8 \setbeamercovered{transparent}
11 %\usepackage{setspace} %for "\begin{onehalfspace}"
12 \usepackage[english]{babel}
15 \usepackage[utf8]{inputenc}
19 \usepackage[T1]{fontenc}
20 % Or whatever. Note that the encoding and the font should match. If T1
21 % does not look nice, try deleting the line with the fontenc.
23 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
24 \def\sisac{{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}}
26 \title[TODO] % (optional, use only with long paper titles)
31 \author[Rocnik] % (optional, use only with lots of authors)
33 % - Give the names in the same order as the appear in the paper.
34 % - Use the \inst{?} command only if the authors have different
37 \institute % (optional, but mostly needed)
39 Technische Universit\"at Graz\\
42 % - Use the \inst command only if there are several affiliations.
43 % - Keep it simple, no one is interested in your street address.
45 % \date[CFP 2003] % (optional, should be abbreviation of conference name)
46 % {Conference on Fabulous Presentations, 2003}
47 % - Either use conference name or its abbreviation.
48 % - Not really informative to the audience, more for people (including
49 % yourself) who are reading the slides online
51 % \subject{Theoretical Computer Science}
52 % This is only inserted into the PDF information catalog. Can be left
57 % If you have a file called "university-logo-filename.xxx", where xxx
58 % is a graphic format that can be processed by latex or pdflatex,
59 % resp., then you can add a logo as follows:
61 % \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
62 % \logo{\pgfuseimage{university-logo}}
66 % Delete this, if you do not want the table of contents to pop up at
67 % the beginning of each subsection:
70 \begin{frame}<beamer>{Outline}
71 \tableofcontents[currentsection,currentsubsection]
76 % If you wish to uncover everything in a step-wise fashion, uncomment
77 % the following command:
79 %\beamerdefaultoverlayspecification{<+->}
84 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
86 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
92 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
93 %% Table of Contents %%
94 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
96 \begin{frame}{Outline}
98 % You might wish to add the option [pausesections]
101 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
102 %%---------------------------------------------------------------%%
103 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
105 \section[Fourier]{Fourier transform}
107 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
109 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
111 \begin{frame}\frametitle{Fourier Transformation: Introduction}
113 \item Transform operation by using property-tables $\rightarrow$ \emph{easy}
114 \item Transform operation by using integral $\rightarrow$ \emph{difficult}
115 \item No math \emph{tricks}
116 \item Important: Visualisation?!
120 \subsection[simple]{Fourier transform Example 1}
122 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
123 %% Transform expl 1 SPEC %%
124 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
126 \begin{frame}\frametitle{Fourier Transform 1: Specification}
130 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
131 \>given \>:\> Time continiues, not periodic Signal \\
132 \> \> \> \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\
133 \>precond \>:\> TODO\\
134 \>find \>:\> $X(j\cdot\omega)$\\
135 \>postcond \>:\> TODO\\
140 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
141 %% Transform expl 1 CALC %%
142 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
144 \begin{frame}\frametitle{Fourier Transformation 1: Calculation}
145 TODO: Bernhard fragen ob Tabelle oder Rechnung
148 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
149 %% Transform expl 1 REQ %%
150 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
152 \begin{frame}\frametitle{Fourier Transform 1: Development effort}
155 \begin{tabular}{l|l|r}
156 requirements & comments &effort\\ \hline\hline
157 solving Intrgrals & simple via propertie table & 20\\
158 & \emph{real} & MT\\ \hline
159 transformation table & simple transform & 20\\ \hline
160 example collection & with explanations & 20\\ \hline\hline
164 effort --- in 45min units\\
165 MT --- thesis ``Integrals'' (mathematics)
169 \subsection[difficult]{Fourier transform Example 2}
171 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
172 %% Transform expl 2 SPEC %%
173 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
175 \begin{frame}\frametitle{Fourier Transform 2: Specification}
179 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
180 \>given \>:\> piecewise\_function \\
181 \> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\
182 %?(iterativer) datentyp in Isabelle/HOL
183 \> \> \> translation $T=2$\\
184 %WN these 2 inputs calculated to [(0,-\infty<t<-1), (1,-1\leq t\leq 1), (0, 1<t<\infty)]
185 %WN translation helpful only, if solution available from other calculation - DROP T ???
186 \>precond \>:\> TODO\\
187 \>find \>:\> $X(j\cdot\omega)$\\
188 \>postcond \>:\> TODO\\
194 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
195 %% Transform expl 2 CALC %%
196 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
198 \begin{frame}\frametitle{Fourier Transform 2: Calculation}
203 01 \> ${\cal F}\;(x(t-2)) =$\\
204 \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\
205 02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\
206 \`definition $X\;(j\cdot\omega)$\\
207 03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\
208 \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\
209 04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\
210 \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\
211 05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\
212 %\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\
213 \` pbl: integration in $\cal C$\\
214 06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\
215 \` $f\;t\;|_a^b = f\;b-f\;a$\\
216 07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} - \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\
217 \vdots\` pbl: simplification+factorization in $\cal C$\\
218 08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\
220 09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\
222 10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$
227 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
228 %% Transform expl 2 REQ %%
229 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
232 \begin{frame}\frametitle{Fourier Transform 2: Development effort}
235 \begin{tabular}{l|l|r}
236 requirements & comments &effort\\ \hline\hline
237 solving Intrgrals & simple via propertie table & 20\\
238 & \emph{real} & MT\\ \hline
239 transformation table & simple transform & 20\\ \hline
240 visualisation & backend & 10\\ \hline
241 example collection & with explanations & 20\\ \hline\hline
245 effort --- in 45min units\\
246 MT --- thesis ``Integrals'' (mathematics)
249 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
250 %-----------------------------------------------------------------%
251 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
253 \section[Discrete time]{Discrete-time systems}
254 \subsection[Convolution]{Convolution}
256 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
258 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
260 \begin{frame}\frametitle{Convolution: Introduction}
262 \item Calculation\ldots
263 \item Visualisation\ldots
266 \ldots of parallel filter structures
270 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
272 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
274 \begin{frame}\frametitle{Convolution: Specification}
278 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
279 \>given \>:\> Signals h1[n], h2[n] \\
280 \> \> \> \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\
281 %?(iterativer) datentyp in Isabelle/HOL
282 \>precond \>:\> TODO\\
283 \>find \>:\> $h1[n]\,*\,h2[n]$\\
284 \>postcond \>:\> TODO\\
290 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
292 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
294 \begin{frame}\frametitle{Convolution: Calculation}
298 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
300 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
302 \begin{frame}\frametitle{Convolution: Development effort}
305 \begin{tabular}{l|l|r}
306 requirements & comments &effort\\ \hline\hline
307 simplify rationals & \sisac & 0\\ \hline
308 define $\sum\limits_{i=0}^{n}i$ & partly \sisac & 10\\ \hline
309 simplify sum & termorder & 10\\
310 & simplify rules & 20\\
311 & use simplify rationals& 0\\ \hline
312 index adjustments & with unit step & 10\\ \hline
313 example collection & with explanations & 20\\ \hline\hline
317 effort --- in 45min units\\
320 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
321 %-----------------------------------------------------------------%
322 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
324 \section[Z-transform]{Z-Transform}
325 \subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform}
327 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
328 %% Z-Transform INTRO %%
329 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
331 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction}
333 \item Pure Transformation is simple to realise with Z-Transform Properties (Table)
334 \item Partial Fraction are just math simplifications
339 %\subsection[]{Indextranformation}
340 %\begin{frame}\frametitle{TODO}
344 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
345 %% Z-Transform SPEC %%
346 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
348 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification}
352 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
353 \>given \>:\> Expression of z \\
354 \> \> \> \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\
355 \>precond \>:\> TODO\\
356 \>find \>:\> Expression of n\\
358 \>postcond \>:\> TODO\\
364 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
366 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
368 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation}
372 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
374 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
377 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Development effort}
380 \begin{tabular}{l|l|r}
381 requirements & comments &effort\\ \hline\hline
382 solve for part.fract. & \sisac: degree 2 & 0\\
383 & complex nomminators & 30\\
384 & degree > 2 & MT\\ \hline
385 simplify polynomial & \sisac & 0\\
386 simplify rational & \sisac & 0\\ \hline
387 part.fract.decomposition& degree 2 & \\
388 & specification, method& 30\\ \hline
389 ${\cal Z}^{-1}$ table & & 20\\
390 & explanations, figures& 20\\ \hline
391 example collection & with explanations & 20\\ \hline\hline
396 effort --- in 45min units\\
397 MT --- thesis ``factorization'' (mathematics)