\documentclass[12pt,numbers,sort&compress]{article}

%% Use the option review to obtain double line spacing
%% \documentclass[authoryear,preprint,review,12pt]{elsarticle}

%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
%% for a journal layout:
%% \documentclass[final,1p,times]{elsarticle}
%% \documentclass[final,1p,times,twocolumn]{elsarticle}
%% \documentclass[final,3p,times]{elsarticle}
%% \documentclass[final,3p,times,twocolumn]{elsarticle}
%% \documentclass[final,5p,times]{elsarticle}
%% \documentclass[final,5p,times,twocolumn]{elsarticle}

%% if you use PostScript figures in your article
%% use the graphics package for simple commands
%% \usepackage{graphics}
%% or use the graphicx package for more complicated commands
%% \usepackage{graphicx}
%% or use the epsfig package if you prefer to use the old commands
%% \usepackage{epsfig}

%% The amssymb package provides various useful mathematical symbols
\usepackage[usenames,dvipsnames,pdftex]{color}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{siunitx}
%\usepackage{subeqnarray}
\usepackage[hang]{subfigure}
\usepackage{verbatim}
\usepackage{bm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{booktabs}

\usepackage{graphicx}
\newcommand{\pathToFigures}{./figures}
\graphicspath{{\pathToFigures/}}
\DeclareGraphicsExtensions{.pdf,.png}

\usepackage[pdftex,                                     % hyper-references for pdftex
bookmarksnumbered=true,%                       % generate bookmarks with numbers
pagebackref=true,%                                      % generate backref in biblio
colorlinks=true,%
]{hyperref}%

%% The amsthm package provides extended theorem environments
%% \usepackage{amsthm}

%% The lineno packages adds line numbers. Start line numbering with
%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
%% for the whole article with \linenumbers.
%% \usepackage{lineno}

\newlength{\diagramsize}
\setlength{\diagramsize}{0.4\textwidth}

\newcommand{\question}[1]{\textcolor{Red}{#1}}
\newcommand{\note}[1]{\textcolor{CornflowerBlue}{#1}}
\newcommand{\term}[1]{\textsc{#1}}

\newcommand{\eref}[1]{Eq.~\eqref{#1}}
\newcommand{\Eref}[1]{Eq.~\eqref{#1}}
\newcommand{\erefs}[1]{Eqs.~\eqref{#1}}
\newcommand{\Erefs}[1]{Eqs.~\eqref{#1}}
\newcommand{\fref}[1]{Fig.~\ref{#1}}
\newcommand{\Fref}[1]{Fig.~\ref{#1}}
\newcommand{\frefs}[1]{Figs.~\ref{#1}}
\newcommand{\Frefs}[1]{Figs.~\ref{#1}}
\newcommand{\tref}[1]{Tab.~\ref{#1}}
\newcommand{\Tref}[1]{Tab.~\ref{#1}}
\newcommand{\trefs}[1]{Tabs.~\ref{#1}}
\newcommand{\Trefs}[1]{Tabs.~\ref{#1}}

\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\cf}{\textit{cf.}}
\newcommand{\Euler}{\textsc{Euler}}
\newcommand{\Gauss}{\textsc{Gauss}}

\newcommand{\kB}{\ensuremath{k_\text{B}}}

\newcommand{\transpose}[1]{\ensuremath{{#1}^{\mathrm T}}}
\newcommand{\inverse}[1]{\ensuremath{{#1}^{-1}}}
\newcommand{\invtranspose}[1]{\ensuremath{{#1}^{\mathrm{-T}}}}
\newcommand{\sign}[1]{\ensuremath{\operatorname{sgn}\left({#1}\right)}}
\newcommand{\grad}[1][]{\ensuremath{\operatorname{grad}{#1}}}
\newcommand{\Grad}[1][]{\ensuremath{\operatorname{Grad}{#1}}}
\newcommand{\divergence}[1][]{\ensuremath{\operatorname{div}{#1}}}
\newcommand{\Divergence}[1][]{\ensuremath{\operatorname{Div}{#1}}}
\newcommand{\totalder}[2]{\ensuremath{\frac{\inc{#1}}{\inc{#2}}}}
\newcommand{\partialder}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}}
\newcommand{\inc}[1]{\ensuremath{\text d{#1}}}
\newcommand{\abs}[1]{\ensuremath{\left|{#1}\right|}}
\newcommand{\norm}[1]{\ensuremath{\left|\left|{#1}\right|\right|}}

\newcommand{\avg}[1]{\ensuremath{\bar{#1}}}
\newcommand{\fluct}[1]{\ensuremath{\tilde{#1}}}
\newcommand{\FT}[1]{\ensuremath{\hat{#1}}}

\newcommand{\domain}[1]{\ensuremath{\mathcal{#1}}}
\newcommand{\tnsrfour}[1]{\ensuremath{\mathbb{#1}}}
\newcommand{\tnsr}[1]{\ensuremath{\bm{#1}}}
\newcommand{\vctr}[1]{\ensuremath{\bm{#1}}}

\newcommand{\eyetwo}{\ensuremath{\tnsr I}}
\newcommand{\eyefour}{\ensuremath{\tnsrfour I}}

\newcommand{\stiffness}{\ensuremath{\tnsrfour D}}
\newcommand{\refStiffness}{\ensuremath{\avg{\tnsrfour D}}}
\newcommand{\fPK}{\ensuremath{\tnsr P}}
\newcommand{\sPK}{\ensuremath{\tnsr S}}
\newcommand{\F}[1][]{\ensuremath{\tnsr F^{#1}}}
\newcommand{\Favg}{\ensuremath{\avg{\F}}}
\newcommand{\Ffluct}{\ensuremath{\fluct{\F}}}
\newcommand{\Fp}[1][]{\ensuremath{\tnsr F_\text{p}^{#1}}}
\newcommand{\Fe}[1][]{\ensuremath{\tnsr F_\text{e}^{#1}}}
\newcommand{\Lp}{\ensuremath{\tnsr L_\text{p}}}
\newcommand{\Q}[1]{\ensuremath{\tnsr Q^{(#1)}}}
\newcommand{\x}[2][]{\ensuremath{\vctr x^{(#2)}_\text{#1}}}
\newcommand{\dg}[2][]{\ensuremath{\Delta\vctr g^{(#2)}_\text{#1}}}
\newcommand{\g}[1][]{\ensuremath{\vctr g_\text{#1}}}
\newcommand{\A}[2][]{\ensuremath{A^{(#2)}_\text{#1}}}
\newcommand{\N}[2]{\ensuremath{\varrho^{(#1)}_\text{#2}}}
\newcommand{\Burgers}[1]{\ensuremath{\vctr s^{(#1)}}}
\newcommand{\n}[1]{\ensuremath{\vctr n^{(#1)}}}
\newcommand{\m}[2]{\ensuremath{\vctr m^{(#1)}_{#2}}}
\newcommand{\ld}[1]{\ensuremath{\vctr p^{(#1)}}}
\newcommand{\velocity}[2]{\ensuremath{v^{(#1)}_\text{#2}}}
\newcommand{\avgvelocity}[2]{\ensuremath{{\bar v}^{(#1)}_ \text{#2}}}
\newcommand{\flux}[2]{\ensuremath{\vctr f^{(#1)}_ \text{#2}}}
\newcommand{\averageflux}[2]{\ensuremath{\bar{\vctr f}^{(#1)}_ \text{#2}}}
\newcommand{\interfaceflux}[2]{\ensuremath{\tilde{\vctr f}^{(#1)}_ \text{#2}}}
\newcommand{\transmissivity}[1]{\ensuremath{\chi^{(#1)}}}
\newcommand{\galpha}{\ensuremath{\gamma^{(\alpha)}}}
\newcommand{\dotgalpha}{\ensuremath{\dot{\gamma}^{(\alpha)}}}
\newcommand{\taualpha}{\ensuremath{\tau^{(\alpha)}}}
\newcommand{\taualphamax}{\ensuremath{\hat\tau^{(\alpha)}}}
\newcommand{\density}[2]{\ensuremath{\varrho^{(#1)}_ \text{#2}}}
\newcommand{\densityfunc}[2]{\ensuremath{{\tilde\varrho}^{(#1)}_ \text{#2}}}
\newcommand{\avgdensity}[2]{\ensuremath{{\bar\varrho}^{(#1)}_ \text{#2}}}
\newcommand{\dotdensity}[2]{\ensuremath{\dot{\varrho}^{(#1)}_ \text{#2}}}
\newcommand{\densityexcess}[2]{\ensuremath{\Delta\varrho^{(#1)}_ \text{#2}}}
\newcommand{\cs}[2][]{\ensuremath{\sigma^{(#1)}_ \text{#2}}}




%% Title, authors and addresses

%% use the tnoteref command within \title for footnotes;
%% use the tnotetext command for theassociated footnote;
%% use the fnref command within \author or \address for footnotes;
%% use the fntext command for theassociated footnote;
%% use the corref command within \author for corresponding author footnotes;
%% use the cortext command for theassociated footnote;
%% use the ead command for the email address,
%% and the form \ead[url] for the home page:
%% \title{Title\tnoteref{label1}}
%% \tnotetext[label1]{}
%% \author{Name\corref{cor1}\fnref{label2}}
%% \ead{email address}
%% \ead[url]{home page}
%% \fntext[label2]{}
%% \cortext[cor1]{}
%% \address{Address\fnref{label3}}
%% \fntext[label3]{}

\title{Fourier Transforms}

%% use optional labels to link authors explicitly to addresses:
%% \author[label1,label2]{}
%% \address[label1]{}
%% \address[label2]{}

\author{M.~Diehl}

%% \linenumbers

% main text
\begin{document}
\maketitle
% ----------------------------------------------------------------------------------------------------------------------------
\section{Discrete vs. continuous FT}
% ----------------------------------------------------------------------------------------------------------------------------
continuous Fourier transform
\begin{equation}
\hat{f}(k) = \int \limits_{-\pi}^{\pi} f(x) \cdot e^{-2\pi i k x} \inc x
\end{equation}
discrete Fourier transform
\begin{align}
\hat{f}_k &=  \frac{1}{d} \sum\limits_{n=0}^{N-1} f \left( x = \frac{n}{N}d \right) \cdot e^{\left(-2 \pi i \cdot \frac{k}{d} \cdot \frac{n}{N} \cdot d \right)} \cdot \frac{d}{N}\\
&= \frac{1}{N} \sum \limits_{n=0}^{N-1} f \left( x = \frac{n}{N} d \right) \cdot e^{-\frac{2 \pi i}{N} \cdot k \cdot n} 
\end{align}

% ----------------------------------------------------------------------------------------------------------------------------
\section{Differentation}
% ----------------------------------------------------------------------------------------------------------------------------
Expression in frequency and angular frequency
\begin{align}
\hat{f}(k) &= \frac{1}{d} \int \limits_0^d f(x) e^{\frac{-2 \pi i}{d} k x}\\
&= \frac{1}{d} \int \limits_0^d f(x) e^{-2 \pi i \xi x} \inc x
\end{align}
\begin{align}
\hat{f}'&= \frac{\partial}{\partial x} \left( \int \limits_{-\infty }^{\infty} \hat{f}(x) \cdot e^{i \xi x} \inc k \right)\\
&=  \int \limits_{-\infty}^{\infty} \frac{\partial}{\partial x} \left( \hat{f}(x) \cdot e^{i \xi x} \right) \inc k\\
&= \int \limits_{-\infty}^{\infty} i \xi \cdot \hat{f}(x) \cdot e^{i \xi x} \inc k
\end{align}

\section{Transform}
example with  $N=4$ and $x = \mathrm{sin}\left( \frac{n}{N} \cdot 2 \pi \right)$
\begin{align}
X_k &= \sum \limits_{n=0}^{N-1} x_n \cdot e ^{- \frac{2 \pi i }{N} \cdot  k \cdot n};~~~k = 0;1;..;N-1\\
&= \sum \limits_{n=0}^{N-1} x_n \cdot \left(\mathrm{cos}\left(- \frac{2 \pi}{N}\cdot k \cdot n \right) + i\cdot \mathrm{sin}\left(- \frac{2 \pi}{N} \cdot k \cdot n\right) \right)\\
X_0 &= \sum \limits_{n=0}^{N-1} x_n e ^\\
&= 0 + 1 + 0 +(-1) = 0 \\
X_1 &= \sum \limits_{n=0}^{N-1} x_n e^{-i \frac{2\pi}{N} \cdot 1 \cdot n} = 0 + e ^ {-i \frac{2\pi}{N} \cdot 1 \cdot 1} + 0 +e ^ {-i \frac{2\pi}{N} \cdot 1 \cdot 3}\\
&= 0 + (- 2i)\\
X_2 &= 0 + 0i\\
X_3 &= 0 + 2i
\end{align}
$X_2$ is Nyquist frequency and has only a real part, $X_3$ is conjugate complex of $X_1$ for real only input.
\section{Inverse Transform}
\begin{align}
x_n &= \frac{1}{N} \sum \limits_{k=0}^{N-1} X_k e^{\frac{2\pi i}{N} \cdot k \cdot n}\\
x_0 &= \frac{1}{4}\left(0 - 2ie^0 + 0 + 2ie^0 \right) = 0\\
x_1 &= \frac{1}{4}\left(0 - 2ie^{\frac{2\pi i}{4}\cdot 1 \cdot 1} + 0 + 2ie^{\frac{2\pi i}{4}\cdot 3 \cdot 1} \right) = 1\\
x_2 &= 0\\
x_1 &= -1
\end{align}
\end{document}
\endinput
