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

4、9;);title('Angle Part');ylabel('Radians')subplot(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(

5、9;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 = angle(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

6、);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(2,2,4);plot(k/500,imagX);gridxlabel('frequency in pi units');title('Imaginary Part')2n = 0:10; x = (

7、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 Part')subplot(2,1,2);plot(w/pi,angX/pi);gridxlabel('frequen

8、cy 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('frequency in units of pi&#

9、39;);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

10、 = x1 * (exp(-j*pi/500).(n'*k);X2 = 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 * (

11、exp(-j*pi/500).(m'*k);Y_check = (exp(-j*2).w).*X;error = 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(-1,1,0,6

12、0)xlabel('frequency in units of pi');ylabel('|X|')title('Magnitude of X')subplot(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,

13、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)xlabel('frequency in units of pi');ylabel('randiands/pi')title('Angle of Y')4.（4）n = -5:10; x = rand(1,length(n);k =

14、 -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 = -0.005: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

15、'*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),Xa*1000);xlabel('Frequency in KHz');ylabel('Xa(jW)&

16、#39;*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 = fliplr(X),X(2:K+1);subplot(1,1,1)

17、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（