Plot phase on FM modulation on time domain

I have message signal m(t) which is a data set and i want to find of ∅(𝑡) values after that plot ∅(𝑡) values on time domain. where formula of ∅(𝑡) is ∅(𝑡)=2𝜋50∫𝑚(𝜏)𝑑𝜏.

How i found the m(t) is shown down below.

Fs = 1000; % Sampling frequency
t = -10:1/Fs:10-1/Fs; % Time interval
f = -length(t)/2:1:length(t)/2-1; % Frequcny interval
fc = 250; % Hz
Kf = 50; 

Tau = 2; center_1 = 1; center_2 = 3; % Recuired values for creating of "m(t)"


square_wave1 = square_wave_device(Tau, center_1, t); % First square wave
square_wave2 = square_wave_device(Tau, center_2, t); % Second square wave

mt = square_wave1 - square_wave2; % Message Signal

%-------------------- Creating the square waves -----------------------%
function square_output = square_wave_device(T, x, t)
    x = x - 1 ; % Calculating the center of the positive side of the wave
    square_output = (square((1/2*pi).*(t-x).*(T/2))+1)/2; 
end

Answer 1

[Hassaan]

% Sampling frequency and Time interval
Fs = 1000; % Sampling frequency
t = -10:1/Fs:10-1/Fs; % Time interval

% Frequency interval
f = -length(t)/2:1:length(t)/2-1; % Frequency interval

% Required values for creating "m(t)"
Tau = 2; center_1 = 1; center_2 = 3;

% Creating square waves
square_wave1 = square_wave_device(Tau, center_1, t); % First square wave
square_wave2 = square_wave_device(Tau, center_2, t); % Second square wave

% Message Signal m(t)
mt = square_wave1 - square_wave2;

% Plotting m(t)
figure;
plot(t, mt);
title('Message Signal m(t) vs Time');
xlabel('Time (seconds)');
ylabel('m(t)');
grid on;

% Calculating phi(t)
Kf = 50; % Given constant
dt = 1/Fs; % Time step (delta t)
phi_t = 2 * pi * Kf * cumsum(mt) * dt; % Integration of m(t)

% Plotting phi(t)
figure;
plot(t, phi_t);
title('\phi(t) vs Time');
xlabel('Time (seconds)');
ylabel('\phi(t)');
grid on;

% Function for generating square waves
function square_output = square_wave_device(T, x, t)
    x = x - 1; % Calculating the center of the positive side of the wave
    square_output = (square((1/2*pi).*(t-x).*(T/2))+1)/2; 
end

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[Emir Dönmez]

But how can the all values of integral can be negative ?

My m(t) graph is like this:

[Hassaan]

@Emir Dönmez Updated my answer. Thank you.

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If you find the solution helpful and it resolves your issue, it would be greatly appreciated if you could accept the answer. Also, leaving an upvote and a comment are also wonderful ways to provide feedback.

Professional Interests

• Technical Services and Consulting
• Embedded Systems | Firmware Developement | Simulations
• Electrical and Electronics Engineering

Feel free to contact me.

[Emir Dönmez]

Everything is the same and when i plot m(t) from 0 to 4 the graph of "phi" becomes like down below:

In my opinion max value of the phi should be 1 or 2 not 600, am i right ?

[Hassaan]

Yes you can adjust as per your needs.

------------------------------------------------------------------------------------------------------------------------------------------------ If you find the solution helpful and it resolves your issue, it would be greatly appreciated if you could accept the answer. Also, leaving an upvote and a comment are also wonderful ways to provide feedback. Professional Interests Technical Services and Consulting Embedded Systems | Firmware Developement | Simulations Electrical and Electronics Engineering Feel free to contact me.

[Emir Dönmez]

after that how can i integrate the phi value to make FM modulation ?

fc = 250hz, Ac = 5