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\usepackage{endnotes}

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\usepackage{setspace}

\usepackage{isabelle}

\usepackage{isabellesym}

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\bibliographystyle{alpha}

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

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

\begin{document}

\title{

	\Large{

  	\bf Interactive Course Material for Signal Processing based on Isabelle/\isac\\~\\

  }

	\sisac-Team in Cooperation with \\~\\

	Institute for Software Technology\\

	Institute of Signal Processing and Speech Communication\\~\\

	Graz University of Technology\\

	\vspace{0.7cm}

	\normalsize{

		Supervisor: Univ.-Prof. Dipl.-Ing. Dr.techn. Wotawa Franz

	}

}

\author{Jan Simon Rocnik\\\href{mailto:student.tugraz.at}{\tt jan.rocnik@student.tugraz.at}}

\date{\today}

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\maketitle

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\begin{center}

Special Thanks to\\

\hfill \\

\emph{Dr.techn. Walther Neuper}\\

\emph{Dipl.-Ing. Bernhard Geiger}

\end{center}

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\begin{abstract}

The Baccalaureate Thesis creates interactive course material for Signal Processing based on the experimental math assistant Isabelle/\sisac.

The content of the course material is defined together with the Signal Processing and Speech Communication Laboratory (SPSC Lab) of Graz University of Technology (TUG). The content is planned to be used in specific lectures and labs of the SPSC and thus is thoroughly concerned with underlying mathematical and physical theory.

One challenge of this thesis is, that theory is not yet mechanized in Computer Theorem Provers (CTP); so this thesis will provide preliminary definitions in so-called \emph{theories} of the CTP Isabelle and theorems without proofs.

Another callenge is the implementation of interactive courses: this is done within the educational math assistant Isabelle/\sisac, which is under development at TU Graz. The present state of \sisac{} happens to provide the {\em first} occasion for authoring by a non-member of the \sisac{} developer team. So this challenge involves  alpha-testing of the underlying ``CTP-based programming language'', because error messages are still not user-friendly and need frequent contact with \sisac-developers.

So the practical outcome of this thesis is twofold:

(1) interactive course material hopefully useful in education within the SPSC Lab and within STEOP, the introductory orientation phase at TUG, as a preview for students in Telematics on later application of math knowledge introduced in the first semester and

(2) a detailed description of technicalities in programming implemented as an interactive Isabelle/Isar theory, providing future programmers with guidelines and \sisac-developers with feedback in usability of the CTP-based program language. 

\end{abstract}\clearpage

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This thesis is structured as follows:

\section{Introduction --TODO--}

The motivation to this thesis mainly takes it source from thr practice of mathematics learning and further ... STEOP

\textbf{mathematics applied} in signal processing (SP)

mathematics mechanized in Computer Theorem Provers \textbf{(CTP)} ... (almost) traditional mathematical notation (predicate calculus) for axioms, definitions, lemmas, theorems. Recent developments provide also proofs in a humand readable format TODO

This thesis tries to \textbf{connect these two worlds} ... this trial is one of the first; others see related work

the major challenge of the practical part is, that ``connecting the two worlds'' involves programming in a CTP-based programming language which is in a very early state of prototyping.

\subsection{Mechanization of Mathematics}

todo

hughe theories of mathematics

still a hugh gap between these theories and ``real applications'' e.g. in SP

? academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modelling --- mechanized math starts from mathematical models and (hopefully !) proceeds to match physics.

CTP Isabelle ... survey of knowledge, links to knowledge

\paragraph{\sisac}

TODO

adds an ``application'' axis (formal specifications of problems) and an ``algorithmic'' axis to the ``deductive'' axis of knowledge ... 3-dimensional universe of mathematics.

\subsection{Goals of the Thesis}

todo

\subsection{Milestones for the Project}

Die Planung des Projekts teilt sich in folgende Iterationen:

\begin{enumerate}

\item \textbf{Sammeln von Informationen "uber Themengebiete und deren Realisierbarkeit } (29.06. -- 27.07.)

identify problems relevant for certain SP lectures

estimate chances to realized them within the scope of this thesis

order for implementing the problems negotiated with lecturers

\item \textbf{1. Prsentation - Auswhlen der realisierbaren Themengebiete} (27.07.)

\item \textbf{Ausarbeiten der Aufgaben in \isac} (01.09. -- 11.11.)

\item \textbf{Dokumentation der Aufgaben} (14.11. -- 02.12.)

\item \textbf{Ausarbeitung in Latex, Bakkarbeit} (05.12. -- todo)

\item \textbf{2. Prsentation - Abschluss der Arbeit} (todo)

\end{enumerate}

\subsection{Structure of the Thesis}

todo

\section{Mechanization of Mathematics for SP Problems}

todo

\subsection{Relevant Knowledge available in Isabelle}

todo

\paragraph{example FFT}, describe in detail !!!! 

? different meaning: FFT in Maple

gap between what is available and what is required (@)!

traditional notation ?

\subsection{Relevant Knowledge available in isac}

todo

specifications (``application axis'') and methods (``algorithmic axis'')

partial fractions, cancellation of multivariate rational terms, ...

\subsection{Survey: Available Knowledge and Selected Problems}

todo

estimate gap (@) for each problem (tables)

conclusion: following order for implementing the problems ...

\subsection{Formalization of missing knowledge in Isabelle}

todo

axiomatization ... where ... and

\subsection{Notes on Problems with Traditional Notation}

todo

u[n] !!

f x =  why not f(x) ?!?!

...

terms are not full simplified in traditional notations, in isac we have to simplify them complete to check weather results are compatible or not. in e.g. the solutions of an second order linear equation is an rational in isac but in tradition we keep fractions as long as possible and as long as they are 'beautiful' (1/8, 5/16,...)

\section{Implementation of Certain SP Problems}

todo

\subsection{Formal Specification of Problems}

todo

\subsection{Methods Solving the Problems}

todo

\subsection{Integration of Subproblems available in isac}

todo

\subsection{Examples and Multimedia Content}

todo

\section{Related Work and Open Questions}

todo

See ``introduction'': This thesis tries to connect these two worlds ... this trial is one of the first; others see related work

\section{Beschreibung der Meilensteine}\label{ms-desc}

todo

\section{Bericht zum Projektverlauf}

todo

\section{Abschliesende Bemerkungen}

todo

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%\bibliography{bib}

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\appendix

%\section*{Anhang}

\section{Demobeispiel}

%\input{./Inverse_Z_Transform/document/Inverse_Z_Transform.tex}

\begin{verbatim}

bsp

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\section{Stundenliste}

\subsection*{Voraussetzungen zum Arbeitsbeginn schaffen}

\begin{tabular}[t]{lll}

    {\bf Datum} & {\bf Stunden} & {\bf Beschreibung} \\

    10.02.2011 & 2:00 & Besprechung der Problemstellung \\

\end{tabular}

\section{Calculations}

\input{calulations}

\end{document}