\documentclass[11pt,twoside,a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%
\author{Thomas~Sailer,~HB9JNX and Alexander~Kurpiers,~DL8AAU}
\title{Channel Simulator Gaussian Noise Generator Formulas}
\date{\today}

\begin{document}
\maketitle

\newcommand{\fourierpair}{\circ\cdots\bullet}

\begin{equation}
  \text{Fourier:}\quad
  e^{-a t^2}\quad\fourierpair\quad
  \sqrt{\frac{\pi}{a}}e^{-\frac{\omega^2}{4a}}
\end{equation}

\begin{equation}
  S_{xx}(f) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{f}{\sigma}\right)^2}
\end{equation}

\begin{equation}
  f_{3dB} = 2\sigma\sqrt{\ln 2}
  \quad\longrightarrow\quad
  \sigma = \frac{f_{3dB}}{2\sqrt{\ln 2}}
\end{equation}

\begin{equation}
  H(\omega) = \frac{1}{\sqrt{\sigma}\sqrt{\sqrt{2\pi}}}e^{-\frac{1}{4}\left(\frac{\omega}{2\pi\sigma}\right)^2}
\end{equation}

\begin{equation}
  h(t) = \frac{1}{\sqrt{2\sigma}\sqrt{\sqrt{2\pi}}}e^{-\left(2\pi\sigma{}t\right)^2}
\end{equation}

\end{document}
