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1、信号与系统实验报告总结 信号与系统实验报告 实验信号抽样及信号重建一、实验目的 1、进一步理解信号的抽样及抽样定理; 2、进一步掌握抽样信号的频谱分析; 3、掌握和理解信号抽样以及信号重建的原理; 二、实验内容及步骤 练习1、什么是抽样定理,信号采样后重建的步骤，抽样频率如何设置, 答:(1). 抽样,Sampling,:就是从连续时间信号中抽取一系列的信号样本，从而，得到 一个离散时间序列(Discrete-time sequence). 抽样定理:设时间连续信号f(t)，其最高截止频率为f m ，如果用时间间隔为 Ts/2 则理想取样后的频谱就不会产生混叠,故有: 让取样信号x(t)通过一
2、带宽等于折叠频率的理想低通滤波器: ,H(j)=T |s/2 滤波器只允许通过基带频谱,即原信号频谱,故: ,Y(j)=X(j)H(j)=Xa(j) 因此在滤波器的输出得到了恢复的原模拟信号: y(t)=xa(t) 从时域上看,上述理想的低通滤波器的脉冲响应为: 根据卷积公式可求得理想低通滤波器的输出为: 由上式显然可得: 则: 上式表明只要满足取样频率高于两倍信号最高频率,连续时间函数xa(t)就可用他的取 样值xa(nT)来表达而不损失任何信息，这时只要把每一个取样瞬时值与内插函数式相乘求 和即可得出xa(t),在每一取样点上,由于只有该取样值所对应的内插函数式不为零,所以各 个取样点上的
3、信号值不变。 (3). 频率设置:根据抽样定理 ws/wm的值必须大于或等于2 练习2、给范例程序Program4_1加注释。 % Program clear, close all, tmax = 4; dt = 0.01; t = 0:dt:tmax; Ts = 1/10; % Sampling period ws = 2*pi/Ts; % Sampling frequency w0 = 20*pi; dw = 0.1; % The frequency of x(t) w = -w0:dw:w0; n = 0:1:tmax/Ts; % Make the time variable to be
4、 discrete x = exp(-4*t).*u(t); xn = exp(-4*n*Ts); % The sampled version of x(t) subplot(221) % Plot the original signal x(t) plot(t,x), title(A continuous-time signal x(t), xlabel(Time t), axis(0,tmax,0,1), grid on subplot(223) % Plot xn stem(n,xn,.), title(The sampled version xn of x(t), xlabel(Tim
5、e index n), axis(0,tmax/Ts,0,1), grid on Xa = x*exp(-j*t*w)*dt; X = 0; for k = -8:8; % Periodically extend X to form a periodic signal X = X + x*exp(-j*t*(w-k*ws)*dt; end subplot(222) % Plot xa plot(w,abs(Xa) title(Magnitude spectrum of x(t), grid on axis(-60,60,0,1.8*max(abs(Xa) subplot(224) plot(w
6、,abs(X) title(Magnitude spectrum of xn), xlabel(Frequency in radians/s),grid on axis(-60,60,0,1.8*max(abs(Xa) 练习3、分别进行设置ws/wm= 2，ws/wm= 1，ws/wm= 3，并运行抽样信号重建程序， 并根据抽样定理及重建条件分析三种设置情况下的结果。 % The original signal is the raised cosin signal: x(t) = 1+cos(pi*t).*u(t+1)-u(t-1). clear; close all, wm = 2*pi;
7、% The highest frequency of x(t) a = input(Type in the frequency rate ws/wm=:); % ws is the sampling frequency wc = wm; % The cutoff frequency of the ideal lowpass filter t0 = 2; t = -t0:0.01:t0; x = (1+cos(pi*t).*(u(t+1)-u(t-1); subplot(221); % Plot the original signal x(t) plot(t,x); grid on, axis(
8、-2,2,-0.5,2.5); title(Original signal x(t);xlabel(Time t); ws = a*wm; % Sampling frequency Ts = 2*pi/ws; % Sampling period N = fix(t0/Ts); % Determine the number of samplers n = -N:N; nTs = n*Ts; % The discrete time variable xs = (1+cos(pi*nTs).*(u(nTs+1)-u(nTs-1); % The sampled version of x(t) subp
9、lot(2,2,2) % Plot xs stem(n,xs,.); xlabel(Time index n); grid on, title(Sampled version xn); xr = zeros(1,length(t); % Specify a memory to save the reconstructed signal L = length(-N:N); xa = xr; figure(2); % Open a new figure window to see the demo of signal reconstruction stem(nTs,xs,.); xlabel(Ti
10、me index n); grid on;hold on for i = 1:L m = (L-1)/2+1-i; xa = Ts*(wc)*xs(i)*sinc(wc)*(t+m*Ts)/pi)/pi; plot(t,xa,b:);axis(-2,2,-0.5,2.5); hold on pause xr = xr+xa; % Interpolation end plot(t,xr,r); axis(-2,2,-0.5,2.5); hold on figure(1); subplot(223) plot(t,xr,r);axis(-2,2,-0.5,2.5); xlabel(Time t);
11、grid on title(Reconstructed signal xr(t); % Compute the error between the reconstructed signal and the original signal error = abs(xr-x); subplot(2,2,4) plot(t,error);grid on title(Error);xlabel(Time t) 程序运行结果图: 当ws/wm= 1时 当ws/wm= 2时 当ws/wm= 3时 分析:当ws/wm=2时才能无失真的重建原来的图形。即满足抽样定理ws/wm的值必须大于或等于2重建原图形。
