訊號調變

The general AM equation:
$$ y(t) = \big[A + m(t)\big] \cdot \cos(2\pi f_c t) $$

If the modulating signal is a sine wave:
$ m(t) = A_m \cos(2\pi f_m t) $
Then:
$$ y(t) = \big[A + A_m \cos(2\pi f_m t)\big] \cdot \cos(2\pi f_c t) $$
Where:

• $ A $ = carrier amplitude
• $ A_m $ = modulating amplitude
• $ f_c $ = carrier frequency (Hz)
• $ f_m $ = modulating frequency (Hz)

Modulation index $ m = A_m / A $ (must be ≤ 1 for no overmodulation).
Expanded form:
$$ y(t) = A\cos(2\pi f_c t) + \frac{A_m}{2} \cos\big(2\pi (f_c+f_m)t\big) + \frac{A_m}{2} \cos\big(2\pi (f_c-f_m)t\big) $$

For FM, the instantaneous frequency varies sinusoidally:
$$ y(t) = A \cos\left(2\pi f_c t + \frac{\Delta f}{f_m} \sin(2\pi f_m t)\right) $$ Where:

• $ \Delta f $ = peak frequency deviation
• Modulation index $ \beta = \Delta f / f_m $

So: $$ y(t) = A \cos\big(2\pi f_c t + \beta \sin(2\pi f_m t)\big) $$

For a sine wave carrier modulated by a sine wave message:
$$ y(t) = A_c \cos\big[2\pi f_c t + \phi(t)\big] $$ Where $\phi(t)$ is the phase deviation proportional to the message signal.

With sine wave message:
$$ m(t) = A_m \cos(2\pi f_m t) $$

The phase deviation is: $$ \phi(t) = k_p \cdot m(t) = k_p A_m \cos(2\pi f_m t) = \Delta \phi \cdot \cos(2\pi f_m t) $$ Where:

• $ k_p $ = phase sensitivity (rad/volt)
• $ \Delta \phi $ = peak phase deviation (radians) = $ k_p A_m $

Final PM signal: $$ y(t) = A_c \cos\big[2\pi f_c t + \Delta \phi \cos(2\pi f_m t)\big] $$