实验：用双线性变换法设计IIR数字滤波器

实验三：用双线性变换法设计IIR数字滤波器（设计性4学时）一.实验目的：(1）熟悉用双线性变换法设计IIR数字滤波器的原理与方法。(2）掌握数字滤波器的计算机仿真方法。(3）通过观察对实际心电图信号的滤波作用，获得数字滤波的感性知识。二.实验内容及步骤:(1)用双线性变换法设计一个巴特沃斯低通IIR数字滤波器，设计指标参数为：在通带内频率低于0.2pi时,最大衰减小于1dB;在阻带内[0.3pi,pi]频率区间上，,最小衰减大于15dB；(2)以0.02pi为采样间隔，打印出数字滤波器在频率区间[0,0.5pi]上的幅频响应特性曲线；(3)用所设计的滤波器对实际心电图信号采样序列(在本实验后面给出)进行仿真滤波处理，并分别打印出滤波前后的心电图波形图，观察总结滤波作用与效果。（4）采用不同阶数的Butterworth低通滤波器，比较滤波效果。三.实验步骤：（1）复习有关巴特沃斯模拟滤波器设计和用双线性变换法设计IIR数字滤波器的内容，按照教材例6.4.2，用双线性变换法设计数字滤波器系统函数H（z）。方法一：教材例6.4.2种已求出满足本实验要求的数字滤波器系统函数：方法二：根据设计指标，调用MATLAB信号处理工具箱函数buttord和butter，也可得到H（z）。（2）编写滤波器仿真程序，计算H(z)对心电图信号采样序列x(n)的相应序列y(n)。（3）在通过计算机上运行仿真滤波程序，并调用通用绘图子程序，完成实验内容（2）和（3）。本实验要用的MATLAB绘图函数参阅教材。四.,思考题:用双线性变换法设计数字滤波器过程中,变换公式：s=11z1z1T2中T的取值,对设计结果有无影响?为什么?五.实验报告要求（1）简述实验目的及原理；（2）由所打印的特性曲线及设计过程简述双线性变换法的特点；（3）对比滤波前后的心电图信号波形,说明数字滤波器的滤波过程与滤波作用；(4)简要回答思考题.六:心电图信号采样序列x(n)：人体心电图信号在测量过程中往往受到工业高频干扰，所以必须经过低通滤波处理后，才能作为判断心脏功能的有用信息。下面给出一实际心电图信号采样序列样式本x(n)，其中存在高频干扰，在实验中，以x(n)作为输入序列，滤除其中的干扰成分。{x(n)}={-4,-2,0,-4,-6,-4,-2,-4,-6,-6,-4,-4,-6,-6,-2,6,12,8,0,-16-38,-60,-84,-90,-66,-32,-4,-2,-4,8，12,12,10,6,6,6,4,0,0,00,0,-2,-4,0,0,0,-2,-2,0，0,-2,,-2,-2,-2,0}参考程序：一；T=1;Fs=1/T;wpz=0.2;wsz=0.3;wp=2*tan(wpz*pi/2);ws=2*tan(wsz*pi/2);rp=1;rs=15;[N,wc]=buttord(wp,ws,rp,rs,'s');[B,A]=butter(N,wc,'s');fk=0:1/512:1;wk=2*pi*fk;Hk=freqs(B,A,wk);subplot(2,2,1);plot(fk,20*log10(abs(Hk)));gridon;xlabel('\omega/\pi');ylabel('幅度(dB)');axis([0,1,-100,5]);title('(b)');[N,wc]=buttord(wpz,wsz,rp,rs);[Bz,Az]=butter(N,wc);wk=0:pi/512:pi;Hz=freqz(Bz,Az,wk);subplot(2,2,4);plot(wk/pi,20*log10(abs(Hz)));gridon;xlabel('\omega/\pi');ylabel('幅度(dB)');axis([0,1,-100,5]);title('(b)');二；x=[-4,-2,0,-4,-6,-4,-2,-4,-6,-6,-4,-4,-6,-6,-2,6,12,8,0,-16,-38,-60,-84,-90,-66,-32,-4,-2,-4,8,12,12,10,6,6,6,4,0,0,0,0,0,-2,-4,0,0,0,-2,-2,0,0,-2,-2,-2,-2,0];subplot(2,2,1);n=0:55;stem(n,x,'.');xlabel('n');ylabel('x(n)');title('x(n)的脉冲响应');A=0.09036;b1=[A,2*A,A];a1=[1,-1.2686,0.7051];h1=filter(b1,a1,x);[H1,w]=freqz(b1,a1,100);b2=[A,2*A,A];a2=[1,-1.0106,0.3583];h2=filter(b2,a2,h1);[H2,w]=freqz(b2,a2,100);b3=[A,2*A,A];a3=[1,-0.9044,0.2155];h3=filter(b3,a3,h2);[H3,w]=freqz(b3,a3,100);subplot(2,2,2);stem(n,h3,'.');xlabel('n');ylabel('y(n)');title('通过滤波器H1(z),H2(z),H3(z)后的y3(n)函数');subplot(2,2,3);H4=H1.*(H2);H=H4.*(H3);mag=abs(H);db=20*log10((mag+eps)/max(mag));plot(w/pi,db);xlabel('ω/π');ylabel('20log[Ha3(ejw)]');title('通过滤波器H1(z),H2(z),H3(z)后的对数频率响应20log[Ha3(ejw)]函数');grid;figure(2);N=1024;n=0:N/2-1;Xk=fft(x,N);AXk=abs(Xk(1:N/2));f=(0:N/2-1)*Fs/N;f=f/Fs;subplot(211);plot(f,AXk);title('x(n)的频谱');xlabel('f');ylabel('|X(k)|');axis([0,0.5,0,400]);Yk=fft(y,N);AYk=abs(Yk(1:N/2));subplot(212);plot(f,AYk);title('y(n)的频谱');xlabel('f');ylabel('|Y(k)|');axis([0,0.5,0,400])x=[-4,-2,0,-4,-6,-4,-2,-4,-6,-6,-4,-4,-6,-6,-2,6,12,8,0,-16,-38,-60,-84,-90,-66,-32,-4,-2,-4,8,12,12,10,6,6,6,4,0,0,0,0,0,-2,-4,0,0,0,-2,-2,0,0,-2,-2,-2,-2,0];subplot(2,2,1);n=0:55;stem(n,x,'.');xlabel('n');ylabel('x(n)');title('x(n)的脉冲响应');A=0.09036;b1=[A,2*A,A];a1=[1,-1.2686,0.7051];h1=filter(b1,a1,x);[H1,w]=freqz(b1,a1,100);b2=[A,2*A,A];a2=[1,-1.0106,0.3583];h2=filter(b2,a2,h1);[H2,w]=freqz(b2,a2,100);b3=[A,2*A,A];a3=[1,-0.9044,0.2155];h3=filter(b3,a3,h2);[H3,w]=freqz(b3,a3,100);subplot(2,2,2);stem(n,h3,'.');xlabel('n');ylabel('y(n)');title('通过滤波器H1(z),H2(z),H3(z)后的y3(n)函数');subplot(2,2,3);H4=H1.*(H2);H=H4.*(H3);mag=abs(H);db=20*log10((mag+eps)/max(mag));plot(w/pi,db);xlabel('ω/π');ylabel('20log[Ha3(ejw)]');title('通过滤波器H1(z),H2(z),H3(z)后的对数频率响应20log[Ha3(ejw)]函数');grid;%x(n)的心电脉冲函数x=[-4,-2,0,-4,-6,-4,-2,-4,-6,-6,-4,-4,-6,-6,-2,6,12,8,0,-16,-38,-60,-84,-90,-66,-32,-4,-2,-4,8,12,12,10,6,6,6,4,0,0,0,0,0,-2,-4,0,0,0,-2,-2,0,0,-2,-2,-2,-2,0]subplot(2,2,1);n=0:55;stem(n,x,'.');xlabel('n');ylabel('x(n)');title('x(n)的心电脉冲函数');%通过滤波器H1(z)后的y1(n)函数A=0.09036;b1=[A,2*A,A];a1=[1,-1.2686,0.7051];h1=filter(b1,a1,x);[H1,w]=freqz(b1,a1,100);%通过滤波器H1(z),H2(Z)后的y2(n)函数b2=[A,2*A,A];a2=[1,-1.0106,0.3583];h2=filter(b2,a2,h1);[H2,w]=freqz(b2,a2,100);%通过滤波器H1(z),H2(Z),H3(Z)后的y3(n)函数b3=[A,2*A,A];a3=[1,-0.9044,0.2155];h3=filter(b3,a3,h2);[H3,w]=freqz(b3,a3,100);subplot(2,2,2);stem(n,h3,'.');xlabel('n');ylabel('y(n)');title('通过滤波器H1(Z),H2(Z),H3(Z)后的y3(n)函数');subplot(2,2,3);H4=H1.*(H2);H=H4.*(H3);mag=abs(H);db=20*log10((mag+eps)/max(mag));plot(w/pi,db);xlabel('w/pi');ylabel('20log[Ha3(ejw)]');title('通过滤波器H1(z)、H2(z)、H3(z)后的对数频率响应20log[Ha3(ejw)]函数');grid;