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Aufgaben:Aufgabe 3.10: Berechnung der Rauschleistungen - Versionsgeschichte
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Guenter am 28. März 2020 um 16:47 Uhr
2020-03-28T16:47:37Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 28. März 2020, 16:47 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l49" >Zeile 49:</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>===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; Das Signal–zu–Rausch–Leistungsverhältnis (Sinken–SNR) $ \rho_{v }$ ist der Quotient aus <del class="diffchange diffchange-inline">der </del>Nutzleistung $P_{\rm S}$ und <del class="diffchange diffchange-inline">der </del>Rauschleistung $P_{\rm R}$. <del class="diffchange diffchange-inline"><br>Speziell bei der </del>Phasenmodulation gilt:</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; Das Signal–zu–Rausch–Leistungsverhältnis<ins class="diffchange diffchange-inline">&nbsp; </ins>(Sinken–SNR)<ins class="diffchange diffchange-inline">&nbsp; </ins>$ \rho_{v }$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist der Quotient aus Nutzleistung<ins class="diffchange diffchange-inline">&nbsp; </ins>$P_{\rm S}$<ins class="diffchange diffchange-inline">&nbsp; </ins>und Rauschleistung<ins class="diffchange diffchange-inline">&nbsp; </ins>$P_{\rm R}$.<ins class="diffchange diffchange-inline">&nbsp; Für die </ins>Phasenmodulation gilt:</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>:$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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>:$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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 Messung mit $f_{\rm N} = f_5 = 5 \ \rm kHz$ hat das SNR $ \rho_{v } = 10^5$ $($entsprechend $10 · \lg ρ_v =50\ \rm dB)$ ergeben. </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 Messung mit<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{\rm N} = f_5 = 5 \ \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>hat das SNR<ins class="diffchange diffchange-inline">&nbsp; </ins>$ \rho_{v } = 10^5$<ins class="diffchange diffchange-inline">&nbsp; </ins>$($entsprechend<ins class="diffchange diffchange-inline">&nbsp; </ins>$10 · \lg ρ_v =50\ \rm dB)$<ins class="diffchange diffchange-inline">&nbsp; </ins>ergeben. </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 doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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>*Die doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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;"><div>:$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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;"><div>Dieses Ergebnis lässt sich auch über die Beziehung &nbsp;$ρ_v = η^2/2 · ξ$&nbsp; herleiten. </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>Dieses Ergebnis lässt sich auch über die Beziehung &nbsp;$ρ_v = η^2/2 · ξ$&nbsp; herleiten. </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>*Bei Phasenmodulation ist $η$ unabhängig von der Nachrichtenfrequenz. </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>*Bei Phasenmodulation ist<ins class="diffchange diffchange-inline">&nbsp; </ins>$η$<ins class="diffchange diffchange-inline">&nbsp; </ins>unabhängig von der Nachrichtenfrequenz. </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>*Der SNR–Verlust geht darauf zurück, dass nun die Leistungskenngröße &nbsp;$ξ = P_{\rm S}/(N_0 · f_{\rm N})$&nbsp; halbiert 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>*Der SNR–Verlust geht darauf zurück, dass nun die Leistungskenngröße &nbsp;$ξ = P_{\rm S}/(N_0 · f_{\rm N})$&nbsp; halbiert 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;"></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>'''(2)'''&nbsp; Bei Frequenzmodulation und der Nachrichtenfrequenz $f_{\rm N} = 5 \ \rm kHz$ erhält man für die Rauschleistung</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> </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>'''(2)'''&nbsp; Bei Frequenzmodulation und der Nachrichtenfrequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{\rm N} = 5 \ \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>erhält man für die Rauschleistung<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>:$$P_{\rm R} = \int_{-f_{\rm N}}^{ + f_{\rm N}} {\it \Phi}_{v {\rm , \hspace{0.08cm}FM} } (f)\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0}{\Delta f_{\rm A}^{\hspace{0.1cm}2}} \cdot \int_{0}^{ f_{\rm N}} f^2\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0 \cdot f_{\rm N}^{\hspace{0.1cm}3}}{3 \cdot \Delta f_{\rm A}^2} \hspace{0.05cm}.$$</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>:$$P_{\rm R} = \int_{-f_{\rm N}}^{ + f_{\rm N}} {\it \Phi}_{v {\rm , \hspace{0.08cm}FM} } (f)\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0}{\Delta f_{\rm A}^{\hspace{0.1cm}2}} \cdot \int_{0}^{ f_{\rm N}} f^2\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0 \cdot f_{\rm N}^{\hspace{0.1cm}3}}{3 \cdot \Delta f_{\rm A}^2} \hspace{0.05cm}.$$</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>Unter Berücksichtigung des Frequenzhubs $Δf_{\rm A} = η · f_{\rm N}$ ergibt sich somit:</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>Unter Berücksichtigung des Frequenzhubs<ins class="diffchange diffchange-inline">&nbsp; </ins> $Δf_{\rm A} = η · f_{\rm N}$<ins class="diffchange diffchange-inline">&nbsp; </ins>ergibt sich somit:</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm N}}{3 \cdot \eta^2} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v }= \frac{3 \cdot \eta^2 \cdot P_{\rm S}}{2 \cdot N_0 \cdot f_{\rm N}} = 3 \cdot \rho_{v {\rm , \hspace{0.08cm}PM}}\hspace{0.05cm}.$$</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm N}}{3 \cdot \eta^2} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v }= \frac{3 \cdot \eta^2 \cdot P_{\rm S}}{2 \cdot N_0 \cdot f_{\rm N}} = 3 \cdot \rho_{v {\rm , \hspace{0.08cm}PM}}\hspace{0.05cm}.$$</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>Das heißt: Die Frequenzmodulation ist um den Faktor $3$ (oder $4.77 \ \rm dB$<del class="diffchange diffchange-inline">) </del>besser als die Phasenmodulation:</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>Das heißt:<ins class="diffchange diffchange-inline">&nbsp; </ins>Die Frequenzmodulation ist um den Faktor<ins class="diffchange diffchange-inline">&nbsp; </ins>$3<ins class="diffchange diffchange-inline">$&nbsp; </ins>$(<ins class="diffchange diffchange-inline">$</ins>oder $4.77 \ \rm dB<ins class="diffchange diffchange-inline">)</ins>$<ins class="diffchange diffchange-inline">&nbsp; </ins>besser als die Phasenmodulation:</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm dB}}\hspace{0.05cm}.$$</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>'''(3)'''&nbsp; Entsprechend dem Ergebnis der Teilaufgabe '''(2)''' erhält man mit $f_{10} = 10 \ \rm kHz$:</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; Entsprechend dem Ergebnis der Teilaufgabe<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(2)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>erhält man mit<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{10} = 10 \ \rm kHz$:</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm 10}}{3 \cdot \eta_{10}^{\hspace{0.1cm}2}} = \frac{ f_{\rm 10} \cdot \eta_{5}^{\hspace{0.1cm}2}}{ 3 \cdot f_{\rm 5} \cdot \eta_{10}^{\hspace{0.1cm}2}}\cdot \frac{2 \cdot N_0 \cdot f_{\rm 5}}{\eta_{5}^{\hspace{0.1cm}2}} \hspace{0.05cm}.$$</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm 10}}{3 \cdot \eta_{10}^{\hspace{0.1cm}2}} = \frac{ f_{\rm 10} \cdot \eta_{5}^{\hspace{0.1cm}2}}{ 3 \cdot f_{\rm 5} \cdot \eta_{10}^{\hspace{0.1cm}2}}\cdot \frac{2 \cdot N_0 \cdot f_{\rm 5}}{\eta_{5}^{\hspace{0.1cm}2}} \hspace{0.05cm}.$$</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>Der zweite Term gibt die Rauschleistung des Vergleichssystems $($PM, $f_{\rm N} = f_5)$ an, die zum Ergebnis $10 · \lg ρ_v = 50\ \rm dB$ geführt hat.</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>Der zweite Term gibt die Rauschleistung des Vergleichssystems<ins class="diffchange diffchange-inline">&nbsp; </ins>$($PM, $f_{\rm N} = f_5)$<ins class="diffchange diffchange-inline">&nbsp; </ins>an,<ins class="diffchange diffchange-inline">&nbsp; </ins>die zum Ergebnis<ins class="diffchange diffchange-inline">&nbsp; </ins>$10 · \lg ρ_v = 50\ \rm dB$<ins class="diffchange diffchange-inline">&nbsp; </ins>geführt hat.</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>*Bei Frequenzmodulation ist nun jedoch der Modulationsindex $η$ umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient $η_5^2/η_{10}^2 = 4$ 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>*Bei Frequenzmodulation ist nun jedoch der Modulationsindex<ins class="diffchange diffchange-inline">&nbsp; </ins>$η$<ins class="diffchange diffchange-inline">&nbsp; </ins>umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient<ins class="diffchange diffchange-inline">&nbsp; </ins>$η_5^2/η_{10}^2 = 4$<ins class="diffchange diffchange-inline">&nbsp; </ins>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>*Somit ergibt sich für den Vorfaktor $8/3$. Aufgrund der größeren Rauschleistung ist das SNR kleiner:</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>*Somit ergibt sich für den Vorfaktor<ins class="diffchange diffchange-inline">&nbsp; </ins>$8/3$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Aufgrund der größeren Rauschleistung ist das SNR kleiner:</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} - 10 \cdot {\rm lg} \hspace{0.15cm}({8}/{3})\hspace{0.15cm}\underline {\approx 45.74\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} - 10 \cdot {\rm lg} \hspace{0.15cm}({8}/{3})\hspace{0.15cm}\underline {\approx 45.74\,{\rm dB}}\hspace{0.05cm}.$$</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><del style="font-weight: bold; text-decoration: none;">Bei gleicher Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ist nun die FM um $1.25 \ \rm dB$ schlechter als die PM, da sich nun die Halbierung von $η$ – nach Quadrierung der Faktor $4$ – stärker auswirkt als der systembedingte Faktor $3$, um den die FM gegenüber der PM überlegen ist.</del></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="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>*Der Vergleich der Teilaufgaben '''(2)''' und '''(3)''' zeigt einen Unterschied um den Faktor $8$ bzw. $9.03 \ \rm dB$. </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 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>*Der ungünstigere Wert für die größere Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor $4$ – und die doppelte Rauschbandbreite.</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">Bei gleicher Nachrichtenfrequenz&nbsp; $f_{\rm N} = 10 \ \rm kHz$&nbsp; ist nun die FM um&nbsp; $1.25 \ \rm dB$&nbsp; schlechter als die PM, da sich nun die Halbierung von&nbsp; $η$&nbsp; – nach Quadrierung der Faktor&nbsp; $4$&nbsp; –&nbsp; stärker auswirkt als der systembedingte Faktor&nbsp; $3$, um den die FM gegenüber der PM überlegen ist.</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> </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>*Der Vergleich der Teilaufgaben<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(2)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(3)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>zeigt einen Unterschied um den Faktor<ins class="diffchange diffchange-inline">&nbsp; </ins>$8$<ins class="diffchange diffchange-inline">&nbsp; </ins>bzw.<ins class="diffchange diffchange-inline">&nbsp; </ins>$9.03 \ \rm dB$. </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>*Der ungünstigere Wert für die größere Nachrichtenfrequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{\rm N} = 10 \ \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor<ins class="diffchange diffchange-inline">&nbsp; </ins>$4$<ins class="diffchange diffchange-inline">&nbsp; </ins>– und die doppelte Rauschbandbreite.</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>{{ML-Fuß}}</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-Fuß}}</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=30004&oldid=prev
Guenter am 28. März 2020 um 16:34 Uhr
2020-03-28T16:34:10Z
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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 28. März 2020, 16:34 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_ID1115__Mod_A_3_9.png|right|frame|Rauschleistungsdichten von PM und FM]]</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_ID1115__Mod_A_3_9.png|right|frame|Rauschleistungsdichten von PM und FM]]</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz $f_{\rm N}$. Zunächst gelte für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = f_5 = 5 \ \rm kHz$&nbsp; und der Modulationsindex (Phasenhub) sei &nbsp;$η = 5$.</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f_{\rm N}$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Zunächst gelte für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = f_5 = 5 \ \rm kHz$&nbsp; und der Modulationsindex (Phasenhub) sei &nbsp;$η = 5$.</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte &nbsp;$N_0$&nbsp; ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte &nbsp;${\it \Phi}_{v {\rm , \hspace{0.08cm}PM} }(f) = {\it \Phi}_0$, die auch vom Modulationsindex &nbsp;$η$&nbsp; abhängt:</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte &nbsp;$N_0$&nbsp; ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte &nbsp;${\it \Phi}_{v {\rm , \hspace{0.08cm}PM} }(f) = {\it \Phi}_0$, die auch vom Modulationsindex &nbsp;$η$&nbsp; abhängt:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14" >Zeile 14:</td>
<td colspan="2" class="diff-lineno">Zeile 14:</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>*Gegeben ist der Rauschabstand &nbsp;$10 · \lg ρ_v = 50 \ \rm dB$&nbsp; für Phasenmodulation und &nbsp;$f_{\rm N} = 5 \ \rm kHz$. </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>*Gegeben ist der Rauschabstand &nbsp;$10 · \lg ρ_v = 50 \ \rm dB$&nbsp; für Phasenmodulation und &nbsp;$f_{\rm N} = 5 \ \rm kHz$. </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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = 5 \ \rm kHz$&nbsp; sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = f_{10} = 10 \ \rm kHz$.</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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = 5 \ \rm kHz$&nbsp; sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz &nbsp;$f_{\rm N} = f_{10} = 10 \ \rm kHz$.</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>
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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><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="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>{Welcher Rauschabstand ergibt sich bei Phasenmodulation und &nbsp;$f_{\rm N} = 10 \ \rm kHz$? Interpretieren Sie das 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>{Welcher Rauschabstand ergibt sich bei<ins class="diffchange diffchange-inline">&nbsp; <u></ins>Phasenmodulation<ins class="diffchange diffchange-inline"></u>&nbsp; </ins>und &nbsp;$f_{\rm N} = 10 \ \rm kHz$?<ins class="diffchange diffchange-inline">&nbsp; </ins>Interpretieren Sie das 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>|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>$10 · \lg ρ_v \ = \ $ { 46.99 3% } $\ \rm dB$ </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>$10 · \lg ρ_v \ = \ $ { 46.99 3% } $\ \rm dB$ </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>{Berechnen Sie den Rauschabstand bei Frequenzmodulation und &nbsp;$f_{\rm N} = 5 \ \rm kHz$. Wie groß ist der Modulationsindex bei dieser Konstellation?</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 den Rauschabstand bei<ins class="diffchange diffchange-inline">&nbsp; <u></ins>Frequenzmodulation<ins class="diffchange diffchange-inline"></u>&nbsp; </ins>und &nbsp;$f_{\rm N} = 5 \ \rm kHz$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Wie groß ist der Modulationsindex bei dieser Konstellation?</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>$10 · \lg ρ_v \ = \ $ { 54.77 3% } $\ \rm dB$</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>$10 · \lg ρ_v \ = \ $ { 54.77 3% } $\ \rm dB$</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 den Rauschabstand bei Frequenzmodulation und &nbsp;$f_{\rm N} = 10 \ \rm kHz$. Interpretieren Sie das Ergebnis im Vergleich zu '''(1)''' und '''(2)'''.</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 den Rauschabstand bei<ins class="diffchange diffchange-inline">&nbsp; <u></ins>Frequenzmodulation<ins class="diffchange diffchange-inline"></u>&nbsp; </ins>und &nbsp;$f_{\rm N} = 10 \ \rm kHz$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Interpretieren Sie das Ergebnis im Vergleich zu<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(1)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(2)'''.</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>$10 · \lg ρ_v \ = \ $ { 45.74 3% } $\ \rm dB$ </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>$10 · \lg ρ_v \ = \ $ { 45.74 3% } $\ \rm dB$ </div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=26936&oldid=prev
Guenter am 20. Dezember 2018 um 15:45 Uhr
2018-12-20T15:45:59Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 20. Dezember 2018, 15:45 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l46" >Zeile 46:</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>===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; Das Signal–zu–Rausch–Leistungsverhältnis (Sinken–SNR) $ \rho_{v }$ ist der Quotient aus der Nutzleistung $P_{\rm S}$ und der Rauschleistung $P_{\rm <del class="diffchange diffchange-inline">S</del>}$. Speziell bei der Phasenmodulation gilt:</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; Das Signal–zu–Rausch–Leistungsverhältnis (Sinken–SNR) $ \rho_{v }$ ist der Quotient aus der Nutzleistung $P_{\rm S}$ und der Rauschleistung $P_{\rm <ins class="diffchange diffchange-inline">R</ins>}$. <ins class="diffchange diffchange-inline"><br></ins>Speziell bei der Phasenmodulation gilt:</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>:$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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>:$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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 Messung mit $f_{\rm N} = f_5 = 5 \ \rm kHz$ hat das SNR $ \rho_{v } = 10^5$ (entsprechend $10 · \lg ρ_v =50 dB$<del class="diffchange diffchange-inline">) </del>ergeben. </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 Messung mit $f_{\rm N} = f_5 = 5 \ \rm kHz$ hat das SNR $ \rho_{v } = 10^5<ins class="diffchange diffchange-inline">$ </ins>$(<ins class="diffchange diffchange-inline">$</ins>entsprechend $10 · \lg ρ_v =50<ins class="diffchange diffchange-inline">\ \rm </ins>dB<ins class="diffchange diffchange-inline">)</ins>$ ergeben. </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 doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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>*Die doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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;"><div>:$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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="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>Dieses Ergebnis lässt sich auch über die Beziehung $ρ_v = η^2/2 · ξ$ herleiten. </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>Dieses Ergebnis lässt sich auch über die Beziehung <ins class="diffchange diffchange-inline">&nbsp;</ins>$ρ_v = η^2/2 · ξ$<ins class="diffchange diffchange-inline">&nbsp; </ins>herleiten. </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>*Bei Phasenmodulation ist $η$ unabhängig von der Nachrichtenfrequenz. </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>*Bei Phasenmodulation ist $η$ unabhängig von der Nachrichtenfrequenz. </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>*Der SNR–Verlust geht darauf zurück, dass nun die Leistungskenngröße $ξ = P_{\rm S}/(N_0 · f_{\rm N})$ halbiert 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>*Der SNR–Verlust geht darauf zurück, dass nun die Leistungskenngröße <ins class="diffchange diffchange-inline">&nbsp;</ins>$ξ = P_{\rm S}/(N_0 · f_{\rm N})$<ins class="diffchange diffchange-inline">&nbsp; </ins>halbiert 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;"></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-l64" >Zeile 64:</td>
<td colspan="2" class="diff-lineno">Zeile 64:</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm dB}}\hspace{0.05cm}.$$</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="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; Entsprechend dem Ergebnis der Teilaufgabe (2) erhält man mit $f_{10} = 10 \ \rm kHz$:</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; Entsprechend dem Ergebnis der Teilaufgabe <ins class="diffchange diffchange-inline">'''</ins>(2)<ins class="diffchange diffchange-inline">''' </ins>erhält man mit $f_{10} = 10 \ \rm kHz$:</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm 10}}{3 \cdot \eta_{10}^{\hspace{0.1cm}2}} = \frac{ f_{\rm 10} \cdot \eta_{5}^{\hspace{0.1cm}2}}{ 3 \cdot f_{\rm 5} \cdot \eta_{10}^{\hspace{0.1cm}2}}\cdot \frac{2 \cdot N_0 \cdot f_{\rm 5}}{\eta_{5}^{\hspace{0.1cm}2}} \hspace{0.05cm}.$$</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>:$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm 10}}{3 \cdot \eta_{10}^{\hspace{0.1cm}2}} = \frac{ f_{\rm 10} \cdot \eta_{5}^{\hspace{0.1cm}2}}{ 3 \cdot f_{\rm 5} \cdot \eta_{10}^{\hspace{0.1cm}2}}\cdot \frac{2 \cdot N_0 \cdot f_{\rm 5}}{\eta_{5}^{\hspace{0.1cm}2}} \hspace{0.05cm}.$$</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>Der zweite Term gibt die Rauschleistung des Vergleichssystems (PM, $f_{\rm N} = f_5$<del class="diffchange diffchange-inline">) </del>an, die zum Ergebnis $10 · \lg ρ_v = 50\ \rm dB$ geführt hat.</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>Der zweite Term gibt die Rauschleistung des Vergleichssystems <ins class="diffchange diffchange-inline">$</ins>(<ins class="diffchange diffchange-inline">$</ins>PM, $f_{\rm N} = f_5<ins class="diffchange diffchange-inline">)</ins>$ an, die zum Ergebnis $10 · \lg ρ_v = 50\ \rm dB$ geführt hat.</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>Bei Frequenzmodulation ist nun jedoch der Modulationsindex $η$ umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient $η_5^2/η_{10}^2 = 4$ ist. Somit ergibt sich für den Vorfaktor $8/3$. Aufgrund der größeren Rauschleistung ist das SNR kleiner:</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>Bei Frequenzmodulation ist nun jedoch der Modulationsindex $η$ umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient $η_5^2/η_{10}^2 = 4$ ist. </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>Somit ergibt sich für den Vorfaktor $8/3$. Aufgrund der größeren Rauschleistung ist das SNR kleiner:</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} - 10 \cdot {\rm lg} \hspace{0.15cm}({8}/{3})\hspace{0.15cm}\underline {\approx 45.74\,{\rm dB}}\hspace{0.05cm}.$$</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>:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} - 10 \cdot {\rm lg} \hspace{0.15cm}({8}/{3})\hspace{0.15cm}\underline {\approx 45.74\,{\rm dB}}\hspace{0.05cm}.$$</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>Bei gleicher Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ist nun die FM um $1.25 \ \rm dB$ schlechter als die PM, da sich nun die Halbierung von $η$ – nach Quadrierung der Faktor $4$ – stärker auswirkt als der systembedingte Faktor $3$, um den die FM gegenüber der PM überlegen 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>Bei gleicher Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ist nun die FM um $1.25 \ \rm dB$ schlechter als die PM, da sich nun die Halbierung von $η$ – nach Quadrierung der Faktor $4$ – stärker auswirkt als der systembedingte Faktor $3$, um den die FM gegenüber der PM überlegen 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="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>*Der Vergleich der Teilaufgaben (2) und (3) zeigt einen Unterschied um den Faktor $8$ bzw. $9.03 \ \rm dB$. </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>*Der Vergleich der Teilaufgaben <ins class="diffchange diffchange-inline">'''</ins>(2)<ins class="diffchange diffchange-inline">''' </ins>und <ins class="diffchange diffchange-inline">'''</ins>(3)<ins class="diffchange diffchange-inline">''' </ins>zeigt einen Unterschied um den Faktor $8$ bzw. $9.03 \ \rm dB$. </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>*Der ungünstigere Wert für die größere Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor $4$ – und die doppelte Rauschbandbreite.</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>*Der ungünstigere Wert für die größere Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor $4$ – und die doppelte Rauschbandbreite.</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>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=26935&oldid=prev
Guenter am 20. Dezember 2018 um 15:37 Uhr
2018-12-20T15:37:03Z
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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 20. Dezember 2018, 15:37 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_ID1115__Mod_A_3_9.png|right|frame|Rauschleistungsdichten von PM und FM]]</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_ID1115__Mod_A_3_9.png|right|frame|Rauschleistungsdichten von PM und FM]]</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz $f_{\rm N}$. Zunächst gelte für die Nachrichtenfrequenz $f_{\rm N} = f_5 = 5 \ \rm kHz$ und der Modulationsindex (Phasenhub) sei $η = 5$.</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz $f_{\rm N}$. Zunächst gelte für die Nachrichtenfrequenz <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = f_5 = 5 \ \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>und der Modulationsindex (Phasenhub) sei <ins class="diffchange diffchange-inline">&nbsp;</ins>$η = 5$.</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte $N_0$ ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte ${\it \Phi}_{v {\rm , \hspace{0.08cm}<del class="diffchange diffchange-inline">FM</del>} }(f) = {\it \Phi}_0$, die auch vom Modulationsindex abhängt:</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte <ins class="diffchange diffchange-inline">&nbsp;</ins>$N_0$<ins class="diffchange diffchange-inline">&nbsp; </ins>ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte <ins class="diffchange diffchange-inline">&nbsp;</ins>${\it \Phi}_{v {\rm , \hspace{0.08cm}<ins class="diffchange diffchange-inline">PM</ins>} }(f) = {\it \Phi}_0$, die auch vom Modulationsindex <ins class="diffchange diffchange-inline">&nbsp;$η$&nbsp; </ins>abhängt:</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>:$${\it \Phi}_0 = \frac{N_0}{\eta^2} \hspace{0.05cm}.$$</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>:$${\it \Phi}_0 = \frac{N_0}{\eta^2} \hspace{0.05cm}.$$</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>Für die Berechnung der Rauschleistung $P_{\rm R}$ ist lediglich der Frequenzbereich von $±f_{\rm N}$ relevant (siehe Grafik).</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>Für die Berechnung der Rauschleistung <ins class="diffchange diffchange-inline">&nbsp;</ins>$P_{\rm R}$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist lediglich der Frequenzbereich von <ins class="diffchange diffchange-inline">&nbsp;</ins>$±f_{\rm N}$<ins class="diffchange diffchange-inline">&nbsp; </ins>relevant (siehe Grafik).</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 Rauschleistungsdichte nach der FM–Demodulation lautet mit dem Frequenzhub $Δf_{\rm A}$:</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 Rauschleistungsdichte nach der FM–Demodulation lautet mit dem Frequenzhub <ins class="diffchange diffchange-inline">&nbsp;</ins>$Δf_{\rm A}$:</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>:$${\it \Phi}_{v {\rm , \hspace{0.08cm}FM} } (f) = N_0 \cdot \left(\frac{f}{\Delta f_{\rm A}}\right)^2 \hspace{0.05cm}.$$</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>:$${\it \Phi}_{v {\rm , \hspace{0.08cm}FM} } (f) = N_0 \cdot \left(\frac{f}{\Delta f_{\rm A}}\right)^2 \hspace{0.05cm}.$$</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>*Gegeben ist der Rauschabstand $10 · \lg ρ_v = 50 \ \rm dB$ für Phasenmodulation und $<del class="diffchange diffchange-inline">f_N </del>= 5 kHz$. </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>*Gegeben ist der Rauschabstand <ins class="diffchange diffchange-inline">&nbsp;</ins>$10 · \lg ρ_v = 50 \ \rm dB$<ins class="diffchange diffchange-inline">&nbsp; </ins>für Phasenmodulation und <ins class="diffchange diffchange-inline">&nbsp;</ins>$<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= 5 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$. </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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz $f_{\rm N} = 5 \ \rm kHz$<del class="diffchange diffchange-inline">) </del>sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz $f_{\rm N} = f_{10} = 10 \ \rm kHz$.</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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = 5 \ \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = f_{10} = 10 \ \rm kHz$.</div></td></tr>
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<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="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>''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>''Hinweise:''</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 Aufgabe gehört zum Kapitel [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 Aufgabe gehört zum Kapitel<ins class="diffchange diffchange-inline">&nbsp; </ins>[[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 den Abschnitt [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation#Systemvergleich_von_AM.2C_PM_und_FM_hinsichtlich_Rauschen|Systemvergleich von AM, PM und FM hinsichtlich Rauschen]].</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 den Abschnitt<ins class="diffchange diffchange-inline">&nbsp; </ins>[[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation#Systemvergleich_von_AM.2C_PM_und_FM_hinsichtlich_Rauschen|Systemvergleich von AM, PM und FM hinsichtlich Rauschen]].</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> </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> </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 colspan="2" class="diff-lineno" id="mw-diff-left-l25" >Zeile 25:</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;"></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 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="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>{Welcher Rauschabstand ergibt sich bei Phasenmodulation und $f_{\rm N} = 10 \ \rm kHz$? Interpretieren Sie das 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>{Welcher Rauschabstand ergibt sich bei Phasenmodulation und <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = 10 \ \rm kHz$? Interpretieren Sie das 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>|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>$10 · \lg ρ_v \ = \ $ { 46.99 3% } $\ \rm dB$ </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>$10 · \lg ρ_v \ = \ $ { 46.99 3% } $\ \rm dB$ </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>{Berechnen Sie den Rauschabstand bei Frequenzmodulation und $f_{\rm N} = 5 \ \rm kHz$. Wie groß ist der Modulationsindex bei dieser Konstellation?</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 den Rauschabstand bei Frequenzmodulation und <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = 5 \ \rm kHz$. Wie groß ist der Modulationsindex bei dieser Konstellation?</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>$10 · \lg ρ_v \ = \ $ { 54.77 3% } $\ \rm dB$</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>$10 · \lg ρ_v \ = \ $ { 54.77 3% } $\ \rm dB$</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 den Rauschabstand bei Frequenzmodulation und $f_{\rm N} = 10 \ \rm kHz$. Interpretieren Sie das Ergebnis im Vergleich zu <del class="diffchange diffchange-inline"> </del>(1) und (2).</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 den Rauschabstand bei Frequenzmodulation und <ins class="diffchange diffchange-inline">&nbsp;</ins>$f_{\rm N} = 10 \ \rm kHz$. Interpretieren Sie das Ergebnis im Vergleich zu <ins class="diffchange diffchange-inline">'''</ins>(1)<ins class="diffchange diffchange-inline">''' </ins>und <ins class="diffchange diffchange-inline">'''</ins>(2)<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>|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>$10 · \lg ρ_v \ = \ $ { 45.74 3% } $\ \rm <del class="diffchange diffchange-inline">kHz</del>$ </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>$10 · \lg ρ_v \ = \ $ { 45.74 3% } $\ \rm <ins class="diffchange diffchange-inline">dB</ins>$ </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;"><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_3.10:_Berechnung_der_Rauschleistungen&diff=24873&oldid=prev
Mwiki-lnt: Textersetzung - „*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.“ durch „ “
2018-05-29T12:02:47Z
<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>
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<td colspan="2" class="diff-lineno">Zeile 19:</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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 den Abschnitt [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation#Systemvergleich_von_AM.2C_PM_und_FM_hinsichtlich_Rauschen|Systemvergleich von AM, PM und FM hinsichtlich Rauschen]].</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 den Abschnitt [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation#Systemvergleich_von_AM.2C_PM_und_FM_hinsichtlich_Rauschen|Systemvergleich von AM, PM und FM hinsichtlich Rauschen]].</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>
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Mwiki-lnt
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=22013&oldid=prev
Guenter: Guenter verschob die Seite Aufgaben:3.10 Berechnung der Rauschleistungen nach Aufgaben:Aufgabe 3.10: Berechnung der Rauschleistungen
2018-01-03T14:26:32Z
<p>Guenter verschob die Seite <a href="/Aufgaben:3.10_Berechnung_der_Rauschleistungen" class="mw-redirect" title="Aufgaben:3.10 Berechnung der Rauschleistungen">3.10 Berechnung der Rauschleistungen</a> nach <a href="/Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen" title="Aufgaben:Aufgabe 3.10: Berechnung der Rauschleistungen">Aufgabe 3.10: Berechnung der Rauschleistungen</a></p>
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Version vom 3. Januar 2018, 14:26 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
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=14133&oldid=prev
Guenter am 1. August 2017 um 16:04 Uhr
2017-08-01T16:04:41Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 1. August 2017, 16:04 Uhr</td>
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<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>{{quiz-Header|Buchseite=Modulationsverfahren/<del class="diffchange diffchange-inline">Rauscheinfluss bei PM und FM</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>{{quiz-Header|Buchseite=Modulationsverfahren/<ins class="diffchange diffchange-inline">Rauscheinfluss_bei_Winkelmodulation</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>}}</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>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=13839&oldid=prev
Guenter am 17. Juli 2017 um 11:09 Uhr
2017-07-17T11:09:35Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 17. Juli 2017, 11:09 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;"><div>''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>''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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 den Abschnitt [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation#Systemvergleich_von_AM.2C_PM_und_FM_hinsichtlich_Rauschen|Systemvergleich von AM, PM und FM hinsichtlich Rauschen]].</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>*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.</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>*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; 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> </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>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=13838&oldid=prev
Guenter am 17. Juli 2017 um 10:02 Uhr
2017-07-17T10:02:52Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 17. Juli 2017, 10:02 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l41" >Zeile 41:</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>===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<del class="diffchange diffchange-inline">.</del>''' Das Signal–zu–Rausch–Leistungsverhältnis (Sinken–SNR) <del class="diffchange diffchange-inline">ρυ </del>ist der Quotient aus der Nutzleistung $<del class="diffchange diffchange-inline">P_S</del>$ und der Rauschleistung $<del class="diffchange diffchange-inline">P_R</del>$. Speziell bei der Phasenmodulation gilt:</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>1<ins class="diffchange diffchange-inline">)</ins>'''<ins class="diffchange diffchange-inline">&nbsp; </ins>Das Signal–zu–Rausch–Leistungsverhältnis (Sinken–SNR) <ins class="diffchange diffchange-inline">$ \rho_{v }$ </ins>ist der Quotient aus der Nutzleistung $<ins class="diffchange diffchange-inline">P_{\rm S}</ins>$ und der Rauschleistung $<ins class="diffchange diffchange-inline">P_{\rm S}</ins>$. Speziell bei der Phasenmodulation gilt:</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>$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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><ins class="diffchange diffchange-inline">:</ins>$$ \rho_{v } = \frac{P_{\rm S}}{P_{\rm R}} = \frac{P_{\rm S}}{{\it \Phi}_0 \cdot 2 f_{\rm N} } =\frac{\eta^2}{2} \cdot \frac{P_{\rm S}}{N_0 \cdot f_{\rm N} }\hspace{0.05cm}.$$</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 Messung mit $<del class="diffchange diffchange-inline">f_N </del>= f_5 = 5 kHz$ hat das SNR $<del class="diffchange diffchange-inline">ρ_υ </del>= <del class="diffchange diffchange-inline">105</del>$ (entsprechend $50 dB$) ergeben. Die doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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><ins class="diffchange diffchange-inline">*</ins>Die Messung mit $<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= f_5 = 5 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$ hat das SNR $ <ins class="diffchange diffchange-inline">\rho_{v } </ins>= <ins class="diffchange diffchange-inline">10^5</ins>$ (entsprechend $<ins class="diffchange diffchange-inline">10 · \lg ρ_v =</ins>50 dB$) ergeben. </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>$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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><ins class="diffchange diffchange-inline">*</ins>Die doppelte Nachrichtenfrequenz führt zum halben SNR, da nun die doppelte Rauschleistung wirksam 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><del class="diffchange diffchange-inline">Dieses Ergebnis lässt sich auch über die Beziehung $ρ_υ = η^2/2 · ξ$ herleiten. Bei PM ist η unabhängig von der Nachrichtenfrequenz. Der SNR–Verlust geht darauf zurück, dass nun $ξ = P_S/(N_0 · f_N)$ halbiert wird.</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>$$ \rho_{v }= 0.5 \cdot 10^5 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v } \hspace{0.15cm}\underline {\approx 46.99\,{\rm dB}}\hspace{0.05cm}.$$</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 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;">Dieses Ergebnis lässt sich auch über die Beziehung $ρ_v = η^2/2 · ξ$ herleiten. </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;">*Bei Phasenmodulation ist $η$ unabhängig von der Nachrichtenfrequenz. </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;">*Der SNR–Verlust geht darauf zurück, dass nun die Leistungskenngröße $ξ = P_{\rm S}/(N_0 · f_{\rm N})$ halbiert wird.</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><del style="font-weight: bold; text-decoration: none;">'''2.''' Bei Frequenzmodulation und der Nachrichtenfrequenz $f_N = 5 kHz$ erhält man für die Rauschleist</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;">$$P_{\rm R} = \int_{-f_{\rm N}}^{ + f_{\rm N}} {\it \Phi}_{v {\rm , \hspace{0.08cm}FM} } (f)\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0}{\Delta f_{\rm A}^{\hspace{0.1cm}2}} \cdot \int_{0}^{ f_{\rm N}} f^2\hspace{0.1cm}{\rm d}f = \frac{2 \cdot N_0 \cdot f_{\rm N}^{\hspace{0.1cm}3}}{3 \cdot \Delta f_{\rm A}^2} \hspace{0.05cm}.$$</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;">Unter Berücksichtigung von $Δf_A = η · f_N$ (Frequenzhub) ergibt sich somit:</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;">$$P_{\rm R} = \frac{2 \cdot N_0 \cdot f_{\rm N}}{3 \cdot \eta^2} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v }= \frac{3 \cdot \eta^2 \cdot P_{\rm S}}{2 \cdot N_0 \cdot f_{\rm N}} = 3 \cdot \rho_{v {\rm , \hspace{0.08cm}PM}}\hspace{0.05cm}.$$</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;">Das heißt: Die Frequenzmodulation ist um den Faktor 3 (oder 4.77 dB) besser als die PM:</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;">$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm dB}}\hspace{0.05cm}.$$</del></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="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">3.</del>''' <del class="diffchange diffchange-inline"> Entsprechend dem Ergebnis aus b) erhält man mit </del>$f_{<del class="diffchange diffchange-inline">10</del>} = <del class="diffchange diffchange-inline">10 </del>kHz$:</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">(2)</ins>'''<ins class="diffchange diffchange-inline">&nbsp; Bei Frequenzmodulation und der Nachrichtenfrequenz </ins>$f_{<ins class="diffchange diffchange-inline">\rm N</ins>} = <ins class="diffchange diffchange-inline">5 \ \rm </ins>kHz$ <ins class="diffchange diffchange-inline">erhält man für die Rauschleistung</ins></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>$$P_{\rm R} = \<del class="diffchange diffchange-inline">frac</del>{<del class="diffchange diffchange-inline">2 </del>\<del class="diffchange diffchange-inline">cdot N_0 </del>\<del class="diffchange diffchange-inline">cdot f_</del>{\rm <del class="diffchange diffchange-inline">10</del>}}{<del class="diffchange diffchange-inline">3 </del>\cdot \<del class="diffchange diffchange-inline">eta_</del>{<del class="diffchange diffchange-inline">10</del>}^{\hspace{0.1cm}2}} <del class="diffchange diffchange-inline">= </del>\<del class="diffchange diffchange-inline">frac</del>{ f_{\rm <del class="diffchange diffchange-inline">10</del>} \cdot \<del class="diffchange diffchange-inline">eta_</del>{<del class="diffchange diffchange-inline">5</del>}^{\hspace{0.1cm}<del class="diffchange diffchange-inline">2</del>}}{ 3 \cdot f_{\rm <del class="diffchange diffchange-inline">5</del>} \cdot \<del class="diffchange diffchange-inline">eta_</del>{<del class="diffchange diffchange-inline">10</del>}^{\hspace{0.<del class="diffchange diffchange-inline">1cm</del>}2<del class="diffchange diffchange-inline">}}</del>\cdot \<del class="diffchange diffchange-inline">frac</del>{2 \cdot N_0 \cdot f_{\rm <del class="diffchange diffchange-inline">5</del>}}<del class="diffchange diffchange-inline">{</del>\<del class="diffchange diffchange-inline">eta_</del>{<del class="diffchange diffchange-inline">5}^</del>{\hspace{0.<del class="diffchange diffchange-inline">1cm</del>}<del class="diffchange diffchange-inline">2</del>}} \hspace{0.05cm}.$$</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>:$$P_{\rm R} = \<ins class="diffchange diffchange-inline">int_{-f_{\rm N}}^{ + f_{\rm N}} </ins>{\<ins class="diffchange diffchange-inline">it </ins>\<ins class="diffchange diffchange-inline">Phi}_{v </ins>{\rm <ins class="diffchange diffchange-inline">, \hspace{0.08cm}FM} } (f)\hspace{0.1cm</ins>}<ins class="diffchange diffchange-inline">{\rm d</ins>}<ins class="diffchange diffchange-inline">f = \frac</ins>{<ins class="diffchange diffchange-inline">2 </ins>\cdot <ins class="diffchange diffchange-inline">N_0}{</ins>\<ins class="diffchange diffchange-inline">Delta f_</ins>{<ins class="diffchange diffchange-inline">\rm A</ins>}^{\hspace{0.1cm}2}} \<ins class="diffchange diffchange-inline">cdot \int_{0}^</ins>{ f_{\rm <ins class="diffchange diffchange-inline">N}} f^2\hspace{0.1cm}{\rm d</ins>}<ins class="diffchange diffchange-inline">f = \frac{2 </ins>\cdot <ins class="diffchange diffchange-inline">N_0 </ins>\<ins class="diffchange diffchange-inline">cdot f_</ins>{<ins class="diffchange diffchange-inline">\rm N</ins>}^{\hspace{0.1cm}<ins class="diffchange diffchange-inline">3</ins>}}{3 \cdot <ins class="diffchange diffchange-inline">\Delta f_{\rm A}^2} \hspace{0.05cm}.$$</ins></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">Der zweite Term gibt die Rauschleistung des Vergleichssystems </del>(<del class="diffchange diffchange-inline">Phasenmodulation, </del>$<del class="diffchange diffchange-inline">f_N = f_5</del>$) <del class="diffchange diffchange-inline">an, </del>die <del class="diffchange diffchange-inline">zum Ergebnis </del>$10 <del class="diffchange diffchange-inline">· </del>lg <del class="diffchange diffchange-inline">ρ_υ </del>= 50 dB$ <del class="diffchange diffchange-inline">geführt hat.</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">Unter Berücksichtigung des Frequenzhubs $Δf_{\rm A} = η · </ins>f_{\rm <ins class="diffchange diffchange-inline">N}$ ergibt sich somit:</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">:$$P_{\rm R</ins>} <ins class="diffchange diffchange-inline">= \frac{2 </ins>\cdot <ins class="diffchange diffchange-inline">N_0 </ins>\<ins class="diffchange diffchange-inline">cdot f_</ins>{<ins class="diffchange diffchange-inline">\rm N}</ins>}<ins class="diffchange diffchange-inline">{3 \cdot \eta</ins>^<ins class="diffchange diffchange-inline">2} \hspace</ins>{<ins class="diffchange diffchange-inline">0.3cm}\Rightarrow </ins>\hspace{0.<ins class="diffchange diffchange-inline">3cm} \rho_{v </ins>}<ins class="diffchange diffchange-inline">= \frac{3 \cdot \eta^</ins>2 \cdot <ins class="diffchange diffchange-inline">P_{</ins>\<ins class="diffchange diffchange-inline">rm S}}</ins>{2 \cdot N_0 \cdot f_{\rm <ins class="diffchange diffchange-inline">N</ins>}} <ins class="diffchange diffchange-inline">= 3 \cdot </ins>\<ins class="diffchange diffchange-inline">rho_</ins>{<ins class="diffchange diffchange-inline">v </ins>{<ins class="diffchange diffchange-inline">\rm , </ins>\hspace{0.<ins class="diffchange diffchange-inline">08cm</ins>}<ins class="diffchange diffchange-inline">PM</ins>}}\hspace{0.05cm}.$$</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">Das heißt: Die Frequenzmodulation ist um den Faktor $3$ </ins>(<ins class="diffchange diffchange-inline">oder </ins>$<ins class="diffchange diffchange-inline">4.77 \ \rm dB</ins>$) <ins class="diffchange diffchange-inline">besser als </ins>die <ins class="diffchange diffchange-inline">Phasenmodulation:</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>$10 <ins class="diffchange diffchange-inline">\cdot {\rm </ins>lg<ins class="diffchange diffchange-inline">} \hspace{0.15cm}\rho_{v }</ins>= 50<ins class="diffchange diffchange-inline">\,{\rm dB} + 10 \cdot {\rm lg} \hspace{0.15cm}{3}\hspace{0.15cm}\underline {\approx 54.77\,{\rm </ins>dB<ins class="diffchange diffchange-inline">}}\hspace{0.05cm}.$</ins>$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Bei FM ist nun jedoch </del>der <del class="diffchange diffchange-inline">Modulationsindex umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient </del>$<del class="diffchange diffchange-inline">η_5^2/η_</del>{10}<del class="diffchange diffchange-inline">^2 </del>= <del class="diffchange diffchange-inline">4</del>$ <del class="diffchange diffchange-inline">ist. Somit ergibt sich für den Vorfaktor 8/3. Aufgrund der größeren Rauschleistung ist das SNR kleiner</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">'''(3)'''&nbsp; Entsprechend dem Ergebnis </ins>der <ins class="diffchange diffchange-inline">Teilaufgabe (2) erhält man mit </ins>$<ins class="diffchange diffchange-inline">f_</ins>{10} = <ins class="diffchange diffchange-inline">10 \ \rm kHz</ins>$:</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>$$<del class="diffchange diffchange-inline">10 </del>\cdot {\rm <del class="diffchange diffchange-inline">lg</del>} \hspace{0.<del class="diffchange diffchange-inline">15cm</del>}<del class="diffchange diffchange-inline">\rho_{v </del>}= <del class="diffchange diffchange-inline">50</del>\<del class="diffchange diffchange-inline">,</del>{\rm <del class="diffchange diffchange-inline">dB</del>} <del class="diffchange diffchange-inline">- 10 </del>\cdot {<del class="diffchange diffchange-inline">\rm lg</del>} \hspace{0.<del class="diffchange diffchange-inline">15cm</del>}<del class="diffchange diffchange-inline">(</del>{<del class="diffchange diffchange-inline">8</del>}<del class="diffchange diffchange-inline">/</del>{<del class="diffchange diffchange-inline">3</del>}<del class="diffchange diffchange-inline">)</del>\hspace{0.<del class="diffchange diffchange-inline">15cm</del>}\<del class="diffchange diffchange-inline">underline </del>{\<del class="diffchange diffchange-inline">approx 45.74</del>\<del class="diffchange diffchange-inline">,</del>{\rm <del class="diffchange diffchange-inline">dB</del>}}\hspace{0.05cm}.$$</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><ins class="diffchange diffchange-inline">:</ins>$$<ins class="diffchange diffchange-inline">P_{\rm R} = \frac{2 \cdot N_0 </ins>\cdot <ins class="diffchange diffchange-inline">f_</ins>{\rm <ins class="diffchange diffchange-inline">10}}{3 \cdot \eta_{10</ins>}<ins class="diffchange diffchange-inline">^{</ins>\hspace{0.<ins class="diffchange diffchange-inline">1cm}2</ins>}} = \<ins class="diffchange diffchange-inline">frac{ f_</ins>{\rm <ins class="diffchange diffchange-inline">10</ins>} \cdot <ins class="diffchange diffchange-inline">\eta_</ins>{<ins class="diffchange diffchange-inline">5</ins>}<ins class="diffchange diffchange-inline">^{</ins>\hspace{0.<ins class="diffchange diffchange-inline">1cm</ins>}<ins class="diffchange diffchange-inline">2}}{ 3 \cdot f_</ins>{<ins class="diffchange diffchange-inline">\rm 5</ins>} <ins class="diffchange diffchange-inline">\cdot \eta_</ins>{<ins class="diffchange diffchange-inline">10</ins>}<ins class="diffchange diffchange-inline">^{</ins>\hspace{0.<ins class="diffchange diffchange-inline">1cm}2}</ins>}\<ins class="diffchange diffchange-inline">cdot \frac</ins>{<ins class="diffchange diffchange-inline">2 </ins>\<ins class="diffchange diffchange-inline">cdot N_0 </ins>\<ins class="diffchange diffchange-inline">cdot f_</ins>{\rm <ins class="diffchange diffchange-inline">5}}{\eta_{5}^{\hspace{0.1cm}2</ins>}} \hspace{0.05cm}.$$</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><del class="diffchange diffchange-inline">Bei gleicher Nachrichtenfrequenz </del>$<del class="diffchange diffchange-inline">f_N </del>= <del class="diffchange diffchange-inline">10 kHz</del>$ <del class="diffchange diffchange-inline">ist nun </del>die <del class="diffchange diffchange-inline">FM um 1.25 </del>dB <del class="diffchange diffchange-inline">schlechter als die PM, da sich nun die Halbierung von η – nach Quadrierung der Faktor 4 – stärker auswirkt als der systembedingte Faktor 3, um den die FM gegenüber der PM überlegen ist</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">Der zweite Term gibt die Rauschleistung des Vergleichssystems (PM, </ins>$<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= <ins class="diffchange diffchange-inline">f_5</ins>$<ins class="diffchange diffchange-inline">) an, </ins>die <ins class="diffchange diffchange-inline">zum Ergebnis $10 · \lg ρ_v = 50\ \rm </ins>dB<ins class="diffchange diffchange-inline">$ geführt hat</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>Der Vergleich der Teilaufgaben <del class="diffchange diffchange-inline">b</del>) und <del class="diffchange diffchange-inline">c</del>) zeigt einen Unterschied um den Faktor 8 bzw. 9.03 dB. Der ungünstigere Wert für die größere Nachrichtenfrequenz $<del class="diffchange diffchange-inline">f_N </del>= 10 kHz$ ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor 4 – und die doppelte Rauschbandbreite.</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">Bei Frequenzmodulation ist nun jedoch der Modulationsindex $η$ umgekehrt proportional zur Nachrichtenfrequenz, so dass der Quotient $η_5^2/η_{10}^2 = 4$ ist. Somit ergibt sich für den Vorfaktor $8/3$. Aufgrund der größeren Rauschleistung ist das SNR kleiner:</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">:$$10 \cdot {\rm lg} \hspace{0.15cm}\rho_{v }= 50\,{\rm dB} - 10 \cdot {\rm lg} \hspace{0.15cm}({8}/{3})\hspace{0.15cm}\underline {\approx 45.74\,{\rm dB}}\hspace{0.05cm}.$$</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">Bei gleicher Nachrichtenfrequenz $f_{\rm N} = 10 \ \rm kHz$ ist nun die FM um $1.25 \ \rm dB$ schlechter als die PM, da sich nun die Halbierung von $η$ – nach Quadrierung der Faktor $4$ – stärker auswirkt als der systembedingte Faktor $3$, um den die FM gegenüber der PM überlegen ist.</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> </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>Der Vergleich der Teilaufgaben <ins class="diffchange diffchange-inline">(2</ins>) und <ins class="diffchange diffchange-inline">(3</ins>) zeigt einen Unterschied um den Faktor <ins class="diffchange diffchange-inline">$</ins>8<ins class="diffchange diffchange-inline">$ </ins>bzw. <ins class="diffchange diffchange-inline">$</ins>9.03 <ins class="diffchange diffchange-inline">\ \rm </ins>dB<ins class="diffchange diffchange-inline">$</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>Der ungünstigere Wert für die größere Nachrichtenfrequenz $<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= 10 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$ ergibt sich durch den nur halb so großen Modulationsindex – nach Quadrierung Faktor <ins class="diffchange diffchange-inline">$</ins>4<ins class="diffchange diffchange-inline">$ </ins>– und die doppelte Rauschbandbreite.</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>{{ML-Fuß}}</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-Fuß}}</div></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_3.10:_Berechnung_der_Rauschleistungen&diff=13837&oldid=prev
Guenter am 17. Juli 2017 um 09:44 Uhr
2017-07-17T09:44:43Z
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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 17. Juli 2017, 09:44 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6" >Zeile 6:</td>
<td colspan="2" class="diff-lineno">Zeile 6:</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz $f_{\rm N}$. Zunächst gelte für die Nachrichtenfrequenz $f_{\rm N} = f_5 = 5 \ \rm kHz$ und der Modulationsindex (Phasenhub) sei $η = 5$.</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 werden die Phasen– und Frequenzmodulation einer Cosinusschwingung mit der Frequenz $f_{\rm N}$. Zunächst gelte für die Nachrichtenfrequenz $f_{\rm N} = f_5 = 5 \ \rm kHz$ und der Modulationsindex (Phasenhub) sei $η = 5$.</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte $N_0$ ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte ${\it \Phi}_{<del class="diffchange diffchange-inline">υ</del>, <del class="diffchange diffchange-inline">PM</del>}(f) = {\it \Phi}_0$, die auch vom Modulationsindex abhängt:</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>Bei Vorhandensein von additivem Gaußschen Rauschen mit der Rauschleistungsdichte $N_0$ ergibt sich nach dem PM–Demodulator eine konstante Rauschleistungsdichte ${\it \Phi}_{<ins class="diffchange diffchange-inline">v {\rm </ins>, <ins class="diffchange diffchange-inline">\hspace{0.08cm}FM} </ins>}(f) = {\it \Phi}_0$, die auch vom Modulationsindex abhängt:</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>:$${\it \Phi}_0 = \frac{N_0}{\eta^2} \hspace{0.05cm}.$$</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>:$${\it \Phi}_0 = \frac{N_0}{\eta^2} \hspace{0.05cm}.$$</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>Für die Berechnung der Rauschleistung $P_{\rm R}$ ist lediglich der Frequenzbereich von $±f_{\rm N}$ relevant (siehe Grafik).</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>Für die Berechnung der Rauschleistung $P_{\rm R}$ ist lediglich der Frequenzbereich von $±f_{\rm N}$ relevant (siehe Grafik).</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14" >Zeile 14:</td>
<td colspan="2" class="diff-lineno">Zeile 14:</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>*Gegeben ist der Rauschabstand $10 · \lg ρ_v = 50 \ \rm dB$ für Phasenmodulation und $f_N = 5 kHz$. </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>*Gegeben ist der Rauschabstand $10 · \lg ρ_v = 50 \ \rm dB$ für Phasenmodulation und $f_N = 5 kHz$. </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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz $f_{\rm N} = 5 \ \rm kHz$) sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz $f_{\rm N} = f_{10} = 10 \ \rm kHz$.</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>*Gesucht sind in dieser Aufgabe der Rauschabstand bei FM für die Nachrichtenfrequenz $f_{\rm N} = 5 \ \rm kHz$) sowie die sich ergebenden Rauschabstände von PM und FM für die Nachrichtenfrequenz $f_{\rm N} = f_{10} = 10 \ \rm kHz$.</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 style="font-weight: bold; text-decoration: none;"></del></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>''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>''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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 [[Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation|Rauscheinfluss bei Winkelmodulation]].</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 style="font-weight: bold; text-decoration: none;">*Bezug genommen wird aber auch auf das Kapitel [[Modulationsverfahren/Phasenmodulation_(PM)|Phasenmodulation]] und insbesondere auf den Abschnitt [[Modulationsverfahren/Frequenzmodulation_(FM)#Signalverl.C3.A4ufe_bei_Frequenzmodulation|Signalverläufe bei Frequenzmodulation]].</del></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>*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.</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>*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.</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">*Gehen Sie bei der Berechnung von folgenden Werten der Besselfunktion aus:</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="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">:$${\rm J}_0 (2.4) \approx 0, \hspace{0.2cm}{\rm J}_1 (2.4) = -{\rm J}_{-1} (2.4)\approx 0.52, \hspace{0.2cm}{\rm J}_2 (2.4) = {\rm J}_{-2} (2.4)\approx 0.43.$$</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> </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 class="diffchange diffchange-inline">'''Hinweis:''' Diese Aufgabe bezieht sich auf den Theorieteil von [http://www.lntwww.de/Modulationsverfahren/Rauscheinfluss_bei_Winkelmodulation Kapitel 3.3]. </del></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;"><div>===Fragebogen===</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>===Fragebogen===</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 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="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>{Welcher Rauschabstand ergibt sich bei $<del class="diffchange diffchange-inline">f_N </del>= 10 kHz$ <del class="diffchange diffchange-inline">und PM</del>? Interpretieren Sie das 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>{Welcher Rauschabstand ergibt sich bei <ins class="diffchange diffchange-inline">Phasenmodulation und </ins>$<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= 10 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$? Interpretieren Sie das 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>|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>$<del class="diffchange diffchange-inline">PM, f_N = 10 kHz: </del>10 · lg <del class="diffchange diffchange-inline">ρ_υ</del>$ <del class="diffchange diffchange-inline">= </del>{ 46.99 3% } $dB$ </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>$10 · <ins class="diffchange diffchange-inline">\</ins>lg <ins class="diffchange diffchange-inline">ρ_v \ = \ </ins>$ { 46.99 3% } $<ins class="diffchange diffchange-inline">\ \rm </ins>dB$ </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>{Berechnen Sie den Rauschabstand <del class="diffchange diffchange-inline">für </del>$<del class="diffchange diffchange-inline">f_N </del>= 5 kHz$ <del class="diffchange diffchange-inline">und FM</del>. Wie groß ist der Modulationsindex bei dieser Konstellation?</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 den Rauschabstand <ins class="diffchange diffchange-inline">bei Frequenzmodulation und </ins>$<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= 5 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$. Wie groß ist der Modulationsindex bei dieser Konstellation?</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>$<del class="diffchange diffchange-inline">FM, f_N = 5 kHz: </del>10 · lg <del class="diffchange diffchange-inline">ρ_υ</del>$ <del class="diffchange diffchange-inline">= </del>{ 54.77 3% } $dB$</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>$10 · <ins class="diffchange diffchange-inline">\</ins>lg <ins class="diffchange diffchange-inline">ρ_v \ = \ </ins>$ { 54.77 3% } $<ins class="diffchange diffchange-inline">\ \rm </ins>dB$</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 den Rauschabstand <del class="diffchange diffchange-inline">für </del>$<del class="diffchange diffchange-inline">f_N </del>= 10 kHz$ <del class="diffchange diffchange-inline">und FM</del>. Interpretieren Sie das Ergebnis im Vergleich zu <del class="diffchange diffchange-inline">den Teilfragen a</del>) und <del class="diffchange diffchange-inline">b</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>{Berechnen Sie den Rauschabstand <ins class="diffchange diffchange-inline">bei Frequenzmodulation und </ins>$<ins class="diffchange diffchange-inline">f_{\rm N} </ins>= 10 <ins class="diffchange diffchange-inline">\ \rm </ins>kHz$. Interpretieren Sie das Ergebnis im Vergleich zu <ins class="diffchange diffchange-inline"> (1</ins>) und <ins class="diffchange diffchange-inline">(2</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>|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>$ <del class="diffchange diffchange-inline">FM, f_N = 10 kHz: </del>10 · lg <del class="diffchange diffchange-inline">ρ_υ</del>$ <del class="diffchange diffchange-inline">= </del>{ 45.74 3% } $<del class="diffchange diffchange-inline">KHz</del>$ </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>$10 · <ins class="diffchange diffchange-inline">\</ins>lg <ins class="diffchange diffchange-inline">ρ_v \ = \ </ins>$ { 45.74 3% } $<ins class="diffchange diffchange-inline">\ \rm kHz</ins>$ </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;"><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