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.

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.

Proof:

Will be added later.