103 %\beamerdefaultoverlayspecification{<+->} |
79 %\beamerdefaultoverlayspecification{<+->} |
104 |
80 |
105 |
81 |
106 \begin{document} |
82 \begin{document} |
107 |
83 |
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84 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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85 %% Title Page %% |
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86 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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87 |
108 \begin{frame} |
88 \begin{frame} |
109 \titlepage |
89 \titlepage |
110 \end{frame} |
90 \end{frame} |
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91 |
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92 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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93 %% Table of Contents %% |
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94 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
111 |
95 |
112 \begin{frame}{Outline} |
96 \begin{frame}{Outline} |
113 \tableofcontents |
97 \tableofcontents |
114 % You might wish to add the option [pausesections] |
98 % You might wish to add the option [pausesections] |
115 \end{frame} |
99 \end{frame} |
116 |
100 |
117 |
101 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
118 % Structuring a talk is a difficult task and the following structure |
102 %%---------------------------------------------------------------%% |
119 % may not be suitable. Here are some rules that apply for this |
103 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
120 % solution: |
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121 |
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122 % - Exactly two or three sections (other than the summary). |
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123 % - At *most* three subsections per section. |
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124 % - Talk about 30s to 2min per frame. So there should be between about |
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125 % 15 and 30 frames, all told. |
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126 |
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127 % - A conference audience is likely to know very little of what you |
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128 % are going to talk about. So *simplify*! |
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129 % - In a 20min talk, getting the main ideas across is hard |
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130 % enough. Leave out details, even if it means being less precise than |
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131 % you think necessary. |
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132 % - If you omit details that are vital to the proof/implementation, |
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133 % just say so once. Everybody will be happy with that. |
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134 |
104 |
135 \section[Fourier]{Fourier transform} |
105 \section[Fourier]{Fourier transform} |
136 \subsection[Series]{Fourier series} |
106 |
137 \begin{frame}\frametitle{TODO} |
107 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
138 TODO |
108 %% Fourier INTRO %% |
139 \end{frame} |
109 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
140 |
110 |
141 \subsection[Transform]{Fourier transform} |
111 \begin{frame}\frametitle{Fourier Transformation: Introduction} |
142 \begin{frame}\frametitle{FT expl 1} |
112 \begin{itemize} |
143 TODO |
113 \item Transform operation by using property-tables $\rightarrow$ \emph{easy} |
144 \end{frame} |
114 \item Transform operation by using integral $\rightarrow$ \emph{difficult} |
145 |
115 \item No math \emph{tricks} |
146 \begin{frame}\frametitle{FT expl 2a} |
116 \item Important: Visualisation?! |
147 TODO |
117 \end{itemize} |
148 \end{frame} |
118 \end{frame} |
149 |
119 |
150 \begin{frame}\frametitle{FT expl 2b} |
120 \subsection[simple]{Fourier transform Example 1} |
151 Problem (from Bernhard) |
121 |
152 \end{frame} |
122 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
153 |
123 %% Transform expl 1 SPEC %% |
154 \begin{frame}\frametitle{FT expl 2b: specification } |
124 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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125 |
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126 \begin{frame}\frametitle{Fourier Transform 1: Specification} |
155 {\footnotesize\it |
127 {\footnotesize\it |
156 fourier transform |
128 Fourier Transform |
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129 \begin{tabbing} |
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130 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
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131 \>given \>:\> Time continiues, not periodic Signal \\ |
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132 \> \> \> \>$(x (t::real), exp(-\,(\alpha::real\,+\,\alpha::imag)\,*\,t::real)*u(t::real))$\\ |
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133 \>precond \>:\> TODO\\ |
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134 \>find \>:\> $X(j\cdot\omega)$\\ |
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135 \>postcond \>:\> TODO\\ |
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136 \end{tabbing} |
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137 |
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138 } |
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139 \end{frame} |
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140 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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141 %% Transform expl 1 CALC %% |
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142 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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143 |
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144 \begin{frame}\frametitle{Fourier Transformation 1: Calculation} |
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145 TODO: Bernhard fragen ob Tabelle oder Rechnung |
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146 \end{frame} |
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147 |
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148 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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149 %% Transform expl 1 REQ %% |
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150 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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151 |
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152 \begin{frame}\frametitle{Fourier Transform 1: Development effort} |
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153 {\small |
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154 \begin{center} |
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155 \begin{tabular}{l|l|r} |
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156 requirements & comments &effort\\ \hline\hline |
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157 solving Intrgrals & simple via propertie table & 20\\ |
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158 & \emph{real} & MT\\ \hline |
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159 transformation table & simple transform & 20\\ \hline |
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160 example collection & with explanations & 20\\ \hline\hline |
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161 & & 60-80\\ |
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162 \end{tabular} |
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163 \end{center} |
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164 effort --- in 45min units\\ |
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165 MT --- thesis ``Integrals'' (mathematics) |
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166 } |
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167 \end{frame} |
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168 |
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169 \subsection[difficult]{Fourier transform Example 2} |
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170 |
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171 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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172 %% Transform expl 2 SPEC %% |
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173 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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174 |
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175 \begin{frame}\frametitle{Fourier Transform 2: Specification} |
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176 {\footnotesize\it |
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177 Fourier Transform |
157 \begin{tabbing} |
178 \begin{tabbing} |
158 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
179 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
159 \>given \>:\> piecewise\_function \\ |
180 \>given \>:\> piecewise\_function \\ |
160 \> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\ |
181 \> \> \> \>$(x (t::real), [(0,-\infty<t<1), (1,1\leq t\leq 3), (0, 3<t<\infty)])$\\ |
161 %?(iterativer) datentyp in Isabelle/HOL |
182 %?(iterativer) datentyp in Isabelle/HOL |
162 \> \> \> translation $T=2$\\ |
183 \> \> \> translation $T=2$\\ |
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184 %WN these 2 inputs calculated to [(0,-\infty<t<-1), (1,-1\leq t\leq 1), (0, 1<t<\infty)] |
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185 %WN translation helpful only, if solution available from other calculation - DROP T ??? |
163 \>precond \>:\> TODO\\ |
186 \>precond \>:\> TODO\\ |
164 \>find \>:\> $X(j\cdot\omega)$\\ |
187 \>find \>:\> $X(j\cdot\omega)$\\ |
165 \>postcond \>:\> TODO\\ |
188 \>postcond \>:\> TODO\\ |
166 \end{tabbing} |
189 \end{tabbing} |
167 |
190 |
168 } |
191 } |
169 \end{frame} |
192 \end{frame} |
170 |
193 |
171 \begin{frame}\frametitle{FT expl 2b: calculation} |
194 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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195 %% Transform expl 2 CALC %% |
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196 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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197 |
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198 \begin{frame}\frametitle{Fourier Transform 2: Calculation} |
172 \footnotesize{ |
199 \footnotesize{ |
173 \begin{tabbing} |
200 \begin{tabbing} |
174 000\=\kill |
201 000\=\kill |
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202 %WN first |
175 01 \> ${\cal F}\;(x(t-2)) =$\\ |
203 01 \> ${\cal F}\;(x(t-2)) =$\\ |
176 \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\ |
204 \`${\cal F}\;(x(t-T)) = e^{-j\cdot\omega\cdot T}\cdot X\;j\cdot\omega$\\ |
177 02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\ |
205 02 \> $e^{-j\cdot\omega\cdot 2}\cdot X\;(j\cdot\omega)$\\ |
178 \`definition $X\;(j\cdot\omega)$\\ |
206 \`definition $X\;(j\cdot\omega)$\\ |
179 03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
207 03 \> $e^{-j\cdot\omega\cdot 2}\cdot \int_{-\infty}^\infty x\;t\;\cdot e^{-j\cdot\omega\cdot t} d t$\\ |
194 10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$ |
222 10 \> $e^{-j\cdot\omega\cdot 2}\cdot 2\cdot\frac{\sin\;\omega}{\omega}$ |
195 \end{tabbing} |
223 \end{tabbing} |
196 } |
224 } |
197 \end{frame} |
225 \end{frame} |
198 |
226 |
199 \begin{frame}\frametitle{FT expl 2b} |
227 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
200 prerequisites |
228 %% Transform expl 2 REQ %% |
201 \end{frame} |
229 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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230 |
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231 %WN ... |
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232 \begin{frame}\frametitle{Fourier Transform 2: Development effort} |
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233 {\small |
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234 \begin{center} |
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235 \begin{tabular}{l|l|r} |
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236 requirements & comments &effort\\ \hline\hline |
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237 solving Intrgrals & simple via propertie table & 20\\ |
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238 & \emph{real} & MT\\ \hline |
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239 transformation table & simple transform & 20\\ \hline |
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240 visualisation & backend & 10\\ \hline |
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241 example collection & with explanations & 20\\ \hline\hline |
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242 & & 70-80\\ |
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243 \end{tabular} |
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244 \end{center} |
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245 effort --- in 45min units\\ |
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246 MT --- thesis ``Integrals'' (mathematics) |
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247 } |
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248 \end{frame} |
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249 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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250 %-----------------------------------------------------------------% |
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251 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
202 |
252 |
203 \section[Discrete time]{Discrete-time systems} |
253 \section[Discrete time]{Discrete-time systems} |
204 \subsection[Convolution]{Convolution} |
254 \subsection[Convolution]{Convolution} |
205 \begin{frame}\frametitle{Convolution (Faltung)} |
255 |
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256 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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257 %% DTS INTRO %% |
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258 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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259 |
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260 \begin{frame}\frametitle{Convolution: Introduction} |
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261 \begin{itemize} |
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262 \item Calculation\ldots |
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263 \item Visualisation\ldots |
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264 \end{itemize} |
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265 \begin{center} |
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266 \ldots of parallel filter structures |
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267 \end{center} |
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268 \end{frame} |
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269 |
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270 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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271 %% DTS SPEC %% |
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272 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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273 |
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274 \begin{frame}\frametitle{Convolution: Specification} |
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275 {\footnotesize\it |
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276 Convolution |
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277 \begin{tabbing} |
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278 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
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279 \>given \>:\> Signals h1[n], h2[n] \\ |
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280 \> \> \> \>((h1 (n::real),(3/5)\textasciicircum{}n\,u(n::real)),\,(h2 (n::real),(-2/3)\textasciicircum{}n\,u(n::real)))\\ |
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281 %?(iterativer) datentyp in Isabelle/HOL |
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282 \>precond \>:\> TODO\\ |
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283 \>find \>:\> $h1[n]\,*\,h2[n]$\\ |
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284 \>postcond \>:\> TODO\\ |
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285 \end{tabbing} |
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286 |
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287 } |
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288 \end{frame} |
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289 |
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290 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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291 %% DTS CALC %% |
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292 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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293 |
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294 \begin{frame}\frametitle{Convolution: Calculation} |
206 TODO |
295 TODO |
207 \end{frame} |
296 \end{frame} |
208 |
297 |
209 \section[${\cal Z}$ transform]{${\cal Z}$ transform} |
298 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
210 %\subsection[]{} |
299 %% DTS REQ %% |
211 \begin{frame}\frametitle{TODO} |
300 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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301 |
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302 \begin{frame}\frametitle{Convolution: Development effort} |
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303 {\small |
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304 \begin{center} |
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305 \begin{tabular}{l|l|r} |
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306 requirements & comments &effort\\ \hline\hline |
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307 simplify rationals & \sisac & 0\\ \hline |
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308 define $\sum\limits_{i=0}^{n}i$ & partly \sisac & 10\\ \hline |
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309 simplify sum & termorder & 10\\ |
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310 & simplify rules & 20\\ |
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311 & use simplify rationals& 0\\ \hline |
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312 index adjustments & with unit step & 10\\ \hline |
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313 example collection & with explanations & 20\\ \hline\hline |
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314 & & 70-90\\ |
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315 \end{tabular} |
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316 \end{center} |
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317 effort --- in 45min units\\ |
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318 } |
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319 \end{frame} |
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320 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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321 %-----------------------------------------------------------------% |
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322 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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323 |
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324 \section[Z-transform]{Z-Transform} |
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325 \subsection[(Inverse) Z-Transform]{(Inverse) Z-Transform} |
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326 |
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327 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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328 %% Z-Transform INTRO %% |
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329 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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330 |
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331 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Introduction} |
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332 \begin{itemize} |
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333 \item Pure Transformation is simple to realise with Z-Transform Properties (Table) |
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334 \item Partial Fraction are just math simplifications |
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335 \end{itemize} |
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336 \end{frame} |
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337 |
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338 % does not fit?! |
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339 %\subsection[]{Indextranformation} |
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340 %\begin{frame}\frametitle{TODO} |
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341 %TODO |
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342 %\end{frame} |
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343 |
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344 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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345 %% Z-Transform SPEC %% |
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346 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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347 |
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348 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Specification} |
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349 {\footnotesize\it |
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350 Convolution |
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351 \begin{tabbing} |
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352 1\=postcond \=: \= \= $\;\;\;\;$\=\kill |
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353 \>given \>:\> Expression of z \\ |
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354 \> \> \> \>(X (z::real\,+z::imag),3/(z-1/4-1/8\,z\textasciicircum{}(-1)))\\ |
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355 \>precond \>:\> TODO\\ |
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356 \>find \>:\> Expression of n\\ |
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357 \> \> \> \>$h[n]$\\ |
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358 \>postcond \>:\> TODO\\ |
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359 \end{tabbing} |
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360 |
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361 } |
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362 \end{frame} |
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363 |
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364 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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365 %% Z expl CALC %% |
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366 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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367 |
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368 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Calculation} |
212 TODO |
369 TODO |
213 \end{frame} |
370 \end{frame} |
214 |
371 |
215 \subsection[]{Indextranformation} |
372 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
216 \begin{frame}\frametitle{TODO} |
373 %% Z expl REQ %% |
217 TODO |
374 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
218 \end{frame} |
375 |
219 |
376 |
220 \subsection[Inverse ${\cal Z}$]{Inverse ${\cal Z}$ transform} |
377 \begin{frame}\frametitle{(Inverse) ${\cal Z}$-Transformation: Development effort} |
221 \begin{frame}\frametitle{Development effort} |
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222 {\small |
378 {\small |
223 \begin{center} |
379 \begin{center} |
224 \begin{tabular}{l|l|r} |
380 \begin{tabular}{l|l|r} |
225 requirements & comments &effort\\ \hline\hline |
381 requirements & comments &effort\\ \hline\hline |
226 solve for part.fract. & \sisac: degree 2 & 0\\ |
382 solve for part.fract. & \sisac: degree 2 & 0\\ |