z变换的初值终值定理，及matlab示例

z变换的初值定理：

证明：

示例：



matlab代码：

%1
x=[1+1j 2+5j 3 4-2j];
m=100;
n=20;
w=exp(-j*2*pi/m);
rec1=zeros(10,m);
rec2=zeros(10,m);
for i=1:n
    y = czt(x,m,w,i);
    rec1(i,:)=imag(y);
    rec2(i,:)=real(y);
end
xx=1:n;
yy=1:m;
[xx,yy]=meshgrid(xx,yy);
subplot(2,1,1)
surf(xx,yy,rec1');
title('初值定理示例','FontName','黑体','FontSize',20);
xlabel('|z|的值','FontName','黑体','FontSize',20);
yticks([0 25 50 75 100]);
yticklabels({'0','0.5\pi','1.5\pi','2\pi'});
ylabel('\omega','FontName','黑体','FontSize',20);
zlabel('X(z)的虚部','FontName','黑体','FontSize',20);
subplot(2,1,2)
surf(xx,yy,rec2');
xlabel('|z|的值','FontName','黑体','FontSize',20);
yticks([0 25 50 75 100]);
yticklabels({'0','0.5\pi','1.5\pi','2\pi'});
ylabel('\omega','FontName','黑体','FontSize',20);
zlabel('X(z)的实部','FontName','黑体','FontSize',20);
%2
subplot(2,1,1)
stem(real(x))
title('因果x[n]序列','FontName','黑体','FontSize',15)
xlabel('n','FontName','黑体','FontSize',15);
xticks([1 2 3 4]);
xticklabels({'0','1','2','3'});
ylabel('x的实部','FontName','黑体','FontSize',15);
grid on
subplot(2,1,2)
stem(imag(x))
xlabel('n','FontName','黑体','FontSize',15);
xticks([1 2 3 4]);
xticklabels({'0','1','2','3'});
ylabel('x的虚部','FontName','黑体','FontSize',15);
grid on

z变换的终值定理：

证明：

示例：



matlab代码：

%3
alpha=0.7+0.2j;
l=30;
x=ones(l,1);
for i=1:l
    x(i)=(alpha^(i-1)+1);
end
subplot(2,1,1)
stem(real(x))
xlabel('n','FontName','黑体','FontSize',15);
xticks(1:l);
xticklabels({'0','1','2','3','···'});
ylabel('x的实部','FontName','黑体','FontSize',15);
title('(\alpha^n+1)\mu[n]','FontName','Times New Roman','FontSize',15)
subplot(2,1,2)
stem(imag(x))
xlabel('n','FontName','黑体','FontSize',15);
xticks(1:l);
xticklabels({'0','1','2','3','···'});
ylabel('x的虚部','FontName','黑体','FontSize',15);
%4
alpha=0.7+0.2j;
m=100;
k=0:m-1;
w=exp(-j*2*k*pi/m);
rec1=zeros(10,m);
rec2=zeros(10,m);
r=1.01:0.01:1.20;
for i=1:20
    z=r(i)*w;
    f=(z-1).*(z./(z-1)+z./(z-alpha));
    rec1(i,:)=real(f);
    rec2(i,:)=imag(f);
end
yy=1:100;
[xx,yy]=meshgrid(r,yy);
subplot(2,1,1)
surf(xx,yy,rec1');
title('终值定理示例','FontName','黑体','FontSize',20);
xlabel('|z|的值','FontName','黑体','FontSize',20);
yticks([0 25 50 75 100]);
yticklabels({'0','0.5\pi','1.5\pi','2\pi'});
ylabel('\omega','FontName','黑体','FontSize',20);
zlabel('X(z)的实部','FontName','黑体','FontSize',20);
subplot(2,1,2)
title('终值定理示例')
surf(xx,yy,rec2');
xlabel('|z|的值','FontName','黑体','FontSize',20);
yticks([0 25 50 75 100]);
yticklabels({'0','0.5\pi','1.5\pi','2\pi'});
ylabel('\omega','FontName','黑体','FontSize',20);
zlabel('X(z)的虚部','FontName','黑体','FontSize',20);