# -*- mode: org; -*-

#+TITLE: Quadrature Amplitude Modulation
#+SUBTITLE: A method for digital modulation widely used in modern telecommunication.
#+AUTHOR: Marius Peter
#+DATE: <2022-03-31 Thu>

#+DESCRIPTION: Quadrature Ampliture Modulation: a method for digital modulation widely used in modern telecommunication.
#+MACRO: QAM quadrature amplitude modulation


#+begin_abstract
Quadrature Amplitude Modulation is widely used.
#+end_abstract


* Complex numbers

Euler equation forming the basis of {{{QAM}}}.

#+NAME: euler
#+CAPTION: This is a caption.
\begin{equation}
e^{j\theta} = \cos(\theta) + j\sin(\theta)
\end{equation}

Equation [[euler]] tells us that any complex phasor can be decomposed into
the sum of a cosine and sine component.


** Cosine---the in-phase component

#+NAME: cosine
\begin{equation}
\cos(2\pi f_0 t) = \frac{e^{j2\pi f_0 t} + e^{-j2\pi f_0 t}}{2}
= \frac{e^{j2\pi f_0 t}}{2}
+ \frac{e^{-j2\pi f_0 t}}{2}
\end{equation}

We see that this equation features the following elements:

|               <r> |                     |
|   \( -j2\pi f_0 t \) | Negative frequency  |
|    \( j2\pi f_0 t \) | Positive frequency  |
| \( \frac{1}{2} \) | Component magnitude |


** Sine---the quadrature component

#+NAME: sine
\begin{equation}
\sin(2\pi f_0 t) = \frac{e^{j2\pi f_0 t} - e^{-j2\pi f_0 t}}{2}
= \frac{e^{j2\pi f_0 t}}{2}
- \frac{e^{-j2\pi f_0 t}}{2}
\end{equation}


* The constellation


#+begin_insight
QAM is based on imaginary numbers.
#+end_insight


| Real | Imaginary |
|------+-----------|
|    1 | i         |
|    2 | -i        |
|      |           |