232 lines
9.7 KiB
TeX
232 lines
9.7 KiB
TeX
\documentclass[12pt,numbers,sort&compress]{article}
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%% Use the option review to obtain double line spacing
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%% \documentclass[authoryear,preprint,review,12pt]{elsarticle}
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%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
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%% for a journal layout:
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%% \documentclass[final,1p,times]{elsarticle}
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%% \documentclass[final,1p,times,twocolumn]{elsarticle}
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%% \documentclass[final,3p,times]{elsarticle}
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%% \documentclass[final,3p,times,twocolumn]{elsarticle}
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%% \documentclass[final,5p,times]{elsarticle}
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%% \documentclass[final,5p,times,twocolumn]{elsarticle}
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%% if you use PostScript figures in your article
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%% use the graphics package for simple commands
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%% \usepackage{graphics}
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%% or use the graphicx package for more complicated commands
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%% \usepackage{graphicx}
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%% or use the epsfig package if you prefer to use the old commands
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%% \usepackage{epsfig}
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%% The amssymb package provides various useful mathematical symbols
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\usepackage[usenames,dvipsnames,pdftex]{color}
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\usepackage{amsmath,amssymb,amsfonts}
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\usepackage{siunitx}
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%\usepackage{subeqnarray}
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\usepackage[hang]{subfigure}
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\usepackage{verbatim}
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\usepackage{bm}
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\usepackage{tikz}
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\usetikzlibrary{arrows}
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\usepackage{booktabs}
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\usepackage{graphicx}
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\newcommand{\pathToFigures}{./figures}
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\graphicspath{{\pathToFigures/}}
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\DeclareGraphicsExtensions{.pdf,.png}
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\usepackage[pdftex, % hyper-references for pdftex
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bookmarksnumbered=true,% % generate bookmarks with numbers
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pagebackref=true,% % generate backref in biblio
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colorlinks=true,%
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]{hyperref}%
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%% The amsthm package provides extended theorem environments
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%% \usepackage{amsthm}
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%% The lineno packages adds line numbers. Start line numbering with
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%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
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%% for the whole article with \linenumbers.
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%% \usepackage{lineno}
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\newlength{\diagramsize}
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\setlength{\diagramsize}{0.4\textwidth}
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\newcommand{\question}[1]{\textcolor{Red}{#1}}
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\newcommand{\note}[1]{\textcolor{CornflowerBlue}{#1}}
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\newcommand{\term}[1]{\textsc{#1}}
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\newcommand{\eref}[1]{Eq.~\eqref{#1}}
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\newcommand{\Eref}[1]{Eq.~\eqref{#1}}
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\newcommand{\erefs}[1]{Eqs.~\eqref{#1}}
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\newcommand{\Erefs}[1]{Eqs.~\eqref{#1}}
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\newcommand{\fref}[1]{Fig.~\ref{#1}}
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\newcommand{\Fref}[1]{Fig.~\ref{#1}}
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\newcommand{\frefs}[1]{Figs.~\ref{#1}}
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\newcommand{\Frefs}[1]{Figs.~\ref{#1}}
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\newcommand{\tref}[1]{Tab.~\ref{#1}}
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\newcommand{\Tref}[1]{Tab.~\ref{#1}}
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\newcommand{\trefs}[1]{Tabs.~\ref{#1}}
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\newcommand{\Trefs}[1]{Tabs.~\ref{#1}}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\cf}{\textit{cf.}}
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\newcommand{\Euler}{\textsc{Euler}}
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\newcommand{\Gauss}{\textsc{Gauss}}
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\newcommand{\kB}{\ensuremath{k_\text{B}}}
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\newcommand{\transpose}[1]{\ensuremath{{#1}^{\mathrm T}}}
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\newcommand{\inverse}[1]{\ensuremath{{#1}^{-1}}}
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\newcommand{\invtranspose}[1]{\ensuremath{{#1}^{\mathrm{-T}}}}
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\newcommand{\sign}[1]{\ensuremath{\operatorname{sgn}\left({#1}\right)}}
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\newcommand{\grad}[1][]{\ensuremath{\operatorname{grad}{#1}}}
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\newcommand{\Grad}[1][]{\ensuremath{\operatorname{Grad}{#1}}}
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\newcommand{\divergence}[1][]{\ensuremath{\operatorname{div}{#1}}}
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\newcommand{\Divergence}[1][]{\ensuremath{\operatorname{Div}{#1}}}
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\newcommand{\totalder}[2]{\ensuremath{\frac{\inc{#1}}{\inc{#2}}}}
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\newcommand{\partialder}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}}
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\newcommand{\inc}[1]{\ensuremath{\text d{#1}}}
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\newcommand{\abs}[1]{\ensuremath{\left|{#1}\right|}}
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\newcommand{\norm}[1]{\ensuremath{\left|\left|{#1}\right|\right|}}
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\newcommand{\avg}[1]{\ensuremath{\bar{#1}}}
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\newcommand{\fluct}[1]{\ensuremath{\tilde{#1}}}
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\newcommand{\FT}[1]{\ensuremath{\hat{#1}}}
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\newcommand{\domain}[1]{\ensuremath{\mathcal{#1}}}
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\newcommand{\tnsrfour}[1]{\ensuremath{\mathbb{#1}}}
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\newcommand{\tnsr}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\vctr}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\eyetwo}{\ensuremath{\tnsr I}}
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\newcommand{\eyefour}{\ensuremath{\tnsrfour I}}
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\newcommand{\stiffness}{\ensuremath{\tnsrfour D}}
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\newcommand{\refStiffness}{\ensuremath{\avg{\tnsrfour D}}}
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\newcommand{\fPK}{\ensuremath{\tnsr P}}
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\newcommand{\sPK}{\ensuremath{\tnsr S}}
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\newcommand{\F}[1][]{\ensuremath{\tnsr F^{#1}}}
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\newcommand{\Favg}{\ensuremath{\avg{\F}}}
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\newcommand{\Ffluct}{\ensuremath{\fluct{\F}}}
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\newcommand{\Fp}[1][]{\ensuremath{\tnsr F_\text{p}^{#1}}}
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\newcommand{\Fe}[1][]{\ensuremath{\tnsr F_\text{e}^{#1}}}
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\newcommand{\Lp}{\ensuremath{\tnsr L_\text{p}}}
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\newcommand{\Q}[1]{\ensuremath{\tnsr Q^{(#1)}}}
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\newcommand{\x}[2][]{\ensuremath{\vctr x^{(#2)}_\text{#1}}}
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\newcommand{\dg}[2][]{\ensuremath{\Delta\vctr g^{(#2)}_\text{#1}}}
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\newcommand{\g}[1][]{\ensuremath{\vctr g_\text{#1}}}
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\newcommand{\A}[2][]{\ensuremath{A^{(#2)}_\text{#1}}}
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\newcommand{\N}[2]{\ensuremath{\varrho^{(#1)}_\text{#2}}}
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\newcommand{\Burgers}[1]{\ensuremath{\vctr s^{(#1)}}}
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\newcommand{\n}[1]{\ensuremath{\vctr n^{(#1)}}}
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\newcommand{\m}[2]{\ensuremath{\vctr m^{(#1)}_{#2}}}
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\newcommand{\ld}[1]{\ensuremath{\vctr p^{(#1)}}}
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\newcommand{\velocity}[2]{\ensuremath{v^{(#1)}_\text{#2}}}
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\newcommand{\avgvelocity}[2]{\ensuremath{{\bar v}^{(#1)}_ \text{#2}}}
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\newcommand{\flux}[2]{\ensuremath{\vctr f^{(#1)}_ \text{#2}}}
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\newcommand{\averageflux}[2]{\ensuremath{\bar{\vctr f}^{(#1)}_ \text{#2}}}
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\newcommand{\interfaceflux}[2]{\ensuremath{\tilde{\vctr f}^{(#1)}_ \text{#2}}}
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\newcommand{\transmissivity}[1]{\ensuremath{\chi^{(#1)}}}
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\newcommand{\galpha}{\ensuremath{\gamma^{(\alpha)}}}
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\newcommand{\dotgalpha}{\ensuremath{\dot{\gamma}^{(\alpha)}}}
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\newcommand{\taualpha}{\ensuremath{\tau^{(\alpha)}}}
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\newcommand{\taualphamax}{\ensuremath{\hat\tau^{(\alpha)}}}
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\newcommand{\density}[2]{\ensuremath{\varrho^{(#1)}_ \text{#2}}}
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\newcommand{\densityfunc}[2]{\ensuremath{{\tilde\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\avgdensity}[2]{\ensuremath{{\bar\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\dotdensity}[2]{\ensuremath{\dot{\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\densityexcess}[2]{\ensuremath{\Delta\varrho^{(#1)}_ \text{#2}}}
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\newcommand{\cs}[2][]{\ensuremath{\sigma^{(#1)}_ \text{#2}}}
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%% Title, authors and addresses
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%% use the tnoteref command within \title for footnotes;
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%% use the tnotetext command for theassociated footnote;
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%% use the fnref command within \author or \address for footnotes;
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%% use the fntext command for theassociated footnote;
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%% use the corref command within \author for corresponding author footnotes;
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%% use the cortext command for theassociated footnote;
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%% use the ead command for the email address,
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%% and the form \ead[url] for the home page:
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%% \title{Title\tnoteref{label1}}
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%% \tnotetext[label1]{}
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%% \author{Name\corref{cor1}\fnref{label2}}
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%% \ead{email address}
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%% \ead[url]{home page}
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%% \fntext[label2]{}
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%% \cortext[cor1]{}
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%% \address{Address\fnref{label3}}
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%% \fntext[label3]{}
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\title{Fourier Transforms}
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%% use optional labels to link authors explicitly to addresses:
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%% \author[label1,label2]{}
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%% \address[label1]{}
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%% \address[label2]{}
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\author{M.~Diehl}
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%% \linenumbers
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% main text
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\begin{document}
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\maketitle
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% ----------------------------------------------------------------------------------------------------------------------------
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\section{Discrete vs. continuous FT}
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% ----------------------------------------------------------------------------------------------------------------------------
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continuous Fourier transform
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\begin{equation}
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\hat{f}(k) = \int \limits_{-\pi}^{\pi} f(x) \cdot e^{-2\pi i k x} \inc x
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\end{equation}
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discrete Fourier transform
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\begin{align}
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\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}\\
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&= \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}
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\end{align}
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% ----------------------------------------------------------------------------------------------------------------------------
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\section{Differentation}
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% ----------------------------------------------------------------------------------------------------------------------------
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Expression in frequency and angular frequency
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\begin{align}
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\hat{f}(k) &= \frac{1}{d} \int \limits_0^d f(x) e^{\frac{-2 \pi i}{d} k x}\\
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&= \frac{1}{d} \int \limits_0^d f(x) e^{-2 \pi i \xi x} \inc x
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\end{align}
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\begin{align}
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\hat{f}'&= \frac{\partial}{\partial x} \left( \int \limits_{-\infty }^{\infty} \hat{f}(x) \cdot e^{i \xi x} \inc k \right)\\
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&= \int \limits_{-\infty}^{\infty} \frac{\partial}{\partial x} \left( \hat{f}(x) \cdot e^{i \xi x} \right) \inc k\\
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&= \int \limits_{-\infty}^{\infty} i \xi \cdot \hat{f}(x) \cdot e^{i \xi x} \inc k
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\end{align}
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\section{Transform}
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example with $N=4$ and $x = \mathrm{sin}\left( \frac{n}{N} \cdot 2 \pi \right)$
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\begin{align}
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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\\
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&= \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)\\
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X_0 &= \sum \limits_{n=0}^{N-1} x_n e ^\\
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&= 0 + 1 + 0 +(-1) = 0 \\
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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}\\
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&= 0 + (- 2i)\\
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X_2 &= 0 + 0i\\
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X_3 &= 0 + 2i
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\end{align}
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$X_2$ is Nyquist frequency and has only a real part, $X_3$ is conjugate complex of $X_1$ for real only input.
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\section{Inverse Transform}
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\begin{align}
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x_n &= \frac{1}{N} \sum \limits_{k=0}^{N-1} X_k e^{\frac{2\pi i}{N} \cdot k \cdot n}\\
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x_0 &= \frac{1}{4}\left(0 - 2ie^0 + 0 + 2ie^0 \right) = 0\\
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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\\
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x_2 &= 0\\
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x_1 &= -1
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\end{align}
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\end{document}
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\endinput
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