)使SKIPIF10(3)計(jì)算SKIPIF10(4)加窗SKIPIF10.例1用窗函數(shù)法設(shè)計(jì)線性相位高通FIRDF,指標(biāo)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(biāo),可選漢寧和海明窗,我們選海明窗,由SKIPIF10,SKIPIF10,對(duì)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(jiàn)書(shū),略)上述過(guò)程可用Matlab中的命令fir1來(lái)實(shí)現(xiàn).格式1:hn=fir1(N,wc,ftype,window(N+1);ftype可選high,stop;window窗名,默認(rèn)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(zhǎng)度為N+1)當(dāng)wc=wc1,wc2時(shí),為帶通濾波器.例如上例的命令為(注設(shè)計(jì)時(shí),對(duì)作歸一化)wc=29/66;N=32;%N=h(n)的長(zhǎng)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(duì)高通,帶阻,階數(shù)必須為偶數(shù).

13、例2用窗函數(shù)法設(shè)計(jì)一個(gè)FIR帶通濾波器,指標(biāo)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(biāo),選blackman窗,由過(guò)度帶寬SKIPIF10,得SKIPIF10,通帶區(qū)間約定用SKIPIF10表示,計(jì)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(shè)計(jì)FIR低通濾波器,實(shí)現(xiàn)對(duì)模擬信號(hào)采樣后進(jìn)行數(shù)字低通濾波,對(duì)模擬信號(hào)的指標(biāo)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(huà)出幅頻衰減曲線和相頻特性曲線.解(1)轉(zhuǎn)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(guò)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(duì)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(duì)稱性數(shù)據(jù)P1