
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{setspace} %for "\begin{onehalfspace}"
neuper@42074: \usepackage[english]{babel}
neuper@42074: % or whatever
neuper@42074: 
neuper@42074: \usepackage[utf8]{inputenc}
neuper@42074: % or whatever
neuper@42074: 
neuper@42074: \usepackage{times}
neuper@42074: \usepackage[T1]{fontenc}
neuper@42074: % Or whatever. Note that the encoding and the font should match. If T1
neuper@42074: % does not look nice, try deleting the line with the 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: 
neuper@42074: \author[Rocnik] % (optional, use only with lots of authors)
neuper@42074: {Jan~Rocnik}
neuper@42074: % - Give the names in the same order as the appear in the paper.
neuper@42074: % - Use the \inst{?} command only if the authors have different
neuper@42074: %   affiliation.
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: % - Use the \inst command only if there are several affiliations.
neuper@42074: % - Keep it simple, no one is interested in your street address.
neuper@42074: 
neuper@42074: % \date[CFP 2003] % (optional, should be abbreviation of conference name)
neuper@42074: % {Conference on Fabulous Presentations, 2003}
neuper@42074: % - Either use conference name or its abbreviation.
neuper@42074: % - Not really informative to the audience, more for people (including
neuper@42074: %   yourself) who are reading the slides online
neuper@42074: 
neuper@42074: % \subject{Theoretical Computer Science}
neuper@42074: % This is only inserted into the PDF information catalog. Can be left
neuper@42074: % out.
neuper@42074: 
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: 
neuper@42074: % If you wish to uncover everything in a step-wise fashion, uncomment
neuper@42074: % the following command:
neuper@42074: 
neuper@42074: %\beamerdefaultoverlayspecification{<+->}
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@42163: \begin{frame}{Introduction}
jan@42163: Issues to be accomplished in this thesis:
jan@42163: 
jan@42163: \begin{itemize}
jan@42163: 
jan@42163: \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@42163: \item How do calculations look like, if using mechanised knowledge?
jan@42163: \item What are the problems and subproblems to be solved?
jan@42163: \item Which problems are already implemented in \sisac?
jan@42163: \item How are the new Problems specified rigorously (\sisac)?
jan@42163: \item Which variantes of programms in \sisac{} solving 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: 
neuper@42075: \section[Fourier]{Fourier transform}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Fourier INTRO                          %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transformation: Introduction}
neuper@42122: \begin{itemize}
neuper@42122: \item Transform operation by using property-tables $\rightarrow$ \emph{easy}
neuper@42122: \item Transform operation by using integral $\rightarrow$ \emph{difficult}
neuper@42122: \item No math \emph{tricks}
neuper@42122: \item Important: Visualisation?!
neuper@42122: \end{itemize}
neuper@42074: \end{frame}
neuper@42074: 
neuper@42122: \subsection[simple]{Fourier transform Example 1}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Transform expl 1 SPEC                  %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transform 1: Specification}
neuper@42122: {\footnotesize\it
neuper@42122: Fourier Transform
jrocnik@42159: 
neuper@42122: \begin{tabbing}
neuper@42122: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42122: \>given    \>:\>  Time continiues, not periodic Signal \\
neuper@42122: \>         \> \>  \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\
neuper@42122: \>precond  \>:\>  TODO\\
neuper@42122: \>find     \>:\>  $X(j\cdot\omega)$\\
neuper@42122: \>postcond \>:\>  TODO\\
neuper@42122: \end{tabbing}
neuper@42122: 
neuper@42122: }
neuper@42122: \end{frame}
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Transform expl 1 CALC                  %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transformation 1: Calculation}
neuper@42122: TODO: Bernhard fragen ob Tabelle oder Rechnung
neuper@42074: \end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Transform expl 1 REQ                  %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transform 1: 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: example collection      & with explanations    &    20\\ \hline\hline
neuper@42122:                         &                      & 60-80\\
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}
neuper@42074: 
neuper@42122: \subsection[difficult]{Fourier transform Example 2}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%										Transform expl 2 SPEC                      %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transform 2: Specification}
neuper@42076: {\footnotesize\it
jrocnik@42159: 
jrocnik@42159: \textbf{(a)} 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: 
neuper@42076: \begin{tabbing}
neuper@42076: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42076: \>given    \>:\>  piecewise\_function \\
neuper@42076: \>         \> \>  \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\
neuper@42076:                         %?(iterativer) datentyp in Isabelle/HOL
neuper@42076: \>         \> \>  translation $T=2$\\
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: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Transform expl  2 CALC                 %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jrocnik@42159: %\begin{frame}\frametitle{Fourier Transform 2: Calculation}
jrocnik@42159: %\footnotesize{
jrocnik@42159: %\begin{tabbing}
jrocnik@42159: %000\=\kill
jrocnik@42159: %01 \> ${\cal F}\;(x(t-2)) =$\\
jrocnik@42159: %      \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\
jrocnik@42159: %02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\
jrocnik@42159: %      \`definition $X\;(j\cdot\omega)$\\
jrocnik@42159: %03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\
jrocnik@42159: %      \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\
jrocnik@42159: %04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\
jrocnik@42159: %      \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\
jrocnik@42159: %05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\
jrocnik@42159: %      %\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\
jrocnik@42159: %       \` pbl: integration in $\cal C$\\
jrocnik@42159: %06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\
jrocnik@42159: %      \` $f\;t\;|_a^b = f\;b-f\;a$\\
jrocnik@42159: %07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} -  \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\
jrocnik@42159: %\vdots\` pbl: simplification+factorization in $\cal C$\\
jrocnik@42159: %08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\
jrocnik@42159: %      \` trick~!\\
jrocnik@42159: %09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\
jrocnik@42159: %      \` table\\
jrocnik@42159: %10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$
jrocnik@42159: %\end{tabbing}
jrocnik@42159: %}
jrocnik@42159: %\end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Transform expl 2 REQ                   %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Fourier Transform 2: 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@42163: %\subsection[Summary]{Summary}
jan@42163: 
jan@42163: \begin{frame}{Summary}
jan@42163: todo
jan@42163: 
jan@42163: \begin{itemize}
jan@42163: 
jan@42163: \item todo
jan@42163: 
jan@42163: \end{itemize}
jan@42163: \end{frame}
jan@42163: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %-----------------------------------------------------------------%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42074: 
neuper@42076: \section[Discrete time]{Discrete-time systems}
neuper@42076: \subsection[Convolution]{Convolution}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												DTS INTRO				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Convolution: Introduction}
neuper@42122: \begin{itemize}
neuper@42122: \item Calculation\ldots
neuper@42122: \item Visualisation\ldots
neuper@42122: \end{itemize}
neuper@42122: \begin{center}
neuper@42122: \ldots of parallel filter structures
neuper@42122: \end{center}
neuper@42122: \end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												DTS SPEC				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{Convolution: Specification}
neuper@42122: {\footnotesize\it
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: 
neuper@42122: \begin{tabbing}
neuper@42122: 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
neuper@42122: \>given    \>:\>  Signals h1[n], h2[n] \\
neuper@42122: \>         \> \>  \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\
neuper@42122:                         %?(iterativer) datentyp in Isabelle/HOL
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: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												DTS CALC				                       %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jrocnik@42159: %\begin{frame}\frametitle{Convolution: Calculation}
jrocnik@42159: %TODO
jrocnik@42159: %\end{frame}
neuper@42075: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												DTS 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}
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: % does not fit?!
neuper@42122: %\subsection[]{Indextranformation}
neuper@42122: %\begin{frame}\frametitle{TODO}
neuper@42122: %TODO
neuper@42122: %\end{frame}
neuper@42122: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Z-Transform  SPEC                      %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification}
neuper@42122: {\footnotesize\it
jrocnik@42159: 
jrocnik@42159: Determine the inverse $\cal{z}$ 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: 
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		CALC                         %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: 
jrocnik@42159: %\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation}
jrocnik@42159: %TODO
jrocnik@42159: %\end{frame}
neuper@42074: 
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42122: %%												Z expl		REQ	                         %%
neuper@42122: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
neuper@42076: 
neuper@42122: 
neuper@42122: \begin{frame}\frametitle{(Inverse) ${\cal Z}$-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: 
neuper@42074: \end{document}
neuper@42074: 
neuper@42074: 




