抽样定理与信号重建

1.抽样定理:
若 f(t)是带限信号，带宽为ω_m，f(t)经采样后的频谱F_s(ω),此时要满足Nyquist取样定理

2.信号重建:
f_s(t)经过理想低通滤波h(t)后可重建f(t)

注：可参考郭宝龙教授的教材

例5-1Sa(t)的临界采样及信号重构

wm=1; %信号带宽
wc=wm; %滤波器截止频率
Ts=pi/wm; %采样间隔
ws=2pi/Ts; %采样角频率
n=-100:100; %时域采样电数
nTs=nTs %时域采样点
f=sinc(nTs/pi);
Dt=0.005;t=-15:Dt:15;
fa=fTswc/pisinc((wc/pi)(ones(length(nTs),1)*t-nTs’*ones(1,length(t)))); %信号重构
t1=-15:0.5:15;
f1=sinc(t1/pi);
subplot(211);
stem(t1,f1);
xlabel(‘kTs’);
ylabel(‘f(kTs)’);
title(‘sa(t)=sinc(t/pi)的临界采样信号’);
subplot(212);
plot(t,fa)
xlabel(‘t’);
ylabel(‘fa(t)’);
title(‘由 sa(t)=sinc(t/pi)的临界采样信号重构 sa(t)’);
grid;

例5-2 Sa(t)的过采样及信号重构

Sa(t)的过采样及信号重构和绝对误差分析程序和例 5-1 类似，将采样间隔改成 Ts=0.7π/ω_m , 滤波器截止频率该成 ω_c=1.1ω_m ，添加一个误差函数

wm=1;
wc=1.1wm;
Ts=0.7pi/wm;
ws=2pi/Ts;
n=-100:100;
nTs=nTs
f=sinc(nTs/pi);
Dt=0.005;t=-15:Dt:15;
fa=fTswc/pisinc((wc/pi)(ones(length(nTs),1)*t-nTs’*ones(1,length(t))));
error=abs(fa-sinc(t/pi)); %重构信号与原信号误差
t1=-15:0.5:15;
f1=sinc(t1/pi);
subplot(311);
stem(t1,f1);
xlabel(‘kTs’);ylabel(‘f(kTs)’);
title(‘sa(t)=sinc(t/pi)的采样信号’);
subplot(312);
plot(t,fa)
xlabel(‘t’);
ylabel(‘fa(t)’);
title(‘由 sa(t)=sinc(t/pi)的过采样信号重构 sa(t)’);
grid;
subplot(313);
plot(t,error);
xlabel(‘t’);
ylabel(‘error(t)’);
title(‘过采样信号与原信号的误差 error(t)’);

例 5-3 Sa(t)的欠采样及信号重构和绝对误差分析

程序和例 5-2 类似，将采样间隔改成 Ts=1.5*π/ω_m , 滤波器截止频率该成 ω_c =ω_m =1

课后习题

2.1

wm=2;
wc=1.2wm;
Ts=1;
ws=2pi/Ts;%取样频率
n = -100 : 100;
nTs=nTs;
f = 0.5 * ( 1 + cos( n * Ts ) );
Dt=0.005;t=-15:Dt:15;
fa=fTswc/pisinc((wc/pi)*(ones(length(nTs),1)*t-nTs’*ones(1,length(t))));
error=abs( fa - 0.5 * ( 1 + cos( t ) ) ); %重构信号与原信号误差
t1=-15:0.5:15;
f1= 0.5 * ( 1 + cos( t1 ) ) ;
subplot(311);
stem(t1,f1);
xlabel(‘kTs’);ylabel(‘f(kTs)’);
title(‘f( t ) = f( n * Ts )的采样信号’);
subplot(312);
plot(t,fa)
xlabel(‘t’);
ylabel(‘fa(t)’);
title(‘由 f( t ) = f( n * Ts )的过采样信号重构 f(t)’);
grid;
subplot(313);
plot(t,error);
xlabel(‘t’);
ylabel(‘error( t )’);
title(‘过采样信号与原信号的误差 error( t )’);

2.2

wm=2;
wc=2;
Ts=2.5;
ws=2pi/Ts;%取样频率
n = -100 : 100;
nTs=nTs;
f = 0.5 * ( 1 + cos( n * Ts ) );
Dt=0.005;t=-15:Dt:15;
fa=fTswc/pisinc((wc/pi)(ones(length(nTs),1)*t-nTs’*ones(1,length(t))));
error=abs( fa - 0.5 * ( 1 + cos( t ) ) ); %重构信号与原信号误差
t1=-15:0.5:15;
f1= 0.5 * ( 1 + cos( t1 ) ) ;
subplot(311);
stem(t1,f1);
xlabel(‘kTs’);ylabel(‘f(kTs)’);
title(‘f( t ) = f( n * Ts )的采样信号’);
subplot(312);
plot(t,fa)
xlabel(‘t’);
ylabel(‘fa(t)’);
title(‘由 f( t ) = f( n * Ts )的过采样信号重构 f(t)’);
grid;
subplot(313);
plot(t,error);
xlabel(‘t’);
ylabel(‘error( t )’);
title(‘过采样信号与原信号的误差 error( t )’);