IIR數(shù)字濾波器設(shè)計上機

1、 Butterworth模擬低通濾波器設(shè)計 3. 利用MATLAB設(shè)計BW LP num,den=butter(N,wc,s)確定階數(shù)為N，3-dB截頻為wc(radian/s)的Butterworth filter分子和分母多項式。s 表示模擬域。 N,wc=buttord(wp,ws,Ap,As,s)確定模擬Butterworth filter的階數(shù)N和3-dB截頻wc。wc是由阻帶參數(shù)確定的。 s 表示模擬域。 z,p,k=buttap(N)確定N階歸一化的Butterworth filter 的零點、極點和增益(gain)1 例：設(shè)計滿足下列條件的模擬Butterworth低通濾波器

2、fp=1kHz, fs=2kHz, Ap=1dB, As=40dBWp=2*pi*1000;Ws=2*pi*2000;Ap=1;As=40;N,Wc=buttord(Wp,Ws,Ap,As,s);fprintf(Order of the filter=%.0fn,N)num,den = butter(N,Wc,s);disp(Numerator polynomial);fprintf(%.4en,num);disp(Denominator polynomial);fprintf(%.4en,den);omega=Wp Ws;h = freqs(num,den,omega);fprintf(Ap

3、= %.4fn,-20*log10(abs(h(1);fprintf(As= %.4fn,-20*log10(abs(h(2);omega = 0: 200: 12000*pi;h = freqs(num,den,omega);gain=20*log10(abs(h);plot(omega/(2*pi),gain);xlabel(Frequency in Hz);ylabel(Gain in dB);2 例：設(shè)計滿足下列條件的模擬Butterworth低通濾波器 fp=1kHz, fs=2kHz, Ap=1dB, As=40dBAp=0.62dB, As=40dB050010001500200

4、025003000-80-60-40-200Frequency in HzGain in dBBW型: N=83例：設(shè)計滿足下列條件的模擬CB I型低通濾波器 fp=1kHz, fs=2kHz, Ap=1dB, As=40dB%filter specificationWp=2*pi*1000;Ws=2*pi*2000;Ap=1;As=40;%Computer filter orderN,Wc=cheb1ord(Wp,Ws,Ap,As,s);fprintf(Order of the filter=%.0fn,N)%compute filter coefficientsnum,den = cheb

5、y1(N,Ap,Wc,s);disp(Numerator polynomial);fprintf(%.4en,num);disp(Denominator polynomial);fprintf(%.4en,den);4例：設(shè)計滿足下列條件的模擬CB I型低通濾波器 fp=1kHz, fs=2kHz, Ap=1dB, As=40dB%Compute Ap and As of designed filter omega=Wp Ws; h = freqs(num,den,omega); fprintf(Ap= %.4fn,-20*log10(abs(h(1); fprintf(As= %.4fn,-

6、20*log10(abs(h(2);5例：設(shè)計滿足下列條件的模擬CB I型低通濾波器 fp=1kHz, fs=2kHz, Ap=1dB, As=40dBAp=1.00dB, As=45dB050010001500200025003000-70-60-50-40-30-20-10Frequency in HzGain in dBBW型:N=8CB 型:N=56切比雪夫II （CB II）型模擬低通濾波器3. 利用MATLAB設(shè)計CB II LP N,wc=cheb2ord(wp,ws,Ap,As,s) 確定模擬切比雪夫II型濾波器的階數(shù)N。 num,den=cheby2(N,As,wc,s) 確

7、定階數(shù)為N，阻帶衰減為As dB的切比雪夫II型濾波器的分子和分母多項式。wc由cheb2ord函數(shù)確定。7* 橢圓低通濾波器3. 利用MATLAB設(shè)計橢圓低通濾波器 N,wc=ellipord(wp,ws,Ap,As,s) num,den=ellip(N,Ap,As,wc,s)確定橢圓濾波器的階數(shù)N。wc=wp。 確定階數(shù)為N，通帶衰減為Ap dB，阻帶衰減為As dB的橢圓濾波器的分子和分母多項式。wc是橢圓濾波器的通帶截頻。8例：設(shè)計滿足下列指標(biāo)的模擬橢圓低通濾波器 fp=1kHz, fs=2kHz, Ap=1dB, As=40dB050010001500200025003000-100

8、-80-60-40-200Frequency in HzGain in dBAp=1.00dB, As=40dBBW型 :N=8CB I型:N=5橢圓型:N=49模擬高通濾波器的設(shè)計 MATLAB實現(xiàn)numt,dent = lp2hp(num,den,W0)10%高通濾波器的設(shè)計wp=1/(2*pi*5000);ws=1/(2*pi*1000);Ap=1;As=40;N,Wc=buttord(wp,ws,Ap,As,s);num,den = butter(N,Wc,s);disp(LP 分子多項式);fprintf(%.4en,num);disp(LP 分母多項式);fprintf(%.4en

9、,den);numt,dent = lp2hp(num,den,1);disp(HP 分子多項式);fprintf(%.4en,numt);disp(HP 分母多項式);fprintf(%.4en,dent);例： 設(shè)計滿足下列條件的模擬BW型高通濾波器 fp=5kHz, fs=1kHz, Ap1dB, As 40dB。11Ap= 40.0000 As= 0.10980100020003000400050006000-70-60-50-40-30-20-100Frequency in HzGain in dB例： 設(shè)計滿足下列條件的模擬BW型高通濾波器 fp=5kHz, fs=1kHz, Ap

10、1dB, As 40dB12模擬帶通濾波器的設(shè)計 MATLAB實現(xiàn)numt,dent = lp2bp(num,den,W0,B)13例: 試設(shè)計一個滿足下列指標(biāo)的BW型帶通濾波器 wp1=6 rad/s, wp2=8 rad/s, ws1=4 rad/s, ws2=11 rad/s, Ap1 dB, As 32dB。Ap=1;As=32;wp1=6;wp2=8;ws1=4;ws2=11;B=wp2-wp1;w0=sqrt(wp1*wp2);wp=1;wLs1=(ws1*ws1-w0*w0)/B/ws1;wLs2=(ws2*ws2-w0*w0)/B/ws2;wLs=min(abs(wLs1),a

11、bs(wLs2);ws=wLs;N,Wc=buttord(wp,ws,Ap,As,s);num,den = butter(N,Wc,s);numt,dent = lp2bp(num,den,w0,B);w=linspace(2,12,1000);h=freqs(numt,dent,w);plot(w,20*log10(abs(h) ; grid ; xlabel(Frequency in rad/s);ylabel(Gain in dB)14例: 試設(shè)計一個滿足下列指標(biāo)的BW型帶通濾波器 wp1=6 rad/s, wp2=8 rad/s, ws1=4 rad/s, ws2=11 rad/s,

12、Ap1 dB, As 32dB。15模擬帶阻濾波器的設(shè)計 MATLAB實現(xiàn)numt,dent = lp2bs(num,den,W0,B)16例：試設(shè)計一個滿足下列指標(biāo)的BW型帶阻濾波器 Ap=1dB;As=20dB;wp1=10;wp2=30;ws1=19;ws2=21。Ap=1;As=20;wp1=10;wp2=30;ws1=19;ws2=21;B=ws2-ws1;w0=sqrt(ws1*ws2);wLp1=B*wp1/(w0*w0-wp1*wp1);wLp2=B*wp2/(w0*w0-wp2*wp2);wLp=max(abs(wLp1),abs(wLp2);N,Wc=buttord(wLp

13、,1,Ap,As,s)num,den = butter(N,Wc,s ); numt,dent=lp2bs(num,den,w0,B); w=linspace(5,35,1000);h=freqs(numt,dent,w);plot(w,20*log10(abs(h);w=wp1 ws1 ws2 wp2;set(gca,xtick,w);grid;h=freqs(numt,dent,w);A=-20*log10(abs(h)17例：試設(shè)計一個滿足下列指標(biāo)的BW型帶阻濾波器 Ap=1dB;As=20dB;wp1=10;wp2=30;ws1=19;ws2=21。10192130-100-80-60

14、-40-20018脈沖響應(yīng)不變法設(shè)計DF的步驟numd,dend = impinvar(num,den,Fs)num,den：AF分子、分母多項式的系數(shù)向量Fs=1/T：抽樣頻率numd,dend：DF分子、分母多項式的系數(shù)向量脈沖響應(yīng)不變法的MATLAB實現(xiàn)19例：利用AF-BW filter及脈沖響應(yīng)不變法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB 。%Design DF BW low-pass filter using impulse invariance%DF BW LP specficationWp=0.2*pi; Ws=0.6*pi; Ap=2

15、; As=15;Fs=1; %Sampling frequency(Hz)%Analog Butterworth specficationwp=Wp*Fs; ws=Ws*Fs;%determine the order of AF filterN=buttord(wp,ws,Ap,As,s);%determine the 3-db cutoff frequency of BW filter from pass-band specficationwc=wp/(10(0.1*Ap)-1)(1/N/2);%determine the AF-BW filter numa,dena=butter(N,wc

16、,s);20例：利用AF-BW filter及脈沖響應(yīng)不變法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB 。 %determine the DF filter numd,dend=impinvar(numa,dena,Fs); %plot the frequency response w=linspace(0,pi,1024); h=freqz(numd,dend,w); norm=max(abs(h); numd=numd/norm; plot(w/pi,20*log10(abs(h/norm); xlabel(Normalized frequency);

17、 ylabel(Gain,dB); %computer Ap As of the designed filter w=Wp Ws; h=freqz(numd,dend,w); fprintf(Ap= %.4fn,-20*log10( abs(h(1); fprintf(As= %.4fn,-20*log10( abs(h(2);21例：利用AF-BW filter及脈沖響應(yīng)不變法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB 。00.10.20.30.40.50.60.70.80.91-18-16-14-12-10-8-6-4-20Normalized fre

18、quencyGain,dBAp = 1.72dBAs = 14.2dB22例：用雙線性變換法和一階巴特沃思低通濾波器，設(shè)計一個3dB截頻為Wp的數(shù)字濾波器，并與脈沖響應(yīng)不變法設(shè)計的DF比較。00.6100.71Normalized frequencyAmplitude脈沖響應(yīng)不變法雙線性變換法Wp =0.6p 脈沖響應(yīng)不變法存在頻譜混疊，所設(shè)計的DF不滿足給定指標(biāo)。而雙線性變換法不存在頻譜混疊，所設(shè)計的DF滿足給定指標(biāo)。3dB23例：用雙線性變換法和一階巴特沃思低通濾波器，設(shè)計一個3dB截頻為Wp的數(shù)字濾波器，并與脈沖響應(yīng)不變法設(shè)計的DF比較。H雙(z)和H脈(z)幅度響應(yīng)比較的MATLAB實

19、現(xiàn)Wp=0.6*pi;b=1-exp(-Wp);b1=tan(Wp/2)*1 1;a=1 -exp(-Wp);a1=1+tan(Wp/2) tan(Wp/2)-1;w=linspace(0,pi,512);h=freqz(b,a,w);h1=freqz(b1,a1,w);plot(w/pi,(abs(h),w/pi,(abs(h1) );xlabel(Normalized frequency);ylabel(Amplitude);set(gca,ytick,0 0.7 1);set(gca,xtick,0 Wp/pi 1);grid;24例：利用AF-BW filter及雙線性變換法設(shè)計一DF

20、，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB %Design DF BW low-pass filter using impulse invariance%DF BW LP specficationWp=0.2*pi; Ws=0.6*pi; Ap=2; As=15;T=2;Fs=1/T; %Sampling frequency(Hz)%Analog Butterworth specficationwp=2*tan(Wp/2)/T;ws=2*tan(Ws/2)/T;%determine the order of AF filter and the 3-dB cutoff

21、 frequency N,wc=buttord(wp,ws,Ap,As,s)%determine the AF-BW filternuma,dena=butter(N,wc,s)25例：利用AF-BW filter及雙線性變換法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB %determine the DF filternumd,dend=bilinear(numa,dena,Fs)%plot the frequency responsew=linspace(0,pi,1024);h=freqz(numd,dend,w);plot(w/pi,20*log10

22、(abs(h);axis(0 1 -50 0);grid;xlabel(Normalized frequency);ylabel(Gain,dB);%computer Ap As of the designed filterw=Wp Ws;h=freqz(numd,dend,w);fprintf(Ap= %.4fn,-20*log10( abs(h(1);fprintf(As= %.4fn,-20*log10( abs(h(2);26例：利用AF-BW filter及雙線性變換法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB Ap= 0.3945As= 15.

23、000027例：利用AF-BW filter及雙線性變換法設(shè)計一DF，滿足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB 將雙線性變換法與脈沖響應(yīng)不變法所設(shè)計DF的結(jié)果比較。雙線性變換Ap= 0.3945As= 15.0000脈沖響應(yīng)不變法雙線性變換法脈沖響應(yīng)不變Ap= 1.1187As= 12.362828利用MATLAB實現(xiàn)IIR數(shù)字濾波器 確定數(shù)字濾波器的階數(shù)及3dB截頻WcN, Wc = buttord(Wp, Ws, Ap, As) 其中Wp, Ws為歸一化角頻率。 例Wp=0.1p, 則Wp=0.1 若為帶通或帶阻濾波器，則Wp=Wp1, Wp2; Ws=Ws1,

24、 Ws2 BW型數(shù)字濾波器29利用MATLAB實現(xiàn)IIR數(shù)字濾波器 BW型數(shù)字濾波器 確定DF系統(tǒng)函數(shù)分子、分母多項式 低通 num,den = butter(N,Wc) 高通 num,den = butter(N,Wc,high) 帶通 num,den = butter(N,Wc) 其中Wc=W1, W2 帶阻 num,den = butter(N,Wc,stop) 其中Wc=W1, W230利用MATLAB實現(xiàn)IIR數(shù)字濾波器 CB I型數(shù)字濾波器 確定數(shù)字濾波器的階數(shù)及參數(shù)WcN, Wc = cheb1ord(Wp, Ws, Ap, As) DF系統(tǒng)函數(shù)分子、分母多項式的確定 低通 nu

25、m,den = cheby1(N,Ap,Wc) 高通 num,den = cheby1(N, Ap, Wc,high) 帶通 num,den = cheby1(N, Ap, Wc) 帶阻 num,den = cheby1(N, Ap, Wc,stop) * 帶通、帶阻中W=W1, W231利用MATLAB實現(xiàn)IIR數(shù)字濾波器 CB II型數(shù)字濾波器 確定數(shù)字濾波器的階數(shù)及參數(shù)WcN, Wc = cheb2ord(Wp, Ws, Ap, As) DF系統(tǒng)函數(shù)分子、分母多項式的確定 低通 num,den = cheby2(N,As,Wc) 高通 num,den = cheby2(N, As, Wc,high) 帶通 num,den = cheby2(N, As, Wc) 帶阻 num,den = cheby2(N, As, Wc,stop) * 帶通、帶阻中W=W1, W232利用MATLAB實現(xiàn)IIR數(shù)字濾波器 橢圓(C)型數(shù)字濾波器 確定數(shù)字濾波器的階數(shù)及參數(shù)WcN, Wc = ellipord(Wp, Ws, Ap, As) DF系統(tǒng)函數(shù)分子、分母多項式的確定