Quadrature Amplitude Modulation

Author

Marius Peter

Date

<2022-03-31 Thu>

# -*- mode: org; -*-

#+SUBTITLE: A method for digital modulation widely used in modern telecommunication.

#+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 | | | |