實(shí)驗(yàn)2離散時(shí)間傅里葉變換

1、電 子 科 技 大 學(xué)實(shí) 驗(yàn) 報(bào) 告學(xué)生姓名：項(xiàng)陽(yáng) 學(xué) 號(hào)： 2010231060011 指導(dǎo)教師：鄧建一、實(shí)驗(yàn)項(xiàng)目名稱(chēng)：離散時(shí)間傅里葉變換二、實(shí)驗(yàn)?zāi)康模菏煜ば蛄械母盗⑷~變換、傅立葉變換的性質(zhì)、連續(xù)信號(hào)經(jīng)理想采樣后進(jìn)行重建，加深對(duì)時(shí)域采樣定理的理解。三、實(shí)驗(yàn)內(nèi)容： 1. 求下列序列的離散時(shí)間傅里葉變換 (a) (b) 2. 設(shè)畫(huà)出并觀察其周期性。 3. 設(shè)畫(huà)出并觀察其共軛對(duì)稱(chēng)性。 4. 驗(yàn)證離散時(shí)間傅里葉變換的線性、時(shí)移、頻移、反轉(zhuǎn)（翻褶）性質(zhì)。 5. 已知連續(xù)時(shí)間信號(hào)為，求：(a) 的傅里葉變換；(b) 采樣頻率為5000Hz，繪出,用理想內(nèi)插函數(shù)重建，并對(duì)結(jié)果進(jìn)行討論；(c) 采樣頻率為1

2、000Hz，繪出，用理想內(nèi)插函數(shù)重建，并對(duì)結(jié)果進(jìn)行討論。四、實(shí)驗(yàn)原理：1. 離散時(shí)間傅里葉變換(DTFT)的定義： 2周期性：是周期為的函數(shù) 3對(duì)稱(chēng)性：對(duì)于實(shí)值序列，是共軛對(duì)稱(chēng)函數(shù)。 4線性：對(duì)于任何，有 5時(shí)移6頻移 7反轉(zhuǎn)（翻褶）五、實(shí)驗(yàn)器材（設(shè)備、元器件）： PC機(jī)、Windows XP、MatLab 7.1六、實(shí)驗(yàn)步驟：本實(shí)驗(yàn)要求學(xué)生運(yùn)用MATLAB編程產(chǎn)生一些基本的離散時(shí)間信號(hào)，并通過(guò)MATLAB的幾種繪圖指令畫(huà)出這些圖形，以加深對(duì)相關(guān)教學(xué)內(nèi)容的理解，同時(shí)也通過(guò)這些簡(jiǎn)單的函數(shù)練習(xí)了MATLAB的使用。七、實(shí)驗(yàn)源代碼：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、實(shí)驗(yàn)數(shù)據(jù)及結(jié)果分析：2.1.a 2.1.b2.22.32.42.5a2.5bc九、實(shí)驗(yàn)結(jié)論：離散時(shí)間傅里葉變換（DTFT，Discrete-time Fourier Transform）是傅里葉變換的一種。它將以離散時(shí)間nT（