n = -5:10; x = rand(1,length(n));

k = -100:100; w = (pi/100)*k;                 % freqency between -pi and +pi  ,   [0,pi] axis divided into 101 points.

X = x * (exp(-j*pi/100)) .^ (n'*k);           % DTFT of x

% folding property

y = fliplr(x); m = -fliplr(n);                         % signal folding

Y = y * (exp(-j*pi/100)) .^ (m'*k);           % DTFT of y 

magX = abs(X); angX = angle(X); realX = real(X); imagX = imag(X);

magY = abs(Y); angY = angle(Y); realY = real(Y); imagY = imag(Y);

%verification

Y_check = fliplr(X);                                   % X(-w)

error = max(abs(Y-Y_check));                  % Difference

figure('NumberTitle', 'off', 'Name', 'x & y sequence')

set(gcf,'Color','white');

subplot(2,1,1); stem(n,x); title('x sequence ');         xlabel('n'); ylabel('x(n)'); grid on;

%subplot(2,2,2); stem(n,imag(x)); title('x sequence Imaginary Part');    xlabel('n'); ylabel('Imaginary x(n))'); grid on;

subplot(2,1,2); stem(m,y); title('y sequence ');         xlabel('n'); ylabel('y(n)'); grid on;

%subplot(2,2,4); stem(n,imag(y)); title('y sequence Imaginary Part');    xlabel('n'); ylabel('Imaginary y(n))'); grid on;

%% --------------------------------------------------------------

%%        START X's  mag ang real imag

%% --------------------------------------------------------------

figure('NumberTitle', 'off', 'Name', 'X its Magnitude and Angle, Real and Imaginary Part');

set(gcf,'Color','white');

subplot(2,2,1); plot(w/pi,magX); grid on;  % axis([-2,2,0,15]);

title('Magnitude Part');

xlabel('frequency in \pi units'); ylabel('Magnitude  |X|');

subplot(2,2,3); plot(w/pi, angX/pi); grid on;  % axis([-2,2,-1,1]);

title('Angle Part');

xlabel('frequency in \pi units'); ylabel('Radians/\pi');

subplot('2,2,2'); plot(w/pi, realX); grid on;

title('Real Part');

xlabel('frequency in \pi units'); ylabel('Real');

subplot('2,2,4'); plot(w/pi, imagX); grid on;

title('Imaginary Part');

xlabel('frequency in \pi units'); ylabel('Imaginary');

%% --------------------------------------------------------------

%%        END X's  mag ang real imag

%% --------------------------------------------------------------

%% --------------------------------------------------------------

%%        START Y's  mag ang real imag

%% --------------------------------------------------------------

figure('NumberTitle', 'off', 'Name', 'Y its Magnitude and Angle, Real and Imaginary Part');

set(gcf,'Color','white');

subplot(2,2,1); plot(w/pi,magY); grid on;  % axis([-2,2,0,15]);

title('Magnitude Part');

xlabel('frequency in \pi units'); ylabel('Magnitude  |Y|');

subplot(2,2,3); plot(w/pi, angY/pi); grid on;  % axis([-2,2,-1,1]);

title('Angle Part');

xlabel('frequency in \pi units'); ylabel('Radians/\pi');

subplot('2,2,2'); plot(w/pi, realY); grid on;

title('Real Part');

xlabel('frequency in \pi units'); ylabel('Real');

subplot('2,2,4'); plot(w/pi, imagY); grid on;

title('Imaginary Part');

xlabel('frequency in \pi units'); ylabel('Imaginary');

%% --------------------------------------------------------------

%%        END Y's  mag ang real imag

%% --------------------------------------------------------------

%% ----------------------------------------------------------------

%%        START Graphical verification

%% ----------------------------------------------------------------

figure('NumberTitle', 'off', 'Name', 'X Y compare theirs Magnitude and Angle');

set(gcf,'Color','white');

subplot(2,2,1); plot(w/pi,magX); grid on; %axis([-1,1,0,12]);

xlabel('frequency in \pi units'); ylabel('|X|'); title('Magnitude of X ');

subplot(2,2,2); plot(w/pi,angX/pi); grid on; axis([-1,1,-1,1]);

xlabel('frequency in \pi units'); ylabel('Radians/\pi'); title('Angle of X ');

subplot(2,2,3); plot(w/pi,magY); grid on; %axis([-1,1,0,12]);

xlabel('frequency in \pi units'); ylabel('|Y|'); title('Magnitude of Y ');

subplot(2,2,4); plot(w/pi,angY/pi); grid on; axis([-1,1,-1,1]);

xlabel('frequency in \pi units'); ylabel('Radians/\pi'); title('Angle of Y ');

%% ----------------------------------------------------------------

%%        END Graphical verification

%% ----------------------------------------------------------------