Demonstration of Z-transform and ROC in time series analysis

Exploration of z - transform and region of convergence in the both continuous and discrete time series to apply in real world applications.

Objectives:

•To demonstrate the computation of Z-transform for both continuous and discrete signals.

•To illustrate the concept of the region of convergence and its implications in signal analysis.

•To provide practical examples and applications showcasing the importance of Z-transform and ROC in time series analysis.

%continuous time series
syms t; 
x = exp(-2*t) * (heaviside(t) - heaviside(t-1)); 
Ts = 0.1; 
n = 0:20; 
xn = subs(x, t, n*Ts); 
syms z; 
X = ztrans(xn, z);
disp('Z-transform of x[n]:');
disp(X);
roc = abs(z) > exp(2*Ts);
figure;
fimplicit(roc, 'b', 'LineWidth', 1.5);
xlabel('Re(z)');
ylabel('Im(z)');
title('Region of Convergence in continuous time (ROC) for Z-transform');
disp(['Region of Convergence in continuous time (ROC): |z| > ', num2str(exp(2*Ts))]);
%discrete time series
n=0:20;
x=0.5.^n;
syms z; 
X = ztrans(x, z);
disp('Z-transform of x[n]:');
disp(X);
roc = abs(z) > 0.5;
figure;
fimplicit(roc, 'b', 'LineWidth', 1.5);
xlabel('Re(z)');
ylabel('Im(z)');
title('Region of Convergence in discrete time (ROC) for Z-transform');
disp('Region of Convergence in discrete time (ROC): |z| > 0.5');