
neuper@42074: 
neuper@42074: \documentclass{beamer}
neuper@42074: 
neuper@42074: 
neuper@42074: \mode<presentation>
neuper@42074: {
neuper@42074:   \usetheme{Hannover}
neuper@42074:   \setbeamercovered{transparent}
neuper@42074: }
neuper@42074: 
neuper@42074: \usepackage[english]{babel}
neuper@42074: \usepackage[utf8]{inputenc}
neuper@42074: \usepackage{times}
neuper@42074: \usepackage[T1]{fontenc}
neuper@42074: 
neuper@42074: \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
neuper@42074: \def\sisac{{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}}
neuper@42074: 
jan@42163: \title[SPSC in \isac] % (optional, use only with long paper titles)
jan@42163: {Interactive Course Material\\ for Signal Processing\\ based on Isabelle/\isac}
neuper@42074: 
jan@42163: \subtitle{Baccalaureate Thesis}
neuper@42074: 
jan@42175: \author[Ro\v{c}nik]
jan@42175: {Jan Rocnik}
neuper@42074: 
neuper@42074: \institute % (optional, but mostly needed)
neuper@42074: {
neuper@42074:   Technische Universit\"at Graz\\
neuper@42074:   Institut f\"ur TODO
neuper@42074: }
neuper@42074: 
neuper@42074: % If you have a file called "university-logo-filename.xxx", where xxx
neuper@42074: % is a graphic format that can be processed by latex or pdflatex,
neuper@42074: % resp., then you can add a logo as follows:
neuper@42074: 
neuper@42074: % \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
neuper@42074: % \logo{\pgfuseimage{university-logo}}
neuper@42074: 
neuper@42074: 
neuper@42074: 
neuper@42074: % Delete this, if you do not want the table of contents to pop up at
neuper@42074: % the beginning of each subsection:
neuper@42074: \AtBeginSubsection[]
neuper@42074: {
neuper@42074:   \begin{frame}<beamer>{Outline}
neuper@42074:     \tableofcontents[currentsection,currentsubsection]
neuper@42074:   \end{frame}
neuper@42074: }
neuper@42074: 
neuper@42074: \begin{document}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Title Page                             %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42074: \begin{frame}
neuper@42074:   \titlepage
neuper@42074: \end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Table of Contents                      %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42074: \begin{frame}{Outline}
neuper@42074:   \tableofcontents
neuper@42074:   % You might wish to add the option [pausesections]
neuper@42074: \end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%---------------------------------------------------------------%%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42074: 
jan@42163: \section[Intro]{Introduction}
jan@42163: 
jan@42175: \begin{frame}{Issues to be Accomplished}
jan@42163: 
jan@42163: \begin{itemize}
jan@42163: 
jan@42183: \item What knowledge is already mechanised in \emph{Isabelle}?
jan@42163: \item How can missing theorems and definitions be mechanised?
jan@42163: \item What is the effort for such mechanisation?
jan@42175: \item How do calculations look like, by using mechanised knowledge?
jan@42175: \item What problems and subproblems have to be solved?
jan@42163: \item Which problems are already implemented in \sisac?
jan@42175: \item How are the new problems specified (\sisac)?
jan@42175: \item Which variantes of programms in \sisac\ solve the problems?
jan@42163: \item What is the contents of the interactiv course material (Figures, etc.)?
jan@42163: 
jan@42163: \end{itemize}
jan@42163: \end{frame}
jan@42163: 
jan@42163: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42163: %%---------------------------------------------------------------%%
jan@42163: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42163: 
jan@42175: \section[Fourier]{Fourier transformation}
jan@42175: \subsection[Fourier]{Fourier transform}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Fourier INTRO                          %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transformation: Introduction}
jan@42191: Possibilities:
neuper@42122: \begin{itemize}
jan@42175: \item Transform operation by using property-tables
jan@42175: \item Transform operation by using integral
jan@42191: \end{itemize}
jan@42191: Also Important:
jan@42191: \begin{itemize}
jan@42191: \item Visualisation?!
neuper@42122: \end{itemize}
neuper@42074: \end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: %%										Transform expl   SPEC                      %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jan@42175: \begin{frame}\frametitle{Fourier Transformation: Specification}
jan@42175: {\footnotesize
jrocnik@42159: 
jan@42175: Determine the fourier transform for the given rectangular impulse:
jrocnik@42159: 
jrocnik@42159: \begin{center}
jrocnik@42159: $x(t)= \left\{
jrocnik@42159:      \begin{array}{lr}
jan@42163:        1 & -1\leq t\leq1\\
jrocnik@42159:        0 & else
jrocnik@42159:      \end{array}
jrocnik@42159:    \right.$
jrocnik@42159: \end{center}
jrocnik@42159: 
jan@42175: \hrulefill
jan@42175: 
neuper@42076: \begin{tabbing}
neuper@42076: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42076: \>given    \>:\>  piecewise\_function \\
jan@42191: \>         \> \>  \>$fun (x (t::real),\ x=1\ ((t>=-1)\ \&\ (t<=1)),\ x=0)$\\
neuper@42076: \>precond  \>:\>  TODO\\
neuper@42076: \>find     \>:\>  $X(j\cdot\omega)$\\
neuper@42076: \>postcond \>:\>  TODO\\
neuper@42076: \end{tabbing}
neuper@42076: 
neuper@42076: }
neuper@42076: \end{frame}
neuper@42076: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: %%												Transform expl   REQ                   %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jan@42175: \begin{frame}\frametitle{Fourier Transform: Development effort}
neuper@42122: {\small
neuper@42122: \begin{center}
neuper@42122: \begin{tabular}{l|l|r}
neuper@42122: requirements            & comments             &effort\\ \hline\hline
neuper@42122: solving Intrgrals		    & simple via propertie table     &     20\\
neuper@42122:                         & \emph{real}          &    MT\\ \hline
neuper@42122: transformation table    & simple transform     &    20\\ \hline
neuper@42122: visualisation						& backend							 &    10\\ \hline
neuper@42122: example collection      & with explanations    &    20\\ \hline\hline
neuper@42122:                         &                      & 70-80\\
neuper@42122: \end{tabular}
neuper@42122: \end{center}
neuper@42122: effort --- in 45min units\\
neuper@42122: MT --- thesis ``Integrals'' (mathematics)
neuper@42122: }
neuper@42074: \end{frame}
jan@42163: 
jan@42163: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42163: %%--------------------FOURIER---Conclusion-----------------------%%
jan@42163: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42163: 
jan@42175: \begin{frame}{Fourier Transformation: Summary}
jan@42163: \begin{itemize}
jan@42163: 
jan@42175: \item Standard integrals can be solved with tables
jan@42183: \item No real integration (yet avaible)
jan@42175: \item Math \emph{tricks} difficult to implement
jan@42175: 
jan@42163: 
jan@42163: \end{itemize}
jan@42163: \end{frame}
jan@42163: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %-----------------------------------------------------------------%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42074: 
jan@42175: \section[LTI Systems]{LTI systems}
jan@42183: \subsection[Convolution]{Convolution (Faltung)}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: %%												LTI INTRO				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Convolution: Introduction}
neuper@42122: \begin{itemize}
jan@42191: \item Calculation include sums
jan@42191: \item Demonstrative examples
jan@42191: \item Visualisation is important
neuper@42122: \end{itemize}
neuper@42122: \end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: %%												LTI SPEC				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Convolution: Specification}
jan@42175: {\footnotesize
jrocnik@42159: 
jrocnik@42159: Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response:
jrocnik@42159: 
jrocnik@42159: \begin{center}
jrocnik@42159: $h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\
jrocnik@42159: $h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$
jrocnik@42159: \end{center}
jrocnik@42159: 
jrocnik@42159: The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$.
jrocnik@42159: 
jan@42173: \hrulefill
jan@42191: 
neuper@42122: \begin{tabbing}
neuper@42122: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42122: \>given    \>:\>  Signals h1[n], h2[n] \\
jan@42191: \>         \> \>  \>((h1[n]=(3/5)\textasciicircum{}n*u[n]),\,h2[n]=(-2/3)\textasciicircum{}n*u[n]))\\
jan@42191:                         
neuper@42122: \>precond  \>:\>  TODO\\
neuper@42122: \>find     \>:\>  $h1[n]\,*\,h2[n]$\\
neuper@42122: \>postcond \>:\>  TODO\\
neuper@42122: \end{tabbing}
neuper@42122: 
neuper@42122: }
neuper@42122: \end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: %%												LTI REQ  				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Convolution: Development effort}
neuper@42122: {\small
neuper@42122: \begin{center}
neuper@42122: \begin{tabular}{l|l|r}
neuper@42122: requirements            & comments             &effort\\ \hline\hline
neuper@42122: simplify rationals      & \sisac               &     0\\ \hline
neuper@42122: define $\sum\limits_{i=0}^{n}i$ & partly \sisac  &    10\\ \hline
neuper@42122: simplify sum			      & termorder            &    10\\
neuper@42122:                         & simplify rules       &    20\\
neuper@42122:                         & use simplify rationals&     0\\ \hline
neuper@42122: index adjustments       & with unit step       &      10\\ \hline
neuper@42122: example collection      & with explanations    &    20\\ \hline\hline
neuper@42122:                         &                      & 70-90\\
neuper@42122: \end{tabular}
neuper@42122: \end{center}
neuper@42122: effort --- in 45min units\\
neuper@42122: }
neuper@42122: \end{frame}
jan@42173: 
jan@42173: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: %%--------------------LTI-------Conclusion-----------------------%%
jan@42173: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: 
jan@42175: \begin{frame}{Convolution: Summary}
jan@42173: \begin{itemize}
jan@42173: 
jan@42175: \item Standard example
jan@42183: \item Straight forward
jan@42175: \item Challenge are sum limits
jan@42173: 
jan@42173: \end{itemize}
jan@42173: \end{frame}
jan@42173: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %-----------------------------------------------------------------%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \section[Z-transform]{Z-Transform}
neuper@42122: \subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Z-Transform  INTRO                     %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction}
neuper@42122: \begin{itemize}
neuper@42122: \item Pure Transformation is simple to realise with Z-Transform Properties (Table)
neuper@42122: \item Partial Fraction are just math simplifications
neuper@42122: \end{itemize}
neuper@42122: \end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Z-Transform  SPEC                      %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jan@42173: \begin{frame}\frametitle{(Inverse) Z-Transformation: Specification}
jan@42175: {\footnotesize
jrocnik@42159: 
jan@42175: Determine the inverse z transform of the following expression. Hint: applay the partial fraction expansion.
jrocnik@42159: 
jrocnik@42159: \begin{center}
jrocnik@42159: $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable
jrocnik@42159: \end{center}
jrocnik@42159: 
jan@42175: 
jan@42173: \hrulefill
jan@42173: 
neuper@42122: \begin{tabbing}
neuper@42122: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42122: \>given    \>:\>  Expression of z \\
neuper@42122: \>         \> \>  \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\
neuper@42122: \>precond  \>:\>  TODO\\
neuper@42122: \>find     \>:\>  Expression of n\\
neuper@42122: \>         \> \>  \>$h[n]$\\
neuper@42122: \>postcond \>:\>  TODO\\
neuper@42122: \end{tabbing}
neuper@42122: 
neuper@42122: }
neuper@42122: \end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Z expl		REQ	                         %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42076: 
neuper@42122: 
jan@42173: \begin{frame}\frametitle{(Inverse) Z-Transformation: Development effort}
neuper@42075: {\small
neuper@42075: \begin{center}
neuper@42075: \begin{tabular}{l|l|r}
neuper@42075: requirements            & comments             &effort\\ \hline\hline
neuper@42075: solve for part.fract.   & \sisac: degree 2     &     0\\
neuper@42075:                         & complex nomminators  &    30\\
neuper@42075:                         & degree > 2           &    MT\\ \hline
neuper@42075: simplify polynomial     & \sisac               &     0\\
neuper@42075: simplify rational       & \sisac               &     0\\ \hline
neuper@42075: part.fract.decomposition& degree 2             &      \\
neuper@42075:                         & specification, method&    30\\ \hline
neuper@42075: ${\cal Z}^{-1}$ table    &                       &   20\\
neuper@42075:                         & explanations, figures&    20\\ \hline
neuper@42075: example collection      & with explanations    &    20\\ \hline\hline
neuper@42075:                         &                      & 90-120\\
neuper@42075: %                        &                      & 1 MT
neuper@42075: \end{tabular}
neuper@42075: \end{center}
neuper@42075: effort --- in 45min units\\
neuper@42075: MT --- thesis ``factorization'' (mathematics)
neuper@42075: }
neuper@42075: \end{frame}
neuper@42075: 
jan@42173: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: %%--------------------Z-TRANS---Conclusion-----------------------%%
jan@42173: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42173: 
jan@42175: \begin{frame}{(Inverse) Z-Transformation: Summary}
jan@42173: \begin{itemize}
jan@42173: 
jan@42175: \item No \emph{higher} math operations
jan@42175: \item Different subproblems of math (equation systems, etc.)
jan@42175: \item Both directions have the same effort
jan@42173: 
jan@42173: \end{itemize}
jan@42173: \end{frame}
jan@42173: 
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: %-----------------------------------------------------------------%
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: 
jan@42183: \section[Conclusions]{Conclusions}
jan@42175: 
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42191: %--------------------------DEMONSTRATION--------------------------%
jan@42191: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42191: 
jan@42191: \begin{frame}{Demonstration}
jan@42191: 
jan@42191: \centering{Demonstration}
jan@42191: 
jan@42191: \end{frame}
jan@42191: 
jan@42191: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: %--------------------------CONCLUSION-----------------------------%
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: 
jan@42175: \begin{frame}{Conclusions}
jan@42175: 
jan@42175: Design Challanges:
jan@42175: 
jan@42175: {\small
jan@42175: \begin{itemize}
jan@42175: 
jan@42175: \item Pre and Post conditions
jan@42175: \item Exact mathematic behind functions
jan@42175: \item accurate mathematic notation
jan@42175: 
jan@42175: \end{itemize}
jan@42175: }
jan@42175: 
jan@42175: Goals:
jan@42175: {\small
jan@42175: \begin{itemize}
jan@42175: 
jan@42175: \item Spot the power of \sisac
jan@42175: \item Implementation of generell but simple math problems
jan@42175: \item Setting up a good first guideline (documentation) for furher problem implemenations
jan@42175: 
jan@42175: \end{itemize}
jan@42175: 
jan@42175: \centering{Efforts are only approximations, due we have no \emph{real} experience data!}
jan@42175: }
jan@42175: 
jan@42175: \end{frame}
jan@42175: 
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: %--------------------------TIME LINE------------------------------%
jan@42175: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
jan@42175: 
jan@42175: \begin{frame}{Comming up}
jan@42175: 
jan@42175: {\small
jan@42175: \begin{tabular}{l r}
jan@42175: 
jan@42175: Juli 2011 & project startup\\
jan@42175: Juli 2011 & information collection, 1st presentation\\
jan@42175: August 2011 & extern traineeship\\
jan@42175: September 2011 & main work\\
jan@42175: after Oktober & finishing, documentation\\
jan@42175: 
jan@42175: \end{tabular}
jan@42175: }
jan@42175: 
jan@42175: \end{frame}
jan@42175: 
jan@42175: 
neuper@42074: \end{document}
neuper@42074: 
neuper@42074: 




