How can I draw the mt sign? Frequency Modulation

The message signal 𝑚(𝑡)=∏((𝑡−1)/2))−∏(𝑡−3)/2),

How can I draw the mt sign?

The message signal 𝑚(𝑡)=∏((𝑡−1)/2))−∏(𝑡−3)/2), c(𝑡) = 5𝐶𝑜𝑠(500𝜋𝑡) to obtain FM signal 𝑦(𝑡) = 5𝐶𝑜𝑠(500𝜋𝑡 + ∅(𝑡)), where ∅(𝑡) = 2𝜋𝑘𝑓 ∫ 𝑚(𝜏)𝑑𝜏 with 𝑘𝑓 = 50. The demodulator consists of a differentiator, a diode (assume ideal), a low pass filter (LPF) and a capacitor for DC blocking, successively. 1) Plot the message signal. Obtain and plot the magnitude spectrum of the message signal. 2) Obtain and plot the time-domain representation and magnitude spectrum of the phase ∅(𝑡). 3) Obtain and plot the time-domain representation and magnitude spectrum of the modulated signal 𝑦(𝑡).

Answer 1

[Hassaan]

% Parameters
fs = 8000;  % Sampling frequency for decent visualization
t = 0:1/fs:5;  % Time vector from 0 to 5 seconds
f_c = 500;  % Carrier frequency
k_f = 50;  % Frequency sensitivity

% Create the message signal
message = message_signal(t);

% Carrier signal
carrier = cos(2 * pi * f_c * t);

% Frequency Modulate the carrier with the message
integral_message = cumtrapz(t, message);  % Integrate message signal with respect to time
fm_signal = cos(2 * pi * f_c * t + k_f * integral_message);  % FM signal

% Plot the message signal
figure;
subplot(2, 1, 1);
plot(t, message);
title('Message Signal m(t)');
xlabel('Time [s]');
ylabel('Amplitude');

% Plot the magnitude spectrum of the message signal
subplot(2, 1, 2);
fft_message = fft(message);
fft_freq = linspace(-fs/2, fs/2, length(message));
plot(fft_freq, fftshift(abs(fft_message)));
title('Magnitude Spectrum of Message Signal m(t)');
xlabel('Frequency [Hz]');
ylabel('Magnitude');

% Plot the modulated signal y(t) and its magnitude spectrum
figure;

% Time domain representation of modulated signal y(t)
subplot(3, 1, 1);
plot(t, fm_signal);
title('Time-domain Representation of Modulated Signal y(t)');
xlabel('Time [s]');
ylabel('Amplitude');

% Magnitude spectrum of modulated signal y(t)
subplot(3, 1, 2);
fft_fm = fft(fm_signal);
plot(fft_freq, fftshift(abs(fft_fm)));
title('Magnitude Spectrum of Modulated Signal y(t)');
xlabel('Frequency [Hz]');
ylabel('Magnitude');

% Phase spectrum of modulated signal y(t)
subplot(3, 1, 3);
phase_fm = angle(fft_fm);
plot(fft_freq, fftshift(phase_fm));
title('Phase Spectrum of Modulated Signal y(t)');
xlabel('Frequency [Hz]');
ylabel('Phase [Radians]');

% Define the message signal m(t)
function mt = message_signal(t)
    mt = (t - (1/2)).^2 .* (sinc(t - (1/2))).^2 - (t - (3/2)).^2 .* (sinc(t - (3/2))).^2;
end

Please note that MATLAB's sinc function is normalized differently from the mathematical definition of sinc used in signal processing. MATLAB defines sinc(x) = sin(pi*x)/(pi*x), while the standard signal processing definition is sinc(x) = sin(x)/x. If you need the unnormalized version, you should replace sinc(t - (1/2)) with sin(pi*(t - (1/2)))/(pi*(t - (1/2))) and similarly for the other terms.

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