《DSP using MATLAB》Problem 5.18

代码：

%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 5.18 \n\n');

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

% -------------------------------------------------------------------------------------
%          X(k) is 20-point DFTs of Complex-valued sequence x(n)
%          X(k) = [3cos(0.2pi*k) + j4sin(0.1pi*k)][u(k)-u(k-20)]
%               N = 20       k=[0:19]
%
%            xccs = [x(n)+ x*((-n))]/2      xcca = [x(n) - x*((-n))]/2
%            DFT[xccs] = real(X(k))         DFT[xcca] = j*imag(X(k))
% -------------------------------------------------------------------------------------

    k1 = [0:19];
Xk_DFT = (3*cos(0.2*pi*k1) + j*4*sin(0.1*pi*k1)) .* (stepseq(0,min(k1),max(k1))-stepseq(20,min(k1),max(k1)));
    N1 = length(Xk_DFT);                                              % length is 10

    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', 'P5.18 X(k), DFT of x(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(k1, magXk_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude DFT of x(n), N=20');  grid on;
subplot(2,2,3); stem(k1, angXk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle DFT of x(n), N=20');  grid on;
subplot(2,2,2); stem(k1, realXk_DFT);
xlabel('k'); ylabel('real (k)');
title('real DFT of x(n), N=20');  grid on;
subplot(2,2,4); stem(k1, imagXk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of x(n), N=20');  grid on;

[xn] = idft(Xk_DFT, N1);                     % Complex-valued sequence
n = [0 : N1-1];

% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%        x(n) decomposition into circular-conjugate-symmetric  and
%                                circular-conjugate-antisymmetric parts
% +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[xccs, xcca] = circevod_cv(xn);

% +++++++++++++++++++++++++++++++++++++++++++++++++++++++
%           DFT(k) of xccs and xcca, k=[0:N1-1]
% +++++++++++++++++++++++++++++++++++++++++++++++++++++++
k1 = [0:19];

Xk_CCS_DFT = dft(xccs, length(xccs));
Xk_CCA_DFT = dft(xcca, length(xcca));

    N1 = length(Yk_DFT);                                    % length is 10

    magXk_CCS_DFT = abs( [ Xk_CCS_DFT ] );                          % DFT magnitude
    angXk_CCS_DFT = angle( [Xk_CCS_DFT] )/pi;                       % DFT angle
   realXk_CCS_DFT = real(Xk_CCS_DFT);
   imagXk_CCS_DFT = imag(Xk_CCS_DFT);

    magXk_CCA_DFT = abs( [ Xk_CCA_DFT ] );                          % DFT magnitude
    angXk_CCA_DFT = angle( [Xk_CCA_DFT] )/pi;                       % DFT angle
   realXk_CCA_DFT = real(Xk_CCA_DFT);
   imagXk_CCA_DFT = imag(Xk_CCA_DFT);

figure('NumberTitle', 'off', 'Name', 'P5.18 DFT(k) of xccs(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(k1, magXk_CCS_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude DFT of xccs(n), N=20');  grid on;
subplot(2,2,3); stem(k1, angXk_CCS_DFT);
xlabel('k'); ylabel('angle(k)');
title('angle DFT of xccs(n), N=20');  grid on;
subplot(2,2,2); stem(k1, realXk_CCS_DFT);
xlabel('k'); ylabel('real (k)');
title('real DFT of xccs(n), N=20');  grid on;
subplot(2,2,4); stem(k1, imagXk_CCS_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of xccs(n), N=20');  grid on;

figure('NumberTitle', 'off', 'Name', 'P5.18 DFT(k) of xcca(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(k1, magXk_CCA_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude DFT of xcca(n), N=20');  grid on;
subplot(2,2,3); stem(k1, angXk_CCA_DFT);
xlabel('k'); ylabel('angle(k)');
title('angle DFT of xcca(n), N=20');  grid on;
subplot(2,2,2); stem(k1, realXk_CCA_DFT);
xlabel('k'); ylabel('real (k)');
title('real DFT of xcca(n), N=20');  grid on;
subplot(2,2,4); stem(k1, imagXk_CCA_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of xcca(n), N=20');  grid on;

% --------------------------------------------------------------
%          Verify
%     DFT[xccs] = real(X(k))    DFT[xcca] = j*imag(X(k))
% --------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'P5.18 Verify')
set(gcf,'Color','white');
subplot(2,2,1); stem(k1, realXk_DFT);
xlabel('k'); ylabel('real(k)');
title('real DFT of x(n), N=20');  grid on;
subplot(2,2,3); stem(k1, imagXk_DFT);
xlabel('k'); ylabel('imag(k)');
title('imag DFT of x(n), N=20');  grid on;

subplot(2,2,2); stem(k1, realXk_CCS_DFT);
xlabel('k'); ylabel('real (k)');
title('real DFT of xccs(n), N=20');  grid on;
subplot(2,2,4); stem(k1, imagXk_CCA_DFT);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of xcca(n), N=20');  grid on;

error1 = sum( abs(realXk_DFT - realXk_CCS_DFT) )
error2 = sum( abs(imagXk_DFT - imagXk_CCA_DFT) )

% ----------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'P5.18 x(n) & xccs(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(n, real(xn));
xlabel('n'); ylabel('x(n)');
title('real[x(n)], IDFT of X(k)');  grid on;
subplot(2,2,2); stem(n, real(xccs));
xlabel('n'); ylabel('xccs(n)');
title('real xccs');  grid on;

subplot(2,2,3); stem(n, imag(xn));
xlabel('n'); ylabel('x(n)');
title('imag[x(n)], IDFT of X(k)');  grid on;
subplot(2,2,4); stem(n, imag(xccs));
xlabel('n'); ylabel('xccs(n)');
title('imag xccs');  grid on;

% ---------------------------------------------------------------
figure('NumberTitle', 'off', 'Name', 'P5.18 x(n) & xcca(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(n, real(xn));
xlabel('n'); ylabel('x(n)');
title('real[x(n)], IDFT of X(k)');  grid on;
subplot(2,2,2); stem(n, real(xcca));
xlabel('n'); ylabel('xcca(n)');
title('real xcca');  grid on;

subplot(2,2,3); stem(n, imag(xn));
xlabel('n'); ylabel('x(n)');
title('imag[x(n)], IDFT of X(k)');  grid on;
subplot(2,2,4); stem(n, imag(xcca));
xlabel('n'); ylabel('xcca(n)');
title('imag xcca');  grid on;

　　运行结果：

20点DFT，X(k)

圆周共轭对称序列的DFT

圆周共轭反对称的DFT

从下图看出，序列的DFT X(k)的实部与圆周共轭对称分量的DFT的实部相等；

序列的DFT X(k)的虚部与圆周共轭反对称分量的DFT的虚部相等；

圆周共轭对称序列：xccs（n）

圆周共轭反对称序列xcca（n）