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上篇博文分析了同一有限长序列在不同的N下的DFT之间的不同： MATLAB 】使用 MATLAB 作图讨论有限长序列的 N 点 DFT（强烈推荐）（含MATLAB脚本）
那篇博文中，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。
这篇博文就是为此而写。
案例：
想要基于有限样本数来确定他的频谱。
下面我们分如下几种情况来分别讨论：
a. 求出并画出 ，N = 10 的DFT以及DTFT；
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；
c.求出并画出 ，N = 100 的DFT以及DTFT；
d.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；
e. 比较c和d这两个序列的序列的DFT以及DTFT的异同。
那就干呗！
题解：
a.
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
b.
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
c.
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
太小了，放大看：
d.
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
e.
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 off
局部放大看：
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