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}
131 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
132 \>given \>:\> Time continiues, not periodic Signal \\
133 \> \> \> \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\
134 \>precond \>:\> TODO\\
135 \>find \>:\> $X(j\cdot\omega)$\\
136 \>postcond \>:\> TODO\\
141 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
142 %% Transform expl 1 CALC %%
143 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
145 \begin{frame}\frametitle{Fourier Transformation 1: Calculation}
146 TODO: Bernhard fragen ob Tabelle oder Rechnung
149 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
150 %% Transform expl 1 REQ %%
151 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
153 \begin{frame}\frametitle{Fourier Transform 1: Development effort}
156 \begin{tabular}{l|l|r}
157 requirements & comments &effort\\ \hline\hline
158 solving Intrgrals & simple via propertie table & 20\\
159 & \emph{real} & MT\\ \hline
160 transformation table & simple transform & 20\\ \hline
161 example collection & with explanations & 20\\ \hline\hline
165 effort --- in 45min units\\
166 MT --- thesis ``Integrals'' (mathematics)
170 \subsection[difficult]{Fourier transform Example 2}
172 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
173 %% Transform expl 2 SPEC %%
174 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
176 \begin{frame}\frametitle{Fourier Transform 2: Specification}
179 \textbf{(a)} Determine the fourier transform for the given rectangular impulse:
191 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
192 \>given \>:\> piecewise\_function \\
193 \> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\
194 %?(iterativer) datentyp in Isabelle/HOL
195 \> \> \> translation $T=2$\\
196 \>precond \>:\> TODO\\
197 \>find \>:\> $X(j\cdot\omega)$\\
198 \>postcond \>:\> TODO\\
204 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
205 %% Transform expl 2 CALC %%
206 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
208 %\begin{frame}\frametitle{Fourier Transform 2: Calculation}
212 %01 \> ${\cal F}\;(x(t-2)) =$\\
213 % \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\
214 %02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\
215 % \`definition $X\;(j\cdot\omega)$\\
216 %03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\
217 % \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\
218 %04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\
219 % \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\
220 %05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\
221 % %\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\
222 % \` pbl: integration in $\cal C$\\
223 %06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\
224 % \` $f\;t\;|_a^b = f\;b-f\;a$\\
225 %07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} - \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\
226 %\vdots\` pbl: simplification+factorization in $\cal C$\\
227 %08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\
229 %09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\
231 %10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$
236 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
237 %% Transform expl 2 REQ %%
238 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
240 \begin{frame}\frametitle{Fourier Transform 2: Development effort}
243 \begin{tabular}{l|l|r}
244 requirements & comments &effort\\ \hline\hline
245 solving Intrgrals & simple via propertie table & 20\\
246 & \emph{real} & MT\\ \hline
247 transformation table & simple transform & 20\\ \hline
248 visualisation & backend & 10\\ \hline
249 example collection & with explanations & 20\\ \hline\hline
253 effort --- in 45min units\\
254 MT --- thesis ``Integrals'' (mathematics)
257 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
258 %-----------------------------------------------------------------%
259 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
261 \section[Discrete time]{Discrete-time systems}
262 \subsection[Convolution]{Convolution}
264 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
266 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
268 \begin{frame}\frametitle{Convolution: Introduction}
270 \item Calculation\ldots
271 \item Visualisation\ldots
274 \ldots of parallel filter structures
278 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
280 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
282 \begin{frame}\frametitle{Convolution: Specification}
285 Consider the two discrete-time, linear and time-invariant (LTI) systems with the following impulse response:
288 $h_1[n]=\left(\frac{3}{5}\right)^n\cdot u[n]$\\
289 $h_1[n]=\left(-\frac{2}{3}\right)^n\cdot u[n]$
292 The two systems are cascaded seriell. Derive the impulse respinse of the overall system $h_c[n]$.
295 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
296 \>given \>:\> Signals h1[n], h2[n] \\
297 \> \> \> \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\
298 %?(iterativer) datentyp in Isabelle/HOL
299 \>precond \>:\> TODO\\
300 \>find \>:\> $h1[n]\,*\,h2[n]$\\
301 \>postcond \>:\> TODO\\
307 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
309 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
311 %\begin{frame}\frametitle{Convolution: Calculation}
315 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
319 \begin{frame}\frametitle{Convolution: Development effort}
322 \begin{tabular}{l|l|r}
323 requirements & comments &effort\\ \hline\hline
324 simplify rationals & \sisac & 0\\ \hline
325 define $\sum\limits_{i=0}^{n}i$ & partly \sisac & 10\\ \hline
326 simplify sum & termorder & 10\\
327 & simplify rules & 20\\
328 & use simplify rationals& 0\\ \hline
329 index adjustments & with unit step & 10\\ \hline
330 example collection & with explanations & 20\\ \hline\hline
334 effort --- in 45min units\\
337 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
338 %-----------------------------------------------------------------%
339 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
341 \section[Z-transform]{Z-Transform}
342 \subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform}
344 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
345 %% Z-Transform INTRO %%
346 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
348 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction}
350 \item Pure Transformation is simple to realise with Z-Transform Properties (Table)
351 \item Partial Fraction are just math simplifications
356 %\subsection[]{Indextranformation}
357 %\begin{frame}\frametitle{TODO}
361 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
362 %% Z-Transform SPEC %%
363 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
365 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification}
368 Determine the inverse $\cal{z}$ transform of the following expression. Hint: applay the partial fraction expansion.
371 $X(z)=\frac{3}{z-\frac{1}{4}-\frac{1}{8}z^{-1}},\ \ x[n]$ is absolute summable
375 1\=postcond \=: \= \= $\;\;\;\;$\=\kill
376 \>given \>:\> Expression of z \\
377 \> \> \> \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\
378 \>precond \>:\> TODO\\
379 \>find \>:\> Expression of n\\
381 \>postcond \>:\> TODO\\
387 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
389 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
391 %\begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation}
395 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
397 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
400 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Development effort}
403 \begin{tabular}{l|l|r}
404 requirements & comments &effort\\ \hline\hline
405 solve for part.fract. & \sisac: degree 2 & 0\\
406 & complex nomminators & 30\\
407 & degree > 2 & MT\\ \hline
408 simplify polynomial & \sisac & 0\\
409 simplify rational & \sisac & 0\\ \hline
410 part.fract.decomposition& degree 2 & \\
411 & specification, method& 30\\ \hline
412 ${\cal Z}^{-1}$ table & & 20\\
413 & explanations, figures& 20\\ \hline
414 example collection & with explanations & 20\\ \hline\hline
419 effort --- in 45min units\\
420 MT --- thesis ``factorization'' (mathematics)