notes/docs/physics/mathematical-physics/signal-analysis/amplitude-modulation.md

Amplitude modulation

Theorem: a multiplication of two harmonic functions results in a sum of harmonics withh the sum and difference of the original frequencies. This is called heterodyne.

??? note "Proof:"

Will be added later.

For example if we have a harmonic signal m: \mathbb{R} \to \mathbb{R} with \omega, A \in \mathbb{R} given by

m(t) = A \cos \omega t,

for all t \in \mathbb{R} and a harmonic carrier signal c: \mathbb{R} \to \mathbb{R} with \omega_c \in \mathbb{R} given by

c(t) = \cos \omega_c t.

for all t \in \mathbb{R}. Then the multiplication of both is given by

m(t)c(t) = A \cos (\omega t) \cos (\omega_c t) = \frac{A}{2} \bigg(\cos t(\omega + \omega)c + \cos t(\omega - \omega_c) \bigg),

obtaining heterodyne.

Definition: amplitude modulation makes use of a harmonic carrier signal c: \mathbb{R} \to \mathbb{R} with a reasonable angular frequency \omega_c \in \mathbb{R} given by

c(t) = \cos \omega_c t

for all t \in \mathbb{R} to modulate a signal m: \mathbb{R} \to \mathbb{R}.

Theorem: For the case that the carrier signal is not additionaly transmitted we obtain

m(t) c(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \big(M(\omega + \omega_c) + M(\omega - \omega_c) \big),

for all t, \omega \in \mathbb{R}.

For the case that the carrier signal is additionaly transmitted we obtain

m(t) (1 + c(t)) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \Big(M(\omega + \omega_c) + M(\omega - \omega_c) + \pi \big(\delta(\omega + \omega_c) + \delta(\omega - \omega_c) \big) \Big)

for all t, \omega \in \mathbb{R}.

Therefore multiple bandlimited signals can be transmitted simultaneously in frequency bands.

??? note "Proof:"

Will be added later.