IIR数字滤波器设计上机

1、 Butterworth模拟低通滤波器设计 3. 利用MATLAB设计BW LP num,den=butter(N,wc,s)确定阶数为N，3-dB截频为wc(radian/s)的Butterworth filter分子和分母多项式。s 表示模拟域。 N,wc=buttord(wp,ws,Ap,As,s)确定模拟Butterworth filter的阶数N和3-dB截频wc。wc是由阻带参数确定的。 s 表示模拟域。 z,p,k=buttap(N)确定N阶归一化的Butterworth filter 的零点、极点和增益(gain)1 例：设计满足下列条件的模拟Butterworth低通滤波器

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：设计满足下列条件的模拟Butterworth低通滤波器 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：设计满足下列条件的模拟CB I型低通滤波器 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：设计满足下列条件的模拟CB I型低通滤波器 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：设计满足下列条件的模拟CB I型低通滤波器 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）型模拟低通滤波器3. 利用MATLAB设计CB II LP N,wc=cheb2ord(wp,ws,Ap,As,s) 确定模拟切比雪夫II型滤波器的阶数N。 num,den=cheby2(N,As,wc,s) 确

7、定阶数为N，阻带衰减为As dB的切比雪夫II型滤波器的分子和分母多项式。wc由cheb2ord函数确定。7* 椭圆低通滤波器3. 利用MATLAB设计椭圆低通滤波器 N,wc=ellipord(wp,ws,Ap,As,s) num,den=ellip(N,Ap,As,wc,s)确定椭圆滤波器的阶数N。wc=wp。 确定阶数为N，通带衰减为Ap dB，阻带衰减为As dB的椭圆滤波器的分子和分母多项式。wc是椭圆滤波器的通带截频。8例：设计满足下列指标的模拟椭圆低通滤波器 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椭圆型:N=49模拟高通滤波器的设计 MATLAB实现numt,dent = lp2hp(num,den,W0)10%高通滤波器的设计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 分子多项式);fprintf(%.4en,num);disp(LP 分母多项式);fprintf(%.4en

9、,den);numt,dent = lp2hp(num,den,1);disp(HP 分子多项式);fprintf(%.4en,numt);disp(HP 分母多项式);fprintf(%.4en,dent);例： 设计满足下列条件的模拟BW型高通滤波器 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： 设计满足下列条件的模拟BW型高通滤波器 fp=5kHz, fs=1kHz, Ap

10、1dB, As 40dB12模拟带通滤波器的设计 MATLAB实现numt,dent = lp2bp(num,den,W0,B)13例: 试设计一个满足下列指标的BW型带通滤波器 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: 试设计一个满足下列指标的BW型带通滤波器 wp1=6 rad/s, wp2=8 rad/s, ws1=4 rad/s, ws2=11 rad/s,

12、Ap1 dB, As 32dB。15模拟带阻滤波器的设计 MATLAB实现numt,dent = lp2bs(num,den,W0,B)16例：试设计一个满足下列指标的BW型带阻滤波器 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：试设计一个满足下列指标的BW型带阻滤波器 Ap=1dB;As=20dB;wp1=10;wp2=30;ws1=19;ws2=21。10192130-100-80-60

14、-40-20018脉冲响应不变法设计DF的步骤numd,dend = impinvar(num,den,Fs)num,den：AF分子、分母多项式的系数向量Fs=1/T：抽样频率numd,dend：DF分子、分母多项式的系数向量脉冲响应不变法的MATLAB实现19例：利用AF-BW filter及脉冲响应不变法设计一DF，满足 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及脉冲响应不变法设计一DF，满足 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及脉冲响应不变法设计一DF，满足 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：用双线性变换法和一阶巴特沃思低通滤波器，设计一个3dB截频为Wp的数字滤波器，并与脉冲响应不变法设计的DF比较。00.6100.71Normalized frequencyAmplitude脉冲响应不变法双线性变换法Wp =0.6p 脉冲响应不变法存在频谱混叠，所设计的DF不满足给定指标。而双线性变换法不存在频谱混叠，所设计的DF满足给定指标。3dB23例：用双线性变换法和一阶巴特沃思低通滤波器，设计一个3dB截频为Wp的数字滤波器，并与脉冲响应不变法设计的DF比较。H双(z)和H脉(z)幅度响应比较的MATLAB实

19、现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及双线性变换法设计一DF

20、，满足 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及双线性变换法设计一DF，满足 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及双线性变换法设计一DF，满足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB Ap= 0.3945As= 15.

23、000027例：利用AF-BW filter及双线性变换法设计一DF，满足 Wp=0.2p, Ws=0.6p, Ap2dB, As15dB 将双线性变换法与脉冲响应不变法所设计DF的结果比较。双线性变换Ap= 0.3945As= 15.0000脉冲响应不变法双线性变换法脉冲响应不变Ap= 1.1187As= 12.362828利用MATLAB实现IIR数字滤波器 确定数字滤波器的阶数及3dB截频WcN, Wc = buttord(Wp, Ws, Ap, As) 其中Wp, Ws为归一化角频率。 例Wp=0.1p, 则Wp=0.1 若为带通或带阻滤波器，则Wp=Wp1, Wp2; Ws=Ws1,

24、 Ws2 BW型数字滤波器29利用MATLAB实现IIR数字滤波器 BW型数字滤波器 确定DF系统函数分子、分母多项式 低通 num,den = butter(N,Wc) 高通 num,den = butter(N,Wc,high) 带通 num,den = butter(N,Wc) 其中Wc=W1, W2 带阻 num,den = butter(N,Wc,stop) 其中Wc=W1, W230利用MATLAB实现IIR数字滤波器 CB I型数字滤波器 确定数字滤波器的阶数及参数WcN, Wc = cheb1ord(Wp, Ws, Ap, As) DF系统函数分子、分母多项式的确定 低通 nu

25、m,den = cheby1(N,Ap,Wc) 高通 num,den = cheby1(N, Ap, Wc,high) 带通 num,den = cheby1(N, Ap, Wc) 带阻 num,den = cheby1(N, Ap, Wc,stop) * 带通、带阻中W=W1, W231利用MATLAB实现IIR数字滤波器 CB II型数字滤波器 确定数字滤波器的阶数及参数WcN, Wc = cheb2ord(Wp, Ws, Ap, As) DF系统函数分子、分母多项式的确定 低通 num,den = cheby2(N,As,Wc) 高通 num,den = cheby2(N, As, Wc,high) 带通 num,den = cheby2(N, As, Wc) 带阻 num,den = cheby2(N, As, Wc,stop) * 带通、带阻中W=W1, W232利用MATLAB实现IIR数字滤波器 椭圆(C)型数字滤波器 确定数字滤波器的阶数及参数WcN, Wc = ellipord(Wp, Ws, Ap, As) DF系统函数分子、分母多项式的确定