实验2离散时间傅里叶变换

1、电 子 科 技 大 学实 验 报 告学生姓名：项阳 学 号： 2010231060011 指导教师：邓建一、实验项目名称：离散时间傅里叶变换二、实验目的：熟悉序列的傅立叶变换、傅立叶变换的性质、连续信号经理想采样后进行重建，加深对时域采样定理的理解。三、实验内容： 1. 求下列序列的离散时间傅里叶变换 (a) (b) 2. 设画出并观察其周期性。 3. 设画出并观察其共轭对称性。 4. 验证离散时间傅里叶变换的线性、时移、频移、反转（翻褶）性质。 5. 已知连续时间信号为，求：(a) 的傅里叶变换；(b) 采样频率为5000Hz，绘出,用理想内插函数重建，并对结果进行讨论；(c) 采样频率为1

2、000Hz，绘出，用理想内插函数重建，并对结果进行讨论。四、实验原理：1. 离散时间傅里叶变换(DTFT)的定义： 2周期性：是周期为的函数 3对称性：对于实值序列，是共轭对称函数。 4线性：对于任何，有 5时移6频移 7反转（翻褶）五、实验器材（设备、元器件）： PC机、Windows XP、MatLab 7.1六、实验步骤：本实验要求学生运用MATLAB编程产生一些基本的离散时间信号，并通过MATLAB的几种绘图指令画出这些图形，以加深对相关教学内容的理解，同时也通过这些简单的函数练习了MATLAB的使用。七、实验源代码：1（a）w = 0:1:500*pi/500;x = exp(j*w

3、) ./ (exp(j*w) - 0.5*ones(1,501);magx = abs(x);angx = angle(x);realx = real(x);imagx = imag(x);subplot(2,2,1);plot(w/pi,magx);gridxlabel(frequency in pi units);title(Magnitude Part);ylabel(Magnitude)subplot(2,2,3);plot(w/pi,angx);gridxlabel(frequency in pi units);title(Angle Part);ylabel(Radians)sub

4、plot(2,2,2);plot(w/pi,realx);gridxlabel(frequency in pi units);title(Real Part);ylabel(Real)subplot(2,2,4);plot(w/pi,imagx);gridxlabel(frequency in pi units);title(Imaginary Part);ylabel(Imaginary)1.（b）n = -1:3;x = 1:5;k = 0:500;w = (pi/500)*k;X = x * (exp(-j*pi/500) . (n*k);magX = abs(X);angX = ang

5、le(X);realX = real(X);imagX = imag(X);subplot(2,2,1);plot(k/500,magX);gridxlabel(frequency in pi units);title(magnitude Part)subplot(2,2,3);plot(k/500,angX);gridxlabel(frequency in pi units);title(Angle Part)subplot(2,2,2);plot(k/500,realX);gridxlabel(frequency in pi units);title(Real Part)subplot(2

6、,2,4);plot(k/500,imagX);gridxlabel(frequency in pi units);title(Imaginary Part)2n = 0:10; x = (0.9*exp(j*pi/3).n;k = -200:200;w = (pi/100)*k;X = x * (exp(-j*pi/100) . (n*k);magX = abs(X);angX = angle(X);subplot(2,1,1);plot(w/pi,magX);gridxlabel(frequency in units of pi);ylabel(|x|)title(Magnitude Pa

7、rt)subplot(2,1,2);plot(w/pi,angX/pi);gridxlabel(frequency in units of pi);ylabel(radians/pi)title(Angle Part)3subplot(1,1,1)n = -5:5; x = (-0.9).n;k = -200:200;w = (pi/100)*k;X = x * (exp(-j*pi/100) . (n*k);magX = abs(X);angX = angle(X);subplot(2,1,1);plot(w/pi,magX);gridaxis(-2,2,0,15)xlabel(freque

8、ncy in units of pi);ylabel(|x|)title(Magnitude Part)subplot(2,1,2);plot(w/pi,angX/pi);gridaxis(-2,2,-1,1)xlabel(frequency in units of pi);ylabel(radians/pi)title(Angle Part)4（1）x1 = rand(1,11);x2 = rand(1,11);n = 0:10;alpha = 2; beta = 3;k = 0:500;w = (pi/500)*k;X1 = x1 * (exp(-j*pi/500).(n*k);X2 =

9、x2 * (exp(-j*pi/500).(n*k);x = alpha*x1 + beta*x2;X = x * (exp(-j*pi/500).(n*k);X_check = alpha*X1 + beta*X2;error = max(abs(X - X_check)4.（2）x = rand(1,11);n = 0:10;k = 0:500;w = (pi/500)*k;X = x * (exp(-j*pi/500).(n*k);y = x; m = n+2;Y = y * (exp(-j*pi/500).(m*k);Y_check = (exp(-j*2).w).*X;error =

10、 max(abs(Y - Y_check)4.（3）n = 0:100; x = cos(pi*n/2);k = -100:100;w = (pi/100)*k;X = x * (exp(-j*pi/100) . (n*k);y = exp(j*pi*n/4).*x;Y = y * (exp(-j*pi/100) . (n*k);subplot(1,1,1)subplot(2,2,1);plot(w/pi,abs(X);grid;axis(-1,1,0,60)xlabel(frequency in units of pi);ylabel(|X|)title(Magnitude of X)sub

11、plot(2,2,2);plot(w/pi,angle(X)/pi);grid;axis(-1,1,-1,1)xlabel(frequency in units of pi);ylabel(randiands/pi)title(Angle of X)subplot(2,2,3);plot(w/pi,abs(Y);grid;axis(-1,1,0,60)xlabel(frequency in units of pi);ylabel(|Y|)title(Magnitude of Y)subplot(2,2,4);plot(w/pi,angle(Y)/pi);grid;axis(-1,1,-1,1)

12、xlabel(frequency in units of pi);ylabel(randiands/pi)title(Angle of Y)4.（4）n = -5:10; x = rand(1,length(n);k = -100:100;w = (pi/100)*k;X = x * (exp(-j*pi/100) . (n*k);y = fliplr(x);m = -fliplr(n);Y = y* (exp(-j*pi/100).(m*k);Y_check = fliplr(X);error = max(abs(Y - Y_check)5.（a）Dt = 0.00005; t = -0.0

13、05:Dt:0.005;xa = exp(-1000*abs(t);Wmax = 2*pi*2000;K =500; k = 0:1:K;W = k*Wmax/K;Xa = xa * exp(-j*t*W)*Dt;Xa = real(Xa);W = -fliplr(W),W(2:501);Xa = fliplr(Xa),Xa(2:501);subplot(1,1,1)subplot(2,1,1);plot(t*1000,xa);xlabel(t in msec);ylabel(xa(t)title(Analog Signakl)subplot(2,1,2);plot(W/(2*pi*1000)

14、,Xa*1000);xlabel(Frequency in KHz);ylabel(Xa(jW)*1000)title(Continuous-tine Fouroer Transform)5.（b）（c）Dt = 0.00005; t = -0.005:Dt:0.005;xa = exp(-1000*abs(t);Ts = 0.0002;n = -25:1:25;x = exp(-1000*abs(n*Ts);K =500; k = 0:1:K;w = pi*k/K;X = x * exp(-j*n*w);X = real(X);w = -fliplr(w),w(2:K+1);X = flip

15、lr(X),X(2:K+1);subplot(1,1,1)subplot(2,1,1);plot(t*1000,xa);xlabel(t in msec);ylabel(x1(n)title(Discrete Signal);hold onstem(n*Ts*1000,x);gtext(Ts=0.2 msec);hold offsubplot(2,1,2);plot(w/pi,X);xlabel(Frequency in pi units);ylabel(X1(w)title(Discrete-time Fourier Transform)八、实验数据及结果分析：2.1.a 2.1.b2.22.32.42.5a2.5bc九、实验结论：离散时间傅里叶变换（DTFT，Discrete-time Fourier Transform）是傅里叶变换的一种。它将以离散时间nT（其中，T为采样间隔）作为变量的函数（离散时间信号）f(nT)变换