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 \def\sisac{{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}}

 \title[TODO] % (optional, use only with long paper titles)

 {TODO}

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 \author[Rocnik] % (optional, use only with lots of authors)

 {Jan~Rocnik}

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 %   affiliation.

 \institute % (optional, but mostly needed)

 {

   Technische Universit\"at Graz\\

   Institut f\"ur TODO

 }

 % - Use the \inst command only if there are several affiliations.

 % - Keep it simple, no one is interested in your street address.

 % \date[CFP 2003] % (optional, should be abbreviation of conference name)

 % {Conference on Fabulous Presentations, 2003}

 % - Either use conference name or its abbreviation.

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 %   yourself) who are reading the slides online

 % \subject{Theoretical Computer Science}

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 % the beginning of each subsection:

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   \begin{frame}<beamer>{Outline}

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 % the following command:

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 \begin{document}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Title Page                             %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}

   \titlepage

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Table of Contents                      %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}{Outline}

   \tableofcontents

   % You might wish to add the option [pausesections]

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%---------------------------------------------------------------%%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \section[Fourier]{Fourier transform}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Fourier INTRO                          %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transformation: Introduction}

 \begin{itemize}

 \item Transform operation by using property-tables $\rightarrow$ \emph{easy}

 \item Transform operation by using integral $\rightarrow$ \emph{difficult}

 \item No math \emph{tricks}

 \item Important: Visualisation?!

 \end{itemize}

 \end{frame}

 \subsection[simple]{Fourier transform Example 1}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Transform expl 1 SPEC                  %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transform 1: Specification}

 {\footnotesize\it

 Fourier Transform

 \begin{tabbing}

 1\=postcond \=: \= \= $\;\;\;\;$\=\kill

 \>given    \>:\>  Time continiues, not periodic Signal \\

 \>         \> \>  \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\

 \>precond  \>:\>  TODO\\

 \>find     \>:\>  $X(j\cdot\omega)$\\

 \>postcond \>:\>  TODO\\

 \end{tabbing}

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Transform expl 1 CALC                  %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transformation 1: Calculation}

 TODO: Bernhard fragen ob Tabelle oder Rechnung

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Transform expl 1 REQ                  %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transform 1: Development effort}

 {\small

 \begin{center}

 \begin{tabular}{l|l|r}

 requirements            & comments             &effort\\ \hline\hline

 solving Intrgrals		    & simple via propertie table     &     20\\

                         & \emph{real}          &    MT\\ \hline

 transformation table    & simple transform     &    20\\ \hline

 example collection      & with explanations    &    20\\ \hline\hline

                         &                      & 60-80\\

 \end{tabular}

 \end{center}

 effort --- in 45min units\\

 MT --- thesis ``Integrals'' (mathematics)

 }

 \end{frame}

 \subsection[difficult]{Fourier transform Example 2}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%										Transform expl 2 SPEC                      %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transform 2: Specification}

 {\footnotesize\it

 \textbf{(a)} Determine the fourier transform for the given rectangular impulse:

 \begin{center}

 $x(t)= \left\{

      \begin{array}{lr}

        1 & -1\leq t\geq1\\

        0 & else

      \end{array}

    \right.$

 \end{center}

 \begin{tabbing}

 1\=postcond \=: \= \= $\;\;\;\;$\=\kill

 \>given    \>:\>  piecewise\_function \\

 \>         \> \>  \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\

                         %?(iterativer) datentyp in Isabelle/HOL

 \>         \> \>  translation $T=2$\\

 \>precond  \>:\>  TODO\\

 \>find     \>:\>  $X(j\cdot\omega)$\\

 \>postcond \>:\>  TODO\\

 \end{tabbing}

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Transform expl  2 CALC                 %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %\begin{frame}\frametitle{Fourier Transform 2: Calculation}

 %\footnotesize{

 %\begin{tabbing}

 %000\=\kill

 %01 \> ${\cal F}\;(x(t-2)) =$\\

 %      \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\

 %02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\

 %      \`definition $X\;(j\cdot\omega)$\\

 %03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\

 %      \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\

 %04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\

 %      \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\

 %05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\

 %      %\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\

 %       \` pbl: integration in $\cal C$\\

 %06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\

 %      \` $f\;t\;|_a^b = f\;b-f\;a$\\

 %07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} -  \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\

 %\vdots\` pbl: simplification+factorization in $\cal C$\\

 %08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\

 %      \` trick~!\\

 %09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\

 %      \` table\\

 %10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$

 %\end{tabbing}

 %}

 %\end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Transform expl 2 REQ                   %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Fourier Transform 2: Development effort}

 {\small

 \begin{center}

 \begin{tabular}{l|l|r}

 requirements            & comments             &effort\\ \hline\hline

 solving Intrgrals		    & simple via propertie table     &     20\\

                         & \emph{real}          &    MT\\ \hline

 transformation table    & simple transform     &    20\\ \hline

 visualisation						& backend							 &    10\\ \hline

 example collection      & with explanations    &    20\\ \hline\hline

                         &                      & 70-80\\

 \end{tabular}

 \end{center}

 effort --- in 45min units\\

 MT --- thesis ``Integrals'' (mathematics)

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %-----------------------------------------------------------------%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \section[Discrete time]{Discrete-time systems}

 \subsection[Convolution]{Convolution}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												DTS INTRO				                       %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Convolution: Introduction}

 \begin{itemize}

 \item Calculation\ldots

 \item Visualisation\ldots

 \end{itemize}

 \begin{center}

 \ldots of parallel filter structures

 \end{center}

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												DTS SPEC				                       %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Convolution: Specification}

 {\footnotesize\it

 Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response:

 \begin{center}

 $h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\

 $h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$

 \end{center}

 The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$.

 \begin{tabbing}

 1\=postcond \=: \= \= $\;\;\;\;$\=\kill

 \>given    \>:\>  Signals h1[n], h2[n] \\

 \>         \> \>  \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\

                         %?(iterativer) datentyp in Isabelle/HOL

 \>precond  \>:\>  TODO\\

 \>find     \>:\>  $h1[n]\,*\,h2[n]$\\

 \>postcond \>:\>  TODO\\

 \end{tabbing}

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												DTS CALC				                       %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %\begin{frame}\frametitle{Convolution: Calculation}

 %TODO

 %\end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												DTS REQ  				                       %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{Convolution: Development effort}

 {\small

 \begin{center}

 \begin{tabular}{l|l|r}

 requirements            & comments             &effort\\ \hline\hline

 simplify rationals      & \sisac               &     0\\ \hline

 define $\sum\limits_{i=0}^{n}i$ & partly \sisac  &    10\\ \hline

 simplify sum			      & termorder            &    10\\

                         & simplify rules       &    20\\

                         & use simplify rationals&     0\\ \hline

 index adjustments       & with unit step       &      10\\ \hline

 example collection      & with explanations    &    20\\ \hline\hline

                         &                      & 70-90\\

 \end{tabular}

 \end{center}

 effort --- in 45min units\\

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %-----------------------------------------------------------------%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \section[Z-transform]{Z-Transform}

 \subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Z-Transform  INTRO                     %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction}

 \begin{itemize}

 \item Pure Transformation is simple to realise with Z-Transform Properties (Table)

 \item Partial Fraction are just math simplifications

 \end{itemize}

 \end{frame}

 % does not fit?!

 %\subsection[]{Indextranformation}

 %\begin{frame}\frametitle{TODO}

 %TODO

 %\end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Z-Transform  SPEC                      %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification}

 {\footnotesize\it

 Determine the inverse $\cal{z}$ transform of the following expression. Hint: applay the partial fraction expansion.

 \begin{center}

 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable

 \end{center}

 \begin{tabbing}

 1\=postcond \=: \= \= $\;\;\;\;$\=\kill

 \>given    \>:\>  Expression of z \\

 \>         \> \>  \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\

 \>precond  \>:\>  TODO\\

 \>find     \>:\>  Expression of n\\

 \>         \> \>  \>$h[n]$\\

 \>postcond \>:\>  TODO\\

 \end{tabbing}

 }

 \end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Z expl		CALC                         %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation}

 %TODO

 %\end{frame}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 %%												Z expl		REQ	                         %%

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Development effort}

 {\small

 \begin{center}

 \begin{tabular}{l|l|r}

 requirements            & comments             &effort\\ \hline\hline

 solve for part.fract.   & \sisac: degree 2     &     0\\

                         & complex nomminators  &    30\\

                         & degree > 2           &    MT\\ \hline

 simplify polynomial     & \sisac               &     0\\

 simplify rational       & \sisac               &     0\\ \hline

 part.fract.decomposition& degree 2             &      \\

                         & specification, method&    30\\ \hline

 ${\cal Z}^{-1}$ table    &                       &   20\\

                         & explanations, figures&    20\\ \hline

 example collection      & with explanations    &    20\\ \hline\hline

                         &                      & 90-120\\

 %                        &                      & 1 MT

 \end{tabular}

 \end{center}

 effort --- in 45min units\\

 MT --- thesis ``factorization'' (mathematics)

 }

 \end{frame}

 \end{document}