[Matlab]四種IIR濾波器紋波特性對比

測試代碼：

%FilterCompare.m
%對比相同階數時四種IIR濾波器的波紋特性


close all;
clear;
clc;


fs = 1000; %Hz


%低通濾波器對比
N1 = 8;
wp = 50 / ( fs / 2);
ws = 100 / ( fs / 2);
alpha_p = 1;
alpha_s = 20;

[ b1,a1] = butter( N1 , ws , 'low' );
[ b2,a2] = cheby1( N1 , alpha_p , ws , 'low' );
[ b3,a3] = cheby2( N1 , alpha_s , ws , 'low' );
[ b4,a4] = ellip ( N1 , alpha_p,alpha_s , ws , 'low' );



[H1,w1] = freqz(b1,a1,1024*4);
[H2,w2] = freqz(b2,a2,1024*4);
[H3,w3] = freqz(b3,a3,1024*4);
[H4,w4] = freqz(b4,a4,1024*4);
w1 = w1/pi *fs/2;
w2 = w2/pi *fs/2;
w3 = w3/pi *fs/2;
w4 = w4/pi *fs/2;
figure(1);
plot(w1,abs(H1),'black','linewidth',0.5);
hold on;
plot(w2,abs(H2),'red','linewidth',0.5);
plot(w3,abs(H3),'green','linewidth',0.5);
plot(w4,abs(H4),'blue','linewidth',0.5);
xlim([0 500]);
ylim([0 1.1]);
legend({ 'butter' , 'cheby1' , 'cheby2' , 'ellip'});
title('同階數(N=8),同截止頻率(f_c=100Hz)的低通濾波器對比');


%高通濾波器
N1 = 8;
wp = 400 / ( fs / 2);
ws = 400 / ( fs / 2);
alpha_p = 1;
alpha_s = 20;
[ b1,a1] = butter( N1 , ws , 'high' );
[ b2,a2] = cheby1( N1 , alpha_p , ws , 'high' );
[ b3,a3] = cheby2( N1 , alpha_s , ws , 'high' );
[ b4,a4] = ellip( N1 , alpha_p,alpha_s , ws , 'high' );
[H1,w1] = freqz(b1,a1,1024*4);
[H2,w2] = freqz(b2,a2,1024*4);
[H3,w3] = freqz(b3,a3,1024*4);
[H4,w4] = freqz(b4,a4,1024*4);
w1 = w1/pi *fs/2;
w2 = w2/pi *fs/2;
w3 = w3/pi *fs/2;
w4 = w4/pi *fs/2;
figure(2);
plot(w1,abs(H1),'black','linewidth',0.5);
hold on;
plot(w2,abs(H2),'red','linewidth',0.5);
plot(w3,abs(H3),'green','linewidth',0.5);
plot(w4,abs(H4),'blue','linewidth',0.5);
xlim([0 500]);
ylim([0 1.1]);
legend({ 'butter' , 'cheby1' , 'cheby2' , 'ellip'});
title('同階數(N=8),同截止頻率(f_c=400Hz)的高通濾波器對比');



%帶通濾波器

N1 = 8;
wp = [ 100 , 400 ] / ( fs / 2);
ws = [ 100 , 400 ] / ( fs / 2);
alpha_p = 1;
alpha_s = 20;
[ b1,a1] = butter( N1 , ws , 'bandpass' );
[ b2,a2] = cheby1( N1 , alpha_p , ws , 'bandpass' );
[ b3,a3] = cheby2( N1 , alpha_s , ws , 'bandpass' );
[ b4,a4] = ellip( N1 , alpha_p,alpha_s , ws , 'bandpass' );
[H1,w1] = freqz(b1,a1,1024*4);
[H2,w2] = freqz(b2,a2,1024*4);
[H3,w3] = freqz(b3,a3,1024*4);
[H4,w4] = freqz(b4,a4,1024*4);
w1 = w1/pi *fs/2;
w2 = w2/pi *fs/2;
w3 = w3/pi *fs/2;
w4 = w4/pi *fs/2;
figure(3);
plot(w1,abs(H1),'black','linewidth',0.5);
hold on;
plot(w2,abs(H2),'red','linewidth',0.5);
plot(w3,abs(H3),'green','linewidth',0.5);
plot(w4,abs(H4),'blue','linewidth',0.5);
xlim([0 500]);
ylim([0 1.1]);
legend({ 'butter' , 'cheby1' , 'cheby2' , 'ellip'});
title('同階數(N=8),同截止頻率(f_c_1=100Hz f_c_2 = 400Hz)的帶通濾波器對比');



%帶阻濾波器
N1 = 8;
wp = [ 100 , 400 ] / ( fs / 2);
ws = [ 100 , 400 ] / ( fs / 2);
alpha_p = 1;
alpha_s = 20;
[ b1,a1] = butter( N1 , ws , 'stop' );
[ b2,a2] = cheby1( N1 , alpha_p , ws , 'stop' );
[ b3,a3] = cheby2( N1 , alpha_s , ws , 'stop' );
[ b4,a4] = ellip( N1 , alpha_p,alpha_s , ws , 'stop' );
[H1,w1] = freqz(b1,a1,1024*4);
[H2,w2] = freqz(b2,a2,1024*4);
[H3,w3] = freqz(b3,a3,1024*4);
[H4,w4] = freqz(b4,a4,1024*4);
w1 = w1/pi *fs/2;
w2 = w2/pi *fs/2;
w3 = w3/pi *fs/2;
w4 = w4/pi *fs/2;
figure(4);
plot(w1,abs(H1),'black','linewidth',0.5);
hold on;
plot(w2,abs(H2),'red','linewidth',0.5);
plot(w3,abs(H3),'green','linewidth',0.5);
plot(w4,abs(H4),'blue','linewidth',0.5);
xlim([0 500]);
ylim([0 1.1]);
legend({ 'butter' , 'cheby1' , 'cheby2' , 'ellip'});
title('同階數(N=8),同截止頻率(f_c_1=100Hz f_c_2 = 400Hz)的帶阻濾波器對比');

　　

現象：

低通濾波器

高通濾波器

帶通濾波器

帶阻濾波器

 

驗證結果：

1、巴特沃斯濾波器的通帶和阻帶均是單調變化，頻率響應最為平滑，若通帶邊緣滿足指標要求，則通帶內會有富余量，並不經濟，更有效的方法是將指標的精度要求均勻分布在通帶或阻帶內，這樣能設計出階數較低的濾波器（意會，意會）；

2、切比雪夫Ⅰ型濾波器的幅度特性在通帶范圍內是等波紋的，在阻帶范圍內時單調的；

3、切比雪夫Ⅱ型濾波器的幅度特性在阻帶范圍內是等波紋的，在通帶范圍內時單調的；

4、橢圓濾波器的幅度特性在通帶和阻帶范圍內均是等波紋的。