# 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.