Modulation Techniques

Interactive Modulation Tutorial

1.3.1 Amplitude Modulation (AM)

In Amplitude Modulation, the carrier's amplitude changes with the message signal.

Formula:
$$ s(t) = [1 + m(t)] \cos(2\pi f_c t) $$

Key Concept: Modulation Index

The modulation index (m) determines how much the carrier amplitude varies: $$ m = \frac{A_{max} - A_{min}}{A_{max} + A_{min}} $$ Must be ≤ 1 to avoid distortion (overmodulation).

Carrier Frequency (Hz) 10 Hz

Message Frequency (Hz) 1 Hz

Modulation Index 0.5

Example Calculation:
Suppose \( m(t) = 0.5 \sin(2\pi \times 1 t) \) (1 Hz message), and the carrier is 10 Hz:
$$ s(t) = [1 + 0.5 \sin(2\pi \times 1 t)] \cos(2\pi \times 10 t) $$
For \( t = 0.1 \) s:
$$ m(0.1) = 0.5 \sin(0.2\pi) = 0.5 \times 0.5878 = 0.2939 $$
So:
$$ s(0.1) = [1 + 0.2939] \cos(2\pi) = 1.2939 \times 1 = 1.2939 $$

1.3.2 Frequency Modulation (FM)

In Frequency Modulation, the carrier frequency changes with the message signal.

Formula:
$$ s(t) = \cos\Big( 2\pi f_c t + 2\pi k_f \int_0^t m(\tau) d\tau \Big) $$

Carrier Frequency (Hz) 10 Hz

Message Frequency (Hz) 1 Hz

Frequency Sensitivity (k_f) 5

Example Calculation:
Let \( m(t) = \sin(2\pi \times 1 t) \), \( f_c = 10 \) Hz, \( k_f = 5 \):
$$ \int_0^t \sin(2\pi t) dt = -\frac{1}{2\pi} \cos(2\pi t) + \frac{1}{2\pi} $$
So:
$$ s(t) = \cos\Big( 2\pi \times 10 t - 5 \cos(2\pi t) + 5 \Big) $$ For \( t = 0.1 \):
$$ s(0.1) = \cos( 2\pi \times 1 - 5 \times \cos(0.2\pi) + 5 ) $$

1.3.3 Phase Modulation (PM)

In Phase Modulation, the phase changes directly with the message signal.

Formula:
$$ s(t) = \cos\Big( 2\pi f_c t + k_p m(t) \Big) $$

Carrier Frequency (Hz) 10 Hz

Message Frequency (Hz) 1 Hz

Phase Sensitivity (k_p) 0.5

Example Calculation:
Let \( m(t) = \sin(2\pi \times 1 t) \), \( f_c = 10 \) Hz, \( k_p = \pi/2 \):
$$ s(t) = \cos\Big( 2\pi \times 10 t + \frac{\pi}{2} \sin(2\pi t) \Big) $$ For \( t = 0.1 \):
$$ s(0.1) = \cos( 2\pi + \frac{\pi}{2} \times 0.5878 ) = \cos(0.923) = 0.6037 $$

1.3.4 Quadrature Amplitude Modulation (QAM)

QAM combines both amplitude and phase variation to transmit information.

Formula:
$$ s(t) = I(t) \cos(2\pi f_c t) - Q(t) \sin(2\pi f_c t) $$

Key Concept: Constellation Diagram

The constellation diagram shows the possible symbol states in the complex plane, with I (in-phase) on the x-axis and Q (quadrature) on the y-axis.

Example Calculation:
Suppose we transmit symbol (I,Q) = (1, 1) at \( f_c = 10 \) Hz:
$$ s(t) = \cos(2\pi \times 10 t) - \sin(2\pi \times 10 t) $$ For \( t = 0.1 \):
$$ s(0.1) = \cos(2\pi) - \sin(2\pi) = 1 - 0 = 1 $$