0Modulation helps send information over long distances by changing a carrier signal. It makes communication clearer and stronger.
Modulated Signal = Carrier Signal modified by Message Signal Types: AM (Amplitude Modulation): Change carrier wave height FM (Frequency Modulation): Change carrier wave frequency PM (Phase Modulation): Change carrier wave phase
The carrier signal is a steady wave that carries the message.
The message signal is the information you want to send, like sound or data.
AM: s(t) = [1 + m(t)] * cos(2πf_ct)
FM: s(t) = cos(2πf_ct + 2πk_f ∫m(τ)dτ)
PM: s(t) = cos(2πf_ct + k_p m(t))This code creates a simple message signal and shows how it changes a carrier wave using AM, FM, and PM. The plots help you see the differences.
import numpy as np import matplotlib.pyplot as plt # Time array fs = 1000 # samples per second T = 1 # seconds t = np.linspace(0, T, fs, endpoint=False) # Message signal (simple sine wave) fm = 5 # message frequency in Hz m_t = 0.5 * np.sin(2 * np.pi * fm * t) # Carrier signal frequency fc = 50 # carrier frequency in Hz # AM modulation am = (1 + m_t) * np.cos(2 * np.pi * fc * t) # FM modulation kf = 50 # frequency sensitivity integral_m = np.cumsum(m_t) / fs fm_signal = np.cos(2 * np.pi * fc * t + 2 * np.pi * kf * integral_m) # PM modulation kp = np.pi / 2 # phase sensitivity pm_signal = np.cos(2 * np.pi * fc * t + kp * m_t) # Plotting plt.figure(figsize=(10, 8)) plt.subplot(4,1,1) plt.title('Message Signal (m(t))') plt.plot(t, m_t) plt.ylabel('Amplitude') plt.subplot(4,1,2) plt.title('AM Signal') plt.plot(t, am) plt.ylabel('Amplitude') plt.subplot(4,1,3) plt.title('FM Signal') plt.plot(t, fm_signal) plt.ylabel('Amplitude') plt.subplot(4,1,4) plt.title('PM Signal') plt.plot(t, pm_signal) plt.ylabel('Amplitude') plt.xlabel('Time [s]') plt.tight_layout() plt.show()
AM changes the wave height, so the signal looks like a wave with varying peaks.
FM changes how fast the wave oscillates, making the wave squished or stretched.
PM shifts the wave left or right, changing where the wave starts.
Modulation changes a carrier wave to carry information.
AM changes amplitude, FM changes frequency, PM changes phase.
These methods help send signals clearly over distances.
c(t) = A_c cos(2πf_c t) with message signal m(t)?f_c + k_f m(t).s(t) = A_c cos(2π(f_c + k_f m(t)) t), which correctly represents FM.import numpy as np A_c = 1 k_a = 0.5 f_c = 5 fs = 50 t = np.arange(0, 1, 1/fs) m_t = np.sin(2 * np.pi * 1 * t) s_t = A_c * (1 + k_a * m_t) * np.cos(2 * np.pi * f_c * t) print(np.round(s_t[:5], 3))
m_t is a 1 Hz sine wave sampled at 50 Hz. Carrier frequency is 5 Hz. Calculate first 5 samples of s_t using AM formula.s_t = 1 * (1 + 0.5 * m_t) * cos(2π * 5 * t), the first 5 rounded values are approximately [1.0, 0.309, -0.809, -1.191, -0.691].import numpy as np A_c = 1 k_p = 0.5 f_c = 10 fs = 100 t = np.arange(0, 1, 1/fs) m_t = np.sin(2 * np.pi * 2 * t) s_t = A_c * np.cos(2 * np.pi * f_c * t + k_p * np.sin(m_t)) print(np.round(s_t[:5], 3))
k_p * m_t inside the cosine argument, not k_p * np.sin(m_t).np.sin(m_t) instead of directly using m_t as the phase modulation signal.m(t) that has sudden jumps (discontinuities). Which modulation technique among AM, FM, and PM is best to minimize distortion caused by these jumps, and why?Pass the test and this pattern moves unknown strong while earning +1 Grip toward your certificate.