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\documentclass{beamer}
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\mode<presentation>
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{
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\usetheme{Hannover}
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\setbeamercovered{transparent}
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}
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\usepackage[english]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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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[SPSC in \isac] % (optional, use only with long paper titles)
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{Interactive Course Material\\ for Signal Processing\\ based on Isabelle/\isac}
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\subtitle{Baccalaureate Thesis}
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\author[Ro\v{c}nik]
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{Jan Rocnik}
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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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% If you have a file called "university-logo-filename.xxx", where xxx
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% is a graphic format that can be processed by latex or pdflatex,
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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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\end{frame}
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}
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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[Intro]{Introduction}
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\begin{frame}{Issues to be Accomplished}
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\begin{itemize}
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\item What knowledge is already mechanised in \emph{Isabelle}?
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\item How can missing theorems and definitions be mechanised?
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\item What is the effort for such mechanisation?
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\item How do calculations look like, by using mechanised knowledge?
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\item What problems and subproblems have to be solved?
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\item Which problems are already implemented in \sisac?
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\item How are the new problems specified (\sisac)?
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\item Which variantes of programms in \sisac\ solve the problems?
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\item What is the contents of the interactiv course material (Figures, etc.)?
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\end{itemize}
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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 transformation}
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\subsection[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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Possibilities:
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\begin{itemize}
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\item Transform operation by using property-tables
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\item Transform operation by using integral
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\end{itemize}
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Also Important:
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\begin{itemize}
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\item Visualisation?!
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Transform expl SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transformation: Specification}
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{\footnotesize
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Determine the fourier transform for the given rectangular impulse:
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\begin{center}
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$x(t)= \left\{
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\begin{array}{lr}
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1 & -1\leq t\leq1\\
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0 & else
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\end{array}
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\right.$
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\end{center}
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\hrulefill
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\begin{tabbing}
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1\=postcond \=: \= \= $\;\;\;\;$\=\kill
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\>given \>:\> piecewise\_function \\
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\> \> \> \>$fun (x (t::real),\ x=1\ ((t>=-1)\ \&\ (t<=1)),\ x=0)$\\
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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 REQ %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Fourier Transform: 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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%%--------------------FOURIER---Conclusion-----------------------%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Fourier Transformation: Summary}
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\begin{itemize}
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\item Standard integrals can be solved with tables
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\item No real integration (yet avaible)
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\item Math \emph{tricks} difficult to implement
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-----------------------------------------------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[LTI Systems]{LTI systems}
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\subsection[Convolution]{Convolution (Faltung)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% LTI 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 include sums
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\item Demonstrative examples
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\item Visualisation is important
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% LTI SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{Convolution: Specification}
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{\footnotesize
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Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response:
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\begin{center}
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$h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\
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$h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$
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\end{center}
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The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$.
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\hrulefill
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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]=(3/5)\textasciicircum{}n*u[n]),\,h2[n]=(-2/3)\textasciicircum{}n*u[n]))\\
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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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%% LTI 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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%%--------------------LTI-------Conclusion-----------------------%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Convolution: Summary}
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\begin{itemize}
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\item Standard example
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\item Straight forward
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\item Challenge are sum limits
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\end{itemize}
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% Z-Transform SPEC %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{(Inverse) Z-Transformation: Specification}
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{\footnotesize
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Determine the inverse z transform of the following expression. Hint: applay the partial fraction expansion.
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\begin{center}
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$X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable
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\end{center}
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\hrulefill
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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::complex),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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\> \> \> \>$h[n]$\\
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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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%% Z expl REQ %%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}\frametitle{(Inverse) Z-Transformation: 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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solve for part.fract. & \sisac: degree 2 & 0\\
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& complex nomminators & 30\\
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& degree > 2 & MT\\ \hline
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simplify polynomial & \sisac & 0\\
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simplify rational & \sisac & 0\\ \hline
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part.fract.decomposition& degree 2 & \\
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& specification, method& 30\\ \hline
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${\cal Z}^{-1}$ table & & 20\\
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& explanations, figures& 20\\ \hline
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example collection & with explanations & 20\\ \hline\hline
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& & 90-120\\
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% & & 1 MT
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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 ``factorization'' (mathematics)
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}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%--------------------Z-TRANS---Conclusion-----------------------%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{(Inverse) Z-Transformation: Summary}
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\begin{itemize}
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\item No \emph{higher} math operations
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\item Different subproblems of math (equation systems, etc.)
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\item Both directions have the same effort
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-----------------------------------------------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section[Conclusions]{Conclusions}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%--------------------------DEMONSTRATION--------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Demonstration}
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\centering{Demonstration}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%--------------------------CONCLUSION-----------------------------%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
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\begin{frame}{Conclusions}
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|
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Design Challanges:
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|
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|
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|
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{\small
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|
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\begin{itemize}
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|
388 |
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|
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\item Pre and Post conditions
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|
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\item Exact mathematic behind functions
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|
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\item Accurate mathematic notation
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|
392 |
|
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|
393 |
\end{itemize}
|
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|
394 |
}
|
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|
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|
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|
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Goals:
|
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|
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{\small
|
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|
398 |
\begin{itemize}
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|
399 |
|
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|
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\item Spot the power of \sisac
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|
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\item Implementation of generell but simple math problems
|
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|
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\item Setting up a good first guideline (documentation) for furher problem implemenations
|
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|
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|
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|
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\end{itemize}
|
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|
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|
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|
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\centering{Efforts are only approximations, due we have no \emph{real} experience data!}
|
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|
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}
|
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|
408 |
|
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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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%--------------------------TIME LINE------------------------------%
|
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|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
414 |
|
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|
415 |
\begin{frame}{Comming up}
|
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|
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|
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|
417 |
{\small
|
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|
418 |
\begin{tabular}{l r}
|
neuper@42378
|
419 |
|
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|
420 |
Juli 2011 & project startup\\
|
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|
421 |
Juli 2011 & information collection, 1st presentation\\
|
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|
422 |
August 2011 & extern traineeship\\
|
neuper@42378
|
423 |
September 2011 & main work\\
|
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|
424 |
after Oktober & finishing, documentation\\
|
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|
425 |
|
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|
426 |
\end{tabular}
|
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|
427 |
}
|
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|
428 |
|
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|
429 |
\end{frame}
|
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|
430 |
|
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|
431 |
|
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|
432 |
\end{document}
|
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|
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|
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|
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|