%         ***** MATLAB Code Starts Here
%

%Z_TRANSFORM_01_MAT

%

T = pi/10; % Sample period, seconds

ws = 2*pi/T; % Sample frequency, r/s

%

w1 = 2; w2 = 22; % Frequencies to be studied

%

% Plot the s-plane with the primary strip defined by the imaginary axis

% ranging from -jws/2 to +jws/2 and the two test frequencies, w1 and w2,

% identified.

%

fig_size = [232 84 774 624];

figure(1),clf,plot(0,w1,'bo',0,w2,'rx',[-25 25],[ws/2 ws/2],'r:',[-25 25],[-ws/2 -ws/2],'r:')

axis('square'),axis([-25 25 -25 25]),plotax,xlabel('Real Axis'),ylabel('Imag Axis'),...

title('s-Plane with T = pi/10 s'),set(gcf,'Position',fig_size),...

text(1,w1,['w_1 = ' int2str(w1) ' r/s']),text(1,w2,['w_2 = ' int2str(w2) ' r/s']),...

text(-18,11,'w_s/2'),text(-18,-9,'-w_s/2')

%

% Define the unit circle for use in the z-plane using the function "circle"

% which is provided at the end of this file.

%

[xx,yy] = circle;

%

% Define the points in the s-plane and the z-plane corresponding to the

% test frequencies w1 and w2 and the value of the sampling period T.

%

s1 = j*w1; s2 = j*w2;

z1 = exp(s1*T); z2 = exp(s2*T);

%

% Plot the z-plane with the unit circle and the mappings of the frequencies

% w1 and w2 into the z-plane with the period T.

%

figure(2),clf,...

plot(xx,yy,'k',real(z1),imag(z1),'bo',real(z2),imag(z2),'rx'),grid,...

xlabel('Real Axis'),ylabel('Imag Axis'),title('z-Plane with T = pi/10 s'),...

axis('square'),axis([-2 2 -2 2]),...

set(gcf,'Position',fig_size)

text(0.9,0.6,'z_1 = z_2'),text(-0.95,1.4,'z = exp(s*T)'),...

text(-0.95,1.2,'z_1 = z_2 = 0.8090 + j0.5878')

%

% Compute the continuous-time functions x1(t) and x2(t) corresponding to

% the frequencies w1 and w2, respectively. Compute the values of those

% functions at the sample points separated by T.

%

ts = 0:T:5; t = 0:.01:5;

x1 = cos(w1*t); x2 = cos(w2*t);

x1s = cos(w1*ts); x2s = cos(w2*ts);

%

% Plot the two time functions and their sample values.

%

figure(3),clf,stem(ts,x1s);grid,hold on,plot(t,x1,'k:',t,x2,'r:'),...

hold off,xlabel('Time (s)'),ylabel('Amplitude'),...

title('x_1(t)=cos(2*t) and x_2(t)=cos(22*t) sampled at T = pi/10 s'),...

set(gcf,'Position',fig_size)

%

% Calculate the first several frequencies which appear when the two

% signals x1(t) and x2(t) are sampled with a sampling period T.

%

n = [-1 0 1];

w1 = [-w1 w1]; w2 = [-w2 w2];

%

wx1 = []; wx2 = [];

for i = 1:length(n)

wx1 = [wx1 (w1 - n(i)*ws)];

wx2 = [wx2 (w2 - n(i)*ws)];

end

%

wx1 = sort(wx1);wx2 = sort(wx2);

clear i

%

%
%         ***** MATLAB Code Stops Here





another way of z transform of matlab code

%         ***** MATLAB Code Starts Here
%

%Z_TRANSFORM_03_MAT

%

fig_size = [232 84 774 624];

num = 20*conv([1 -.8],[1 -.7]); % H(z) numerator

den = conv([1 -.5],conv([1 -.6],[1 .8])); % H(z) denominator

%

h = dimpulse(num,den); % Impulse response using "dimpulse"

n = [0:length(h)-1]; % discrete time vector

%

[res,pol,fdth] = residue(num,[den 0]); % PFE of H(z)/z

%

h1 = res(1)*pol(1).^n + res(2)*pol(2).^n + res(3)*pol(3).^n;

h1(1) = h1(1)+res(4); % Impulse response using "residue"

%

figure(1),clf,stem(n,h1);grid,axis([0 25 -25 25]),...

xlabel('Discrete Time Samples'),ylabel('Amplitude'),...

title('Impulse Response h(n)'),...

set(gcf,'Position',fig_size)

%

% Magnitude and Phase of H(z) using "polyval" function

%

wt = [0:pi/20:pi]; % Angle vector (rad)

z = exp(j*wt); % Points on unit circle

%

mag = abs(polyval(num,z)) ./ abs(polyval(den,z));

ph = unwrap(angle(polyval(num,z)) - angle(polyval(den,z)))*180/pi;

%

% Figures of magnitude and phase

%

figure(2),clf,plot(wt,20*log10(mag),wt,20*log10(mag),'ko'),grid,...

xlabel('Angle around unit circle (rad)'),...

ylabel('Magnitude (db)'),title('Frequency Response Magnitude'),...

set(gcf,'Position',fig_size)

%

figure(3),clf,plot(wt,ph,wt,ph,'ko'),grid,...

xlabel('Angle around unit circle (rad)'),...

ylabel('Phase (deg)'),title('Frequency Response Phase'),...

set(gcf,'Position',fig_size)

%

%
%         ***** MATLAB Code Stops Here