1.8 KiB
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 signalm: \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.