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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        |
+|      |           |