Question

Solve it using matlap

Given the following two periodic signals:
x₁(t) = 2cos(10πt)
x₂(t) = 4cos(15πt)
we want to check if x(t) = x₁(t) + x₂(t) is periodic.
using the sampling rate fₛ = 50 Hz, generate x₁ and x₂ for 3 seconds.

Answer 1

To check if the sum of two periodic signals is periodic in MATLAB, define the sampling frequency, generate time vector for 3 seconds, create the signals, and plot them. By using code, determine the periodicity by inspecting if x(t) repeats and if x₁(t) and x₂(t) frequencies have a common multiple.

Step-by-step explanation:

To solve for the periodic signals x₁(t) and x₂(t) using MATLAB and check if their sum x(t) = x₁(t) + x₂(t) is periodic, we can follow these steps:

1. Define the sampling frequency fₛ (50 Hz) and the time vector t for 3 seconds.
2. Generate the signals x₁(t) and x₂(t) using their respective equations.
3. Check for periodicity by observing if the combined signal x(t) repeats itself at regular intervals, also by examining their frequencies.

Here is a MATLAB code snippet that executes these steps:

Fs = 50; % Sampling frequency
T = 1/Fs; % Sampling period
L = 150; % Length of signal (3s * 50Hz)
t = (0:L-1)*T; % Time vector

% Signal Generation
x1 = 2*cos(10*pi*t);
x2 = 4*cos(15*pi*t);

% Combine the signals
x = x1 + x2;

% Plot the signals
figure;
plot(t, x1);
title('Signal x1(t)');
xlabel('Time (seconds)');
ylabel('Amplitude');

figure;
plot(t, x2);
title('Signal x2(t)');
xlabel('Time (seconds)');
ylabel('Amplitude');

figure;
plot(t, x);
title('Combined Signal x(t)');
xlabel('Time (seconds)');
ylabel('Amplitude');

To determine if x(t) is periodic, we could analyze the frequencies of x1 and x2. If these frequencies have a common multiple, the sum x(t) will also be periodic.