代码：
function [X1k, X2k] = real2dft(x1, x2, N)
%% ---------------------------------------------------------------------
%% DFT of two Real-Valued N-Point sequence x1(n) and x2(n)
%% ---------------------------------------------------------------------
%% [X1, X2] = real2dft(x1, x2, N)
%% X1k = n-point DFT of x1
%% X2k = n-point DFT of x2
%% x1 = sequence of length <= N
%% x2 = sequence of length <= N
%% N = length of DFT % ----------------------------------------
% if length of x1 and x2 < N,
% then padding zeros
% ----------------------------------------
if ( length(x1) < N)
x1 = [x1 zeros(1, N-length(x1))];
end if ( length(x2) < N)
x2 = [x2 zeros(1, N-length(x2))];
end x = x1 + j * x2; N = length(x); k = 0:(N-1); Xk_DFT = dft(x, N);
Xk_DFT_fold = Xk_DFT(mod_1(-k,N)+1); Xk_CCS = 0.5*(Xk_DFT + conj(Xk_DFT_fold));
Xk_CCA = 0.5*(Xk_DFT - conj(Xk_DFT_fold)); X1k = Xk_CCS;
X2k = Xk_CCA;
%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
%% Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf(' <DSP using MATLAB> Problem 5.19 \n\n'); banner();
%% ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ % ---------------------------------------------------------------------------------
% X(k) is N-point DFTs of N-point Complex-valued sequence x(n)
% x(n) = xR(n) + j xI(n)
% xR(n) and xI(n) are real and image parts of x(n);
% DFT[xR]=Xccs(k) DFT[j*xI]=Xcca(k)
%
% Xccs = 0.5*[X(k)+ X*((-k))] Xcca = 0.5*[X(k) - X*((-k))]
%
% ---------------------------------------------------------------------------------
n = [0:39];
x1 = cos(0.1*pi*n); % N=40 real-valued sequence
x2 = sin(0.2*pi*n); % N=40 real-valued sequence x = x1 + j * x2; N = length(x); k = 0:(N-1); Xk_DFT = dft(x, N);
Xk_DFT_fold = Xk_DFT(mod_1(-k,N)+1); 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); magXk_DFT_fold = abs( [ Xk_DFT_fold ] ); % DFT magnitude
angXk_DFT_fold = angle( [Xk_DFT_fold] )/pi; % DFT angle
realXk_DFT_fold = real(Xk_DFT_fold);
imagXk_DFT_fold = imag(Xk_DFT_fold); % --------------------------------------------------------
% Calculater one N-point DFT to get
% two N-point DFT
% --------------------------------------------------------
[X1k_DFT, X2k_DFT] = real2dft(x1, x2, N); magX1k_DFT = abs( [ X1k_DFT ] ); % DFT magnitude
angX1k_DFT = angle( [X1k_DFT] )/pi; % DFT angle
realX1k_DFT = real(X1k_DFT);
imagX1k_DFT = imag(X1k_DFT); magX2k_DFT = abs( [ X2k_DFT ] ); % DFT magnitude
angX2k_DFT = angle( [X2k_DFT] )/pi; % DFT angle
realX2k_DFT = real(X2k_DFT);
imagX2k_DFT = imag(X2k_DFT); % -------------------------------------------------------
% Get DFT of xR and xI directorly
% -------------------------------------------------------
XRk_DFT = dft(x1, N);
XIk_DFT = dft(j*x2, N); magXRk_DFT = abs( [ XRk_DFT ] ); % DFT magnitude
angXRk_DFT = angle( [XRk_DFT] )/pi; % DFT angle
realXRk_DFT = real(XRk_DFT);
imagXRk_DFT = imag(XRk_DFT); magXIk_DFT = abs( [ XIk_DFT ] ); % DFT magnitude
angXIk_DFT = angle( [XIk_DFT] )/pi; % DFT angle
realXIk_DFT = real(XIk_DFT);
imagXIk_DFT = imag(XIk_DFT); figure('NumberTitle', 'off', 'Name', 'P5.19 xR(n) and xI(n)')
set(gcf,'Color','white');
subplot(2,1,1); stem(n, x1);
xlabel('n'); ylabel('x1');
title('real part of x(n), cos(0.1\pin), N=40'); grid on;
subplot(2,1,2); stem(n, x2);
xlabel('n'); ylabel('x2');
title('imag part of x(n), sin(0.2\pin), N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 X(k), DFT of x(n)')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magXk_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude DFT of x(n), N=40'); grid on;
subplot(2,2,3); stem(k, angXk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle DFT of x(n), N=40'); grid on;
subplot(2,2,2); stem(k, realXk_DFT);
xlabel('k'); ylabel('real (k)');
title('real DFT of x(n), N=40'); grid on;
subplot(2,2,4); stem(k, imagXk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag DFT of x(n), N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 X((-k))_N')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magXk_DFT_fold);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude X((-k)), N=40'); grid on;
subplot(2,2,3); stem(k, angXk_DFT_fold);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle X((-k)), N=40'); grid on;
subplot(2,2,2); stem(k, realXk_DFT_fold);
xlabel('k'); ylabel('real (k)');
title('real X((-k)), N=40'); grid on;
subplot(2,2,4); stem(k, imagXk_DFT_fold);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag X((-k)), N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 X1(k) by real2dft')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magX1k_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude, N=40'); grid on;
subplot(2,2,3); stem(k, angX1k_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle, N=40'); grid on;
subplot(2,2,2); stem(k, realX1k_DFT);
xlabel('k'); ylabel('real (k)');
title('real, N=40'); grid on;
subplot(2,2,4); stem(k, imagX1k_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag, N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 X2(k) by real2dft')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magX2k_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude, N=40'); grid on;
subplot(2,2,3); stem(k, angX2k_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle, N=40'); grid on;
subplot(2,2,2); stem(k, realX2k_DFT);
xlabel('k'); ylabel('real (k)');
title('real, N=40'); grid on;
subplot(2,2,4); stem(k, imagX2k_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag, N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 XR(k) by direct')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magXRk_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude, N=40'); grid on;
subplot(2,2,3); stem(k, angXRk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle, N=40'); grid on;
subplot(2,2,2); stem(k, realXRk_DFT);
xlabel('k'); ylabel('real (k)');
title('real, N=40'); grid on;
subplot(2,2,4); stem(k, imagXRk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag, N=40'); grid on; figure('NumberTitle', 'off', 'Name', 'P5.19 XI(k) by direct')
set(gcf,'Color','white');
subplot(2,2,1); stem(k, magXIk_DFT);
xlabel('k'); ylabel('magnitude(k)');
title('magnitude, N=40'); grid on;
subplot(2,2,3); stem(k, angXIk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('angle(k)');
title('angle, N=40'); grid on;
subplot(2,2,2); stem(k, realXIk_DFT);
xlabel('k'); ylabel('real (k)');
title('real, N=40'); grid on;
subplot(2,2,4); stem(k, imagXIk_DFT);
%axis([-N/2, N/2, -0.5, 50.5]);
xlabel('k'); ylabel('imag (k)');
title('imag, N=40'); grid on;
运行结果：
复数序列的实部和虚部
复数序列的DFT，X(k)
X((-k))
直接计算实部和虚部的DFT，XR(k)和XI(k)
利用函数real2dft计算实部和虚部对应的DFT，Xccs(k)和Xcca(k)
结论：
如果X(k)是N点复数序列x(n)的N点DFT，x(n)=xR(n)+jxI(n)，那么有
DFT[xR]=Xccs(k) DFT[j*xI]=Xcca(k)
实部序列的DFT是复数序列的DFT的共轭圆周对称分量
虚部序列的DFT是复数序列的DFT的共轭圆周反对称分量。