【 MATLAB 】使用 MATLAB 得到高密度譜（補零得到DFT）和高解析度譜（獲得更多的資料得到DFT）的方式對比（附MATLAB指令碼）
阿新 • • 發佈：2018-12-18
那篇博文中，我們通過補零的方式來增加N，這樣最後的結論是隨著N的不斷增大，我們只會得到DTFT上的更多的取樣點，也就是說頻率取樣率增加了。通過補零，得到高密度譜（DFT），但不能得到高解析度譜，因為補零並沒有任何新的資訊附加到這個訊號上，要想得到高解析度譜，我們就得通過獲得更多的資料來進行求解DFT。
這篇博文就是為此而寫。
案例：
想要基於有限樣本數來確定他的頻譜。
下面我們分如下幾種情況來分別討論：
a. 求出並畫出 ，N = 10 的DFT以及DTFT；
b. 對上一問的x(n)通過補零的方式獲得區間[0,99]上的x(n)，畫出 N = 100點的DFT，並畫出DTFT作為對比；
c.求出並畫出 ，N = 100 的DFT以及DTFT；
d.對c問中的x(n)補零到N = 500，畫出 N = 500點的DFT，並畫出DTFT作為對比；
e. 比較c和d這兩個序列的序列的DFT以及DTFT的異同。
那就幹唄！
題解：
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
區域性放大看：
