实验六
评分人：陈奇分数：9.3
一、实验目的
1.练习并掌握脉冲响应不变法设计IIR数字滤波器的方法；
2.练习并掌握双线性变换法设计IIR数字滤波器的方法；
二、实验原理与方法（同实验五）
[n,wn]=cheb1ord(wp,ws,Rp,Rs,’s’);
[n,wn]=cheb1ord(wp,ws,Rp,Rs);
[z,p,k]=cheb1ap(n,Rp);
[z,p,k]=cheb2ap(n,Rs);
[b,a]=cheby1(n,Rp,wn);
[b,a]=cheby2(n,Rs,wn);
[b,a]=cheby1(n,Rp,wn,’ftype’);
[b,a]=cheby2(n,Rs,wn,’ftype’);
[bz,p,k]=cheby1(n,Rp,wn);
[bz,p,k]=cheby2(n,Rs,wn);
[b,a]=cheby1(n,Rp,wn,’s’);
[b,a]=cheby2(n,Rs,wn,’s’);
[b,a]=cheby1(n,Rp,wn,’ftype’,’s’);
[b,a]=cheby2(n,Rs,wn,’ftype’,’s’);
三、实验内容及步骤
5.用Chebyshev-II实现实验内容1234的滤波器设计并与实验内容1234的结果进行对照,以了解Chebyshev-I/II的区别。
1）
切比雪夫I型
程序：
clear
clc
Wp=0.4pi;Ws=0.6pi;Rp=7;Rs=16;Fs=1000;%数字滤波器指标
%模拟滤波器设计
wp=WpFs;ws=WsFs;Rp=7;Rs=16;%模拟滤波器指标:w=ΩT->Ω=wFs
[n,Wn]=cheb1ord(wp,ws,Rp,Rs,''s'');%阶次估计
[z,p,k]=cheb1ap(n,Rp);%模拟低通原型
[b,a]=zp2tf(z,p,k);%零极点->直接
[b,a]=lp2lp(b,a,Wn);%低通->低通(去归一化)
[H,w]=freqs(b,a);
figure(1)
subplot(2,1,1),plot(w/2/pi,20log10(abs(H)));grid
xlabel(''Frequence(Hz)''),ylabel(''Magnitude(dB)'');
subplot(2,1,2),plot(w/2/pi,angle(H)/pi180);grid
xlabel(''Frequence(Hz)''),ylabel(''Phase(degress)'');
%冲激响应不变法设计数字滤波器
figure(2)
[bz,az]=impinvar(b,a,Fs);
freqz(bz,az,256);
%数字滤波器的直接设计
figure(3)
[n,wn]=cheb1ord(Wp/pi,Ws/pi,Rp,Rs)
[b,a]=cheby1(n,Rp,wn);
freqz(b,a,256);
图像：
切比雪夫II型
程序：
clear
clc
Wp=0.4pi;Ws=0.6pi;Rp=7;Rs=16;Fs=1000;%数字滤波器指标
%模拟滤波器设计
wp=WpFs;ws=WsFs;Rp=7;Rs=16;%模拟滤波器指标:w=ΩT->Ω=wFs
[n,Wn]=cheb2ord(wp,ws,Rp,Rs,''s'');%阶次估计
[z,p,k]=cheb2ap(n,Rp);%模拟低通原型
[b,a]=zp2tf(z,p,k);%零极点->直接
[b,a]=lp2lp(b,a,Wn);%低通->低通(去归一化)
[H,w]=freqs(b,a);
figure(1)
subplot(2,1,1),plot(w/2/pi,20log10(abs(H)));grid
xlabel(''Frequence(Hz)''),ylabel(''Magnitude(dB)'');
subplot(2,1,2),plot(w/2/pi,angle(H)/pi180);grid
xlabel(''Frequence(Hz)''),ylabel(''Phase(degress)'');
%冲激响应不变法设计数字滤波器
figure(2)
[bz,az]=impinvar(b,a,Fs);
freqz(bz,az,256);
%数字滤波器的直接设计
figure(3)
[n,wn]=cheb2ord(Wp/pi,Ws/pi,Rp,Rs)
[b,a]=cheby2(n,Rp,wn);
freqz(b,a,256);
图像：
2）
切比雪夫I型：
程序：
clear
clc
wp=.8pi;ws=.44pi;rp=3;rs=20;Fs=2000;
%Firstlytofinishfrequencyprewarping;
wap=2Fstan(wp/2);was=2Fstan(ws/2);
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);
[b,a]=zp2tf(z,p,k);
[bt,at]=lp2hp(b,a,wap);%低通->高通(去归一化)
w=0:pi:4000pi;
H=freqs(bt,at,w);
subplot(3,1,1),plot(w/2/pi,20log10(abs(H)));grid
title(''High-passAF'');
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz,az]=bilinear(bt,at,Fs);
[h,w]=freqz(bz,az,256,1);
Subplot(312);
plot(w,20log10(abs(h)));grid;
title(''High-passDF(1)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby1(n,rp,wp/pi,''high'');
[h1,w1]=freqz(b,a,256,1);
Subplot(313);
plot(w,20log10(abs(h1)));grid;title(''High-passDF(2)'');
图像：
切比雪夫II型
程序:
clear
clc
wp=.8pi;ws=.44pi;rp=3;rs=20;Fs=2000;
%Firstlytofinishfrequencyprewarping;
wap=2Fstan(wp/2);was=2Fstan(ws/2);
[n,wn]=cheb2ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb2ap(n,rp);
[b,a]=zp2tf(z,p,k);
[bt,at]=lp2hp(b,a,wap);%低通->高通(去归一化)
w=0:pi:4000pi;
H=freqs(bt,at,w);
subplot(3,1,1),plot(w/2/pi,20log10(abs(H)));grid
title(''High-passAF'');
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz,az]=bilinear(bt,at,Fs);
[h,w]=freqz(bz,az,256,1);
Subplot(312);
plot(w,20log10(abs(h)));grid;
title(''High-passDF(1)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb2ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby2(n,rp,wp/pi,''high'');
[h1,w1]=freqz(b,a,256,1);
Subplot(313);
plot(w,20log10(abs(h1)));grid;title(''High-passDF(2)'');
图像：
3）
切比雪夫I型
程序：
clear
clc
fp=[300400];fs=[200500];rp=3;rs=18;Fs=2000;%模拟技术指标
wp=fp2pi/Fs;ws=fs2pi/Fs;%数字技术指标
%模拟低通原型滤波器设计
wap=2Fstan(wp./2);was=2Fstan(ws./2);%预畸变
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);[bp,ap]=zp2tf(z,p,k);[h1,w1]=freqs(bp,ap);
Subplot(411);plot(w1,abs(h1));grid;title(''lowpassG(p)'')
%模拟低通原型->模拟带通滤波器
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bs,as]=lp2bp(bp,ap,w0,bw);
w2=0:pi:4000pi;
h2=freqs(bs,as,w2);
Subplot(412);plot(w2/2/pi,20log10(abs(h2)));grid;title(''bandpassH(p)'')
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz1,az1]=bilinear(bs,as,Fs);
[h3,w3]=freqz(bz1,az1,1024,Fs);
Subplot(413);plot(w3,20log10(abs(h3)));grid;title(''bandpassDFH1(z)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);[bz2,az2]=cheby1(n,rp,wp/pi,''bandpass'');
[h4,w4]=freqz(bz2,az2,1024,Fs);
Subplot(414);plot(w4,20log10(abs(h4)));grid;title(''BandpassDFH2(z)'');
图像：
切比雪夫II型
程序：
clear
clc
fp=[300400];fs=[200500];rp=3;rs=18;Fs=2000;%模拟技术指标
wp=fp2pi/Fs;ws=fs2pi/Fs;%数字技术指标
%模拟低通原型滤波器设计
wap=2Fstan(wp./2);was=2Fstan(ws./2);%预畸变
[n,wn]=cheb2ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb2ap(n,rp);[bp,ap]=zp2tf(z,p,k);[h1,w1]=freqs(bp,ap);
Subplot(411);plot(w1,abs(h1));grid;title(''lowpassG(p)'')
%模拟低通原型->模拟带通滤波器
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bs,as]=lp2bp(bp,ap,w0,bw);
w2=0:pi:4000pi;
h2=freqs(bs,as,w2);
Subplot(412);plot(w2/2/pi,20log10(abs(h2)));grid;title(''bandpassH(p)'')
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz1,az1]=bilinear(bs,as,Fs);
[h3,w3]=freqz(bz1,az1,1024,Fs);
Subplot(413);plot(w3,20log10(abs(h3)));grid;title(''bandpassDFH1(z)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb2ord(wp/pi,ws/pi,rp,rs);[bz2,az2]=cheby2(n,rp,wp/pi,''bandpass'');
[h4,w4]=freqz(bz2,az2,1024,Fs);
Subplot(414);plot(w4,20log10(abs(h4)));grid;title(''BandpassDFH2(z)'');
图像：
4）
切比雪夫I型
程序：
clear
clc
fp=[95105];fs=[99101];rp=3;rs=14;Fs=1000;
wp=fp2pi/Fs;ws=fs2pi/Fs;%wp=[.19pi0.21pi];ws=[.198pi0.202pi];
wap=2Fstan(wp./2);was=2Fstan(ws./2);%frequencyprewarping;
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);[b,a]=zp2tf(z,p,k);
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bt,at]=lp2bs(b,a,w0,bw);
[bz1,az1]=bilinear(bt,at,Fs);
[h,w]=freqz(bz1,az1,256,Fs);
Subplot(221);plot(w,20log10(abs(h)));
grid;title(''bandstopDFH1(z)'');
Subplot(223);plot(w,angle(h));grid;
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby1(n,rp,wp/pi,''stop'')
[h1,w1]=freqz(b,a,256,Fs);
Subplot(222);plot(w1,20log10(abs(h1)));
grid;title(''bandstopDFH2(z)'');
Subplot(224);plot(w,angle(h));grid;
图像：
切比雪夫II型
程序：
clear
clc
fp=[95105];fs=[99101];rp=3;rs=14;Fs=1000;
wp=fp2pi/Fs;ws=fs2pi/Fs;%wp=[.19pi0.21pi];ws=[.198pi0.202pi];
wap=2Fstan(wp./2);was=2Fstan(ws./2);%frequencyprewarping;
[n,wn]=cheb2ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb2ap(n,rp);[b,a]=zp2tf(z,p,k);
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bt,at]=lp2bs(b,a,w0,bw);
[bz1,az1]=bilinear(bt,at,Fs);
[h,w]=freqz(bz1,az1,256,Fs);
Subplot(221);plot(w,20log10(abs(h)));
grid;title(''bandstopDFH1(z)'');
Subplot(223);plot(w,angle(h));grid;
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb2ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby2(n,rp,wp/pi,''stop'')
[h1,w1]=freqz(b,a,256,Fs);
Subplot(222);plot(w1,20log10(abs(h1)));
grid;title(''bandstopDFH2(z)'');
Subplot(224);plot(w,angle(h));grid;
图像：
6.参照实验内容1234,设计如下指标的Chebyshev-I/II滤波器.画出模拟低通、模拟高通/带通/带阻、数字低通/高通/带通/带阻的幅频和相频特性图。
1）低通指标：wp=0.2π,rp=1dB,ws=0.3π,rs=16dB。
切比雪夫I型
程序：
clear
clc
Wp=0.2pi;Ws=0.3pi;Rp=1;Rs=16;Fs=1000;%数字滤波器指标
%模拟滤波器设计
wp=WpFs;ws=WsFs;Rp=7;Rs=16;%模拟滤波器指标:w=ΩT->Ω=wFs
[n,Wn]=cheb1ord(wp,ws,Rp,Rs,''s'');%阶次估计
[z,p,k]=cheb1ap(n,Rp);%模拟低通原型
[b,a]=zp2tf(z,p,k);%零极点->直接
[b,a]=lp2lp(b,a,Wn);%低通->低通(去归一化)
[H,w]=freqs(b,a);
figure(1)
subplot(2,1,1),plot(w/2/pi,20log10(abs(H)));grid
xlabel(''Frequence(Hz)''),ylabel(''Magnitude(dB)'');
subplot(2,1,2),plot(w/2/pi,angle(H)/pi180);grid
xlabel(''Frequence(Hz)''),ylabel(''Phase(degress)'');
%数字滤波器的直接设计
figure(2)
[n,wn]=cheb1ord(Wp/pi,Ws/pi,Rp,Rs)
[b,a]=cheby1(n,Rp,wn);
freqz(b,a,256);
图像：
结果：
n=
2
wn=
0.2000
切比雪夫II型
程序：
clear
clc
Wp=0.2pi;Ws=0.3pi;Rp=1;Rs=16;Fs=1000;%数字滤波器指标
%模拟滤波器设计
wp=WpFs;ws=WsFs;Rp=7;Rs=16;%模拟滤波器指标:w=ΩT->Ω=wFs
[n,Wn]=cheb2ord(wp,ws,Rp,Rs,''s'');%阶次估计
[z,p,k]=cheb2ap(n,Rp);%模拟低通原型
[b,a]=zp2tf(z,p,k);%零极点->直接
[b,a]=lp2lp(b,a,Wn);%低通->低通(去归一化)
[H,w]=freqs(b,a);
figure(1)
subplot(2,1,1),plot(w/2/pi,20log10(abs(H)));grid
xlabel(''Frequence(Hz)''),ylabel(''Magnitude(dB)'');
subplot(2,1,2),plot(w/2/pi,angle(H)/pi180);grid
xlabel(''Frequence(Hz)''),ylabel(''Phase(degress)'');
%数字滤波器的直接设计
figure(2)
[n,wn]=cheb2ord(Wp/pi,Ws/pi,Rp,Rs)
[b,a]=cheby2(n,Rp,wn);
freqz(b,a,256);
图像：
结果：
n=
2
wn=
0.3000
2）高通指标：wp=0.3π,rp=1dB,ws=0.2π,rs=30dB。
切比雪夫I型
程序：
clear
clc
wp=.3pi;ws=.2pi;rp=1;rs=30;Fs=2000;
%Firstlytofinishfrequencyprewarping;
wap=2Fstan(wp/2);was=2Fstan(ws/2);
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);
[b,a]=zp2tf(z,p,k);
[bt,at]=lp2hp(b,a,wap);%低通->高通(去归一化)
w=0:pi:4000pi;
H=freqs(bt,at,w);
subplot(3,1,1),plot(w/2/pi,20log10(abs(H)));grid
title(''High-passAF'');
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz,az]=bilinear(bt,at,Fs);
[h,w]=freqz(bz,az,256,1);
Subplot(312);
plot(w,20log10(abs(h)));grid;
title(''High-passDF(1)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby1(n,rp,wp/pi,''high'');
[h1,w1]=freqz(b,a,256,1);
Subplot(313);
plot(w,20log10(abs(h1)));grid;title(''High-passDF(2)'');
图像：
3）带通指标：
wp=[0.3π0.5π],rp=1dB,ws=[0.20.6],rs=30dB。
切比雪夫I型
程序：
clear
clc
fp=[300500];fs=[200600];rp=1;rs=30;Fs=2000;%模拟技术指标
wp=fp2pi/Fs;ws=fs2pi/Fs;%数字技术指标
%模拟低通原型滤波器设计
wap=2Fstan(wp./2);was=2Fstan(ws./2);%预畸变
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);[bp,ap]=zp2tf(z,p,k);[h1,w1]=freqs(bp,ap);
Subplot(411);plot(w1,abs(h1));grid;title(''lowpassG(p)'')
%模拟低通原型->模拟带通滤波器
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bs,as]=lp2bp(bp,ap,w0,bw);
w2=0:pi:4000pi;
h2=freqs(bs,as,w2);
Subplot(412);plot(w2/2/pi,20log10(abs(h2)));grid;title(''bandpassH(p)'')
%双线性变换法设计数字滤波器
%Note:s=(2/T)(z-1)/(z+1);T=1,thatis2Fs=1,Fs=0.5;
[bz1,az1]=bilinear(bs,as,Fs);
[h3,w3]=freqz(bz1,az1,1024,Fs);
Subplot(413);plot(w3,20log10(abs(h3)));grid;title(''bandpassDFH1(z)'');
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);[bz2,az2]=cheby1(n,rp,wp/pi,''bandpass'');
[h4,w4]=freqz(bz2,az2,1024,Fs);
Subplot(414);plot(w4,20log10(abs(h4)));grid;title(''BandpassDFH2(z)'');
图像：
4）带阻指标：
wp=[0.2π0.6π],rp=1dB,ws=[0.30.5],rs=30dB。
切比雪夫I型
程序：
clear
clc
fp=[100300];fs=[150250];rp=1;rs=30;Fs=1000;
wp=fp2pi/Fs;ws=fs2pi/Fs;%wp=[.19pi0.21pi];ws=[.198pi0.202pi];
wap=2Fstan(wp./2);was=2Fstan(ws./2);%frequencyprewarping;
[n,wn]=cheb1ord(wap,was,rp,rs,''s'');
[z,p,k]=cheb1ap(n,rp);[b,a]=zp2tf(z,p,k);
bw=wap(2)-wap(1);w0=sqrt(wap(1)wap(2));
[bt,at]=lp2bs(b,a,w0,bw);
[bz1,az1]=bilinear(bt,at,Fs);
[h,w]=freqz(bz1,az1,256,Fs);
Subplot(221);plot(w,20log10(abs(h)));
grid;title(''bandstopDFH1(z)'');
Subplot(223);plot(w,angle(h));grid;
%DirectlytodesignH(z)bybutter.m
[n,wn]=cheb1ord(wp/pi,ws/pi,rp,rs);
[b,a]=cheby1(n,rp,wp/pi,''stop'')
[h1,w1]=freqz(b,a,256,Fs);
Subplot(222);plot(w1,20log10(abs(h1)));
grid;title(''bandstopDFH2(z)'');
Subplot(224);plot(w,angle(h));grid;
图像：
结果：
b=
Columns1through5
0.1103-0.33710.8276-1.20811.4721
Columns6through9
-1.20810.8276-0.33710.1103
a=
Columns1through5
1.0000-1.58551.2098-1.11611.3109
Columns6through9
-0.54160.0485-0.22420.1872
总结:
IIR滤波器的设计步骤1）确定数字滤波器的技术指标：通带截止频率ωp(ωl?、ωu)、阻带截止频率ωs(ωs1?、?ωs2)通带衰减αp?、阻带衰减αs。2)?将数字滤波器的技术指标转换成模拟滤波器的技术指标。脉冲响应不变法:?双线性变换法：3)?将模拟滤波器的技术指标转成模拟低通滤波器的技术指标：频率变换4)?设计归一化低通滤波器Ga(p)?。5)?将Ga(p)转成模拟滤波器Ha(s)：去归一化?+?频率变换6)?将模拟滤波器Ha(s)，从s平面转换到z平面，得到数字滤波器系统函数H(z)，脉冲响应不变法或双线性变换法。
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