%用sinc函數進行低通抽樣定理的證明
clc
clear
a=-5;%信號的起始時間
b=5;%信號的終止時間
fc=2;%sinc
ts=1/(16*fc);%采樣時間間隔,該值應設置的較小,使信號能較平滑的畫出
fs=1/ts;%采樣頻率
t=a:ts:b;
f=-fs/2+fs/length(t):fs/length(t):fs/2;
y=sinc(fc*t);
y_spect=fftshift(abs(fft(y)));

%------
fs1=fc*1;%臨界采樣頻率
ts1=1/fs1;%臨界采樣的時間間隔
t1=a:ts1:b;
f1=-fs1/2+fs1/length(t1):fs1/length(t1):fs1/2;
x1=sinc(fc*t1);
y1=0;
for i=1:length(x1)
    y1=y1+x1(i)*sinc((t-t1(i))/ts1);
end
y1_spect=fftshift(abs(fft(y1)));

%-----
fs2=fc*0.5;%欠采樣頻率
ts2=1/fs2;%欠采樣的時間間隔
t2=a:ts2:b;

x2=sinc(fc*t2);
y2=0;
for i=1:length(x2)
    y2=y2+x2(i)*sinc((t-t2(i))/ts2);
end
y2_spect=fftshift(abs(fft(y2)));

%----
fs3=fc*2;%過采樣頻率
ts3=1/fs3;%過采樣的時間間隔
t3=a:ts3:b;
f2=-fs3/2+fs3/length(t3):fs3/length(t3):fs3/2;
x3=sinc(fc*t3);
y3=0;
for i=1:length(x3)
    y3=y3+x3(i)*sinc((t-t3(i))/ts3);
end
y3_spect=fftshift(abs(fft(y3)));

%----
figure(1)
subplot(2,2,1)
plot(t,y,t1,x1,'*black');title('(a)原始信號,黑點為臨界采樣時的抽樣點');grid
subplot(2,2,2)
plot(f,y_spect);title('(b)原始信號的頻譜');grid
subplot(2,2,3)
plot(t,y1);title('(c)用臨界采樣得到的抽樣點重建的信號');grid
subplot(2,2,4)
plot(f,y1_spect);title('(d)重建信號的頻譜');grid

figure(2)
subplot(2,2,1)
plot(t,y,t2,x2,'*black');title('(a)原始信號,黑點為欠采樣時的抽樣點');grid
subplot(2,2,2)
plot(f,y_spect);title('(b)原始信號的頻譜');grid
subplot(2,2,3)
plot(t,y2);title('(c)用欠采樣得到的抽樣點重建的信號');grid
subplot(2,2,4)
plot(f,y2_spect);title('(d)重建信號的頻譜');grid

figure(3)
subplot(2,2,1)
plot(t,y,t3,x3,'*black');title('(a)原始信號,黑點為過采樣時的抽樣點');grid
subplot(2,2,2)
plot(f,y_spect);title('(b)原始信號的頻譜');grid
subplot(2,2,3)
plot(t,y3);title('(c)用過采樣得到的抽樣點重建的信號');grid
subplot(2,2,4)
plot(f,y3_spect);title('(d)重建信號的頻譜');grid