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# -*- 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 |
| | |
|
