port from mathematics-physics notes

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

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+# 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 $m: \mathbb{R} \to \mathbb{R}$. 
+
+<br>
+
+> *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.