\documentclass{beamer}

\mode<presentation>

{

  \usetheme{Hannover}

  \setbeamercovered{transparent}

}

%\usepackage{setspace} %for "\begin{onehalfspace}"

\usepackage[english]{babel}

% or whatever

\usepackage[utf8]{inputenc}

% or whatever

\usepackage{times}

\usepackage[T1]{fontenc}

% Or whatever. Note that the encoding and the font should match. If T1

% does not look nice, try deleting the line with the fontenc.

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

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

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

{TODO}

\subtitle{TODO}

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

{Jan~Rocnik}

% - Give the names in the same order as the appear in the paper.

% - Use the \inst{?} command only if the authors have different

%   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.

% - Not really informative to the audience, more for people (including

%   yourself) who are reading the slides online

% \subject{Theoretical Computer Science}

% This is only inserted into the PDF information catalog. Can be left

% out.

% If you have a file called "university-logo-filename.xxx", where xxx

% is a graphic format that can be processed by latex or pdflatex,

% resp., then you can add a logo as follows:

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}

% \logo{\pgfuseimage{university-logo}}

% Delete this, if you do not want the table of contents to pop up at

% the beginning of each subsection:

\AtBeginSubsection[]

{

  \begin{frame}<beamer>{Outline}

    \tableofcontents[currentsection,currentsubsection]

  \end{frame}

}

% If you wish to uncover everything in a step-wise fashion, uncomment

% the following command:

%\beamerdefaultoverlayspecification{<+->}

\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}