How to define this function?

How can I define the second periodic function(x2(t))?

here is the definition of the first function that I defined.

close all;clear;clc;

Fs = 50;
T = 1/Fs;
t = -2*pi:T:2*pi;
L = length(t);

%peroid 2*pi first function
X1 = 1/2*(1+square(2/3*(t+pi),200/3));
Y1 = fft(X1);

f = Fs*(0:(L-1))/L;

subplot(1,2,1);
plot(t,X1);

subplot(1,2,2);
plot(f,abs(Y1));

[John D'Errico]

x2 is NOT a function of t. Period. As an integral, t goes away.

Answer 1

[Divyam]

Hi @Niloufar,

You can create the following function handle for and calculate using the "integral" function:

% Sample values for T and t
T = 2;
t = 0;

% Function handle for x1(t) 
x1_func = @(t) double(abs(t) <= T/2);

% Calculating x2(t)
x2_func = integral(@(t) x1_func(t), -inf, inf);

% Printing out the values
fprintf("Value of x1(t) at t = %.2f is: %.2f\n", t, x1_func(t));

Value of x1(t) at t = 0.00 is: 1.00

fprintf("Value of x2(t) at t = %.2f is: %.2f\n", t, x2_func);

Value of x2(t) at t = 0.00 is: 2.00

To plot the functions and you can calculate their values for certain interval of t and plot them using the "plot" function:

% Parameters
T = 2;          
dt = 0.01;     
t = -4:dt:4;    

% Calculate x1(t)
x1 = zeros(size(t));
x1(abs(t) <= T/2) = 1;

x1_func = @(t) double(abs(t) <= T/2);

% Calculate x2(t) - integral of x1(t)
x2 = zeros(size(t));
for i = 1:length(t)
    x2(i) = integral(@(t) x1_func(t), -inf, inf);
end

% Create figure with subplots
figure;

% Plot x1(t)
subplot(2,1,1);
plot(t, x1, 'LineWidth', 2);
grid on;
title('x_1(t) - Rectangular Pulse');
xlabel('t');
ylabel('x_1(t)');
ylim([-0.2, 1.2]);

% Add vertical lines to show T/2 and -T/2
hold on;
plot([-T/2 -T/2], [-0.2 1.2], 'r--');
plot([T/2 T/2], [-0.2 1.2], 'r--');
legend('x_1(t)', 'T/2 boundaries');

% Plot x2(t)
subplot(2,1,2);
plot(t, x2, 'LineWidth', 2);
grid on;
title('x_2(t) - Integral of x_1(t)');
xlabel('t');
ylabel('x_2(t)');

For more information regarding the "integral" function, refer to this documentation: https://www.mathworks.com/help/matlab/ref/integral.html