Quadrature Amplitude Modulation
A method for digital modulation widely used in modern telecommunication.
Table of Contents
Quadrature Amplitude Modulation is widely used.
1. Complex numbers
Euler equation forming the basis of quadrature amplitude modulation.
\begin{equation} \label{org46065ff} e^{j\theta} = \cos(\theta) + j\sin(\theta) \end{equation}Equation \eqref{org46065ff} tells us that any complex phasor can be decomposed into the sum of a cosine and sine component.
1.1. Cosine—the in-phase component
\begin{equation}
\label{org999f886}
\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:
| \( -j2\pi f_0 t \) | Negative frequency |
| \( j2\pi f_0 t \) | Positive frequency |
| \( \frac{1}{2} \) | Component magnitude |
1.2. Sine—the quadrature component
\begin{equation}
\label{orgbb2ed96}
\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}
2. The constellation
QAM is based on imaginary numbers.
| Real | Imaginary |
|---|---|
| 1 | i |
| 2 | -i |