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%test
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\documentclass{beamer}
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%\usepackage{setspace} %for "\begin{onehalfspace}"
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\usepackage[english]{babel}
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% or whatever
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\usepackage[utf8]{inputenc}
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% or whatever
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\usepackage{times}
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\usepackage[T1]{fontenc}
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% Or whatever. Note that the encoding and the font should match. If T1
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\def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
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\def\sisac{{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}}
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\title[TODO] % (optional, use only with long paper titles)
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{TODO}
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\subtitle{TODO}
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\author[Rocnik] % (optional, use only with lots of authors)
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{Jan~Rocnik}
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% - Give the names in the same order as the appear in the paper.
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% - Use the \inst{?} command only if the authors have different
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% affiliation.
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\institute % (optional, but mostly needed)
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{
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Technische Universit\"at Graz\\
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Institut f\"ur TODO
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}
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% - Use the \inst command only if there are several affiliations.
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% - Keep it simple, no one is interested in your street address.
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% \date[CFP 2003] % (optional, should be abbreviation of conference name)
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% {Conference on Fabulous Presentations, 2003}
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% - Either use conference name or its abbreviation.
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% - Not really informative to the audience, more for people (including
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% yourself) who are reading the slides online
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% \subject{Theoretical Computer Science}
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% This is only inserted into the PDF information catalog. Can be left
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% out.
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% If you have a file called "university-logo-filename.xxx", where xxx
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% resp., then you can add a logo as follows:
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% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
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% \logo{\pgfuseimage{university-logo}}
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% Delete this, if you do not want the table of contents to pop up at
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% the beginning of each subsection:
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\AtBeginSubsection[]
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{
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\begin{frame}<beamer>{Outline}
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\tableofcontents[currentsection,currentsubsection]
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}
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% the following command:
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%\beamerdefaultoverlayspecification{<+->}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Title Page %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
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\titlepage
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Table of Contents %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Outline}
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\tableofcontents
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% You might wish to add the option [pausesections]
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%---------------------------------------------------------------%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[Fourier]{Fourier transform}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Fourier INTRO %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transformation: Introduction}
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\begin{itemize}
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\item Transform operation by using property-tables $\rightarrow$ \emph{easy}
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\item Transform operation by using integral $\rightarrow$ \emph{difficult}
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\item No math \emph{tricks}
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\item Important: Visualisation?!
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\end{itemize}
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\end{frame}
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\subsection[simple]{Fourier transform Example 1}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 1 SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transform 1: Specification}
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{\footnotesize\it
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Fourier Transform
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\begin{tabbing}
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1\=postcond \=: \= \= $\;\;\;\;$\=\kill
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\>given \>:\> Time continiues, not periodic Signal \\
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\> \> \> \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\
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\>precond \>:\> TODO\\
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\>find \>:\> $X(j\cdot\omega)$\\
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\>postcond \>:\> TODO\\
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\end{tabbing}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 1 CALC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transformation 1: Calculation}
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TODO: Bernhard fragen ob Tabelle oder Rechnung
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 1 REQ %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transform 1: Development effort}
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{\small
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\begin{center}
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\begin{tabular}{l|l|r}
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requirements & comments &effort\\ \hline\hline
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solving Intrgrals & simple via propertie table & 20\\
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& \emph{real} & MT\\ \hline
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transformation table & simple transform & 20\\ \hline
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example collection & with explanations & 20\\ \hline\hline
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& & 60-80\\
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\end{tabular}
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\end{center}
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effort --- in 45min units\\
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MT --- thesis ``Integrals'' (mathematics)
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}
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\end{frame}
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\subsection[difficult]{Fourier transform Example 2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 2 SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transform 2: Specification}
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{\footnotesize\it
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Fourier Transform
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\begin{tabbing}
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1\=postcond \=: \= \= $\;\;\;\;$\=\kill
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\>given \>:\> piecewise\_function \\
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\> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\
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%?(iterativer) datentyp in Isabelle/HOL
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\> \> \> translation $T=2$\\
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%WN these 2 inputs calculated to [(0,-\infty<t<-1), (1,-1\leq t\leq 1), (0, 1<t<\infty)]
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%WN translation helpful only, if solution available from other calculation - DROP T ???
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\>precond \>:\> TODO\\
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\>find \>:\> $X(j\cdot\omega)$\\
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\>postcond \>:\> TODO\\
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\end{tabbing}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 2 CALC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transform 2: Calculation}
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\footnotesize{
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\begin{tabbing}
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000\=\kill
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%WN first
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01 \> ${\cal F}\;(x(t-2)) =$\\
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\`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\
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02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\
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\`definition $X\;(j\cdot\omega)$\\
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03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\
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\` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\
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04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\
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\` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\
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05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\
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%\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\
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\` pbl: integration in $\cal C$\\
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06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\
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\` $f\;t\;|_a^b = f\;b-f\;a$\\
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07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} - \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\
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\vdots\` pbl: simplification+factorization in $\cal C$\\
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08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\
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\` trick~!\\
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09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\
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\` table\\
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10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$
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\end{tabbing}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl 2 REQ %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%WN ...
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\begin{frame}\frametitle{Fourier Transform 2: Development effort}
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{\small
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\begin{center}
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\begin{tabular}{l|l|r}
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requirements & comments &effort\\ \hline\hline
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solving Intrgrals & simple via propertie table & 20\\
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& \emph{real} & MT\\ \hline
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transformation table & simple transform & 20\\ \hline
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visualisation & backend & 10\\ \hline
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example collection & with explanations & 20\\ \hline\hline
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& & 70-80\\
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\end{tabular}
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\end{center}
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effort --- in 45min units\\
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MT --- thesis ``Integrals'' (mathematics)
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-----------------------------------------------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[Discrete time]{Discrete-time systems}
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\subsection[Convolution]{Convolution}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% DTS INTRO %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Convolution: Introduction}
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\begin{itemize}
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\item Calculation\ldots
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\item Visualisation\ldots
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\end{itemize}
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\begin{center}
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\ldots of parallel filter structures
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\end{center}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% DTS SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Convolution: Specification}
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{\footnotesize\it
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Convolution
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\begin{tabbing}
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1\=postcond \=: \= \= $\;\;\;\;$\=\kill
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\>given \>:\> Signals h1[n], h2[n] \\
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\> \> \> \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\
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%?(iterativer) datentyp in Isabelle/HOL
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\>precond \>:\> TODO\\
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\>find \>:\> $h1[n]\,*\,h2[n]$\\
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\>postcond \>:\> TODO\\
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\end{tabbing}
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% DTS CALC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Convolution: Calculation}
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TODO
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% DTS REQ %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Convolution: Development effort}
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{\small
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\begin{center}
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\begin{tabular}{l|l|r}
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requirements & comments &effort\\ \hline\hline
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simplify rationals & \sisac & 0\\ \hline
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define $\sum\limits_{i=0}^{n}i$ & partly \sisac & 10\\ \hline
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simplify sum & termorder & 10\\
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& simplify rules & 20\\
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& use simplify rationals& 0\\ \hline
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index adjustments & with unit step & 10\\ \hline
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example collection & with explanations & 20\\ \hline\hline
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& & 70-90\\
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\end{tabular}
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\end{center}
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effort --- in 45min units\\
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-----------------------------------------------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[Z-transform]{Z-Transform}
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\subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Z-Transform INTRO %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction}
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\begin{itemize}
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\item Pure Transformation is simple to realise with Z-Transform Properties (Table)
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\item Partial Fraction are just math simplifications
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\end{itemize}
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\end{frame}
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% does not fit?!
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%\subsection[]{Indextranformation}
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%\begin{frame}\frametitle{TODO}
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%TODO
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%\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Z-Transform SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
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\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification}
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{\footnotesize\it
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Convolution
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\begin{tabbing}
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1\=postcond \=: \= \= $\;\;\;\;$\=\kill
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\>given \>:\> Expression of z \\
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\> \> \> \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\
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\>precond \>:\> TODO\\
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\>find \>:\> Expression of n\\
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|
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\> \> \> \>$h[n]$\\
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|
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\>postcond \>:\> TODO\\
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|
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\end{tabbing}
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|
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|
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}
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|
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\end{frame}
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
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%% Z expl CALC %%
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
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|
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|
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\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation}
|
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|
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TODO
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|
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\end{frame}
|
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|
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|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
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%% Z expl REQ %%
|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
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|
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|
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|
neuper@42122
|
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\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Development effort}
|
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|
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{\small
|
neuper@42075
|
380 |
\begin{center}
|
neuper@42075
|
381 |
\begin{tabular}{l|l|r}
|
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|
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requirements & comments &effort\\ \hline\hline
|
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|
383 |
solve for part.fract. & \sisac: degree 2 & 0\\
|
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|
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& complex nomminators & 30\\
|
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|
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& degree > 2 & MT\\ \hline
|
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|
386 |
simplify polynomial & \sisac & 0\\
|
neuper@42075
|
387 |
simplify rational & \sisac & 0\\ \hline
|
neuper@42075
|
388 |
part.fract.decomposition& degree 2 & \\
|
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|
389 |
& specification, method& 30\\ \hline
|
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|
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${\cal Z}^{-1}$ table & & 20\\
|
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|
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& explanations, figures& 20\\ \hline
|
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|
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example collection & with explanations & 20\\ \hline\hline
|
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|
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& & 90-120\\
|
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|
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% & & 1 MT
|
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|
395 |
\end{tabular}
|
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|
396 |
\end{center}
|
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|
397 |
effort --- in 45min units\\
|
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|
398 |
MT --- thesis ``factorization'' (mathematics)
|
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|
399 |
}
|
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|
400 |
\end{frame}
|
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|
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|
neuper@42074
|
402 |
|
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|
403 |
\end{document}
|
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|
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|
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|
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|