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MATLAB ------- 使用 MATLAB 得到高密度谱（补零得到DFT）和高分辨率谱（获得更多的数据得到DFT）的方式对比（附MATLAB脚本）! n/ O7 I! V' O% y3 a
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) c1 ] F6 ?* e! b) Q2 g6 w! e上篇分析了同一有限长序列在不同的N下的DFT之间的不同： MATLAB ------- 用 MATLAB 作图讨论有限长序列的 N 点 DFT（含MATLAB脚本）" O# q5 D+ k) C1 Z& I
那篇中，我们通过补零的方式来增加N，这样最后的结论是随着N的不断增大，我们只会得到DTFT上的更多的采样点，也就是说频率采样率增加了。通过补零，得到高密度谱（DFT），但不能得到高分辨率谱，因为补零并没有任何新的信息附加到这个信号上，要想得到高分辨率谱，我们就得通过获得更多的数据来进行求解DFT。
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4 P8 E9 o" w" E o/ I$ k @: O4 d这篇就是为此而写。
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5 V; ]" F1 P8 M! m案例：
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% e ?$ Q$ |4 G" U: f0 d0 D$ e/ v想要基于有限样本数来确定他的频谱。
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0 T, m' }6 ?2 H4 W下面我们分如下几种情况来分别讨论：
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9 q4 w% x1 x6 C) F4 m ?a. 求出并画出
，N = 10 的DFT以及DTFT；
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b. 对上一问的x(n)通过补零的方式获得区间[0,99]上的x(n)，画出 N = 100点的DFT，并画出DTFT作为对比；+ l4 A% e7 ~4 A4 G1 \
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，N = 100 的DFT以及DTFT；" j- v, D6 q, Q+ x* q1 C. f
' I" l g$ ]; ^ }1 n2 O: [5 od.对c问中的x(n)补零到N = 500，画出 N = 500点的DFT，并画出DTFT作为对比； I: w2 q4 [8 T6 {+ c5 r' w" w: F
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e. 比较c和d这两个序列的序列的DFT以及DTFT的异同。5 Z5 E9 L ^' I. A3 e: D
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4 T5 \( _* E+ |2 ?3 ~题解：/ z& u6 B5 |4 _# P! [
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; f6 J. h1 r# x8 B" x. ~1 r! R& |* G, r- clc;clear;close all;
- n = 0:99;
- x = cos(0.48*pi*n) + cos(0.52*pi*n);
- n1 = 0:9;
- y1 = x(1:10);
- subplot(2,1,1)
- stem(n1,y1);
- title('signal x(n), 0 <= n <= 9');
- xlabel('n');ylabel('x(n) over n in [0,9]');
- Y1 = dft(y1,10);
- magY1 = abs(Y1);
- k1 = 0:1:9;
- N = 10;
- w1 = (2*pi/N)*k1;
- subplot(2,1,2);
- stem(w1/pi,magY1);
- title('DFT of x(n) in [0,9]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = y1*exp(-j*n1'*w);
- magX = abs(X);
- hold on
- plot(w/pi,magX);
- hold off
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b.
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- clc;clear;close all;
- n = 0:99;
- x = cos(0.48*pi*n) + cos(0.52*pi*n);
- % zero padding into N = 100
- n1 = 0:99;
- y1 = [x(1:10),zeros(1,90)];
- subplot(2,1,1)
- stem(n1,y1);
- title('signal x(n), 0 <= n <= 99');
- xlabel('n');ylabel('x(n) over n in [0,99]');
- Y1 = dft(y1,100);
- magY1 = abs(Y1);
- k1 = 0:1:99;
- N = 100;
- w1 = (2*pi/N)*k1;
- subplot(2,1,2);
- stem(w1/pi,magY1);
- title('DFT of x(n) in [0,9]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = y1*exp(-j*n1'*w);
- % w = [-fliplr(w),w(2:K+1)]; %plot DTFT in [-pi,pi]
- % X = [fliplr(X),X(2:K+1)]; %plot DTFT in [-pi,pi]
- magX = abs(X);
- % angX = angle(X)*180/pi;
- % figure
- % subplot(2,1,1);
- hold on
- plot(w/pi,magX);
- % title('Discrete-time Fourier Transform in Magnitude Part');
- % xlabel('w in pi units');ylabel('Magnitude of X');
- % subplot(2,1,2);
- % plot(w/pi,angX);
- % title('Discrete-time Fourier Transform in Phase Part');
- % xlabel('w in pi units');ylabel('Phase of X ');
- hold off
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- clc;clear;close all;
- n = 0:99;
- x = cos(0.48*pi*n) + cos(0.52*pi*n);
- % n1 = 0:99;
- % y1 = [x(1:10),zeros(1,90)];
- subplot(2,1,1)
- stem(n,x);
- title('signal x(n), 0 <= n <= 99');
- xlabel('n');ylabel('x(n) over n in [0,99]');
- Xk = dft(x,100);
- magXk = abs(Xk);
- k1 = 0:1:99;
- N = 100;
- w1 = (2*pi/N)*k1;
- subplot(2,1,2);
- stem(w1/pi,magXk);
- title('DFT of x(n) in [0,99]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = x*exp(-j*n'*w);
- % w = [-fliplr(w),w(2:K+1)]; %plot DTFT in [-pi,pi]
- % X = [fliplr(X),X(2:K+1)]; %plot DTFT in [-pi,pi]
- magX = abs(X);
- % angX = angle(X)*180/pi;
- % figure
- % subplot(2,1,1);
- hold on
- plot(w/pi,magX);
- % title('Discrete-time Fourier Transform in Magnitude Part');
- % xlabel('w in pi units');ylabel('Magnitude of X');
- % subplot(2,1,2);
- % plot(w/pi,angX);
- % title('Discrete-time Fourier Transform in Phase Part');
- % xlabel('w in pi units');ylabel('Phase of X ');
- hold off
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- clc;clear;close all;
- n = 0:99;
- x = cos(0.48*pi*n) + cos(0.52*pi*n);
- % n1 = 0:99;
- % y1 = [x(1:10),zeros(1,90)];
- %zero padding into N = 500
- n1 = 0:499;
- x1 = [x,zeros(1,400)];
- subplot(2,1,1)
- stem(n1,x1);
- title('signal x(n), 0 <= n <= 499');
- xlabel('n');ylabel('x(n) over n in [0,499]');
- Xk = dft(x1,500);
- magXk = abs(Xk);
- k1 = 0:1:499;
- N = 500;
- w1 = (2*pi/N)*k1;
- subplot(2,1,2);
- % stem(w1/pi,magXk);
- stem(w1/pi,magXk);
- title('DFT of x(n) in [0,499]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = x1*exp(-j*n1'*w);
- % w = [-fliplr(w),w(2:K+1)]; %plot DTFT in [-pi,pi]
- % X = [fliplr(X),X(2:K+1)]; %plot DTFT in [-pi,pi]
- magX = abs(X);
- % angX = angle(X)*180/pi;
- % figure
- % subplot(2,1,1);
- hold on
- plot(w/pi,magX,'r');
- % title('Discrete-time Fourier Transform in Magnitude Part');
- % xlabel('w in pi units');ylabel('Magnitude of X');
- % subplot(2,1,2);
- % plot(w/pi,angX);
- % title('Discrete-time Fourier Transform in Phase Part');
- % xlabel('w in pi units');ylabel('Phase of X ');
- hold off
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e.
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3 g; A1 q( }' }: h9 N- clc;clear;close all;
- n = 0:99;
- x = cos(0.48*pi*n) + cos(0.52*pi*n);
- subplot(2,1,1)
- % stem(n,x);
- % title('signal x(n), 0 <= n <= 99');
- % xlabel('n');ylabel('x(n) over n in [0,99]');
- Xk = dft(x,100);
- magXk = abs(Xk);
- k1 = 0:1:99;
- N = 100;
- w1 = (2*pi/N)*k1;
- stem(w1/pi,magXk);
- title('DFT of x(n) in [0,99]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = x*exp(-j*n'*w);
- % w = [-fliplr(w),w(2:K+1)]; %plot DTFT in [-pi,pi]
- % X = [fliplr(X),X(2:K+1)]; %plot DTFT in [-pi,pi]
- magX = abs(X);
- % angX = angle(X)*180/pi;
- % figure
- % subplot(2,1,1);
- hold on
- plot(w/pi,magX);
- % title('Discrete-time Fourier Transform in Magnitude Part');
- % xlabel('w in pi units');ylabel('Magnitude of X');
- % subplot(2,1,2);
- % plot(w/pi,angX);
- % title('Discrete-time Fourier Transform in Phase Part');
- % xlabel('w in pi units');ylabel('Phase of X ');
- hold off
- % clc;clear;close all;
- %
- % n = 0:99;
- % x = cos(0.48*pi*n) + cos(0.52*pi*n);
- % n1 = 0:99;
- % y1 = [x(1:10),zeros(1,90)];
- %zero padding into N = 500
- n1 = 0:499;
- x1 = [x,zeros(1,400)];
- subplot(2,1,2);
- % subplot(2,1,1)
- % stem(n1,x1);
- % title('signal x(n), 0 <= n <= 499');
- % xlabel('n');ylabel('x(n) over n in [0,499]');
- Xk = dft(x1,500);
- magXk = abs(Xk);
- k1 = 0:1:499;
- N = 500;
- w1 = (2*pi/N)*k1;
- stem(w1/pi,magXk);
- title('DFT of x(n) in [0,499]');
- xlabel('frequency in pi units');
- %In order to clearly see the relationship between DTFT and DFT, we draw DTFT on the same picture.
- %Discrete-time Fourier Transform
- K = 500;
- k = 0:1:K;
- w = 2*pi*k/K; %plot DTFT in [0,2pi];
- X = x1*exp(-j*n1'*w);
- % w = [-fliplr(w),w(2:K+1)]; %plot DTFT in [-pi,pi]
- % X = [fliplr(X),X(2:K+1)]; %plot DTFT in [-pi,pi]
- magX = abs(X);
- % angX = angle(X)*180/pi;
- % figure
- % subplot(2,1,1);
- hold on
- plot(w/pi,magX,'r');
- % title('Discrete-time Fourier Transform in Magnitude Part');
- % xlabel('w in pi units');ylabel('Magnitude of X');
- % subplot(2,1,2);
- % plot(w/pi,angX);
- % title('Discrete-time Fourier Transform in Phase Part');
- % xlabel('w in pi units');ylabel('Phase of X ');
- hold off8 c" B! c, o: U8 N, m( `( Y
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发表于 2019-12-9 11:11
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