59 lines
1.8 KiB
Markdown
59 lines
1.8 KiB
Markdown
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# Amplitude modulation
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> *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*.
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??? note "*Proof*:"
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Will be added later.
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For example if we have a harmonic signal $m: \mathbb{R} \to \mathbb{R}$ with $\omega, A \in \mathbb{R}$ given by
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$$
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m(t) = A \cos \omega t,
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$$
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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
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$$
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c(t) = \cos \omega_c t.
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$$
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for all $t \in \mathbb{R}$. Then the multiplication of both is given by
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$$
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m(t)c(t) = A \cos (\omega t) \cos (\omega_c t) = \frac{A}{2} \bigg(\cos t(\omega + \omega)c + \cos t(\omega - \omega_c) \bigg),
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$$
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obtaining heterodyne.
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> *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
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>
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> $$
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> c(t) = \cos \omega_c t
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> $$
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>
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> for all $t \in \mathbb{R}$ to modulate a signal $m: \mathbb{R} \to \mathbb{R}$.
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<br>
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> *Theorem*: For the case that the carrier signal is not additionaly transmitted we obtain
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>
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> $$
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> m(t) c(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \big(M(\omega + \omega_c) + M(\omega - \omega_c) \big),
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> $$
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>
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> for all $t, \omega \in \mathbb{R}$.
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>
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> For the case that the carrier signal is additionaly transmitted we obtain
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>
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> $$
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> m(t) (1 + c(t)) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \Big(M(\omega + \omega_c) + M(\omega - \omega_c) + \pi \big(\delta(\omega + \omega_c) + \delta(\omega - \omega_c) \big) \Big)
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> $$
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>
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> for all $t, \omega \in \mathbb{R}$.
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>
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> Therefore multiple bandlimited signals can be transmitted simultaneously in frequency bands.
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??? note "*Proof*:"
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Will be added later.
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