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Aufgaben:Aufgabe 1.4Z: Darstellungsformen von Schwingungen - Versionsgeschichte
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Guenter am 23. November 2021 um 14:45 Uhr
2021-11-23T14:45:10Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 23. November 2021, 14:45 Uhr</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">''</del>Anmerkung zur Nomenklatur:<del class="diffchange diffchange-inline">''</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Anmerkung zur Nomenklatur:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir &nbsp;$\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$.&nbsp; Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise &nbsp;$\phi$&nbsp; vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir &nbsp;$\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$.&nbsp; Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise &nbsp;$\phi$&nbsp; vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td></tr>
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<td colspan="2" class="diff-lineno">Zeile 27:</td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">''</del>Weitere Hinweise:<del class="diffchange diffchange-inline">'' </del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Weitere Hinweise: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals|Beschreibung des physikalischen Signals mit Hilfe des äquivalenten Tiefpass-Signals]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals|Beschreibung des physikalischen Signals mit Hilfe des äquivalenten Tiefpass-Signals]].</div></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=29694&oldid=prev
Guenter am 3. März 2020 um 14:55 Uhr
2020-03-03T14:55:32Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 3. März 2020, 14:55 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l77" >Zeile 77:</td>
<td colspan="2" class="diff-lineno">Zeile 77:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(1)'''&nbsp; Aus der grafischen Darstellung der Zeitfunktion $z(t)$ erkennt man </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(1)'''&nbsp; Aus der grafischen Darstellung der Zeitfunktion<ins class="diffchange diffchange-inline">&nbsp; </ins>$z(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>erkennt man </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*die (normierte) Amplitude $A_{\rm T}\hspace{0.15cm}\underline{ = 2}$ und die Periodendauer $T_0=2$ Millisekunden. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*die (normierte) Amplitude<ins class="diffchange diffchange-inline">&nbsp; </ins>$A_{\rm T}\hspace{0.15cm}\underline{ = 2}$<ins class="diffchange diffchange-inline">&nbsp; </ins>und die Periodendauer<ins class="diffchange diffchange-inline">&nbsp; </ins>$T_0=2$<ins class="diffchange diffchange-inline">&nbsp; </ins>Millisekunden. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Deshalb ist die Signalfrequenz $f_{\rm T} = 1/T_0\hspace{0.15cm}\underline{ = 500}$ Hz und die Kreisfrequenz beträgt $ω_{\rm T}= 2πf_{\rm T} \hspace{0.15cm}\underline{ = 3141.5}$ 1/s.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Deshalb ist die Signalfrequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{\rm T} = 1/T_0\hspace{0.15cm}\underline{ = 500}$<ins class="diffchange diffchange-inline">&nbsp; </ins>Hz und die Kreisfrequenz beträgt<ins class="diffchange diffchange-inline">&nbsp; </ins>$ω_{\rm T}= 2πf_{\rm T} \hspace{0.15cm}\underline{ = 3141.5}$<ins class="diffchange diffchange-inline">&nbsp; </ins>1/s.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(2)'''&nbsp; Das analytische Signal lautet allgemein:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(2)'''&nbsp; Das analytische Signal lautet allgemein:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$z_+(t) = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \hspace{0.<del class="diffchange diffchange-inline">03cm</del>}(\omega_{\rm T}\cdot \hspace{0.05cm}t + \phi_{\rm T})} = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \phi_{\rm T}} \cdot {\rm e}^{{\rm j} \cdot \hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t }\hspace{0.05cm}.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$z_+(t) = A_{\rm T} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>} \cdot \hspace{0.<ins class="diffchange diffchange-inline">05cm</ins>}(\omega_{\rm T}\cdot \hspace{0.05cm}t + \phi_{\rm T})} = A_{\rm T} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>} \cdot <ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\phi_{\rm T}} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>} \cdot <ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t }\hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Gleichzeitig gilt der Zusammenhang:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Gleichzeitig gilt der Zusammenhang:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$A_0 = z_+(t = 0) = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \phi_{\rm T}} \hspace{0.05cm}.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$A_0 = z_+(t = 0) = A_{\rm T} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>} \cdot <ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\phi_{\rm T}} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Die komplexe Amplitude $A_0$ kann aus der oberen Grafik abgelesen werden.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Die komplexe Amplitude<ins class="diffchange diffchange-inline">&nbsp; </ins>$A_0$<ins class="diffchange diffchange-inline">&nbsp; </ins>kann aus der oberen Grafik abgelesen werden.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$A_0 = - \sqrt{2} - {\rm j} \cdot \sqrt{2} = A_{\rm 0} \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} 0.75 \pi} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$A_0 = - \sqrt{2} - {\rm j} \cdot \sqrt{2} = A_{\rm 0} \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} 0.75 \pi} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Ein Vergleich beider Gleichungen führt zum Ergebnis:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Ein Vergleich beider Gleichungen führt zum Ergebnis:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ \phi_{\rm T} = - 0.75 \pi \hspace{0.15cm}\underline {= - 135^{\circ}} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ \phi_{\rm T} = - 0.75 \pi \hspace{0.15cm}\underline {= - 135^{\circ}} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Dabei besteht folgender Zusammenhang mit der Laufzeit $τ$:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Dabei besteht folgender Zusammenhang mit der Laufzeit<ins class="diffchange diffchange-inline">&nbsp; </ins>$τ$:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\phi_{\rm T} = - 2 \pi \cdot f_{\rm T} \cdot \tau \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \tau = \frac{-\phi_{\rm T}}{2 \pi \cdot f_{\rm T}} = \frac{0.75 \pi}{2 \pi \cdot 0.5\,{\rm kHz}} \hspace{0.15cm}\underline {= 0.75 \,{\rm ms}} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\phi_{\rm T} = - 2 \pi \cdot f_{\rm T} \cdot \tau \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \tau = \frac{-\phi_{\rm T}}{2 \pi \cdot f_{\rm T}} = \frac{0.75 \pi}{2 \pi \cdot 0.5\,{\rm kHz}} \hspace{0.15cm}\underline {= 0.75 \,{\rm ms}} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Das analytische Signal legt in der Zeit<ins class="diffchange diffchange-inline">&nbsp; </ins>$T_0$<ins class="diffchange diffchange-inline">&nbsp; </ins>genau eine Umdrehung zurück. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Das analytische Signal legt in der Zeit $T_0$ genau eine Umdrehung zurück. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Ausgehend von<ins class="diffchange diffchange-inline">&nbsp; </ins>$A_0$<ins class="diffchange diffchange-inline">&nbsp; </ins>erreicht man somit nach<ins class="diffchange diffchange-inline">&nbsp; </ins>$t_1 = T_0/8\hspace{0.15cm}\underline{ = 0.25}$<ins class="diffchange diffchange-inline">&nbsp; </ins>ms zum ersten Mal, dass das analytische Signal imaginär ist: </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Ausgehend von $A_0$ erreicht man somit nach $t_1 = T_0/8\hspace{0.15cm}\underline{ = 0.25}$ ms zum ersten Mal, dass das analytische Signal imaginär ist: </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z_+(t_1) = - 2 {\rm j}.$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z_+(t_1) = - 2 {\rm j}.$$ </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Wegen der Beziehung $z(t) = {\rm Re}[z_+(t)]$ tritt zu diesem Zeitpunkt $t_1$ auch der erste Nulldurchgang des Signals $z(t)$ auf.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Wegen der Beziehung<ins class="diffchange diffchange-inline">&nbsp; </ins>$z(t) = {\rm Re}[z_+(t)]$<ins class="diffchange diffchange-inline">&nbsp; </ins>tritt zu diesem Zeitpunkt<ins class="diffchange diffchange-inline">&nbsp; </ins>$t_1$<ins class="diffchange diffchange-inline">&nbsp; </ins>auch der erste Nulldurchgang des Signals<ins class="diffchange diffchange-inline">&nbsp; </ins>$z(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>auf.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Mit dem <del class="diffchange diffchange-inline">Ergebnisder </del>Teilaufgabe '''(2)''' erhält man: </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Mit dem <ins class="diffchange diffchange-inline">Ergebnis der </ins>Teilaufgabe<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(2)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>erhält man: </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.<del class="diffchange diffchange-inline">03cm</del>}\omega_{\rm T}\cdot \hspace{0.05cm}t} = A_0 = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \phi_{\rm T}} = {\rm const.}$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.<ins class="diffchange diffchange-inline">05cm</ins>}\omega_{\rm T}\cdot \hspace{0.05cm}t} = A_0 = A_{\rm T} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>}\cdot <ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\phi_{\rm T}} = {\rm const.}$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Somit gilt für alle Zeiten $t$ und damit auch für $t = 1$ ms:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Somit gilt für alle Zeiten<ins class="diffchange diffchange-inline">&nbsp; </ins>$t$<ins class="diffchange diffchange-inline">&nbsp; </ins>und damit auch für<ins class="diffchange diffchange-inline">&nbsp; </ins>$t = 1$ ms:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Re}[z_{\rm TP}(t)] = - \sqrt{2} \hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm},$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Re}[z_{\rm TP}(t)] = - \sqrt{2} \hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm},$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ {\rm Im}[z_{\rm TP}(t)] = - \sqrt{2}\hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ {\rm Im}[z_{\rm TP}(t)] = - \sqrt{2}\hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm}.$$</div></td></tr>
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<td colspan="2" class="diff-lineno">Zeile 110:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(5)'''&nbsp; Richtig sind die <u>Aussagen 1, 3 und 4</u>:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(5)'''&nbsp; Richtig sind die <u>Aussagen 1, 3 und 4</u>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die einzige Diracfunktion von $Z_+(f)$ liegt bei $f = f_{\rm T}$ und nicht bei $–f_{\rm T}$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die einzige Diracfunktion von<ins class="diffchange diffchange-inline">&nbsp; </ins>$Z_+(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>liegt bei<ins class="diffchange diffchange-inline">&nbsp; </ins>$f = f_{\rm T}$<ins class="diffchange diffchange-inline">&nbsp; </ins>und nicht bei<ins class="diffchange diffchange-inline">&nbsp; </ins>$–f_{\rm T}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das analytische Signal einer harmonischen Schwingung ist immer komplex.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das analytische Signal einer harmonischen Schwingung ist immer komplex.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* Das äquivalente TP–Signal einer harmonischen Schwingung ist meistens komplex. Ausnahme: </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Das äquivalente TP–Signal einer harmonischen Schwingung ist meistens komplex.<ins class="diffchange diffchange-inline">&nbsp; </ins>Ausnahme: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = ±A_{\rm T} · \cos(ω_{\rm T} · t) \ \Rightarrow \ z_{\rm TP}(t) = ±A_{\rm T}.$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = ±A_{\rm T} · \cos(ω_{\rm T} · t) \ \Rightarrow \ z_{\rm TP}(t) = ±A_{\rm T}.$$ </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=29693&oldid=prev
Guenter am 3. März 2020 um 14:44 Uhr
2020-03-03T14:44:47Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 3. März 2020, 14:44 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Zeile 4:</td>
<td colspan="2" class="diff-lineno">Zeile 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|Zwei Darstellungen einer harmonischen Schwingung]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|Zwei Darstellungen einer harmonischen Schwingung]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung &nbsp;$z(t)$, die zusammen mit dem zugehörigen analytischen Signal &nbsp;$z_+(t)$&nbsp; in der Grafik dargestellt ist.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung &nbsp;$z(t)$, die zusammen mit dem zugehörigen analytischen Signal &nbsp;$z_+(t)$&nbsp; in der Grafik dargestellt ist.<ins class="diffchange diffchange-inline">&nbsp; </ins>Diese Signale können mathematisch wie folgt beschrieben werden:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Diese Signale können mathematisch wie folgt beschrieben werden:</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})= A_{\rm T} \cdot \cos(2 \pi f_{\rm T}( t - \tau)) \hspace{0.05cm},$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})= A_{\rm T} \cdot \cos(2 \pi f_{\rm T}( t - \tau)) \hspace{0.05cm},$$ </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_+(t) = A_{\rm 0} \cdot {\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t}$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_+(t) = A_{\rm 0} \cdot {\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t}$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Die zwei Amplitudenparameter &nbsp;$A_{\rm T} $&nbsp; und &nbsp;$A_0$&nbsp; sind jeweils dimensionslos, der Phasenwert &nbsp;$ϕ_{\rm T} $&nbsp; soll zwischen &nbsp;$\text{±π}$&nbsp; liegen und die Laufzeit &nbsp;$τ$&nbsp; ist nicht negativ.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Die zwei Amplitudenparameter &nbsp;$A_{\rm T} $&nbsp; und &nbsp;$A_0$&nbsp; sind jeweils dimensionslos, der Phasenwert &nbsp;$ϕ_{\rm T} $&nbsp; soll zwischen &nbsp;$\text{±π}$&nbsp; liegen und die Laufzeit &nbsp;$τ$&nbsp; ist nicht negativ.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Die Teilaufgabe '''(4)''' bezieht sich auf das äquivalente Tiefpass–Signal &nbsp;$z_{\rm TP}(t)$, das mit &nbsp;$z_+(t)$&nbsp; wie folgt zusammenhängt:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Die Teilaufgabe<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(4)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>bezieht sich auf das äquivalente Tiefpass–Signal &nbsp;$z_{\rm TP}(t)$, das mit &nbsp;$z_+(t)$&nbsp; wie folgt zusammenhängt:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}\omega_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beachten Sie weiter, dass &nbsp;$ϕ_{\rm T}$&nbsp; in obiger Gleichung mit positivem Vorzeichen erscheint. Unter Anmerkungen zur Nomenklatur finden Sie eine Begründung für die unterschiedliche Verwendung von &nbsp;$φ_{\rm T}$&nbsp; und &nbsp;$ϕ_{\rm T} = – φ_{\rm T}$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beachten Sie weiter, dass &nbsp;$ϕ_{\rm T}$&nbsp; in obiger Gleichung mit positivem Vorzeichen erscheint.<ins class="diffchange diffchange-inline">&nbsp; </ins>Unter <ins class="diffchange diffchange-inline">&bdquo;</ins>Anmerkungen zur Nomenklatur<ins class="diffchange diffchange-inline">&rdquo; </ins>finden Sie <ins class="diffchange diffchange-inline">unten </ins>eine Begründung für die unterschiedliche Verwendung von &nbsp;$φ_{\rm T}$&nbsp; und &nbsp;$ϕ_{\rm T} = – φ_{\rm T}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Anmerkung zur Nomenklatur:''</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Anmerkung zur Nomenklatur:''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir &nbsp;$\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$. Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise &nbsp;$\phi$&nbsp; vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir &nbsp;$\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise &nbsp;$\phi$&nbsp; vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Phasenwerte &nbsp;$\varphi_{\rm T} = 90^\circ$ und $\phi_{\rm T} = -90^\circ$&nbsp; sind somit äquivalent und stehen beide für die Sinusfunktion:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Phasenwerte &nbsp;$\varphi_{\rm T} = 90^\circ$ und $\phi_{\rm T} = -90^\circ$&nbsp; sind somit äquivalent und stehen beide für die Sinusfunktion:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\cos(2 \pi f_0 t - 90^{\circ}) = \cos(2 \pi f_0 t - \varphi_{\rm T}) = \cos(2 \pi f_0 t + \phi_{\rm T}) = \sin(2 \pi f_0 t ).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\cos(2 \pi f_0 t - 90^{\circ}) = \cos(2 \pi f_0 t - \varphi_{\rm T}) = \cos(2 \pi f_0 t + \phi_{\rm T}) = \sin(2 \pi f_0 t ).$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l28" >Zeile 28:</td>
<td colspan="2" class="diff-lineno">Zeile 29:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Weitere Hinweise:'' </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Weitere Hinweise:'' </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#<del class="diffchange diffchange-inline">Beschreibung_von_s</del>.<del class="diffchange diffchange-inline">28t</del>.<del class="diffchange diffchange-inline">29_mit_Hilfe_des_analytischen_Signals</del>|Beschreibung <del class="diffchange diffchange-inline">von ''s''(''t'') </del>mit Hilfe des <del class="diffchange diffchange-inline">analytischen </del>Signals]].</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#<ins class="diffchange diffchange-inline">Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_</ins>.<ins class="diffchange diffchange-inline">C3</ins>.<ins class="diffchange diffchange-inline">A4quivalenten_TP-Signals</ins>|Beschreibung <ins class="diffchange diffchange-inline">des physikalischen Signals </ins>mit Hilfe des <ins class="diffchange diffchange-inline">äquivalenten Tiefpass-</ins>Signals]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln des Buches „Signaldarstellung”: </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln des Buches „Signaldarstellung”: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp; [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp; [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35" >Zeile 35:</td>
<td colspan="2" class="diff-lineno">Zeile 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In unserem Tutorial $\rm LNTwww$ wird die Darstellung des analytischen Signals &nbsp;$s_+(t)$&nbsp; in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals &nbsp;$s_{\rm TP}(t)$&nbsp; angibt. Wir verweisen auf die entsprechenden interaktiven Applets </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In unserem Tutorial $\rm LNTwww$ wird die Darstellung des analytischen Signals &nbsp;$s_+(t)$&nbsp; in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals &nbsp;$s_{\rm TP}(t)$&nbsp; angibt. Wir verweisen auf die entsprechenden interaktiven Applets </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp;[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & <del class="diffchange diffchange-inline">analytisches </del>Signal ]],</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp;[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & <ins class="diffchange diffchange-inline">Analytisches </ins>Signal ]],</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>::(2) &nbsp;[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physikalisches Signal & <del class="diffchange diffchange-inline">äquivalentes </del>TP-Signal]].</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>::(2) &nbsp;[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physikalisches Signal & <ins class="diffchange diffchange-inline">Äquivalentes </ins>TP-Signal]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l59" >Zeile 59:</td>
<td colspan="2" class="diff-lineno">Zeile 60:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$t_1 \ = \ $ { 0.25 3% } $\ \text{ms}$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$t_1 \ = \ $ { 0.25 3% } $\ \text{ms}$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Wie lautet das äquivalente Tiefpass–Signal &nbsp;$z_{\rm TP}(t)$? Geben Sie zur Kontrolle den Wert bei &nbsp;$t = 1 \text{ ms}$ ein.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Wie lautet das äquivalente Tiefpass–Signal &nbsp;$z_{\rm TP}(t)$?<ins class="diffchange diffchange-inline">&nbsp; </ins>Geben Sie zur Kontrolle den Wert bei &nbsp;$t = 1 \text{ ms}$ ein.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"} </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"} </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>${\rm Re}\big[z_{\rm TP}(t = 1\ \rm ms)\big] \ = \ $ { -1.454--1.374 } </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>${\rm Re}\big[z_{\rm TP}(t = 1\ \rm ms)\big] \ = \ $ { -1.454--1.374 } </div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=26787&oldid=prev
Guenter am 4. Dezember 2018 um 12:13 Uhr
2018-12-04T12:13:35Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 4. Dezember 2018, 12:13 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l79" >Zeile 79:</td>
<td colspan="2" class="diff-lineno">Zeile 79:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*die (normierte) Amplitude $A_{\rm T}\hspace{0.15cm}\underline{ = 2}$ und die Periodendauer $T_0=2$ Millisekunden. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*die (normierte) Amplitude $A_{\rm T}\hspace{0.15cm}\underline{ = 2}$ und die Periodendauer $T_0=2$ Millisekunden. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Deshalb ist die Signalfrequenz $f_{\rm T} = 1/T_0\hspace{0.15cm}\underline{ = 500}$ Hz und die Kreisfrequenz beträgt $ω_{\rm T}= 2πf_{\rm T} \hspace{0.15cm}\underline{ = 3141.5}$ 1/s.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Deshalb ist die Signalfrequenz $f_{\rm T} = 1/T_0\hspace{0.15cm}\underline{ = 500}$ Hz und die Kreisfrequenz beträgt $ω_{\rm T}= 2πf_{\rm T} \hspace{0.15cm}\underline{ = 3141.5}$ 1/s.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l93" >Zeile 93:</td>
<td colspan="2" class="diff-lineno">Zeile 94:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Das analytische Signal legt in der Zeit $T_0$ genau eine Umdrehung zurück. Ausgehend von $A_0$ erreicht man somit nach $t_1 = T_0/8\hspace{0.15cm}\underline{ = 0.25}$ ms zum ersten Mal, dass das analytische Signal imaginär ist: $z_+(<del class="diffchange diffchange-inline">t1</del>) = <del class="diffchange diffchange-inline">– </del>2 j$<del class="diffchange diffchange-inline">. </del>Wegen der Beziehung $z(t) = {\rm Re}[z_+(t)]$ tritt zu diesem Zeitpunkt $t_1$ auch der erste Nulldurchgang des Signals $z(t)$ auf.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Das analytische Signal legt in der Zeit $T_0$ genau eine Umdrehung zurück. </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Ausgehend von $A_0$ erreicht man somit nach $t_1 = T_0/8\hspace{0.15cm}\underline{ = 0.25}$ ms zum ersten Mal, dass das analytische Signal imaginär ist<ins class="diffchange diffchange-inline">: </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins class="diffchange diffchange-inline">$</ins>$z_+(<ins class="diffchange diffchange-inline">t_1</ins>) = <ins class="diffchange diffchange-inline">- </ins>2 <ins class="diffchange diffchange-inline">{\rm </ins>j<ins class="diffchange diffchange-inline">}.</ins>$<ins class="diffchange diffchange-inline">$ </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Wegen der Beziehung $z(t) = {\rm Re}[z_+(t)]$ tritt zu diesem Zeitpunkt $t_1$ auch der erste Nulldurchgang des Signals $z(t)$ auf.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Mit dem Ergebnisder Teilaufgabe (2) erhält man: </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Mit dem Ergebnisder Teilaufgabe <ins class="diffchange diffchange-inline">'''</ins>(2)<ins class="diffchange diffchange-inline">''' </ins>erhält man: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t} = A_0 = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \phi_{\rm T}} = {\rm const.}$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t} = A_0 = A_{\rm T} \cdot {\rm e}^{{\rm j} \cdot \phi_{\rm T}} = {\rm const.}$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Somit gilt für alle Zeiten $t$ und damit auch für $t = 1$ ms:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Somit gilt für alle Zeiten $t$ und damit auch für $t = 1$ ms:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Re}[z_{\rm TP}(t)] = - \sqrt{2} \hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm},$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Re}[z_{\rm TP}(t)] = - \sqrt{2} \hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm},$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ {\rm Im}[z_{\rm TP}(t)] = - \sqrt{2}\hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$ {\rm Im}[z_{\rm TP}(t)] = - \sqrt{2}\hspace{0.15cm}\underline {= -1.414} \hspace{0.05cm}.$$</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=26786&oldid=prev
Guenter am 4. Dezember 2018 um 12:07 Uhr
2018-12-04T12:07:09Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 4. Dezember 2018, 12:07 Uhr</td>
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<td colspan="2" class="diff-lineno">Zeile 3:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|<del class="diffchange diffchange-inline">Darstellung </del>einer harmonischen Schwingung]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|<ins class="diffchange diffchange-inline">Zwei Darstellungen </ins>einer harmonischen Schwingung]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung &nbsp;$z(t)$, die zusammen mit dem zugehörigen analytischen Signal &nbsp;$z_+(t)$&nbsp; in der Grafik dargestellt ist.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung &nbsp;$z(t)$, die zusammen mit dem zugehörigen analytischen Signal &nbsp;$z_+(t)$&nbsp; in der Grafik dargestellt ist.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l44" >Zeile 44:</td>
<td colspan="2" class="diff-lineno">Zeile 44:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><quiz display=simple></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><quiz display=simple></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Berechnen Sie die Signalparameter $A_{\rm T}$, $f_{\rm T}$ und $ω_{\rm T}$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Berechnen Sie die Signalparameter <ins class="diffchange diffchange-inline">&nbsp;</ins>$A_{\rm T}$, <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm T}$<ins class="diffchange diffchange-inline">&nbsp; </ins>und <ins class="diffchange diffchange-inline">&nbsp;</ins>$ω_{\rm T}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$A_{\rm T} \ = \ $ { 2 3% }</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$A_{\rm T} \ = \ $ { 2 3% }</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l50" >Zeile 50:</td>
<td colspan="2" class="diff-lineno">Zeile 50:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\omega_{\rm T} \ = \ $ { 3141.5 3% } $\ \text{1/s}$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\omega_{\rm T} \ = \ $ { 3141.5 3% } $\ \text{1/s}$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Bestimmen Sie die Phase $\phi_{\rm T}$ (zwischen <del class="diffchange diffchange-inline">±180°</del>) und die Laufzeit $τ$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Bestimmen Sie die Phase <ins class="diffchange diffchange-inline">&nbsp;</ins>$\phi_{\rm T}<ins class="diffchange diffchange-inline">$&nbsp; </ins>$(<ins class="diffchange diffchange-inline">$</ins>zwischen <ins class="diffchange diffchange-inline">$±180^\circ</ins>)<ins class="diffchange diffchange-inline">$ </ins>und die Laufzeit <ins class="diffchange diffchange-inline">&nbsp;</ins>$τ$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\phi_{\rm T} \ = \ $ { -139--131 } $\ \text{Grad}$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\phi_{\rm T} \ = \ $ { -139--131 } $\ \text{Grad}$ </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$τ \ = \ $ { 0.75 3% } $\ \text{ms}$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$τ \ = \ $ { 0.75 3% } $\ \text{ms}$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Zu welcher Zeit $t_1 > 0$ ist das analytische Signal $z_+(t)$ erstmalig imaginär?</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Zu welcher Zeit <ins class="diffchange diffchange-inline">&nbsp;</ins>$t_1 > 0$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist das analytische Signal <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_+(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>erstmalig imaginär?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$t_1 \ = \ $ { 0.25 3% } $\ \text{ms}$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$t_1 \ = \ $ { 0.25 3% } $\ \text{ms}$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Wie lautet das äquivalente Tiefpass–Signal $z_{\rm TP}(t)$? Geben Sie zur Kontrolle den Wert bei $t = 1$ <del class="diffchange diffchange-inline">ms </del>ein.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Wie lautet das äquivalente Tiefpass–Signal <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_{\rm TP}(t)$? Geben Sie zur Kontrolle den Wert bei <ins class="diffchange diffchange-inline">&nbsp;</ins>$t = 1 <ins class="diffchange diffchange-inline">\text{ ms}</ins>$ ein.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"} </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="{}"} </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>${\rm Re}[z_{\rm TP}(t = 1\ \rm ms)] \ = \ $ { -1.454--1.374 } </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>${\rm Re}<ins class="diffchange diffchange-inline">\big</ins>[z_{\rm TP}(t = 1\ \rm ms)<ins class="diffchange diffchange-inline">\big</ins>] \ = \ $ { -1.454--1.374 } </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>${\rm Im}[z_{\rm TP}(t = 1\ \rm ms)] \ = \ $ { -1.454--1.374 }</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>${\rm Im}<ins class="diffchange diffchange-inline">\big</ins>[z_{\rm TP}(t = 1\ \rm ms)<ins class="diffchange diffchange-inline">\big</ins>] \ = \ $ { -1.454--1.374 }</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{Welche der Aussagen gelten für alle harmonischen Schwingungen?</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{Welche der Aussagen gelten für alle harmonischen Schwingungen?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="[]"}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|type="[]"}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>+ Das Spektrum $Z(f)$ besteht aus zwei Diracfunktionen bei $±f_{\rm T}$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>+ Das Spektrum <ins class="diffchange diffchange-inline">&nbsp;</ins>$Z(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>besteht aus zwei Diracfunktionen bei <ins class="diffchange diffchange-inline">&nbsp;</ins>$±f_{\rm T}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>- Das Spektrum $Z_+(f)$ weist eine Diracfunktion bei $–f_{\rm T}$ auf.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>- Das Spektrum <ins class="diffchange diffchange-inline">&nbsp;</ins>$Z_+(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>weist eine Diracfunktion bei <ins class="diffchange diffchange-inline">&nbsp;</ins>$–f_{\rm T}$ auf.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>+ Das Spektrum $Z_{\rm TP}(f)$ beinhaltet eine Diracfunktion bei $f = 0$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>+ Das Spektrum <ins class="diffchange diffchange-inline">&nbsp;</ins>$Z_{\rm TP}(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>beinhaltet eine Diracfunktion bei <ins class="diffchange diffchange-inline">&nbsp;</ins>$f = 0$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>+ Das analytische Signal $z_+(t)$ ist stets komplex.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>+ Das analytische Signal <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_+(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist stets komplex.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>- Das äquivalente TP–Signal $z_{\rm TP}(t)$ ist stets komplex.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>- Das äquivalente TP–Signal <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_{\rm TP}(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist stets komplex.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></quiz></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></quiz></div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=26785&oldid=prev
Guenter am 4. Dezember 2018 um 11:58 Uhr
2018-12-04T11:58:17Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 4. Dezember 2018, 11:58 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Zeile 38:</td>
<td colspan="2" class="diff-lineno">Zeile 38:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(2) &nbsp;[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physikalisches Signal & äquivalentes TP-Signal]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(2) &nbsp;[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physikalisches Signal & äquivalentes TP-Signal]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">''Weitere Hinweise:'' </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*Die Aufgabe gehört zum Kapitel [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*Bezug genommen wird insbesondere auf die Seite [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.29_mit_Hilfe_des_analytischen_Signals|Beschreibung von ''s''(''t'') mit Hilfe des analytischen Signals]].</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], [[Signaldarstellung/Analytisches_Signal_und_zugeh%C3%B6rige_Spektralfunktion|Analytisches Signal und zugehörige Spektralfunktion]] und [[Signaldarstellung/%C3%84quivalentes_Tiefpass-Signal_und_zugeh%C3%B6rige_Spektralfunktion| Äquivalentes Tiefpass-Signal und zugehörige Spektralfunktion]] des Buches „Signaldarstellung”.</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*In unserem Tutorial LNTwww wird die Darstellung des analytischen Signals $s_+(t)$ in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals $s_{\rm TP}(t)$ angibt. Wir verweisen auf die entsprechenden Interaktionsmodule [[Zeigerdiagramm – Darstellung des analytischen Signals]] sowie [[Ortskurve – Verlauf des äquivalenten Tiefpass-Signals]].</del></div></td><td colspan="2"> </td></tr>
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</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=26784&oldid=prev
Guenter am 4. Dezember 2018 um 11:57 Uhr
2018-12-04T11:57:27Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 4. Dezember 2018, 11:57 Uhr</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>''Hinweise:'' </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>''<ins class="diffchange diffchange-inline">Weitere </ins>Hinweise:'' </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.<del class="diffchange diffchange-inline">29_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals</del>|Beschreibung von ''s''(''t'') mit Hilfe des <del class="diffchange diffchange-inline">äquivalenten Tiefpass-</del>Signals]].</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.<ins class="diffchange diffchange-inline">29_mit_Hilfe_des_analytischen_Signals</ins>|Beschreibung von ''s''(''t'') mit Hilfe des <ins class="diffchange diffchange-inline">analytischen </ins>Signals]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln des Buches „Signaldarstellung”: </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln des Buches „Signaldarstellung”: </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp; [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::(1) &nbsp; [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], </div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=26783&oldid=prev
Guenter am 4. Dezember 2018 um 11:31 Uhr
2018-12-04T11:31:41Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 4. Dezember 2018, 11:31 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Zeile 4:</td>
<td colspan="2" class="diff-lineno">Zeile 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|Darstellung einer harmonischen Schwingung]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID969__Mod_Z_1_4.png|right|frame|Darstellung einer harmonischen Schwingung]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung $z(t)$, die zusammen mit dem zugehörigen analytischen Signal $z_+(t)$ in der Grafik dargestellt ist.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Betrachtet wird eine harmonische Schwingung <ins class="diffchange diffchange-inline">&nbsp;</ins>$z(t)$, die zusammen mit dem zugehörigen analytischen Signal <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_+(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>in der Grafik dargestellt ist.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Diese Signale können mathematisch wie folgt beschrieben werden:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Diese Signale können mathematisch wie folgt beschrieben werden:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})= A_{\rm T} \cdot \cos(2 \pi f_{\rm T}( t - \tau)) \hspace{0.05cm},$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$z(t) = A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})= A_{\rm T} \cdot \cos(2 \pi f_{\rm T}( t - \tau)) \hspace{0.05cm},$$ </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_+(t) = A_{\rm 0} \cdot {\rm e}^{{\rm j} \cdot \hspace{0.<del class="diffchange diffchange-inline">03cm</del>}\omega_{\rm T}\cdot \hspace{0.05cm}t}$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$ z_+(t) = A_{\rm 0} \cdot {\rm e}^{{\rm j<ins class="diffchange diffchange-inline">} \hspace{0.05cm</ins>}\cdot \hspace{0.<ins class="diffchange diffchange-inline">05cm</ins>}\omega_{\rm T<ins class="diffchange diffchange-inline">}\hspace{0.05cm</ins>}\cdot \hspace{0.05cm}t}$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Die zwei Amplitudenparameter $A_{\rm T} $ und $A_0$ sind jeweils dimensionslos, der Phasenwert $ϕ_{\rm T} $ soll zwischen $\text{±π}$ liegen und die Laufzeit $τ$ ist nicht negativ.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Die zwei Amplitudenparameter <ins class="diffchange diffchange-inline">&nbsp;</ins>$A_{\rm T} $<ins class="diffchange diffchange-inline">&nbsp; </ins>und <ins class="diffchange diffchange-inline">&nbsp;</ins>$A_0$<ins class="diffchange diffchange-inline">&nbsp; </ins>sind jeweils dimensionslos, der Phasenwert <ins class="diffchange diffchange-inline">&nbsp;</ins>$ϕ_{\rm T} $<ins class="diffchange diffchange-inline">&nbsp; </ins>soll zwischen <ins class="diffchange diffchange-inline">&nbsp;</ins>$\text{±π}$<ins class="diffchange diffchange-inline">&nbsp; </ins>liegen und die Laufzeit <ins class="diffchange diffchange-inline">&nbsp;</ins>$τ$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist nicht negativ.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Die Teilaufgabe (4) bezieht sich auf das äquivalente <del class="diffchange diffchange-inline">TP–Signal </del>$z_{\rm TP}(t)$, das mit $z_+(t)$ wie folgt zusammenhängt:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Die Teilaufgabe <ins class="diffchange diffchange-inline">'''</ins>(4)<ins class="diffchange diffchange-inline">''' </ins>bezieht sich auf das äquivalente <ins class="diffchange diffchange-inline">Tiefpass–Signal &nbsp;</ins>$z_{\rm TP}(t)$, das mit <ins class="diffchange diffchange-inline">&nbsp;</ins>$z_+(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>wie folgt zusammenhängt:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.<del class="diffchange diffchange-inline">03cm</del>}\omega_{\rm T}<del class="diffchange diffchange-inline">\cdot </del>\hspace{0.05cm}<del class="diffchange diffchange-inline">t} </del>\hspace{0.05cm}.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$z_{\rm TP}(t) = z_+(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.<ins class="diffchange diffchange-inline">05cm</ins>}\omega_{\rm T}\hspace{0.05cm}<ins class="diffchange diffchange-inline">\cdot </ins>\hspace{0.05cm<ins class="diffchange diffchange-inline">}t</ins>}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beachten Sie weiter, dass $ϕ_{\rm T}$ in obiger Gleichung mit positivem Vorzeichen erscheint. Unter Anmerkungen zur Nomenklatur finden Sie eine Begründung für die unterschiedliche Verwendung von $φ_{\rm T}$ und $ϕ_{\rm T} = – φ_{\rm T}$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beachten Sie weiter, dass <ins class="diffchange diffchange-inline">&nbsp;</ins>$ϕ_{\rm T}$<ins class="diffchange diffchange-inline">&nbsp; </ins>in obiger Gleichung mit positivem Vorzeichen erscheint. Unter Anmerkungen zur Nomenklatur finden Sie eine Begründung für die unterschiedliche Verwendung von <ins class="diffchange diffchange-inline">&nbsp;</ins>$φ_{\rm T}$<ins class="diffchange diffchange-inline">&nbsp; </ins>und <ins class="diffchange diffchange-inline">&nbsp;</ins>$ϕ_{\rm T} = – φ_{\rm T}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Anmerkung zur Nomenklatur:''</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Anmerkung zur Nomenklatur:''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In diesem Tutorial geht – wie auch in anderer Literatur üblich – bei der Beschreibung von harmonischer Schwingung, Fourierreihe und Fourierintegral die Phase mit negativem Vorzeichen in die Gleichungen ein, während in Zusammenhang mit Modulationsverfahren die Phase stets mit einem Pluszeichen angesetzt wird. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir $\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$. Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise $\phi$ vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Zur Unterscheidung dieser beiden Varianten benutzen wir <ins class="diffchange diffchange-inline">&nbsp;</ins>$\phi_{\rm T}$ und $\varphi_{\rm T} = - \phi_{\rm T}$. Beide Symbole kennzeichnen das kleine griechische „phi”, wobei die Schreibweise <ins class="diffchange diffchange-inline">&nbsp;</ins>$\phi$<ins class="diffchange diffchange-inline">&nbsp; </ins>vorwiegend im anglo-amerikanischen und $\varphi$ im deutschen Sprachraum angewandt wird.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die Phasenwerte $\varphi_{\rm T} = 90^\circ$ und $\phi_{\rm T} = -90^\circ$ sind somit äquivalent und stehen beide für die Sinusfunktion:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Die Phasenwerte <ins class="diffchange diffchange-inline">&nbsp;</ins>$\varphi_{\rm T} = 90^\circ$ und $\phi_{\rm T} = -90^\circ$<ins class="diffchange diffchange-inline">&nbsp; </ins>sind somit äquivalent und stehen beide für die Sinusfunktion:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\cos(2 \pi f_0 t - 90^{\circ}) = \cos(2 \pi f_0 t - \varphi_{\rm T}) = \cos(2 \pi f_0 t + \phi_{\rm T}) = \sin(2 \pi f_0 t ).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\cos(2 \pi f_0 t - 90^{\circ}) = \cos(2 \pi f_0 t - \varphi_{\rm T}) = \cos(2 \pi f_0 t + \phi_{\rm T}) = \sin(2 \pi f_0 t ).$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">''Hinweise:'' </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Die Aufgabe gehört zum Kapitel&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation|Allgemeines Modell der Modulation]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.29_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals|Beschreibung von ''s''(''t'') mit Hilfe des äquivalenten Tiefpass-Signals]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln des Buches „Signaldarstellung”: </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::(1) &nbsp; [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::(2)&nbsp; [[Signaldarstellung/Analytisches_Signal_und_zugeh%C3%B6rige_Spektralfunktion|Analytisches Signal und zugehörige Spektralfunktion]]&nbsp; und </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::(3)&nbsp; [[Signaldarstellung/%C3%84quivalentes_Tiefpass-Signal_und_zugeh%C3%B6rige_Spektralfunktion| Äquivalentes Tiefpass-Signal und zugehörige Spektralfunktion]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*In unserem Tutorial $\rm LNTwww$ wird die Darstellung des analytischen Signals &nbsp;$s_+(t)$&nbsp; in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals &nbsp;$s_{\rm TP}(t)$&nbsp; angibt. Wir verweisen auf die entsprechenden interaktiven Applets </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::(1) &nbsp;[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & analytisches Signal ]],</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">::(2) &nbsp;[[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physikalisches Signal & äquivalentes TP-Signal]].</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=24768&oldid=prev
Mwiki-lnt: Textersetzung - „*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.“ durch „ “
2018-05-29T12:02:38Z
<p>Textersetzung - „*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.“ durch „ “</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 29. Mai 2018, 12:02 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l28" >Zeile 28:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.29_mit_Hilfe_des_analytischen_Signals|Beschreibung von ''s''(''t'') mit Hilfe des analytischen Signals]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Bezug genommen wird insbesondere auf die Seite [[Modulationsverfahren/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.29_mit_Hilfe_des_analytischen_Signals|Beschreibung von ''s''(''t'') mit Hilfe des analytischen Signals]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], [[Signaldarstellung/Analytisches_Signal_und_zugeh%C3%B6rige_Spektralfunktion|Analytisches Signal und zugehörige Spektralfunktion]] und [[Signaldarstellung/%C3%84quivalentes_Tiefpass-Signal_und_zugeh%C3%B6rige_Spektralfunktion| Äquivalentes Tiefpass-Signal und zugehörige Spektralfunktion]] des Buches „Signaldarstellung”.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Weitere Informationen zu dieser Thematik finden Sie in den Kapiteln [[Signaldarstellung/Harmonische_Schwingung|Harmonische Schwingung]], [[Signaldarstellung/Analytisches_Signal_und_zugeh%C3%B6rige_Spektralfunktion|Analytisches Signal und zugehörige Spektralfunktion]] und [[Signaldarstellung/%C3%84quivalentes_Tiefpass-Signal_und_zugeh%C3%B6rige_Spektralfunktion| Äquivalentes Tiefpass-Signal und zugehörige Spektralfunktion]] des Buches „Signaldarstellung”.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In unserem Tutorial LNTwww wird die Darstellung des analytischen Signals $s_+(t)$ in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals $s_{\rm TP}(t)$ angibt. Wir verweisen auf die entsprechenden Interaktionsmodule [[Zeigerdiagramm – Darstellung des analytischen Signals]] sowie [[Ortskurve – Verlauf des äquivalenten Tiefpass-Signals]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In unserem Tutorial LNTwww wird die Darstellung des analytischen Signals $s_+(t)$ in der komplexen Ebene teilweise auch als „Zeigerdiagramm” bezeichnet, während die „Ortskurve” den zeitlichen Verlauf des äquivalenten TP–Signals $s_{\rm TP}(t)$ angibt. Wir verweisen auf die entsprechenden Interaktionsmodule [[Zeigerdiagramm – Darstellung des analytischen Signals]] sowie [[Ortskurve – Verlauf des äquivalenten Tiefpass-Signals]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Mwiki-lnt
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen&diff=21941&oldid=prev
Guenter: Guenter verschob die Seite Aufgaben:1.4Z Darstellungsformen von Schwingungen nach Aufgaben:Aufgabe 1.4Z: Darstellungsformen von Schwingungen
2018-01-03T14:16:38Z
<p>Guenter verschob die Seite <a href="/Aufgaben:1.4Z_Darstellungsformen_von_Schwingungen" class="mw-redirect" title="Aufgaben:1.4Z Darstellungsformen von Schwingungen">1.4Z Darstellungsformen von Schwingungen</a> nach <a href="/Aufgaben:Aufgabe_1.4Z:_Darstellungsformen_von_Schwingungen" title="Aufgaben:Aufgabe 1.4Z: Darstellungsformen von Schwingungen">Aufgabe 1.4Z: Darstellungsformen von Schwingungen</a></p>
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Version vom 3. Januar 2018, 14:16 Uhr</td>
</tr><tr><td colspan="2" class="diff-notice" lang="de"><div class="mw-diff-empty">(kein Unterschied)</div>
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Guenter