极零点性质

1. 极点在单位圆上，系统冲激响应等幅振荡；在单位圆内，系统冲激响应随着频率增大收敛；在单位圆外，系统冲激响应随着频率增大发散

2. 零点越靠近极点，单位脉冲响应的变化越缓慢，零点对极点有抵消作用

zplane(z,p)

$H_1(z)=\frac{z}{z-0.85},H_2(z)=\frac{z}{z-1},H_3=\frac{z}{z-1.5}$

% (1)
z1 = [0]';p1 = [0.85]';k1=1;
[b1,a1] = zp2tf(z1,p1,k1);
figure('color','w');
subplot(321)
zplane(z1,p1);
subplot(322)
impz(b1,a1,20);

z2 = [0]';p2 = [1]';k2 = 1;
[b2,a2] = zp2tf(z2,p2,k2);

subplot(323)
zplane(z2,p2);
subplot(324)
impz(b2,a2,20);

z3 = [0]';p3 = [1.5]';k3 = 1;
[b3,a3] = zp2tf(z3,p3,k3);

subplot(325)
zplane(z3,p3);
subplot(326)
impz(b3,a3,20);

$H_1(z)=\frac{z}{z+0.85},H_2(z)=\frac{z}{z+1},H_3=\frac{z}{z+1.5}$

% (2)
z1 = [0]';p1 = [-0.85]';k1 = 1;
[b1,a1] = zp2tf(z1,p1,k1);

figure('color','w');

subplot(321)
zplane(z1,p1);
subplot(322)
impz(b1,a1,20);

z2 = [0]';p2 = [-1]';k2 = 1;
[b2,a2] = zp2tf(z2,p2,k2);
subplot(323)
zplane(z2,p2);
subplot(324)
impz(b2,a2,20);

z3 = [0]';p3 = [-1.5]';k3 = 1;
[b3,a3] = zp2tf(z3,p3,k3);
subplot(325)
zplane(z3,p3);
subplot(326)
impz(b3,a3,20);

$H_1(z)=\frac{z(z-0.3)}{(z-0.5-0.7j)(z-0.5+0.7j)},H_2(z)=\frac{z(z-0.3)}{(z-0.6-0.8j)(z-0.6+0.8j)},H_3=\frac{z(z-0.3)}{(z-1-1j)(z-1+1j)}$

% (3)
clear;
figure('Color','w');
z1 = [0.3,0]';p1 = [0.5+0.7j,0.5-0.7j]';k1 = 1;
[b1,a1] = zp2tf(z1,p1,k1);
subplot(321)
zplane(z1,p1);
subplot(322)
impz(b1,a1,20);

z2 = [0,0.3]';p2 = [0.6+0.8j,0.6-0.8j]';k2 = 1;
[b2,a2] = zp2tf(z2,p2,k2);
subplot(323)
zplane(z2,p2);
subplot(324)
impz(b2,a2,20);

z3 = [0,0.3]';p3 = [1+1j,1-1j]';k3 = 1;
[b3,a3] = zp2tf(z3,p3,k3);
subplot(325)
zplane(z3,p3);
subplot(326)
impz(b3,a3,20);

$H(z)=\frac{0.2+0.1z^{-1}+0.3z^{-2}+0.1z^{-3}+0.2z^{-4}}{1-1.1z^{-1}+1.5z^{-2}-0.7z^{-3}+0.3z^{-4}}$

roots()

b = [0.2 , 0.1 , 0.3 , 0.1 , 0.2];
a = [1 , -1.1 , 1.5 , -0.7 , 0.3];

[z,p,k] = tf2zp(b,a);

figure('color','w');
subplot(121)
zplane(z,p);title('使用模型转换');
subplot(222)

rz = roots(b);
rp = roots(a);
subplot(122)
zplane(rz,rp);title('使用roots()');