《DSP using MATLAB》示例Example5.7

代码：

x = [1, 1, 1, 1, zeros(1,4)]; 
N = 8;                                                 % zero-padding operation 
X_DFT = dft(x,N);                                      % DFT of x(n)

magX_DFT = abs(X_DFT)
phaX_DFT = angle(X_DFT)*180/pi                         % degrees
realX_DFT = real(X_DFT); imagX_DFT = imag(X_DFT);
angX_DFT = angle(X_DFT);                               % radias

n = 0:(N - 1);
k = 0:1000; w = (pi/500)*k;                            % [0,2pi] axis divided into 501 points.
%k = 0:500; w = (pi/500)*k;                            % [0,pi] axis divided into 501 points.
X_DTFT = x * (exp(-j*pi/500)) .^ (n'*k);               % DTFT of x(n)

magX_DTFT = abs(X_DTFT); angX_DTFT = angle(X_DTFT); realX_DTFT = real(X_DTFT); imagX_DTFT = imag(X_DTFT);

figure('NumberTitle', 'off', 'Name', 'Exameple5.7 x sequence')
set(gcf,'Color','white'); 
stem(n,x); title('x sequence'); axis([0,N,-0.5,1.5]);
xlabel('n'); ylabel('x(n)'); grid on;


%% --------------------------------------------------------------
%%        START X_DTFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'X_DTFT its Magnitude and Angle, Real and Imaginary Part');
set(gcf,'Color','white'); 
subplot(2,2,1); plot(w/pi,magX_DTFT); grid on;  % axis([-2,2,0,15]); 
title('Magnitude Part');
xlabel('frequency in pi units'); ylabel('Magnitude  |X\_DTFT|'); 
subplot(2,2,3); plot(w/pi, angX_DTFT*180/pi); grid on;  % axis([-2,2,-1,1]);
title('Angle Part');
xlabel('frequency in pi units'); ylabel('Degrees'); %axis([-200,200,0,2]);

subplot('2,2,2'); plot(w/pi, realX_DTFT); grid on;
title('Real Part');
xlabel('frequency in pi units'); ylabel('Real');
subplot('2,2,4'); plot(w/pi, imagX_DTFT); grid on;
title('Imaginary Part');
xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DTFT's  mag ang real imag
%% --------------------------------------------------------------


%% --------------------------------------------------------------
%%        START X_DFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Example5.7 X_DFT its Magnitude and Angle');
set(gcf,'Color','white'); 
subplot(2,1,1); stem(n,magX_DFT); hold on; plot(4*w/pi,magX_DTFT,'--'); hold off;
grid on;   axis([-0.5,8.2,-1,5]); 
title('Magnitude Part of the DFT: N = 8');
xlabel('k'); ylabel('Magnitude  |X\_DFT|'); 
subplot(2,1,2); stem(n, phaX_DFT); hold on; plot(4*w/pi,angX_DTFT*180/pi,'--'); hold off;
grid on;   axis([-0.5,8.2,-200,200]);
title('Angle Part of the DFT: N = 8');
xlabel('k'); ylabel('Degrees'); %axis([-200,200,0,2]);

%subplot('2,2,2'); stem(n, realX_DFT); grid on;
%title('Real Part');
%xlabel('frequency in pi units'); ylabel('Real');
%subplot('2,2,4'); stem(n, imagX_DFT); grid on;
%title('Imaginary Part');
%xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DFT's  mag ang real imag
%% --------------------------------------------------------------

　　结果：

将序列末尾补12个零，长度达到16位

代码：

x = [1, 1, 1, 1, zeros(1,12)];                         % append 12 zeros to the end of x(n)
N = 16;                                                % zero-padding operation 
X_DFT = dft(x,N);                                      % DFT of x(n)

magX_DFT = abs(X_DFT)
phaX_DFT = angle(X_DFT)*180/pi                         % degrees
realX_DFT = real(X_DFT); imagX_DFT = imag(X_DFT);
angX_DFT = angle(X_DFT);                               % radias

n = 0:(N - 1);
k = 0:1000; w = (pi/500)*k;                            % [0,2pi] axis divided into 501 points.
%k = 0:500; w = (pi/500)*k;                            % [0,pi] axis divided into 501 points.
X_DTFT = x * (exp(-j*pi/500)) .^ (n'*k);               % DTFT of x(n)

magX_DTFT = abs(X_DTFT); angX_DTFT = angle(X_DTFT); realX_DTFT = real(X_DTFT); imagX_DTFT = imag(X_DTFT);

figure('NumberTitle', 'off', 'Name', 'Exameple5.7 x sequence')
set(gcf,'Color','white'); 
stem(n,x); title('x sequence, N = 16'); axis([0,N,-0.5,1.5]);
xlabel('n'); ylabel('x(n)'); grid on;


%% --------------------------------------------------------------
%%        START X_DTFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'X_DTFT its Magnitude and Angle, Real and Imaginary Part');
set(gcf,'Color','white'); 
subplot(2,2,1); plot(w/pi,magX_DTFT); grid on;  % axis([-2,2,0,15]); 
title('Magnitude Part');
xlabel('frequency in pi units'); ylabel('Magnitude  |X\_DTFT|'); 
subplot(2,2,3); plot(w/pi, angX_DTFT*180/pi); grid on;  % axis([-2,2,-1,1]);
title('Angle Part');
xlabel('frequency in pi units'); ylabel('Degrees'); %axis([-200,200,0,2]);

subplot('2,2,2'); plot(w/pi, realX_DTFT); grid on;
title('Real Part');
xlabel('frequency in pi units'); ylabel('Real');
subplot('2,2,4'); plot(w/pi, imagX_DTFT); grid on;
title('Imaginary Part');
xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DTFT's  mag ang real imag
%% --------------------------------------------------------------


%% --------------------------------------------------------------
%%        START X_DFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Example5.7 X_DFT its Magnitude and Angle');
set(gcf,'Color','white'); 
subplot(2,1,1); stem(n,magX_DFT); hold on; plot(8*w/pi,magX_DTFT,'--'); hold off;
grid on;   axis([-0.5,16.2,-1,5]); 
title('Magnitude Part of the DFT: N = 16');
xlabel('k'); ylabel('Magnitude  |X\_DFT|'); 
subplot(2,1,2); stem(n, phaX_DFT); hold on; plot(8*w/pi,angX_DTFT*180/pi,'--'); hold off;
grid on;   axis([-0.5,16.2,-200,200]);
title('Angle Part of the DFT: N = 16');
xlabel('k'); ylabel('Degrees'); %axis([-200,200,0,2]);

%subplot('2,2,2'); stem(n, realX_DFT); grid on;
%title('Real Part');
%xlabel('frequency in pi units'); ylabel('Real');
%subplot('2,2,4'); stem(n, imagX_DFT); grid on;
%title('Imaginary Part');
%xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DFT's  mag ang real imag
%% --------------------------------------------------------------

　　结果：

继续补零，长度N=128，

代码：

x = [1, 1, 1, 1, zeros(1,124)];                        % append 124 zeros to the end of x(n)
N = 128;                                               % zero-padding operation 
X_DFT = dft(x,N);                                      % DFT of x(n)

magX_DFT = abs(X_DFT)
phaX_DFT = angle(X_DFT)*180/pi                         % degrees
realX_DFT = real(X_DFT); imagX_DFT = imag(X_DFT);
angX_DFT = angle(X_DFT);                               % radias

n = 0:(N - 1);
k = 0:1000; w = (pi/500)*k;                            % [0,2pi] axis divided into 501 points.
%k = 0:500; w = (pi/500)*k;                            % [0,pi] axis divided into 501 points.
X_DTFT = x * (exp(-j*pi/500)) .^ (n'*k);               % DTFT of x(n)

magX_DTFT = abs(X_DTFT); angX_DTFT = angle(X_DTFT); realX_DTFT = real(X_DTFT); imagX_DTFT = imag(X_DTFT);

figure('NumberTitle', 'off', 'Name', 'Exameple5.7 x sequence')
set(gcf,'Color','white'); 
stem(n,x); title('x sequence, N = 128'); axis([0,N,-0.5,1.5]);
xlabel('n'); ylabel('x(n)'); grid on;


%% --------------------------------------------------------------
%%        START X_DTFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'X_DTFT its Magnitude and Angle, Real and Imaginary Part');
set(gcf,'Color','white'); 
subplot(2,2,1); plot(w/pi,magX_DTFT); grid on;  % axis([-2,2,0,15]); 
title('Magnitude Part');
xlabel('frequency in pi units'); ylabel('Magnitude  |X\_DTFT|'); 
subplot(2,2,3); plot(w/pi, angX_DTFT*180/pi); grid on;  % axis([-2,2,-1,1]);
title('Angle Part');
xlabel('frequency in pi units'); ylabel('Degrees'); %axis([-200,200,0,2]);

subplot('2,2,2'); plot(w/pi, realX_DTFT); grid on;
title('Real Part');
xlabel('frequency in pi units'); ylabel('Real');
subplot('2,2,4'); plot(w/pi, imagX_DTFT); grid on;
title('Imaginary Part');
xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DTFT's  mag ang real imag
%% --------------------------------------------------------------


%% --------------------------------------------------------------
%%        START X_DFT's  mag ang real imag
%% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'Example5.7 X_DFT its Magnitude and Angle');
set(gcf,'Color','white'); 
subplot(2,1,1); stem(n,magX_DFT); hold on; plot(64*w/pi,magX_DTFT,'--'); hold off;
grid on;   axis([-0.5,128.2,-1,5]); 
title('Magnitude Part of the DFT: N = 128');
xlabel('k'); ylabel('Magnitude  |X\_DFT|'); 
subplot(2,1,2); stem(n, phaX_DFT); hold on; plot(64*w/pi,angX_DTFT*180/pi,'--'); hold off;
grid on;   axis([-0.5,128.2,-200,200]);
title('Angle Part of the DFT: N = 128');
xlabel('k'); ylabel('Degrees'); %axis([-200,200,0,2]);

%subplot('2,2,2'); stem(n, realX_DFT); grid on;
%title('Real Part');
%xlabel('frequency in pi units'); ylabel('Real');
%subplot('2,2,4'); stem(n, imagX_DFT); grid on;
%title('Imaginary Part');
%xlabel('frequency in pi units'); ylabel('Imaginary');
%% --------------------------------------------------------------
%%        END X_DFT's  mag ang real imag
%% --------------------------------------------------------------

　　运行结果：