《DSP using MATLAB》Problem 5.37

        证明过程：

        代码：

%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 5.37 \n\n');

banner();
%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% -------------------------------------------------------------------------------------
%          X(k) is DFTs of real-valued sequence x(n)             
%           x(n)=A*cos(wn)R(n)   n=[0:N-1]
%             
%             x(n) = sin(5*pi*n/99)    n=[0:100-1]                                     
% -------------------------------------------------------------------------------------
N = 100; A = 1;
n1 = [0:N-1];
x1 = A*sin(5*pi*n1/99);

figure('NumberTitle', 'off', 'Name', 'P5.37 x(n), N=100')
set(gcf,'Color','white'); 
stem(n1, x1); 
xlabel('n'); ylabel('x(n)');
title('x(n)=sin(5\pin/99)');  grid on;

    k1 = [0:N-1];
Xk_DFT = fft(x1, N);
    N1 = length(Xk_DFT);                                              % length 

    magXk_DFT = abs( [ Xk_DFT ] );                                    % DFT magnitude
    angXk_DFT = angle( [Xk_DFT] )/pi;                                 % DFT angle
   realXk_DFT = real(Xk_DFT); imagXk_DFT = imag(Xk_DFT);

figure('NumberTitle', 'off', 'Name', 'P5.37 X(k), DFT of x(n) N=100')
set(gcf,'Color','white'); 
subplot(2,1,1); stem(k1, magXk_DFT); 
xlabel('k'); ylabel('magnitude(k)');
title('magnitude DFT of x(n)');  grid on;
subplot(2,1,2); stem(k1, angXk_DFT);  
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle DFT of x(n)');  grid on;

figure('NumberTitle', 'off', 'Name', 'P5.37 X(k), N=100')
set(gcf,'Color','white'); 
subplot(2,1,1); stem(k1, realXk_DFT); 
xlabel('k'); ylabel('real (k)');
title('real DFT of x(n)');  grid on;
subplot(2,1,2); stem(k1, imagXk_DFT);  
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of x(n)');  grid on;

　　运行结果：

        原始序列

        不同范围的DFT