信号抽样与恢复

实验一信号抽样与恢复一、实验目的学会用MATLAB实现连续信号的采样和重建二、实验原理1．抽样定理若)(tf是带限信号，带宽为m,)(tf经采样后的频谱)(sF就是将)(tf的频谱)(F在频率轴上以采样频率s为间隔进行周期延拓。因此，当s2m时，不会发生频率混叠；而当s2m时将发生频率混叠。2．信号重建经采样后得到信号)(tfs经理想低通)(th则可得到重建信号)(tf，即：)(tf=)(tfs*)(th其中：)(tfs=)(tf)(snTt=)()(ssnTtnTf)()(tSaTthccs所以：)(tf=)(tfs*)(th=)()(ssnTtnTf*)(tSaTccs=csT)]([)(scsnTtSanTf上式表明，连续信号可以展开成抽样函数的无穷级数。利用MATLAB中的tttc)sin()(sin来表示)(tSa，有)(sin)(tctSa，所以可以得到在MATLAB中信号由)(snTf重建)(tf的表达式如下：)(tf=csT)]([sin)(scsnTtcnTf我们选取信号)(tf=)(tSa作为被采样信号，当采样频率s=2m时，称为临界采样。我们取理想低通的截止频率c=m。下面程序实现对信号)(tf=)(tSa的采样及由该采样信号恢复重建)(tSa：三、上机实验内容1．验证实验原理中所述的相关程序；2．设f(t)=0.5*(1+cost)*(u(t+pi)-u(t-pi))，由于不是严格的频带有限信号，但其频谱大部分集中在[0，2]之间，带宽m可根据一定的精度要求做一些近似。试根据以下两种情况用MATLAB实现由f(t)的抽样信号fs(t)重建f(t)并求两者误差，分析两种情况下的结果。(1)m=2,wc=1.2m,Ts=1;(2)m=2,wc=2m,Ts=2.53．对以下simulinkch6example1_He7.mdl低通采样定理以程序实现,具体参数参考框图内参数。实验结果：一、验证试验现象二、(1)wm=2,wc=1.2wm,Ts=1;wm=2;wc=1.2*wm;Ts=1;ws=2*pi/Ts;n=-100:100;nTs=n*Ts;f=0.5*(1+cos(nTs)).*rectpuls(n,2*pi);Dt=0.005;t=-15:Dt:15;fa=f*Ts*wc/pi*sinc((wc/pi)*(ones(length(nTs),1)*t-nTs'*ones(1,length(t))));error=abs(fa-(0.5*(1+cos(t)).*rectpuls(t,2*pi)));%重构信号与原信号误差t1=-15:0.5:15;f1=0.5*(1+cos(t1)).*rectpuls(t1,2*pi);subplot(311);stem(t1,f1);xlabel('kTs');ylabel('f(kTs)');title('sa(t)=0.5*(1+cost)*(u(t+pi)-u(t-pi))的采样信号');subplot(312);plot(t,fa)xlabel('t');ylabel('fa(t)');title('由sa(t)=0.5*(1+cost)*(u(t+pi)-u(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)');(2)wm=2,wc=2,Ts=2.5将频率在上述程序改为：wm=2;wc=wm;Ts=2.5即可;三、将模块改变为程序%Specifytherandomnumberstreamt=linspace(0,1,4000);input=rand(1,4000);figure(1)plot(t,input);%设计50th巴特沃斯模拟低通滤波器Norder=50;fn=150;[b,a]=butter(Norder,fn,'s');%计算H(s)figure(2);freqs(b,a);%也可用指令freqs直接画出H(s)的频率响应曲线。xlabel('频率Hz');ylabel('相角rad');%滤波Hs=tf(b,a);yout1=lsim(Hs,input,t);figure(3);plot(t,yout1);figure(4)ts=1/4000;[Xk,f]=fftseq(yout1,ts);plot(f,20*log10(fftshift(abs(Xk))));title('Theoriginalsignalspectrum');xlabel('Frequency/Hz');ylabel('PSD/w');%pulsegenneratef0=200;yout2=0.5*square(2*pi*f0*t,10)+0.5;figure(5)plot(t,yout2);title('Duty10%');axisauto;%productyout3=yout1'.*yout2;figure(6)plot(t,yout3);%filterfigure(7)yout4=lsim(Hs,yout3,t);plot(t,yout4)%fftyout4figure(8)[Xk_2,f_2]=fftseq(yout4,ts);plot(f_2,20*log10(fftshift(abs(Xk_2))));title('Theoriginalsignalspectrum');xlabel('Frequency/Hz');ylabel('PSD/w');%fftyout3figure(9)[Xk_3,f_3]=fftseq(yout3,ts);plot(f_3,20*log10(fftshift(abs(Xk_3))));title('Theoriginalsignalspectrum');xlabel('Frequency/Hz');ylabel('PSD/w');%fftyout3figure(10)[Xk_4,f_4]=fftseq(yout2,ts);plot(f_4,20*log10(fftshift(abs(Xk_4))));title('Theoriginalsignalspectrum');xlabel('Frequency/Hz');ylabel('PSD/w');%plotfigure(11)plot(t,yout1,'r-');holdonplot(t,yout3,'b-');holdon%plotfigure(12)plot(t,yout4,'r-');holdon（1）产生随机信号（2）滤波器的频响（3）随机信号经过滤波器成为限带信号（4）滤波之后信号的频谱（5）产生脉冲串（6）对限带信号进行采样00.10.20.30.40.50.60.70.80.9100.10.20.30.40.50.60.70.80.91-2000-1500-1000-5000500100015002000-90-80-70-60-50-40-30-20-100TheoriginalsignalspectrumFrequency/HzPSD/w00.10.20.30.40.50.60.70.80.9100.10.20.30.40.50.60.70.80.91Duty10%（7）经低通滤波器对信号进行恢复（8）已恢复信号的频谱（9）已采样信号的频谱出现频谱周期延拓00.10.20.30.40.50.60.70.80.9100.10.20.30.40.50.60.700.10.20.30.40.50.60.70.80.9100.010.020.030.040.050.060.07-2000-1500-1000-5000500100015002000-110-100-90-80-70-60-50-40-30TheoriginalsignalspectrumFrequency/HzPSD/w-2000-1500-1000-5000500100015002000-110-100-90-80-70-60-50-40-30-20TheoriginalsignalspectrumFrequency/HzPSD/w