从傅里叶级数到傅里叶变换
代码如下:
clear
close all
clc
dt=0.001;
t=0:dt:20;
ft=10*sin(10*t)+2*cos(10*t)+5*sin(5*t)+3*cos(5*t)+sin(2*t)+cos(2*t);
figure(1)
plot(t,ft,'.b');
xlabel('t');
ylabel('y');
title('f(t)=10*sin(10*t)+2*cos(10*t)+5*sin(5*t)+3*cos(5*t)+sin(2*t)+cos(2*t)');
w=zeros(1,20);
%% 计算系数
for i1=1:length(w)
k1=0;
for t1=0:dt:20
k1=k1+1;
w(i1)=w(i1)+ft(k1)*exp(-i1*t1*i)/(20/dt);
end
end
w=w*2;
figure(2)
subplot(1,3,1)
fz=sqrt(real(w).^2+imag(w).^2);
plot(1:20,fz);
hold on
plot([10,5,2],fz([10,5,2]),'ro');
xlabel('t');
ylabel('y');
title('幅值');
subplot(1,3,2)
re=real(w);
plot(1:20,re);
hold on
plot([10,5,2],re([10,5,2]),'ro');
xlabel('t');
ylabel('y');
title('实部');
subplot(1,3,3)
ima=-imag(w);
plot(1:20,ima);
hold on
plot([10,5,2],ima([10,5,2]),'ro');
xlabel('t');
ylabel('y');
title('虚部');
%% 还原
ft1=zeros(1,length(ft));
k1=0;
for t1=0:dt:20
k1=k1+1;
for w1=1:20
ft1(k1)=ft1(k1)+w(w1)*exp(i*w1*t1);
end
end
figure(3)
plot(t,ft,'.b');
hold on
plot(t,real(ft1),'r','linewidth',1.5)
xlabel('x')
ylabel('y')
title('傅里叶逆变换');
legend('原函数图像','傅里叶逆变换结果');