137 \begin{frame}\frametitle{TODO} |
137 \begin{frame}\frametitle{TODO} |
138 TODO |
138 TODO |
139 \end{frame} |
139 \end{frame} |
140 |
140 |
141 \subsection[Transform]{Fourier transform} |
141 \subsection[Transform]{Fourier transform} |
142 \begin{frame}\frametitle{Fourier transform expl 1} |
142 \begin{frame}\frametitle{FT expl 1} |
143 TODO |
143 TODO |
144 \end{frame} |
144 \end{frame} |
145 |
145 |
146 \begin{frame}\frametitle{Fourier transform expl 2a} |
146 \begin{frame}\frametitle{FT expl 2a} |
147 TODO |
147 TODO |
148 \end{frame} |
148 \end{frame} |
149 |
149 |
150 \begin{frame}\frametitle{Fourier transform expl 2b} |
150 \begin{frame}\frametitle{FT expl 2b} |
151 Aufgabenstellung von Bernhard |
151 Problem (from Bernhard) |
152 \end{frame} |
152 \end{frame} |
153 |
153 |
154 \begin{frame}\frametitle{Fourier transform expl 2b} |
154 \begin{frame}\frametitle{FT expl 2b: specification } |
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155 {\footnotesize\it |
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156 fourier transform |
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157 \begin{tabbing} |
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158 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
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159 \>given \>:\> piecewise\_function \\ |
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160 \> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\ |
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161 %?(iterativer) datentyp in Isabelle/HOL |
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162 \> \> \> translation $T=2$\\ |
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163 \>precond \>:\> TODO\\ |
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164 \>find \>:\> $X(j\cdot\omega)$\\ |
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165 \>postcond \>:\> TODO\\ |
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166 \end{tabbing} |
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167 |
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168 } |
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169 \end{frame} |
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170 |
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171 \begin{frame}\frametitle{FT expl 2b: calculation} |
155 \footnotesize{ |
172 \footnotesize{ |
156 \begin{tabbing} |
173 \begin{tabbing} |
157 000\=\kill |
174 000\=\kill |
158 01 \> ${\cal F}\;(x(t-2)) =$\\ |
175 01 \> ${\cal F}\;(x(t-2)) =$\\ |
159 \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\ |
176 \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\ |
162 03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
179 03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
163 \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\ |
180 \` $x\;t = 1\;{\it for}\;\{x.\;-1\leq t\;\land\;t\leq 1\}\;{\it and}\;x\;t=0\;{\it otherwise}$\\ |
164 04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
181 04 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-1}^1 1\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
165 \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\ |
182 \` $\int_a^b f\;t\;dt = \int f\;t\;dt\;|_a^b$\\ |
166 05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\ |
183 05 \> $e^{-j\cdot\omega\cdot 2}\cdot \int 1\cdot e^{-j\cdot\omega\cdot t} d t\;|_{-1}^1$\\ |
167 \` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\ |
184 %\` $\int e^{a\cdot t} = \frac{1}{a}\cdot e^{a\cdot t}$\\ |
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185 \` pbl: integration in $\cal C$\\ |
168 06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\ |
186 06 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot t} \;|_{-1}^1)$\\ |
169 \` $f\;t\;|_a^b = f\;b-f\;a$\\ |
187 \` $f\;t\;|_a^b = f\;b-f\;a$\\ |
170 07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} - \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\ |
188 07 \> $e^{-j\cdot\omega\cdot 2}\cdot (\frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot 1} - \frac{1}{-j\cdot\omega}\cdot e^{-j\cdot\omega\cdot -1})$\\ |
171 \vdots\` simplification+factorization in $\cal C$\\ |
189 \vdots\` pbl: simplification+factorization in $\cal C$\\ |
172 08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\ |
190 08 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{-j\cdot\omega}\cdot(e^{j\cdot\omega} - e^{-j\cdot\omega})$\\ |
173 \` trick~!\\ |
191 \` trick~!\\ |
174 09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\ |
192 09 \> $e^{-j\cdot\omega\cdot 2}\cdot \frac{1}{\omega}\cdot(\frac{-e^{j\cdot\omega} + e^{-j\cdot\omega}}{j})$\\ |
175 \` table\\ |
193 \` table\\ |
176 10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$ |
194 10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$ |
177 \end{tabbing} |
195 \end{tabbing} |
178 } |
196 } |
179 \end{frame} |
197 \end{frame} |
180 |
198 |
181 \begin{frame}\frametitle{Fourier transform expl 2b} |
199 \begin{frame}\frametitle{FT expl 2b} |
182 Voraussetzungen |
200 prerequisites |
183 |
201 \end{frame} |
184 |
202 |
185 \end{frame} |
203 \section[Discrete time]{Discrete-time systems} |
186 |
204 \subsection[Convolution]{Convolution} |
187 \begin{frame}\frametitle{Fourier transform expl 2b - table} |
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188 TODO |
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189 \end{frame} |
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190 |
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191 \section[Convolution]{Convolution} |
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192 %\subsection[]{} |
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193 \begin{frame}\frametitle{Convolution (Faltung)} |
205 \begin{frame}\frametitle{Convolution (Faltung)} |
194 TODO |
206 TODO |
195 \end{frame} |
207 \end{frame} |
196 |
208 |
197 \section[${\cal Z}$ transform]{${\cal Z}$ transform} |
209 \section[${\cal Z}$ transform]{${\cal Z}$ transform} |
198 %\subsection[]{} |
210 %\subsection[]{} |
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211 \begin{frame}\frametitle{TODO} |
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212 TODO |
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213 \end{frame} |
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214 |
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215 \subsection[]{Indextranformation} |
199 \begin{frame}\frametitle{TODO} |
216 \begin{frame}\frametitle{TODO} |
200 TODO |
217 TODO |
201 \end{frame} |
218 \end{frame} |
202 |
219 |
203 \subsection[Inverse ${\cal Z}$]{Inverse ${\cal Z}$ transform} |
220 \subsection[Inverse ${\cal Z}$]{Inverse ${\cal Z}$ transform} |