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Aufgaben:Aufgabe 5.8: Matched-Filter für farbige Störung - Versionsgeschichte
2024-03-29T09:27:21Z
Versionsgeschichte dieser Seite in LNTwww
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https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=33043&oldid=prev
Guenter am 22. Februar 2022 um 09:58 Uhr
2022-02-22T09:58:02Z
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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 22. Februar 2022, 09:58 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l69" >Zeile 69:</td>
<td colspan="2" class="diff-lineno">Zeile 69:</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>:$$\rho_{d,\ \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\rho_{d,\ \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</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>\Rightarrow \quad 10\lg \cdot \rho _{d,\ \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\Rightarrow \quad 10\lg \cdot \rho _{d,\ \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</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 colspan="2" class="diff-lineno" id="mw-diff-left-l79" >Zeile 79:</td>
<td colspan="2" class="diff-lineno">Zeile 78:</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>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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 weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den <del class="diffchange diffchange-inline">Nutzimpuls</del>&nbsp; $g(t)$&nbsp; bewirkt. </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 weißem<ins class="diffchange diffchange-inline">&nbsp; </ins>(frequenzunabhängigen)<ins class="diffchange diffchange-inline">&nbsp; </ins>Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben,<ins class="diffchange diffchange-inline">&nbsp; </ins>der die Anpassung an den <ins class="diffchange diffchange-inline">Impuls</ins>&nbsp; $g(t)$&nbsp; bewirkt. </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 farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum&nbsp; ${\it \Phi}_n(f)$&nbsp; werden höhere Frequenzen durch den Korrekturterm&nbsp; $1+\left( f/f_0 \right)^2$&nbsp; angehoben, da in diesem Bereich die Störungen geringer sind. </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 farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum&nbsp; ${\it \Phi}_n(f)$&nbsp; werden höhere Frequenzen durch den Korrekturterm&nbsp; $1+\left( f/f_0 \right)^2$&nbsp; angehoben,<ins class="diffchange diffchange-inline">&nbsp; </ins>da in diesem Bereich die Störungen geringer sind. </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&nbsp; $f = 1/\Delta t = 2f_0 = 1\hspace{0.08cm} \rm kHz$&nbsp; erhält man:</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&nbsp; $f = 1/\Delta t = 2f_0 = 1\hspace{0.08cm} \rm kHz$&nbsp; erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</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>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l95" >Zeile 95:</td>
<td colspan="2" class="diff-lineno">Zeile 94:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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 bestimmte Integral wurde vorne angegeben; es hat den Wert&nbsp; $1$.&nbsp; Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem 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>*Dieses bestimmte Integral wurde vorne angegeben<ins class="diffchange diffchange-inline">;&nbsp</ins>; es hat den Wert&nbsp; $1$.&nbsp; Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem 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>*Damit ergeben sich folgende Gleichungen:</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>*Damit ergeben sich folgende Gleichungen:</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>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l102" >Zeile 102:</td>
<td colspan="2" class="diff-lineno">Zeile 101:</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 _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\rho _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</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><u>Fazit:</u></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><u>Fazit:</u> <ins class="diffchange diffchange-inline">&nbsp; </ins>Es ergibt sich ein um&nbsp; $1.2 \; \rm dB$&nbsp; besseres Ergebnis als bei weißem Rauschen,<ins class="diffchange diffchange-inline">&nbsp; </ins>da hier&nbsp; ${\it \Phi}_n(f)$&nbsp; im gesamten Frequenzbereich mit Ausnahme der Frequenz&nbsp; $f = 0$&nbsp; $($hier gilt das Gleichheitszeichen$)$&nbsp; kleiner ist als&nbsp; $N_0/2$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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">*</del>Es ergibt sich ein um&nbsp; $1.2 \; \rm dB$&nbsp; besseres Ergebnis als bei weißem Rauschen, da hier&nbsp; ${\it \Phi}_n(f)$&nbsp; im gesamten Frequenzbereich mit Ausnahme der Frequenz&nbsp; $f = 0$&nbsp; $($hier gilt das Gleichheitszeichen$)$&nbsp; kleiner ist als&nbsp; $N_0/2$. </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">*</del>Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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>{{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_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=33042&oldid=prev
Guenter am 22. Februar 2022 um 09:52 Uhr
2022-02-22T09:52:52Z
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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 22. Februar 2022, 09:52 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Zeile 13:</td>
<td colspan="2" class="diff-lineno">Zeile 13:</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>Dem Impuls&nbsp; $g(t)$&nbsp; ist eine Störung&nbsp; $n(t)$&nbsp; überlagert, die den Impuls weitgehend überdeckt.&nbsp; </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>Dem Impuls&nbsp; $g(t)$&nbsp; ist eine Störung&nbsp; $n(t)$&nbsp; überlagert,<ins class="diffchange diffchange-inline">&nbsp; </ins>die den Impuls weitgehend überdeckt.&nbsp; </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>Hierfür werden zwei Alternativen betrachtet:</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>Hierfür werden zwei Alternativen betrachtet:</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 zweiseitige Störleistungsdichte sei konstant (nur bei der ersten Teilaufgabe):</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 zweiseitige Störleistungsdichte sei konstant<ins class="diffchange diffchange-inline">&nbsp; </ins>(nur bei der ersten Teilaufgabe):</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das Störsignal&nbsp; $n(t)$&nbsp; sei farbig mit folgender Störleistungsdichte:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das Störsignal&nbsp; $n(t)$&nbsp; sei farbig mit folgender Störleistungsdichte:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23" >Zeile 23:</td>
<td colspan="2" class="diff-lineno">Zeile 23:</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>Dieser zweite LDS-Verlauf kann zum Beispiel aus weißem Rauschen durch ein Formfilter mit dem Frequenzgang</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>Dieser zweite LDS-Verlauf kann zum Beispiel aus weißem Rauschen durch ein Formfilter mit dem Frequenzgang</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>:$$H_{\rm N}(f) = \frac{1}{{1 + {\rm{j}}\cdot f/f_0 }}\quad\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\, \quad h_{\rm N}(t) = 2{\rm{\pi }}f_0 \cdot {\rm{e}}^{ - 2{\rm{\pi }}f_0 t} $$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm N}(f) = \frac{1}{{1 + {\rm{j}}\cdot f/f_0 }}\quad\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\, \quad h_{\rm N}(t) = 2{\rm{\pi }}f_0 \cdot {\rm{e}}^{ - 2{\rm{\pi }}f_0 t} $$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>modelliert werden (Tiefpass erster Ordnung). Weiter soll gelten:</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>modelliert werden<ins class="diffchange diffchange-inline">&nbsp; </ins>(Tiefpass erster Ordnung).<ins class="diffchange diffchange-inline">&nbsp; </ins>Weiter soll gelten:</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>*Das Filter sei jeweils optimal an die Sendeimpulsform&nbsp; $g(t)$&nbsp; und das Störleistungsdichtespektrum&nbsp; ${\it \Phi}_n (f)$&nbsp; angepasst: &nbsp; $H(f) = H_{\rm MF}(f)$<del class="diffchange diffchange-inline">. </del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Das Filter sei jeweils optimal an die Sendeimpulsform&nbsp; $g(t)$&nbsp; und das Störleistungsdichtespektrum&nbsp; ${\it \Phi}_n (f)$&nbsp; angepasst: &nbsp; </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">:$</ins>$H(f) = H_{\rm MF}(f)<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>*Die Filterkonstante&nbsp; $K_{\rm MF}$&nbsp; ist so zu wählen, dass&nbsp; $H(f= 0) =1$&nbsp; gilt. </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 Filterkonstante&nbsp; $K_{\rm MF}$&nbsp; ist so zu wählen, dass&nbsp; $H(f= 0) =1$&nbsp; 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>*Der Detektionszeitpunkt sei vereinfachend&nbsp; $T_{\rm D}= 0$&nbsp; (akausale Systembeschreibung).</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 Detektionszeitpunkt sei vereinfachend&nbsp; $T_{\rm D}= 0$&nbsp; (akausale Systembeschreibung).</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35" >Zeile 35:</td>
<td colspan="2" class="diff-lineno">Zeile 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></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><del class="diffchange diffchange-inline">''</del>Hinweise:<del class="diffchange diffchange-inline">'' </del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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&nbsp; [[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]].</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&nbsp; [[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]]. </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"> </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>*Gegeben ist zudem das folgende bestimmte Integral:</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 zudem das folgende bestimmte Integral:</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l47" >Zeile 47:</td>
<td colspan="2" class="diff-lineno">Zeile 47:</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>{Wie groß ist das S/N&ndash;Verhältnis (in dB) im Fall des weißen Rauschens?</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{Wie groß ist das<ins class="diffchange diffchange-inline">&nbsp; "maximale </ins>S/N&ndash;Verhältnis<ins class="diffchange diffchange-inline">"&nbsp; $\rho_{\rm MF,\hspace{0.08cm} \rm WR}$&nbsp; </ins>(in dB)<ins class="diffchange diffchange-inline">&nbsp; </ins>im Fall des weißen Rauschens?</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 \cdot \lg \hspace{0.1cm} \rho_{<del class="diffchange diffchange-inline">d</del>,\hspace{0.08cm} \rm WR} \ = \ $ { 37.53 3% } $\ \rm dB$</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>$10 \cdot \lg \hspace{0.1cm} \rho_{<ins class="diffchange diffchange-inline">\rm MF</ins>,\hspace{0.08cm} \rm WR} \ = \ $ { 37.53 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 <del class="diffchange diffchange-inline">MF</del>&ndash;Frequenzgang bei den vorliegenden farbigen Störungen.&nbsp; Welchen Wert besitzt&nbsp; $H_\text{MF}(f)$&nbsp; bei der Frequenz&nbsp; $f = 1 \hspace{0.08cm} \rm kHz$&nbsp; betragsmäßig?</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 <ins class="diffchange diffchange-inline">Matched&ndash;Filter</ins>&ndash;Frequenzgang bei den vorliegenden farbigen Störungen.&nbsp; <ins class="diffchange diffchange-inline"><br></ins>Welchen Wert besitzt&nbsp; $H_\text{MF}(f)$&nbsp; bei der Frequenz&nbsp; $f = 1 \hspace{0.08cm} \rm kHz$&nbsp; betragsmäßig?</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>$|H_\text{MF}(f = 1 \hspace{0.08cm} \rm kHz)| \ = \ $ { 0.216 3% }</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$|H_\text{MF}(f = 1 \hspace{0.08cm} \rm kHz)| \ = \ $ { 0.216 3% }</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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein?&nbsp; Begründen Sie das bessere 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>{Welches<ins class="diffchange diffchange-inline">&nbsp; "maximale </ins>S/N&ndash;Verhältnis<ins class="diffchange diffchange-inline">"&nbsp; $\rho_{\rm MF}$&nbsp; </ins>stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein?&nbsp; Begründen Sie das bessere 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>$10 \cdot \lg \hspace{0.1cm} \<del class="diffchange diffchange-inline">rho_d </del>\ = \ $ { 38.73 3% } $\ \rm dB$</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>$10 \cdot \lg \hspace{0.1cm} \<ins class="diffchange diffchange-inline">rho_{\rm MF} </ins>\ = \ $ { 38.73 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>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=29214&oldid=prev
Guenter am 9. Dezember 2019 um 16:58 Uhr
2019-12-09T16:58:10Z
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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 9. Dezember 2019, 16:58 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l71" >Zeile 71:</td>
<td colspan="2" class="diff-lineno">Zeile 71:</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; Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung $T_{\rm D}= 0$:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </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; Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung<ins class="diffchange diffchange-inline">&nbsp; </ins>$T_{\rm D}= 0$:</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>:$$H_\text{MF} (f) = K_\text{MF}\cdot \frac{G^{\star} (f)}{\left| {H_{\rm N} (f)} \right|^2 }\hspace{0.2cm}{\rm mit}\hspace{0.15cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_\text{MF} (f) = K_\text{MF}\cdot \frac{G^{\star} (f)}{\left| {H_{\rm N} (f)} \right|^2 }\hspace{0.2cm}{\rm mit}\hspace{0.15cm}</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>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f/f_0 \right)^2. $$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f/f_0 \right)^2. $$</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;"></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>*Aus der Bedingung &nbsp;$H_\text{MF}(f = 0) = 1$&nbsp; folgt &nbsp;$K_\text{MF} = 1/(g_0 \cdot \Delta t)$. Damit erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Aus der Bedingung &nbsp;$H_\text{MF}(f = 0) = 1$&nbsp; folgt &nbsp;$K_\text{MF} = 1/(g_0 \cdot \Delta t)$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Damit erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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 weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls $g(t)$ bewirkt. </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 weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls<ins class="diffchange diffchange-inline">&nbsp; </ins>$g(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>bewirkt. </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 farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum ${\it \Phi}_n(f)$ werden höhere Frequenzen durch den Korrekturterm $1+\left( f/f_0 \right)^2$ angehoben, da in diesem Bereich die Störungen geringer sind. </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 farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum<ins class="diffchange diffchange-inline">&nbsp; </ins>${\it \Phi}_n(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>werden höhere Frequenzen durch den Korrekturterm<ins class="diffchange diffchange-inline">&nbsp; </ins>$1+\left( f/f_0 \right)^2$<ins class="diffchange diffchange-inline">&nbsp; </ins>angehoben, da in diesem Bereich die Störungen geringer sind. </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 $f = 1/\Delta t = 2f_0 = 1\hspace{0.08cm} \rm kHz$ erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Für<ins class="diffchange diffchange-inline">&nbsp; </ins>$f = 1/\Delta t = 2f_0 = 1\hspace{0.08cm} \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</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>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l87" >Zeile 87:</td>
<td colspan="2" class="diff-lineno">Zeile 89:</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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 erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen. Für den zweiten Summanden erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Der erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen.<ins class="diffchange diffchange-inline">&nbsp; </ins>Für den zweiten Summanden erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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>*Nach der Substitution $x = 2 \cdot \pi^{0.5}\cdot f \cdot \Delta t$ lautet dieses Integral:</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>*Nach der Substitution<ins class="diffchange diffchange-inline">&nbsp; </ins>$x = 2 \cdot \pi^{0.5}\cdot f \cdot \Delta t$<ins class="diffchange diffchange-inline">&nbsp; </ins>lautet dieses Integral:</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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 bestimmte Integral wurde vorne angegeben; es hat den Wert $1$. Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem 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>*Dieses bestimmte Integral wurde vorne angegeben; es hat den Wert<ins class="diffchange diffchange-inline">&nbsp; </ins>$1$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem 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>*Damit ergeben sich folgende Gleichungen:</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>*Damit ergeben sich folgende Gleichungen:</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>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l101" >Zeile 101:</td>
<td colspan="2" class="diff-lineno">Zeile 103:</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><u>Fazit:</u></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><u>Fazit:</u></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Es ergibt sich ein um $1.2 \; \rm dB$ besseres Ergebnis als bei weißem Rauschen, da hier ${\it \Phi}_n(f)$ im gesamten Frequenzbereich mit Ausnahme der Frequenz $f = 0$ (hier gilt das Gleichheitszeichen) kleiner ist als $N_0/2$. </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>*Es ergibt sich ein um<ins class="diffchange diffchange-inline">&nbsp; </ins>$1.2 \; \rm dB$<ins class="diffchange diffchange-inline">&nbsp; </ins>besseres Ergebnis als bei weißem Rauschen, da hier<ins class="diffchange diffchange-inline">&nbsp; </ins>${\it \Phi}_n(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>im gesamten Frequenzbereich mit Ausnahme der Frequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f = 0<ins class="diffchange diffchange-inline">$&nbsp; </ins>$(<ins class="diffchange diffchange-inline">$</ins>hier gilt das Gleichheitszeichen<ins class="diffchange diffchange-inline">$</ins>)<ins class="diffchange diffchange-inline">$&nbsp; </ins>kleiner ist als<ins class="diffchange diffchange-inline">&nbsp; </ins>$N_0/2$. </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>*Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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-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_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=29213&oldid=prev
Guenter am 9. Dezember 2019 um 16:52 Uhr
2019-12-09T16:52:45Z
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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 9. Dezember 2019, 16:52 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >Zeile 3:</td>
<td colspan="2" class="diff-lineno">Zeile 3:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID646__Sto_A_5_8.png|right|frame|Spektrum $G(f)$ des Nutzsignals und LDS ${\it \Phi}_n (f)$ der Störung ]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID646__Sto_A_5_8.png|right|frame|Spektrum<ins class="diffchange diffchange-inline">&nbsp; </ins>$G(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>des Nutzsignals und LDS<ins class="diffchange diffchange-inline">&nbsp; </ins>${\it \Phi}_n (f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>der Störung ]]</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>Am Eingang eines Filters liegt ein Gaußimpuls</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>Am Eingang eines Filters liegt ein Gaußimpuls</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>:$$g(t) = g_0 \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {t/\Delta t} \right)^2 }$$</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>:$$g(t) = g_0 \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {t/\Delta t} \right)^2 }$$</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;"></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>mit Amplitude &nbsp;$g_0 = 2 \hspace{0.08cm}\rm V$&nbsp; und äquivalenter Impulsdauer &nbsp;$\Delta t = 1 \hspace{0.08cm}\rm ms$&nbsp; an. </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>mit Amplitude &nbsp;$g_0 = 2 \hspace{0.08cm}\rm V$&nbsp; und äquivalenter Impulsdauer &nbsp;$\Delta t = 1 \hspace{0.08cm}\rm ms$&nbsp; an. </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 dazugehörige Spektralfunktion $G(f)$ ist oben skizziert. </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 dazugehörige Spektralfunktion<ins class="diffchange diffchange-inline">&nbsp; </ins>$G(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist oben skizziert. </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 Energie dieses Gaußimpulses ist wie folgt gegeben:</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 Energie dieses Gaußimpulses ist wie folgt gegeben:</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>:$$E_g = \int_{ - \infty }^{ + \infty } {g<del class="diffchange diffchange-inline">\left</del>( t \<del class="diffchange diffchange-inline">right)^2 </del>{\rm{d}}t = \frac{g_0 ^2 \cdot \Delta t}{\sqrt 2 }} = 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$E_g = \int_{ - \infty }^{ + \infty } {g<ins class="diffchange diffchange-inline">^2</ins>(t<ins class="diffchange diffchange-inline">) </ins>\ {\rm{d}}t = \frac{g_0 ^2 \cdot \Delta t}{\sqrt 2 }} = 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 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>Dem Impuls $g(t)$ ist eine Störung $n(t)$ überlagert, die den Impuls weitgehend überdeckt. Hierfür werden zwei Alternativen betrachtet:</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>Dem Impuls<ins class="diffchange diffchange-inline">&nbsp; </ins>$g(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist eine Störung<ins class="diffchange diffchange-inline">&nbsp; </ins>$n(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>überlagert, die den Impuls weitgehend überdeckt.<ins class="diffchange diffchange-inline">&nbsp; </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>Hierfür werden zwei Alternativen betrachtet:</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 zweiseitige Störleistungsdichte sei konstant (nur bei der ersten Teilaufgabe):</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 zweiseitige Störleistungsdichte sei konstant (nur bei der ersten Teilaufgabe):</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</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 Störsignal $n(t)$ sei farbig mit folgender Störleistungsdichte:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Das Störsignal<ins class="diffchange diffchange-inline">&nbsp; </ins>$n(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>sei farbig mit folgender Störleistungsdichte:</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>:$${\it \Phi}_n (f) = \frac{N_0 /2}{{1 + \left( {f/f_0 } \right)^2 }},\quad f_0 = 500\;{\rm{Hz}}.$$</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>:$${\it \Phi}_n (f) = \frac{N_0 /2}{{1 + \left( {f/f_0 } \right)^2 }},\quad f_0 = 500\;{\rm{Hz}}.$$</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>Dieser zweite LDS-Verlauf kann zum Beispiel aus weißem Rauschen durch ein Formfilter mit dem Frequenzgang</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>Dieser zweite LDS-Verlauf kann zum Beispiel aus weißem Rauschen durch ein Formfilter mit dem Frequenzgang</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>:$$H_{\rm N}(f) = \frac{1}{{1 + {\rm{j}}f/f_0 }}\quad\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\, \quad h_{\rm N}(t) = 2{\rm{\pi }}f_0 \cdot {\rm{e}}^{ - 2{\rm{\pi }}f_0 t} $$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm N}(f) = \frac{1}{{1 + {\rm{j}}<ins class="diffchange diffchange-inline">\cdot </ins>f/f_0 }}\quad\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\, \quad h_{\rm N}(t) = 2{\rm{\pi }}f_0 \cdot {\rm{e}}^{ - 2{\rm{\pi }}f_0 t} $$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>modelliert werden (Tiefpass erster Ordnung). Weiter soll gelten:</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>modelliert werden (Tiefpass erster Ordnung). Weiter soll gelten:</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>*Das Filter sei jeweils optimal an die Sendeimpulsform $g(t)$ und das Störleistungsdichtespektrum ${\it \Phi}_n (f)$ angepasst: &nbsp; $H(f) = H_{\rm MF}(f)$. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Das Filter sei jeweils optimal an die Sendeimpulsform<ins class="diffchange diffchange-inline">&nbsp; </ins>$g(t)$<ins class="diffchange diffchange-inline">&nbsp; </ins>und das Störleistungsdichtespektrum<ins class="diffchange diffchange-inline">&nbsp; </ins>${\it \Phi}_n (f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>angepasst: &nbsp; $H(f) = H_{\rm MF}(f)$. </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 Filterkonstante $K_{\rm MF}$ ist so zu wählen, dass $H(f= 0) =1$ <del class="diffchange diffchange-inline">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>*Die Filterkonstante<ins class="diffchange diffchange-inline">&nbsp; </ins>$K_{\rm MF}$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist so zu wählen, dass<ins class="diffchange diffchange-inline">&nbsp; </ins>$H(f= 0) =1$<ins class="diffchange diffchange-inline">&nbsp; gilt</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>*Der Detektionszeitpunkt sei vereinfachend $T_{\rm D}= 0$ (akausale Systembeschreibung).</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 Detektionszeitpunkt sei vereinfachend<ins class="diffchange diffchange-inline">&nbsp; </ins>$T_{\rm D}= 0$<ins class="diffchange diffchange-inline">&nbsp; </ins>(akausale Systembeschreibung).</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> </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-l32" >Zeile 32:</td>
<td colspan="2" class="diff-lineno">Zeile 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></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 [[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]].</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>[[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]].</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>*Gegeben ist zudem das folgende bestimmte Integral:</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 zudem das folgende bestimmte Integral:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l48" >Zeile 48:</td>
<td colspan="2" class="diff-lineno">Zeile 52:</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 MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen. <del class="diffchange diffchange-inline"><br></del>Welchen Wert besitzt $H_\text{MF}(f)$ bei der Frequenz $f = 1 \hspace{0.08cm} \rm kHz$ betragsmäßig?</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 MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen.<ins class="diffchange diffchange-inline">&nbsp; </ins>Welchen Wert besitzt<ins class="diffchange diffchange-inline">&nbsp; </ins>$H_\text{MF}(f)$<ins class="diffchange diffchange-inline">&nbsp; </ins>bei der Frequenz<ins class="diffchange diffchange-inline">&nbsp; </ins>$f = 1 \hspace{0.08cm} \rm kHz$<ins class="diffchange diffchange-inline">&nbsp; </ins>betragsmäßig?</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>$|H_\text{MF}(f = 1 \hspace{0.08cm} \rm kHz)| \ = \ $ { 0.216 3% }</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$|H_\text{MF}(f = 1 \hspace{0.08cm} \rm kHz)| \ = \ $ { 0.216 3% }</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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein? Begründen Sie das bessere 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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein?<ins class="diffchange diffchange-inline">&nbsp; </ins>Begründen Sie das bessere 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 \cdot \lg \hspace{0.1cm} \rho_d \ = \ $ { 38.73 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 \cdot \lg \hspace{0.1cm} \rho_d \ = \ $ { 38.73 3% } $\ \rm dB$</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=26224&oldid=prev
Guenter am 27. August 2018 um 07:01 Uhr
2018-08-27T07:01:01Z
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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 27. August 2018, 07:01 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l63" >Zeile 63:</td>
<td colspan="2" class="diff-lineno">Zeile 63:</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="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(1)'''&nbsp; Bei weißem Rauschen gilt nach den allgemeinen Gleichungen im Theorieteil:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(1)'''&nbsp; Bei weißem Rauschen gilt nach den allgemeinen Gleichungen im Theorieteil:</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>:$$\rho_{d, \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\rho_{d,<ins class="diffchange diffchange-inline">\ </ins>\rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</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>\Rightarrow \quad 10\lg \cdot \rho _{d, \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\Rightarrow \quad 10\lg \cdot \rho _{d,<ins class="diffchange diffchange-inline">\ </ins>\rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</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> </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>'''(2)'''&nbsp; Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung $T_{\rm D}= 0$:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(2)'''&nbsp; Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung $T_{\rm D}= 0$:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l70" >Zeile 70:</td>
<td colspan="2" class="diff-lineno">Zeile 71:</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>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f/f_0 \right)^2. $$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f/f_0 \right)^2. $$</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;"></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>Aus der Bedingung $H_\text{MF}(f = 0) = 1$ folgt $K_\text{MF} = 1/(g_0 \cdot \Delta t)$. Damit erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Aus der Bedingung <ins class="diffchange diffchange-inline">&nbsp;</ins>$H_\text{MF}(f = 0) = 1$<ins class="diffchange diffchange-inline">&nbsp; </ins>folgt <ins class="diffchange diffchange-inline">&nbsp;</ins>$K_\text{MF} = 1/(g_0 \cdot \Delta t)$. Damit erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}[ {\Delta t \cdot f} ]^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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 weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls $g(t)$ bewirkt. Bei farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum ${\it \Phi}_n(f)$ werden höhere Frequenzen durch den Korrekturterm $1+\left( f/f_0 \right)^2$ angehoben, da in diesem Bereich die Störungen geringer sind. Für $f = 1/\Delta t = 2f_0 = 1\hspace{0.<del class="diffchange diffchange-inline">05cm</del>} \rm kHz$ erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Bei weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls $g(t)$ bewirkt. </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>Bei farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum ${\it \Phi}_n(f)$ werden höhere Frequenzen durch den Korrekturterm $1+\left( f/f_0 \right)^2$ angehoben, da in diesem Bereich die Störungen geringer sind. </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>Für $f = 1/\Delta t = 2f_0 = 1\hspace{0.<ins class="diffchange diffchange-inline">08cm</ins>} \rm kHz$ erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</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>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Allgemein gilt für das S/N&ndash;Verhältnis am Ausgang des Matched-Filters:</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>'''(3)'''&nbsp; Allgemein gilt für das S/N&ndash;Verhältnis am Ausgang des Matched-Filters:</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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 erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen. Für den zweiten Summanden erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*</ins>Der erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen. Für den zweiten Summanden erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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>Nach der Substitution $x = 2 \cdot \pi^{0.5}\cdot f \cdot \Delta t$ lautet dieses Integral:</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>Nach der Substitution $x = 2 \cdot \pi^{0.5}\cdot f \cdot \Delta t$ lautet dieses Integral:</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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 bestimmte Integral wurde vorne angegeben; es hat den Wert $1$. Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem Rauschen. Damit ergeben sich folgende Gleichungen:</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>Dieses bestimmte Integral wurde vorne angegeben; es hat den Wert $1$. Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem Rauschen. </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>Damit ergeben sich folgende Gleichungen:</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>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</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 _d = \rho _{d,\rm WR} + \Delta \rho _d = \rho _{d, \rm WR} \left( {1 + \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}} \right).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\rho _d = \rho _{d,\rm WR} + \Delta \rho _d = \rho _{d, \rm WR} \left( {1 + \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}} \right).$$</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l91" >Zeile 91:</td>
<td colspan="2" class="diff-lineno">Zeile 96:</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 _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\rho _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</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>Es ergibt sich ein um $1.2 \; \rm dB$ besseres Ergebnis als bei weißem Rauschen, da hier ${\it \Phi}_n(f)$ im gesamten Frequenzbereich mit Ausnahme der Frequenz $f = 0$ (hier gilt das Gleichheitszeichen) kleiner ist als $N_0/2$. Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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"><u>Fazit:</u></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>Es ergibt sich ein um $1.2 \; \rm dB$ besseres Ergebnis als bei weißem Rauschen, da hier ${\it \Phi}_n(f)$ im gesamten Frequenzbereich mit Ausnahme der Frequenz $f = 0$ (hier gilt das Gleichheitszeichen) kleiner ist als $N_0/2$. </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>Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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-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>
<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_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=26223&oldid=prev
Guenter am 27. August 2018 um 06:53 Uhr
2018-08-27T06:53:10Z
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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 27. August 2018, 06:53 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >Zeile 3:</td>
<td colspan="2" class="diff-lineno">Zeile 3:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID646__Sto_A_5_8.png|right|<del class="diffchange diffchange-inline">Zum Matched-Filter bei farbiger </del>Störung ]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID646__Sto_A_5_8.png|right|<ins class="diffchange diffchange-inline">frame|Spektrum $G(f)$ des Nutzsignals und LDS ${\it \Phi}_n (f)$ der </ins>Störung ]]</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>Am Eingang eines Filters liegt ein Gaußimpuls</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>Am Eingang eines Filters liegt ein Gaußimpuls</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>:$$g(t) = g_0 \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {t/\Delta t} \right)^2 }$$</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>:$$g(t) = g_0 \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {t/\Delta t} \right)^2 }$$</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;"></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>mit Amplitude $g_0 = 2 \hspace{0.<del class="diffchange diffchange-inline">05cm</del>}\rm V$ und äquivalenter Impulsdauer $\Delta t = 1 \hspace{0.<del class="diffchange diffchange-inline">05cm</del>}\rm ms$ an. Die dazugehörige Spektralfunktion $G(f)$ ist oben skizziert. Die Energie dieses Gaußimpulses ist wie folgt gegeben:</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>mit Amplitude <ins class="diffchange diffchange-inline">&nbsp;</ins>$g_0 = 2 \hspace{0.<ins class="diffchange diffchange-inline">08cm</ins>}\rm V$<ins class="diffchange diffchange-inline">&nbsp; </ins>und äquivalenter Impulsdauer <ins class="diffchange diffchange-inline">&nbsp;</ins>$\Delta t = 1 \hspace{0.<ins class="diffchange diffchange-inline">08cm</ins>}\rm ms$<ins class="diffchange diffchange-inline">&nbsp; </ins>an. </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>Die dazugehörige Spektralfunktion $G(f)$ ist oben skizziert. </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>Die Energie dieses Gaußimpulses ist wie folgt gegeben:</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>:$$E_g = \int_{ - \infty }^{ + \infty } {g\left( t \right)^2 {\rm{d}}t = \frac{g_0 ^2 \cdot \Delta t}{\sqrt 2 }} = 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$E_g = \int_{ - \infty }^{ + \infty } {g\left( t \right)^2 {\rm{d}}t = \frac{g_0 ^2 \cdot \Delta t}{\sqrt 2 }} = 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}.$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Dem Impuls $g(t)$ ist eine Störung $n(t)$ überlagert, die den Impuls weitgehend überdeckt. Hierfür werden zwei Alternativen betrachtet:</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>Dem Impuls $g(t)$ ist eine Störung $n(t)$ überlagert, die den Impuls weitgehend überdeckt. Hierfür werden zwei Alternativen betrachtet:</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 zweiseitige Störleistungsdichte sei konstant (nur bei der ersten Teilaufgabe):</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 zweiseitige Störleistungsdichte sei konstant (nur bei der ersten Teilaufgabe):</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</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>:$${\it \Phi}_n (f) = \frac{N_0 }{2},\quad N_0 = 10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}.$$</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 Störsignal <del class="diffchange diffchange-inline"><i></del>n<del class="diffchange diffchange-inline"></i></del>(<del class="diffchange diffchange-inline"><i></del>t<del class="diffchange diffchange-inline"></i></del>) sei farbig mit folgender Störleistungsdichte:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Das Störsignal <ins class="diffchange diffchange-inline">$</ins>n(t)<ins class="diffchange diffchange-inline">$ </ins>sei farbig mit folgender Störleistungsdichte:</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>:$${\it \Phi}_n (f) = \frac{N_0 /2}{{1 + \left( {f/f_0 } \right)^2 }},\quad f_0 = 500\;{\rm{Hz}}.$$</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>:$${\it \Phi}_n (f) = \frac{N_0 /2}{{1 + \left( {f/f_0 } \right)^2 }},\quad f_0 = 500\;{\rm{Hz}}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23" >Zeile 23:</td>
<td colspan="2" class="diff-lineno">Zeile 26:</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 Filterkonstante $K_{\rm MF}$ ist so zu wählen, dass $H(f= 0) =1$ 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>*Die Filterkonstante $K_{\rm MF}$ ist so zu wählen, dass $H(f= 0) =1$ wird. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Der Detektionszeitpunkt sei vereinfachend $T_{\rm D}= 0$ (akausale Systembeschreibung).</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 Detektionszeitpunkt sei vereinfachend $T_{\rm D}= 0$ (akausale Systembeschreibung).</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
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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><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l30" >Zeile 30:</td>
<td colspan="2" class="diff-lineno">Zeile 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Gegeben ist zudem das folgende bestimmte Integral:</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 zudem das folgende bestimmte Integral:</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l37" >Zeile 37:</td>
<td colspan="2" class="diff-lineno">Zeile 45:</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>{Wie groß ist das S/N&ndash;Verhältnis (in dB) im Fall des weißen Rauschens?</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>{Wie groß ist das S/N&ndash;Verhältnis (in dB) im Fall des weißen Rauschens?</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 \cdot \lg \hspace{0.1cm} \rho_{d,\hspace{0.08cm} \rm WR} \ = $ { 37.53 3% } $\ \rm dB$</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>$10 \cdot \lg \hspace{0.1cm} \rho_{d,\hspace{0.08cm} \rm WR} \ = <ins class="diffchange diffchange-inline">\ </ins>$ { 37.53 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 MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen. Welchen Wert besitzt <del class="diffchange diffchange-inline"><i>H</i><sub></del>MF<del class="diffchange diffchange-inline"></sub></del>(<del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i></del>) bei <del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i> </del>= 1 kHz betragsmäßig?</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 MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen. <ins class="diffchange diffchange-inline"><br></ins>Welchen Wert besitzt <ins class="diffchange diffchange-inline">$H_\text{</ins>MF<ins class="diffchange diffchange-inline">}</ins>(f)<ins class="diffchange diffchange-inline">$ </ins>bei <ins class="diffchange diffchange-inline">der Frequenz $</ins>f = 1 <ins class="diffchange diffchange-inline">\hspace{0.08cm} \rm </ins>kHz<ins class="diffchange diffchange-inline">$ </ins>betragsmäßig?</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>$|H_\text{MF}(f = 1 \hspace{0.<del class="diffchange diffchange-inline">05cm</del>} \rm kHz)| \ = $ { 0.216 3% }</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>$|H_\text{MF}(f = 1 \hspace{0.<ins class="diffchange diffchange-inline">08cm</ins>} \rm kHz)| \ = <ins class="diffchange diffchange-inline"> \ </ins>$ { 0.216 3% }</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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein? Begründen Sie das bessere Ergebnis.</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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein? Begründen Sie das bessere 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>$10 \cdot \lg \hspace{0.1cm} \rho_d \ = $ { 38.73 3% } $\ \rm dB$</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>$10 \cdot \lg \hspace{0.1cm} \rho_d \ = <ins class="diffchange diffchange-inline"> \ </ins>$ { 38.73 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>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=25093&oldid=prev
Mwiki-lnt: Textersetzung - „\*\s*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0\.” ein.“ durch „ “
2018-05-29T12:03:56Z
<p>Textersetzung - „\*\s*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, 12:03 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27" >Zeile 27:</td>
<td colspan="2" class="diff-lineno">Zeile 27:</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 [[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]].</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 [[Stochastische_Signaltheorie/Matched-Filter|Matched-Filter]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Gegeben ist zudem das folgende bestimmte Integral:</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 zudem das folgende bestimmte Integral:</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</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>:$$\frac{1}{{\sqrt {2{\rm{\pi }}} }}\int_{ - \infty }^{ + \infty } {x^2 \cdot {\rm e}^{ - x^2 /2} \,\,{\rm{d}}x} = 1.$$</div></td></tr>
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Mwiki-lnt
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=21772&oldid=prev
Guenter: Guenter verschob die Seite Aufgaben:5.8 Matched-Filter für farbige Störung nach Aufgaben:Aufgabe 5.8: Matched-Filter für farbige Störung
2018-01-03T13:52:14Z
<p>Guenter verschob die Seite <a href="/Aufgaben:5.8_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung" class="mw-redirect" title="Aufgaben:5.8 Matched-Filter für farbige Störung">5.8 Matched-Filter für farbige Störung</a> nach <a href="/Aufgaben:Aufgabe_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung" title="Aufgaben:Aufgabe 5.8: Matched-Filter für farbige Störung">Aufgabe 5.8: Matched-Filter für farbige Störung</a></p>
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<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="1" style="background-color: #fff; color: #222; text-align: center;">Version vom 3. Januar 2018, 13:52 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_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=12232&oldid=prev
Guenter am 23. April 2017 um 16:01 Uhr
2017-04-23T16:01:57Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 23. April 2017, 16:01 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l54" >Zeile 54:</td>
<td colspan="2" class="diff-lineno">Zeile 54:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:<b></del>1<del class="diffchange diffchange-inline">.</b>&nbsp;</del>&nbsp;Bei weißem Rauschen gilt nach den allgemeinen Gleichungen <del class="diffchange diffchange-inline">von Kapitel 5.4</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>1<ins class="diffchange diffchange-inline">)'''</ins>&nbsp; Bei weißem Rauschen gilt nach den allgemeinen Gleichungen <ins class="diffchange diffchange-inline">im Theorieteil</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>:$$\rho_{d, \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\rho_{d, \rm WR} = \frac{2E_g }{N_0 } = \frac{{2 \cdot 2.83 \cdot 10^{ - 3} \;{\rm{V}}^{\rm{2}} {\rm{s}}}}{{10^{ - 6} \;{\rm{V}}^{\rm{2}}/ {\rm{Hz}}}} = 5.66 \cdot 10^3 \quad</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>\Rightarrow \quad 10\lg \cdot \rho _{d, \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\Rightarrow \quad 10\lg \cdot \rho _{d, \rm WR} \hspace{0.15cm}\underline {= 37.53\;{\rm{dB}}}.$$</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><del class="diffchange diffchange-inline">:<b></del>2<del class="diffchange diffchange-inline">.</b>&nbsp;</del>&nbsp;Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung <del class="diffchange diffchange-inline"><i>T</i><sub></del>D<del class="diffchange diffchange-inline"></sub> </del>= 0:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">'''(</ins>2<ins class="diffchange diffchange-inline">)'''</ins>&nbsp; Für den Frequenzgang bei farbigen Störungen gilt unter der Bedingung <ins class="diffchange diffchange-inline">$T_{\rm </ins>D<ins class="diffchange diffchange-inline">}</ins>= 0<ins class="diffchange diffchange-inline">$</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>:$$H_\text{MF} (f) = K_\text{MF}\cdot \frac{G^{\star} (f)}{\left| {H_{\rm N} (f)} \right|^2 }\hspace{0.2cm}{\rm mit}\hspace{0.15cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_\text{MF} (f) = K_\text{MF}\cdot \frac{G^{\star} (f)}{\left| {H_{\rm N} (f)} \right|^2 }\hspace{0.2cm}{\rm mit}\hspace{0.15cm}</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>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } = 1<del class="diffchange diffchange-inline">\hspace{-0.05cm} </del>+ <del class="diffchange diffchange-inline">\hspace{-0.05cm}</del>\left( <del class="diffchange diffchange-inline">{\frac{</del>f<del class="diffchange diffchange-inline">}{</del>f_0 <del class="diffchange diffchange-inline">}} </del>\right)^2<del class="diffchange diffchange-inline">\hspace{-0.15cm} </del>.$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>G(f) = g_0 \cdot \Delta t \cdot {\rm{e}}^{ - {\rm{\pi }}\left( {\Delta t \hspace{0.03cm}\cdot \hspace{0.03cm}f} \right)^2 } ,\hspace{0.15cm}\frac{1}{\left| {H_{\rm N} (f)} \right|^2 } =1+\left( f<ins class="diffchange diffchange-inline">/</ins>f_0 \right)^2. $$</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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:</del>Aus der Bedingung <del class="diffchange diffchange-inline"><i>H</i><sub></del>MF<del class="diffchange diffchange-inline"></sub> </del>(<del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i> </del>= 0) = 1 folgt <del class="diffchange diffchange-inline"><i>K</i><sub></del>MF<del class="diffchange diffchange-inline"></sub> </del>= 1/(<del class="diffchange diffchange-inline"><i>g</i><sub>0</sub> &middot; &</del>Delta<del class="diffchange diffchange-inline">;<i></del>t<del class="diffchange diffchange-inline"></i></del>). Damit erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Aus der Bedingung <ins class="diffchange diffchange-inline">$H_\text{</ins>MF<ins class="diffchange diffchange-inline">}</ins>(f = 0) = 1<ins class="diffchange diffchange-inline">$ </ins>folgt <ins class="diffchange diffchange-inline">$K_\text{</ins>MF<ins class="diffchange diffchange-inline">} </ins>= 1/(<ins class="diffchange diffchange-inline">g_0 \cdot \</ins>Delta t)<ins class="diffchange diffchange-inline">$</ins>. Damit erhält man:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}<del class="diffchange diffchange-inline">\left( </del>{\Delta t \cdot f} <del class="diffchange diffchange-inline">\right)</del>^2 } \cdot \left( {1 + \left( <del class="diffchange diffchange-inline">{{</del>f<del class="diffchange diffchange-inline">}</del>/<del class="diffchange diffchange-inline">{</del>f_0 <del class="diffchange diffchange-inline">}} </del>\right)^2 } \right).$$</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>:$$H_{\rm MF} (f) = {\rm{e}}^{ - {\rm{\pi }}<ins class="diffchange diffchange-inline">[ </ins>{\Delta t \cdot f} <ins class="diffchange diffchange-inline">]</ins>^2 } \cdot \left( {1 + \left(f/f_0 \right)^2 } \right).$$</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">:</del>Bei weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls <del class="diffchange diffchange-inline"><i></del>g<del class="diffchange diffchange-inline"></i></del>(<del class="diffchange diffchange-inline"><i></del>t<del class="diffchange diffchange-inline"></i></del>) bewirkt. Bei farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum <del class="diffchange diffchange-inline"><i>&</del>Phi<del class="diffchange diffchange-inline">;<sub>n</sub></i></del>(<del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i></del>) werden höhere Frequenzen durch den Korrekturterm <del class="diffchange diffchange-inline"><nobr></del>1 + (<del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"><</del>/<del class="diffchange diffchange-inline">i>/<i>f</i><sub>0</sub></del>)<del class="diffchange diffchange-inline"><sup></del>2<del class="diffchange diffchange-inline"></sup></nobr> </del>angehoben, da in diesem Bereich die Störungen geringer sind. Für <del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i> </del>= 1/<del class="diffchange diffchange-inline">&</del>Delta<del class="diffchange diffchange-inline">;<i></del>t<del class="diffchange diffchange-inline"></i> </del>= <del class="diffchange diffchange-inline">2<i>f</i><sub>0</sub> </del>= 1 kHz erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Bei weißem (frequenzunabhängigen) Rauschen wäre das Matched-Filter allein durch den ersten Term gegeben, der die Anpassung an den Nutzimpuls <ins class="diffchange diffchange-inline">$</ins>g(t)<ins class="diffchange diffchange-inline">$ </ins>bewirkt. Bei farbigen Störungen &nbsp;&#8658;&nbsp; Störleistungsdichtespektrum <ins class="diffchange diffchange-inline">${\it \</ins>Phi<ins class="diffchange diffchange-inline">}_n</ins>(f)<ins class="diffchange diffchange-inline">$ </ins>werden höhere Frequenzen durch den Korrekturterm <ins class="diffchange diffchange-inline">$</ins>1+<ins class="diffchange diffchange-inline">\left</ins>( f/<ins class="diffchange diffchange-inline">f_0 \right</ins>)<ins class="diffchange diffchange-inline">^</ins>2<ins class="diffchange diffchange-inline">$ </ins>angehoben, da in diesem Bereich die Störungen geringer sind. Für <ins class="diffchange diffchange-inline">$</ins>f = 1/<ins class="diffchange diffchange-inline">\</ins>Delta t = <ins class="diffchange diffchange-inline">2f_0 </ins>= 1<ins class="diffchange diffchange-inline">\hspace{0.05cm} \rm </ins>kHz<ins class="diffchange diffchange-inline">$ </ins>erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</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>:$$H_{\rm MF} ( {f = {1}/{\Delta t}} ) = {\rm{e}}^{ - {\rm{\pi }}} \cdot \left( {1 + 2^2 } \right) \hspace{0.15cm}\underline {= 0.216}.$$</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">:<b></del>3<del class="diffchange diffchange-inline">.</b>&nbsp;</del>&nbsp;Allgemein gilt für das S/N&ndash;Verhältnis am Ausgang des Matched-Filters:</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>3<ins class="diffchange diffchange-inline">)'''</ins>&nbsp; Allgemein gilt für das S/N&ndash;Verhältnis am Ausgang des Matched-Filters:</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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>:$$\rho _d = \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{\it{\Phi _n} \left( f \right)}\,\,{\rm{d}}f = } \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \, \,{\rm{d}}f \hspace{0.3cm}+ \hspace{0.3cm} \int_{ - \infty }^{ + \infty } {\frac{\left| {G(f)} \right|^2 }{N_0 /2}} \cdot \frac{f^2 }{f_0 ^2 }\,\,{\rm{d}}f.$$</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">:</del>Der erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen. Für den zweiten Summanden erhält man:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Der erste Summand ist gleich dem S/N&ndash;Verhältnis bei weißem Rauschen. Für den zweiten Summanden erhält man:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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>:$$\Delta \rho _d = \frac{g_0 ^2 \cdot \Delta t^2 }{N_0 /2 \cdot f_0 ^2 }\cdot \int_{ - \infty }^{ + \infty } {f^2 \cdot {\rm{e}}^{ - 2{\rm{\pi }}\left( {f \cdot \Delta t} \right)^2 } }\,\, {\rm{d}}f.$$</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">:</del>Nach der Substitution <del class="diffchange diffchange-inline"><i></del>x<del class="diffchange diffchange-inline"></i> </del>= 2 <del class="diffchange diffchange-inline">&middot; &</del>pi<del class="diffchange diffchange-inline">;<sup>1/2</sup> &middot; <i></del>f<del class="diffchange diffchange-inline"></i> &middot; &</del>Delta<del class="diffchange diffchange-inline">;<i></del>t<del class="diffchange diffchange-inline"></i> </del>lautet dieses Integral:</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>Nach der Substitution <ins class="diffchange diffchange-inline">$</ins>x = 2 <ins class="diffchange diffchange-inline">\cdot \</ins>pi<ins class="diffchange diffchange-inline">^{0.5}\cdot </ins>f <ins class="diffchange diffchange-inline">\cdot \</ins>Delta t<ins class="diffchange diffchange-inline">$ </ins>lautet dieses Integral:</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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>:$$\Delta \rho _d = \frac{\sqrt 2 \cdot g_0 ^2 \cdot \Delta t}{N_0 } \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }} \cdot \int_{ - \infty }^{ + \infty } {\frac{x^2 }{\sqrt {2{\rm{\pi }}} }} \cdot {\rm{e}}^{ - x^2 /2}\,\, {\rm{d}}x.$$</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">:</del>Dieses bestimmte Integral wurde vorne angegeben; es hat den Wert 1. Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem Rauschen. Damit ergeben sich folgende Gleichungen:</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 bestimmte Integral wurde vorne angegeben; es hat den Wert <ins class="diffchange diffchange-inline">$</ins>1<ins class="diffchange diffchange-inline">$</ins>. Der erste Faktor beschreibt wiederum das S/N&ndash;Verhältnis bei weißem Rauschen. Damit ergeben sich folgende Gleichungen:</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>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\Delta \rho _d = \rho _{d,\rm WR} \cdot \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}, \hspace{1cm}</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 _d = \rho _{d,\rm WR} + \Delta \rho _d = \rho _{d, \rm WR} \left( {1 + \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}} \right).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\rho _d = \rho _{d,\rm WR} + \Delta \rho _d = \rho _{d, \rm WR} \left( {1 + \frac{1}{{4{\rm{\pi }}\left( {\Delta t \cdot f_0 } \right)^2 }}} \right).$$</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l83" >Zeile 83:</td>
<td colspan="2" class="diff-lineno">Zeile 83:</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 _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\rho _d = 1.318 \cdot \rho _{d,\rm WR} = 7.46 \cdot 10^3 \hspace{0.3cm} \Rightarrow \quad 10\lg \rho _d \hspace{0.15cm}\underline {= 38.73\;{\rm{dB}}}.$$</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><del class="diffchange diffchange-inline">:</del>Es ergibt sich ein um 1.2 dB besseres Ergebnis als bei weißem Rauschen, da hier <del class="diffchange diffchange-inline"><i>&</del>Phi<del class="diffchange diffchange-inline">;<sub>n</sub></i></del>(<del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i></del>) im gesamten Frequenzbereich mit Ausnahme der Frequenz <del class="diffchange diffchange-inline"><i></del>f<del class="diffchange diffchange-inline"></i> </del>= 0 (hier gilt das Gleichheitszeichen) kleiner ist als <del class="diffchange diffchange-inline"><i>N</i><sub>0</sub></del>/2. Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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>Es ergibt sich ein um <ins class="diffchange diffchange-inline">$</ins>1.2 <ins class="diffchange diffchange-inline">\; \rm </ins>dB<ins class="diffchange diffchange-inline">$ </ins>besseres Ergebnis als bei weißem Rauschen, da hier <ins class="diffchange diffchange-inline">${\it \</ins>Phi<ins class="diffchange diffchange-inline">}_n</ins>(f)<ins class="diffchange diffchange-inline">$ </ins>im gesamten Frequenzbereich mit Ausnahme der Frequenz <ins class="diffchange diffchange-inline">$</ins>f = 0<ins class="diffchange diffchange-inline">$ </ins>(hier gilt das Gleichheitszeichen) kleiner ist als <ins class="diffchange diffchange-inline">$N_0</ins>/2<ins class="diffchange diffchange-inline">$</ins>. Diese Tatsache wird auch vom Matched&ndash;Filter ausgenutzt.</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-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>
<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_5.8:_Matched-Filter_f%C3%BCr_farbige_St%C3%B6rung&diff=12231&oldid=prev
Guenter am 23. April 2017 um 15:26 Uhr
2017-04-23T15:26:50Z
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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 23. April 2017, 15:26 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22" >Zeile 22:</td>
<td colspan="2" class="diff-lineno">Zeile 22:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das Filter sei jeweils optimal an die Sendeimpulsform $g(t)$ und das Störleistungsdichtespektrum ${\it \Phi}_n (f)$ angepasst: &nbsp; $H(f) = H_{\rm MF}(f)$. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Das Filter sei jeweils optimal an die Sendeimpulsform $g(t)$ und das Störleistungsdichtespektrum ${\it \Phi}_n (f)$ angepasst: &nbsp; $H(f) = H_{\rm MF}(f)$. </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 Filterkonstante $K_{\rm MF}$ ist so zu wählen, dass $H(f= 0) =1$ 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>*Die Filterkonstante $K_{\rm MF}$ ist so zu wählen, dass $H(f= 0) =1$ wird. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Der Detektionszeitpunkt sei vereinfachend $T_{\rm D}= 0$ (akausale Systembeschreibung).</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 Detektionszeitpunkt sei vereinfachend $T_{\rm D}= 0$ (akausale Systembeschreibung).</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-l37" >Zeile 37:</td>
<td colspan="2" class="diff-lineno">Zeile 37:</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>{Wie groß ist das S/N&ndash;Verhältnis (in dB) im Fall des weißen Rauschens?</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>{Wie groß ist das S/N&ndash;Verhältnis (in dB) im Fall des weißen Rauschens?</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>$10<del class="diffchange diffchange-inline">\ </del>\cdot \<del class="diffchange diffchange-inline"> lg</del>\ \rho_<del class="diffchange diffchange-inline">\text</del>{d,WR}$ <del class="diffchange diffchange-inline">= </del>{ 37.53 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>$10 \cdot \<ins class="diffchange diffchange-inline">lg </ins>\<ins class="diffchange diffchange-inline">hspace{0.1cm} </ins>\rho_{d,<ins class="diffchange diffchange-inline">\hspace{0.08cm} \rm </ins>WR} <ins class="diffchange diffchange-inline">\ = </ins>$ { 37.53 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="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{Berechnen Sie den MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen. Welchen Wert besitzt <i>H</i><sub>MF</sub>(<i>f</i>) bei <i>f</i> = 1 kHz betragsmäßig?</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>{Berechnen Sie den MF&ndash;Frequenzgang bei den vorliegenden farbigen Störungen. Welchen Wert besitzt <i>H</i><sub>MF</sub>(<i>f</i>) bei <i>f</i> = 1 kHz betragsmäßig?</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>$|H_\text{MF}(f = 1 kHz)|$ <del class="diffchange diffchange-inline">= </del>{ 0.216 3% }</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>$|H_\text{MF}(f = 1 <ins class="diffchange diffchange-inline">\hspace{0.05cm} \rm </ins>kHz)| <ins class="diffchange diffchange-inline">\ = </ins>$ { 0.216 3% }</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>{Welches S/N&ndash;Verhältnis <del class="diffchange diffchange-inline">(in dB) </del>stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein? Begründen Sie das bessere 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>{Welches S/N&ndash;Verhältnis stellt sich im Fall der vorgegebenen farbigen Störung am Empfänger ein? Begründen Sie das bessere 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>$10<del class="diffchange diffchange-inline">\ </del>\cdot \ lg \ \rho_d$ <del class="diffchange diffchange-inline">= </del>{ 38.73 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>$10 \cdot \lg \<ins class="diffchange diffchange-inline">hspace{0.1cm} </ins>\rho_d <ins class="diffchange diffchange-inline">\ = </ins>$ { 38.73 3% } $<ins class="diffchange diffchange-inline">\ \rm </ins>dB$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Guenter