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Aufgaben:Aufgabe 2.6: Zur Huffman-Codierung - Versionsgeschichte
2024-03-29T11:55:16Z
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Guenter am 10. August 2021 um 14:11 Uhr
2021-08-10T14:11:43Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 10. August 2021, 14:11 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l58" >Zeile 58:</td>
<td colspan="2" class="diff-lineno">Zeile 58:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</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>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</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>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf folgende <del class="diffchange diffchange-inline">Interaktionsmodule</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>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf <ins class="diffchange diffchange-inline">das </ins>folgende <ins class="diffchange diffchange-inline">Interaktionsmodul</ins>:&nbsp; [[Applets:Huffman_Shannon_Fano|Huffman- und Shannon-Fano-Codierung&nbsp; &rArr; &nbsp; $\text{SWF}$&ndash;Version]].</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">**&nbsp; [[Applets:Huffman-_und_Shannon-Fano-Codierung|Huffman- und Shannon-Fano-Codierung&nbsp; &rArr; &nbsp; $\text{HTML 5/JS}$&ndash;Version]],</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">**</del>&nbsp; [[Applets:Huffman_Shannon_Fano|Huffman- und Shannon-Fano-Codierung&nbsp; &rArr; &nbsp; $\text{SWF}$&ndash;Version]].</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> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=32106&oldid=prev
Guenter am 10. August 2021 um 11:56 Uhr
2021-08-10T11:56:53Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 10. August 2021, 11:56 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l169" >Zeile 169:</td>
<td colspan="2" class="diff-lineno">Zeile 169:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>dargestellt werden.&nbsp; Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&<del class="diffchange diffchange-inline">dquo</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>dargestellt werden.&nbsp; Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&<ins class="diffchange diffchange-inline">rdquo</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>:$$L_{\rm M} \hspace{0.2cm} = \hspace{0.2cm} (p_{\rm D} + p_{\rm E}) \cdot 2 + (p_{\rm C} + p_{\rm F} + p_{\rm G}) \cdot 3 + p_{\rm B} \cdot 4 +(p_{\rm A} + p_{\rm H}) \cdot 5 = 0.49 \cdot 2 + 0.36 \cdot 3 +0.08 \cdot 4 +0.07 \cdot 5 </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M} \hspace{0.2cm} = \hspace{0.2cm} (p_{\rm D} + p_{\rm E}) \cdot 2 + (p_{\rm C} + p_{\rm F} + p_{\rm G}) \cdot 3 + p_{\rm B} \cdot 4 +(p_{\rm A} + p_{\rm H}) \cdot 5 = 0.49 \cdot 2 + 0.36 \cdot 3 +0.08 \cdot 4 +0.07 \cdot 5 </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\hspace{0.15cm}\underline{= 2.73}\hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\hspace{0.15cm}\underline{= 2.73}\hspace{0.05cm}.$$</div></td></tr>
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Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=32105&oldid=prev
Guenter am 10. August 2021 um 11:50 Uhr
2021-08-10T11:50:30Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 10. August 2021, 11:50 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l58" >Zeile 58:</td>
<td colspan="2" class="diff-lineno">Zeile 58:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</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>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</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>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf <del class="diffchange diffchange-inline">das Interaktionsmodul</del>&nbsp; [[Applets:Huffman-_und_Shannon-Fano-Codierung|Shannon&ndash;Fano&<del class="diffchange diffchange-inline">ndash</del>; <del class="diffchange diffchange-inline">und Huffman</del>&ndash;<del class="diffchange diffchange-inline">Codierung</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>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf <ins class="diffchange diffchange-inline">folgende Interaktionsmodule:</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>&nbsp; [[Applets:Huffman-_und_Shannon-Fano-Codierung|<ins class="diffchange diffchange-inline">Huffman- und </ins>Shannon<ins class="diffchange diffchange-inline">-Fano-Codierung&nbsp; &rArr; &nbsp; $\text{HTML 5/JS}$</ins>&ndash;<ins class="diffchange diffchange-inline">Version]],</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">**&nbsp; [[Applets:Huffman_Shannon_Fano|Huffman- und Shannon-</ins>Fano<ins class="diffchange diffchange-inline">-Codierung&nbsp; &rArr; </ins>&<ins class="diffchange diffchange-inline">nbsp</ins>; <ins class="diffchange diffchange-inline">$\text{SWF}$</ins>&ndash;<ins class="diffchange diffchange-inline">Version</ins>]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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_2.6:_Zur_Huffman-Codierung&diff=32104&oldid=prev
Guenter am 10. August 2021 um 11:43 Uhr
2021-08-10T11:43:45Z
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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 10. August 2021, 11:43 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_ID2451__Inf_A_2_6.png|right|frame|<del class="diffchange diffchange-inline">Baumdiagramm des <br></del>Huffman&ndash;<del class="diffchange diffchange-inline">Verfahrens</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>[[Datei:P_ID2451__Inf_A_2_6.png|right|frame|Huffman&ndash;<ins class="diffchange diffchange-inline">Baumdiagramm</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>Wir betrachten hier eine Quellensymbolfolge&nbsp; $\langle q_\nu \rangle$&nbsp; mit dem Symbolumfang&nbsp; $M = 8$:</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>Wir betrachten hier eine Quellensymbolfolge&nbsp; $\langle q_\nu \rangle$&nbsp; mit dem Symbolumfang&nbsp; $M = 8$:</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>:$$q_{\nu} <del class="diffchange diffchange-inline">= </del>\{ \hspace{0.05cm}q_{\mu} \} = \{ \boldsymbol{\rm A} \hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm B}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm C}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm D}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm E}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm F}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm G}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm H}\hspace{0.05cm}</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$q_{\nu} <ins class="diffchange diffchange-inline">\in </ins>\{ \hspace{0.05cm}q_{\mu} \} = \{ \boldsymbol{\rm A} \hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm B}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm C}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm D}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm E}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm F}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm G}\hspace{0.05cm},\hspace{0.05cm} \boldsymbol{\rm H}\hspace{0.05cm}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\}\hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\}\hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Sind die Symbole gleichwahrscheinlich, gilt also&nbsp; $p_{\rm A} = p_{\rm B} =$ ... $ = p_{\rm H} = 1/M$, so macht Quellencodierung keinen Sinn.&nbsp; Bereits mit dem Dualcode&nbsp; $\rm A$&nbsp; &#8594;&nbsp; <b>000</b>,&nbsp; $\rm B$&nbsp; &#8594;&nbsp; <b>001</b>, ... ,&nbsp; $\rm H$&nbsp; &#8594;&nbsp; <b>111</b>, erreicht nämlich dann die mittlere Codewortlänge&nbsp; $L_{\rm M}$&nbsp; ihre untere Schranke&nbsp; $H$&nbsp; gemäß dem Quellencodierungstheorem&nbsp; $(H$ bezeichnet hierbei die <del class="diffchange diffchange-inline"><i></del>Quellenentropie<del class="diffchange diffchange-inline"></i></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>Sind die Symbole gleichwahrscheinlich,<ins class="diffchange diffchange-inline">&nbsp; </ins>gilt also&nbsp; $p_{\rm A} = p_{\rm B} =$ ... $ = p_{\rm H} = 1/M$,<ins class="diffchange diffchange-inline">&nbsp; </ins>so macht Quellencodierung keinen Sinn.&nbsp; Bereits mit dem Dualcode&nbsp; $\rm A$&nbsp; &#8594;&nbsp; <b>000</b>,&nbsp; $\rm B$&nbsp; &#8594;&nbsp; <b>001</b>, ... ,&nbsp; $\rm H$&nbsp; &#8594;&nbsp; <b>111</b>, erreicht nämlich dann die mittlere Codewortlänge&nbsp; $L_{\rm M}$&nbsp; ihre untere Schranke&nbsp; $H$&nbsp; gemäß dem Quellencodierungstheorem&nbsp; $(H$ bezeichnet hierbei die Quellenentropie$)$:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M,\hspace{0.08cm}min} = H = 3 \hspace{0.15cm}{\rm bit/Quellensymbol} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M,\hspace{0.08cm}min} = H = 3 \hspace{0.15cm}{\rm bit/Quellensymbol} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l31" >Zeile 31:</td>
<td colspan="2" class="diff-lineno">Zeile 31:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Oft wird dieser Algorithmus durch ein Baumdiagramm veranschaulicht. Die <del class="diffchange diffchange-inline">obige </del>Grafik zeigt dieses für den vorliegenden Fall. </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>Oft wird dieser Algorithmus durch ein Baumdiagramm veranschaulicht. Die Grafik zeigt dieses für den vorliegenden Fall. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Sie haben folgende Aufgaben:</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>Sie haben folgende Aufgaben:</div></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>:'''(a)''' &nbsp; Zuordnung der Symbole&nbsp; $\rm A$, ... , $\rm H$&nbsp; zu den mit&nbsp; '''[1]''', ... , '''[8]'''&nbsp; bezeichneten Eingängen.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:'''(a)''' &nbsp; Zuordnung der Symbole&nbsp; $\rm A$, ... , $\rm H$&nbsp; zu den mit&nbsp; '''[1]''', ... , '''[8]'''&nbsp; bezeichneten Eingängen.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>:'''(b)''' &nbsp; Bestimmung der Summenwahrscheinlichkeiten&nbsp; $U$, ... ,&nbsp; $Z$&nbsp; sowie&nbsp; $R$&nbsp; (<del class="diffchange diffchange-inline"><i></del>Root<del class="diffchange diffchange-inline"></i></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>:'''(b)''' &nbsp; Bestimmung der Summenwahrscheinlichkeiten&nbsp; $U$, ... ,&nbsp; $Z$&nbsp; sowie&nbsp; $R$&nbsp; (<ins class="diffchange diffchange-inline">&bdquo;</ins>Root<ins class="diffchange diffchange-inline">&rdquo;</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>:'''(c)''' &nbsp; Zuordnung der Symbole&nbsp; $\rm A$, ... ,&nbsp; $\rm H$&nbsp; zu den entsprechenden Huffman&ndash;Binärfolgen. Eine rote Verbindung entspricht einer&nbsp; <b>1</b>, eine blaue Verbindung einer&nbsp; <b>0</b>.</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>:'''(c)''' &nbsp; Zuordnung der Symbole&nbsp; $\rm A$, ... ,&nbsp; $\rm H$&nbsp; zu den entsprechenden Huffman&ndash;Binärfolgen. Eine rote Verbindung entspricht einer&nbsp; <b>1</b>, eine blaue Verbindung einer&nbsp; <b>0</b>.</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=29355&oldid=prev
Guenter am 25. Januar 2020 um 14:45 Uhr
2020-01-25T14:45:36Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="de">
<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 25. Januar 2020, 14:45 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l58" >Zeile 58:</td>
<td colspan="2" class="diff-lineno">Zeile 58:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</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>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</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>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf das Interaktionsmodul&nbsp; [[Applets:<del class="diffchange diffchange-inline">Huffman_Shannon_Fano</del>|Shannon&ndash;Fano&ndash; und Huffman&ndash;Codierung]].</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf das Interaktionsmodul&nbsp; [[Applets:<ins class="diffchange diffchange-inline">Huffman-_und_Shannon-Fano-Codierung</ins>|Shannon&ndash;Fano&ndash; und Huffman&ndash;Codierung]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=29352&oldid=prev
Guenter am 25. Januar 2020 um 14:35 Uhr
2020-01-25T14:35:27Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="de">
<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 25. Januar 2020, 14:35 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l110" >Zeile 110:</td>
<td colspan="2" class="diff-lineno">Zeile 110:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Musterlösung===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ML-Kopf}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(1)'''&nbsp; Vor dem Huffman&ndash;Algorithmus müssen die Symbole nach ihren Auftrittswahrscheinlichkeiten sortiert werden. Da die zwei unwahrscheinlichsten Symbole schon im ersten Schritt zusammengefasst werden, nehmen die Auftrittswahrscheinlichkeiten von oben nach unten ab (in der <del class="diffchange diffchange-inline">unteren </del>Grafik zu dieser Musterlösung von links nach rechts). Durch Vergleich mit dem Angabenblatt 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>'''(1)'''&nbsp; Vor dem Huffman&ndash;Algorithmus müssen die Symbole nach ihren Auftrittswahrscheinlichkeiten sortiert werden. </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>Da die zwei unwahrscheinlichsten Symbole schon im ersten Schritt zusammengefasst werden, nehmen die Auftrittswahrscheinlichkeiten von oben nach unten ab <ins class="diffchange diffchange-inline"><br></ins>(in der Grafik zu dieser Musterlösung von links nach rechts). </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>Durch Vergleich mit dem Angabenblatt 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>:Symbol $\rm A$: <u>Eingang 7</u>, &nbsp; &nbsp; Symbol $\rm B$: <u>Eingang 6</u>, &nbsp; &nbsp; Symbol $\rm C$: <u>Eingang 3</u>, &nbsp; &nbsp; Symbol $\rm D$: <u>Eingang 1</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>:Symbol $\rm A$:<ins class="diffchange diffchange-inline">&nbsp; </ins> <u>Eingang 7</u>, &nbsp; &nbsp; Symbol $\rm B$:<ins class="diffchange diffchange-inline">&nbsp; </ins> <u>Eingang 6</u>, &nbsp; &nbsp; Symbol $\rm C$:<ins class="diffchange diffchange-inline">&nbsp; </ins> <u>Eingang 3</u>, &nbsp; &nbsp; Symbol $\rm D$:<ins class="diffchange diffchange-inline">&nbsp; </ins> <u>Eingang 1</u>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID2452__Inf_A_2_6a.png|right|right|frame|An die Aufgabe angepasstes Baumdiagramm]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID2452__Inf_A_2_6a.png|right|right|frame|An die Aufgabe angepasstes Baumdiagramm]]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l118" >Zeile 118:</td>
<td colspan="2" class="diff-lineno">Zeile 120:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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; Der Knoten $\rm R$ ist die Baumwurzel (<i>Root</i>). Dieser ist stets mit <del class="diffchange diffchange-inline"><u><i></del>R<del class="diffchange diffchange-inline"></i> </del>= 1<del class="diffchange diffchange-inline"></u> </del>belegt, unabhängig von den Auftrittswahrscheinlichkeiten. </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; Der Knoten<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm R$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist die Baumwurzel<ins class="diffchange diffchange-inline">&nbsp; </ins>(<i>Root</i>).<ins class="diffchange diffchange-inline">&nbsp; </ins>Dieser ist stets mit<ins class="diffchange diffchange-inline">&nbsp; $\underline{</ins>R=1<ins class="diffchange diffchange-inline">}$&nbsp; </ins>belegt, unabhängig von den Auftrittswahrscheinlichkeiten. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Für die weiteren Werte 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>Für die weiteren Werte gilt:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l134" >Zeile 134:</td>
<td colspan="2" class="diff-lineno">Zeile 137:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:Schritt 6:&nbsp;&nbsp;&nbsp; $p_{\rm Z} = p_{\rm X} + p_{\rm D} = 0.29 + 0.25 \hspace{0.15cm}\underline{ =0.54}$.</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>:Schritt 6:&nbsp;&nbsp;&nbsp; $p_{\rm Z} = p_{\rm X} + p_{\rm D} = 0.29 + 0.25 \hspace{0.15cm}\underline{ =0.54}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br clear=all></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><br clear=all></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Den Huffman&ndash;Code für das Symbol $\rm A$ erhält man, wenn man den Weg von der $\rm Root$ (gelber Punkt) zum Symbol $\rm A$ zurückverfolgt und jeder roten Verbindungslinie eine <b>1</b> zuordnet, jeder blauen eine <b>0</b>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(3)'''&nbsp; Den Huffman&ndash;Code für das Symbol<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$<ins class="diffchange diffchange-inline">&nbsp; </ins>erhält man, wenn man den Weg von der<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm Root$<ins class="diffchange diffchange-inline">&nbsp; </ins>(gelber Punkt) zum Symbol<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$<ins class="diffchange diffchange-inline">&nbsp; </ins>zurückverfolgt und jeder roten Verbindungslinie eine<ins class="diffchange diffchange-inline">&nbsp; </ins><b>1</b><ins class="diffchange diffchange-inline">&nbsp; </ins>zuordnet, jeder blauen eine<ins class="diffchange diffchange-inline">&nbsp; </ins><b>0</b>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol $\rm A$:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;rot&ndash;blau&ndash;rot &#8594; <b><u>11101</u></b>,</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>* Symbol $\rm A$:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;rot&ndash;blau&ndash;rot &#8594; <b><u>11101</u></b>,</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l153" >Zeile 153:</td>
<td colspan="2" class="diff-lineno">Zeile 156:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Die Codierung unter Punkt '''(3)''' hat ergeben, dass</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''(4)'''&nbsp; Die Codierung unter Punkt<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(3)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>hat ergeben, dass</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* die Symbole $\rm D$ und $\rm E$ mit zwei Bit,</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 Symbole<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm D$<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm E$<ins class="diffchange diffchange-inline">&nbsp; </ins>mit zwei Bit,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* die Symbole $\rm C$, $\rm F$ und $\rm G$ mit drei Bit,</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 Symbole<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm C$,<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm F$<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm G$<ins class="diffchange diffchange-inline">&nbsp; </ins>mit drei Bit,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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 Symbol $\rm B$ mit vier Bit, und</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 Symbol<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm B$<ins class="diffchange diffchange-inline">&nbsp; </ins> mit vier Bit,<ins class="diffchange diffchange-inline">&nbsp; </ins>und</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* die Symbole $\rm A$ und $\rm H$ jeweils mit fünf Bit</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 Symbole<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm H$<ins class="diffchange diffchange-inline">&nbsp; </ins>jeweils mit fünf Bit</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>dargestellt werden. Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&dquo;):</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>dargestellt werden.<ins class="diffchange diffchange-inline">&nbsp; </ins>Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&dquo;):</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M} \hspace{0.2cm} = \hspace{0.2cm} (p_{\rm D} + p_{\rm E}) \cdot 2 + (p_{\rm C} + p_{\rm F} + p_{\rm G}) \cdot 3 + p_{\rm B} \cdot 4 +(p_{\rm A} + p_{\rm H}) \cdot 5 = 0.49 \cdot 2 + 0.36 \cdot 3 +0.08 \cdot 4 +0.07 \cdot 5 </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M} \hspace{0.2cm} = \hspace{0.2cm} (p_{\rm D} + p_{\rm E}) \cdot 2 + (p_{\rm C} + p_{\rm F} + p_{\rm G}) \cdot 3 + p_{\rm B} \cdot 4 +(p_{\rm A} + p_{\rm H}) \cdot 5 = 0.49 \cdot 2 + 0.36 \cdot 3 +0.08 \cdot 4 +0.07 \cdot 5 </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\hspace{0.15cm}\underline{= 2.73}\hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\hspace{0.15cm}\underline{= 2.73}\hspace{0.05cm}.$$</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins 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 class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(5)'''&nbsp; Richtig ist allein die <u>Antwort 1</u>:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''(5)'''&nbsp; Richtig ist allein die <u>Antwort 1</u>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die mittlere Codewortlänge $L_{\rm M}$ kann nicht kleiner sein als die Quellenentropie $H$. Damit scheiden die Antworten 2 und 3 aus.</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 mittlere Codewortlänge<ins class="diffchange diffchange-inline">&nbsp; </ins>$L_{\rm M}$<ins class="diffchange diffchange-inline">&nbsp; </ins>kann nicht kleiner sein als die Quellenentropie<ins class="diffchange diffchange-inline">&nbsp; </ins>$H$.<ins class="diffchange diffchange-inline">&nbsp; </ins>Damit scheiden die Antworten 2 und 3 aus.</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>*Aus den vorgegebenen Auftrittswahrscheinlichkeiten kann die Quellenentropie zu $H = 2.71$ bit/Quellensymbol berechnet werden. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Aus den vorgegebenen Auftrittswahrscheinlichkeiten kann die Quellenentropie <ins class="diffchange diffchange-inline">tatsächlich </ins>zu<ins class="diffchange diffchange-inline">&nbsp; </ins>$H = 2.71$ bit/Quellensymbol berechnet werden. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Man erkennt, dass diese Huffman&ndash;Codierung die vorgegebene Grenze (Quellencodierungstheorem) $L_{\rm M, \ min } <del class="diffchange diffchange-inline">= </del>H = 2.71$ bit/Quellensymbol nahezu erreicht.</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>*Man erkennt, dass diese Huffman&ndash;Codierung die vorgegebene Grenze<ins class="diffchange diffchange-inline">&nbsp; </ins>(Quellencodierungstheorem)<ins class="diffchange diffchange-inline">&nbsp; </ins>$L_{\rm M, \ min } <ins class="diffchange diffchange-inline">\ge </ins>H = 2.71$<ins class="diffchange diffchange-inline">&nbsp; </ins>bit/Quellensymbol nahezu erreicht.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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_2.6:_Zur_Huffman-Codierung&diff=29351&oldid=prev
Guenter am 25. Januar 2020 um 14:21 Uhr
2020-01-25T14:21:41Z
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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 25. Januar 2020, 14:21 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Zeile 4:</td>
<td colspan="2" class="diff-lineno">Zeile 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID2451__Inf_A_2_6.png|right|frame|Baumdiagramm des <br>Huffman&ndash;Verfahrens]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Datei:P_ID2451__Inf_A_2_6.png|right|frame|Baumdiagramm des <br>Huffman&ndash;Verfahrens]]</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>Wir betrachten hier eine Quellensymbolfolge $\langle q_\nu \rangle$ mit dem Symbolumfang $M = 8$:</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>Wir betrachten hier eine Quellensymbolfolge<ins class="diffchange diffchange-inline">&nbsp; </ins>$\langle q_\nu \rangle$<ins class="diffchange diffchange-inline">&nbsp; </ins>mit dem Symbolumfang<ins class="diffchange diffchange-inline">&nbsp; </ins>$M = 8$:</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>:$$q_{\nu} = \{ \hspace{0.05cm}q_{\mu} \} = \{ \boldsymbol{\rm A} \hspace{0.05cm}, \boldsymbol{\rm B}\hspace{0.05cm}, \boldsymbol{\rm C}\hspace{0.05cm}, \boldsymbol{\rm D}\hspace{0.05cm}, \boldsymbol{\rm E}\hspace{0.05cm}, \boldsymbol{\rm F}\hspace{0.05cm}, \boldsymbol{\rm G}\hspace{0.05cm}, \boldsymbol{\rm H}\hspace{0.05cm}</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$q_{\nu} = \{ \hspace{0.05cm}q_{\mu} \} = \{ \boldsymbol{\rm A} \hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm B}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm C}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm D}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm E}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm F}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm G}\hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.05cm} </ins>\boldsymbol{\rm H}\hspace{0.05cm}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\}\hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\}\hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Sind die Symbole gleichwahrscheinlich, also <del class="diffchange diffchange-inline">gilt </del>$p_{\rm A} = p_{\rm B} =$ ... $ = p_{\rm H} = 1/M$, so macht Quellencodierung keinen Sinn. Bereits mit dem Dualcode $\rm A$ &#8594; <b>000</b>, $\rm B$&nbsp;&#8594;&nbsp;<b>001</b>, ... , $\rm H$ &#8594; <b>111</b>, erreicht nämlich die mittlere Codewortlänge $L_{\rm M}$ ihre untere Schranke $H$ gemäß dem Quellencodierungstheorem (<del class="diffchange diffchange-inline">$</del>H$ bezeichnet hierbei die <i>Quellenentropie</i>):</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>Sind die Symbole gleichwahrscheinlich, <ins class="diffchange diffchange-inline">gilt </ins>also<ins class="diffchange diffchange-inline">&nbsp; </ins>$p_{\rm A} = p_{\rm B} =$ ... $ = p_{\rm H} = 1/M$, so macht Quellencodierung keinen Sinn.<ins class="diffchange diffchange-inline">&nbsp; </ins>Bereits mit dem Dualcode<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$<ins class="diffchange diffchange-inline">&nbsp; </ins>&#8594<ins class="diffchange diffchange-inline">;&nbsp</ins>; <b>000</b>,<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm B$&nbsp; &#8594;&nbsp; <b>001</b>, ... ,<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm H$<ins class="diffchange diffchange-inline">&nbsp; </ins>&#8594<ins class="diffchange diffchange-inline">;&nbsp</ins>; <b>111</b>, erreicht nämlich <ins class="diffchange diffchange-inline">dann </ins>die mittlere Codewortlänge<ins class="diffchange diffchange-inline">&nbsp; </ins>$L_{\rm M}$<ins class="diffchange diffchange-inline">&nbsp; </ins>ihre untere Schranke<ins class="diffchange diffchange-inline">&nbsp; </ins>$H$<ins class="diffchange diffchange-inline">&nbsp; </ins>gemäß dem Quellencodierungstheorem<ins class="diffchange diffchange-inline">&nbsp; $</ins>(H$ bezeichnet hierbei die <i>Quellenentropie</i><ins class="diffchange diffchange-inline">$</ins>)<ins class="diffchange diffchange-inline">$</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>:$$L_{\rm M,\hspace{0.08cm}min} = H = 3 \hspace{0.15cm}{\rm bit/Quellensymbol} \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M,\hspace{0.08cm}min} = H = 3 \hspace{0.15cm}{\rm bit/Quellensymbol} \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Die Symbolwahrscheinlichkeiten seien aber in dieser Aufgabe 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 Symbolwahrscheinlichkeiten seien aber in dieser Aufgabe 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>:$$p_{\rm A} \hspace{-0.05cm}= \hspace{-0.05cm} 0.04 \hspace{0.05cm},\hspace{0.1cm}p_{\rm B} \hspace{-0.05cm}= \hspace{-0.05cm} 0.08 \hspace{0.05cm},\hspace{0.1cm}p_{\rm C} \hspace{-0.05cm}= \hspace{-0.05cm} 0.14 \hspace{0.05cm},\hspace{0.1cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$p_{\rm A} \hspace{-0.05cm}= \hspace{-0.05cm} 0.04 \hspace{0.05cm},\hspace{0.1cm}p_{\rm B} \hspace{-0.05cm}= \hspace{-0.05cm} 0.08 \hspace{0.05cm},\hspace{0.1cm}p_{\rm C} \hspace{-0.05cm}= \hspace{-0.05cm} 0.14 \hspace{0.05cm},\hspace{0.1cm}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>p_{\rm D} \hspace{-0.05cm}= \hspace{-0.05cm} 0.25 \hspace{0.05cm},<del class="diffchange diffchange-inline">$$</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>p_{\rm D} \hspace{-0.05cm}= \hspace{-0.05cm} 0.25 \hspace{0.05cm},<ins class="diffchange diffchange-inline">\hspace{0.1cm} </ins>p_{\rm E} \hspace{-0.05cm}= \hspace{-0.05cm} 0.24 \hspace{0.05cm},\hspace{0.1cm}p_{\rm F} \hspace{-0.05cm}= \hspace{-0.05cm} 0.12 \hspace{0.05cm},\hspace{0.1cm}p_{\rm G} \hspace{-0.05cm}= \hspace{-0.05cm} 0.10 \hspace{0.05cm},\hspace{0.1cm}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:$$</del>p_{\rm E} \hspace{-0.05cm}= \hspace{-0.05cm} 0.24 \hspace{0.05cm},\hspace{0.1cm}p_{\rm F} \hspace{-0.05cm}= \hspace{-0.05cm} 0.12 \hspace{0.05cm},\hspace{0.1cm}p_{\rm G} \hspace{-0.05cm}= \hspace{-0.05cm} 0.10 \hspace{0.05cm},\hspace{0.1cm}</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>p_{\rm H} \hspace{-0.05cm}= \hspace{-0.05cm} 0.03 \hspace{0.05cm}.$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>p_{\rm H} \hspace{-0.05cm}= \hspace{-0.05cm} 0.03 \hspace{0.05cm}.$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Es liegt hier also eine redundante Nachrichtenquelle vor, die man durch Huffman&ndash;Codierung komprimieren kann. <del class="diffchange diffchange-inline">Der </del>Algorithmus wurde 1952 &ndash; also kurz nach Shannons bahnbrechenden Arbeiten zur Informationstheorie &ndash; von [https://de.wikipedia.org/wiki/David_A._Huffman David Albert Huffman] veröffentlicht und erlaubt die Konstruktion von optimalen präfixfreien Codes.</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>Es liegt hier also eine redundante Nachrichtenquelle vor, die man durch Huffman&ndash;Codierung komprimieren kann. </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">*Dieser </ins>Algorithmus wurde 1952 &ndash; also kurz nach Shannons bahnbrechenden Arbeiten zur Informationstheorie &ndash; von<ins class="diffchange diffchange-inline">&nbsp; </ins>[https://de.wikipedia.org/wiki/David_A._Huffman David Albert Huffman]<ins class="diffchange diffchange-inline">&nbsp; </ins>veröffentlicht und erlaubt die Konstruktion von optimalen präfixfreien Codes.</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>Der Algorithmus soll hier ohne Herleitung und Beweis angegeben werden, wobei wir uns auf Binärcodes beschränken &nbsp; &rArr; &nbsp; die Codesymbolfolge besteht nur aus Nullen und Einsen:</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 Algorithmus soll hier ohne Herleitung und Beweis angegeben werden, wobei wir uns auf Binärcodes beschränken &nbsp; &rArr; &nbsp; die Codesymbolfolge besteht nur aus Nullen und Einsen:</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 24:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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; Man fasse die zwei unwahrscheinlichsten Symbole zu einem neuen Symbol zusammen.</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; Man fasse die zwei unwahrscheinlichsten Symbole zu einem neuen Symbol zusammen.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:'''(3)''' &nbsp; Man wiederhole Schritt '''(1)''' und '''(2)''', bis nur zwei (zusammengefasste) Symbole übrig bleiben.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:'''(3)''' &nbsp; Man wiederhole Schritt<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(1)'''<ins class="diffchange diffchange-inline">&nbsp; </ins>und<ins class="diffchange diffchange-inline">&nbsp; </ins>'''(2)''', bis nur zwei (zusammengefasste) Symbole übrig bleiben.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:'''(4)''' &nbsp; Die wahrscheinlichere Symbolmenge wird mit <b>1</b> binär codiert, die andere Menge mit <b>0</b>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:'''(4)''' &nbsp; Die wahrscheinlichere Symbolmenge wird mit<ins class="diffchange diffchange-inline">&nbsp; </ins><b>1</b><ins class="diffchange diffchange-inline">&nbsp; </ins> binär codiert, die andere Menge mit<ins class="diffchange diffchange-inline">&nbsp; </ins><b>0</b>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>:'''(5)''' &nbsp; Man ergänze schrittweise (von unten nach oben) die aufgespaltenen Teilcodes mit <b>1</b> bzw. <b>0</b>.</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>:'''(5)''' &nbsp; Man ergänze schrittweise (von unten nach oben) die aufgespaltenen Teilcodes mit<ins class="diffchange diffchange-inline">&nbsp; </ins><b>1</b><ins class="diffchange diffchange-inline">&nbsp; </ins>bzw.<ins class="diffchange diffchange-inline">&nbsp; </ins><b>0</b>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Oft wird dieser Algorithmus durch ein Baumdiagramm veranschaulicht. Die obige Grafik zeigt dieses für den vorliegenden Fall. Sie haben folgende Aufgaben:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:'''(a)''' &nbsp; Zuordnung der Symbole $\rm A$, </del>.<del class="diffchange diffchange-inline">.. , $\rm H$ zu </del>den <del class="diffchange diffchange-inline">mit '''[1]''', ... , '''[8]''' bezeichneten Eingängen</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">Oft wird dieser Algorithmus durch ein Baumdiagramm veranschaulicht</ins>. <ins class="diffchange diffchange-inline">Die obige Grafik zeigt dieses für </ins>den <ins class="diffchange diffchange-inline">vorliegenden Fall</ins>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">'''(b)''' &nbsp; Bestimmung der Summenwahrscheinlichkeiten $U$, ... , $Z$ sowie $R$ (<i>Root</i>).</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">Sie haben folgende Aufgaben</ins>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:'''(c)''' &nbsp; Zuordnung der Symbole $\rm A$, ... , $\rm H$ zu den entsprechenden Huffman&ndash;Binärfolgen. Eine rote Verbindung <del class="diffchange diffchange-inline">im Baumdiagramm </del>entspricht einer <b>1</b> <del class="diffchange diffchange-inline">und </del>eine blaue Verbindung einer <b>0</b>.</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">:'''(a)''' &nbsp; Zuordnung der Symbole&nbsp; $\rm A$, ... , $\rm H$&nbsp; zu den mit&nbsp; '''[1]''', ... , '''[8]'''&nbsp; bezeichneten Eingängen.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:'''(b)''' &nbsp; Bestimmung der Summenwahrscheinlichkeiten&nbsp; $U$, ... ,&nbsp; $Z$&nbsp; sowie&nbsp; $R$&nbsp; (<i>Root</i>).</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>:'''(c)''' &nbsp; Zuordnung der Symbole<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$, ... ,<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm H$<ins class="diffchange diffchange-inline">&nbsp; </ins>zu den entsprechenden Huffman&ndash;Binärfolgen. Eine rote Verbindung entspricht einer<ins class="diffchange diffchange-inline">&nbsp; </ins><b>1</b><ins class="diffchange diffchange-inline">, </ins>eine blaue Verbindung einer<ins class="diffchange diffchange-inline">&nbsp; </ins><b>0</b>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Sie werden feststellen, dass die mittlere Codewortlänge</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>Sie werden feststellen, dass die mittlere Codewortlänge</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$L_{\rm M} = \sum_{\mu = 1}^{M}\hspace{0.05cm} p_{\mu} \cdot L_{\mu} $$</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>:$$L_{\rm M} = \sum_{\mu = 1}^{M}\hspace{0.05cm} p_{\mu} \cdot L_{\mu} $$</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 Huffman&ndash;Codierung nur unwesentlich größer ist als die Quellenentropie $H$. In dieser Gleichung gelten für den vorliegenden Fall folgende Werte: </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 Huffman&ndash;Codierung nur unwesentlich größer ist als die Quellenentropie<ins class="diffchange diffchange-inline">&nbsp; </ins>$H$.<ins class="diffchange diffchange-inline">&nbsp; </ins>In dieser Gleichung gelten für den vorliegenden Fall folgende Werte: </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>*$M = 8$, $p_1 = p_{\rm A}$, ... , $p_8 = p_{\rm H}$. </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>*$M = 8$,<ins class="diffchange diffchange-inline">&nbsp; </ins> $p_1 = p_{\rm A}$,<ins class="diffchange diffchange-inline">&nbsp; </ins>... ,<ins class="diffchange diffchange-inline">&nbsp; </ins>$p_8 = p_{\rm H}$. </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 jeweilige Bitanzahl der Codesymbole für $\rm A$, ... , $\rm H$ ist mit $L_1$, ... , $L_8$ bezeichnet.</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 jeweilige Bitanzahl der Codesymbole für<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm A$, ... ,<ins class="diffchange diffchange-inline">&nbsp; </ins>$\rm H$<ins class="diffchange diffchange-inline">&nbsp; </ins>ist mit<ins class="diffchange diffchange-inline">&nbsp; </ins>$L_1$, ... ,<ins class="diffchange diffchange-inline">&nbsp; </ins>$L_8$<ins class="diffchange diffchange-inline">&nbsp; </ins>bezeichnet.</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-l48" >Zeile 48:</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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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 [[Informationstheorie/Entropiecodierung_nach_Huffman|Entropiecodierung nach Huffman]].</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>[[Informationstheorie/Entropiecodierung_nach_Huffman|Entropiecodierung nach Huffman]].</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>*Insbesondere wird Bezug genommen auf die Seiten</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>*Insbesondere wird Bezug genommen auf die Seiten<ins class="diffchange diffchange-inline">&nbsp;</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>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</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>** [[Informationstheorie/Entropiecodierung_nach_Huffman#Der_Huffman.E2.80.93Algorithmus|Der Huffman-Algorithmus]],</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</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>**[[Informationstheorie/Entropiecodierung_nach_Huffman#Darstellung_des_Huffman.E2.80.93Codes_als_Baumdiagramm|Darstellung des Huffman-Codes als Baumdiagramm]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf das Interaktionsmodul [[Applets:Huffman_Shannon_Fano|Shannon&ndash;Fano&ndash; und Huffman&ndash;Codierung]].</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Zur Kontrolle Ihrer Ergebnisse verweisen wir auf das Interaktionsmodul<ins class="diffchange diffchange-inline">&nbsp; </ins>[[Applets:Huffman_Shannon_Fano|Shannon&ndash;Fano&ndash; und Huffman&ndash;Codierung]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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 colspan="2" class="diff-lineno" id="mw-diff-left-l92" >Zeile 92:</td>
<td colspan="2" class="diff-lineno">Zeile 98:</td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{Wie groß ist die Quellenentropie $H$? <del class="diffchange diffchange-inline">&nbsp; <i>Hinweis:</i> Es gibt genau eine Lösung.</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>{Wie groß ist die Quellenentropie<ins class="diffchange diffchange-inline">&nbsp; </ins>$H$? </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>|type="<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>|type="<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>+ $H = 2.71\ \rm bit/Quellensymbol.$</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>+ $H = 2.71\ \rm bit/Quellensymbol.$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>- $H = 2.75\ \rm bit/Quellensymbol.$</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>- $H = 2.75\ \rm bit/Quellensymbol.$</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=26351&oldid=prev
Guenter am 28. September 2018 um 08:04 Uhr
2018-09-28T08:04:17Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 28. September 2018, 08:04 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l106" >Zeile 106:</td>
<td colspan="2" class="diff-lineno">Zeile 106:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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; Vor dem Huffman&ndash;Algorithmus müssen die Symbole nach ihren Auftrittswahrscheinlichkeiten sortiert werden. Da die zwei unwahrscheinlichsten Symbole schon im ersten Schritt zusammengefasst werden, nehmen die Auftrittswahrscheinlichkeiten von oben nach unten ab (in der unteren Grafik zu dieser Musterlösung von links nach rechts). Durch Vergleich mit dem Angabenblatt 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>'''(1)'''&nbsp; Vor dem Huffman&ndash;Algorithmus müssen die Symbole nach ihren Auftrittswahrscheinlichkeiten sortiert werden. Da die zwei unwahrscheinlichsten Symbole schon im ersten Schritt zusammengefasst werden, nehmen die Auftrittswahrscheinlichkeiten von oben nach unten ab (in der unteren Grafik zu dieser Musterlösung von links nach rechts). Durch Vergleich mit dem Angabenblatt 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>:Symbol <del class="diffchange diffchange-inline"><b></del>A<del class="diffchange diffchange-inline"></B></del>: <u>Eingang 7</u>, &nbsp; &nbsp; Symbol <del class="diffchange diffchange-inline"><b>B</</del>B<del class="diffchange diffchange-inline">></del>: <u>Eingang 6</u>, &nbsp; &nbsp; Symbol <del class="diffchange diffchange-inline"><b></del>C<del class="diffchange diffchange-inline"></B></del>: <u>Eingang 3</u>, &nbsp; &nbsp; Symbol <del class="diffchange diffchange-inline"><b></del>D<del class="diffchange diffchange-inline"></B></del>: <u>Eingang 1</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>:Symbol <ins class="diffchange diffchange-inline">$\rm </ins>A<ins class="diffchange diffchange-inline">$</ins>: <u>Eingang 7</u>, &nbsp; &nbsp; Symbol <ins class="diffchange diffchange-inline">$\rm </ins>B<ins class="diffchange diffchange-inline">$</ins>: <u>Eingang 6</u>, &nbsp; &nbsp; Symbol <ins class="diffchange diffchange-inline">$\rm </ins>C<ins class="diffchange diffchange-inline">$</ins>: <u>Eingang 3</u>, &nbsp; &nbsp; Symbol <ins class="diffchange diffchange-inline">$\rm </ins>D<ins class="diffchange diffchange-inline">$</ins>: <u>Eingang 1</u>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Datei:P_ID2452__Inf_A_2_6a.png|right|right|frame|An die Aufgabe angepasstes Baumdiagramm]]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Datei:P_ID2452__Inf_A_2_6a.png|right|An die Aufgabe angepasstes Baumdiagramm]]</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">'''(2)'''&nbsp; Der Knoten '''R''' ist die Baumwurzel (<i>Root</i>). Dieser ist stets mit <u><i>R</i> = 1</u> belegt, unabhängig von den Auftrittswahrscheinlichkeiten. </del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Für die weiteren Werte gilt:</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:Schritt 1:&nbsp;&nbsp;</del>&nbsp; <i><del class="diffchange diffchange-inline">U</del></i> <del class="diffchange diffchange-inline">= </del><<del class="diffchange diffchange-inline">i>p</i><sub</del>><del class="diffchange diffchange-inline">A</sub> + </del><i><del class="diffchange diffchange-inline">p</del></i<del class="diffchange diffchange-inline">><sub>H</sub</del>> = <del class="diffchange diffchange-inline">0.04 + 0.03 = <u>0.07</del></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><ins class="diffchange diffchange-inline">'''(2)'''</ins>&nbsp; <ins class="diffchange diffchange-inline">Der Knoten $\rm R$ ist die Baumwurzel (</ins><i><ins class="diffchange diffchange-inline">Root</ins></i><ins class="diffchange diffchange-inline">). Dieser ist stets mit </ins><<ins class="diffchange diffchange-inline">u</ins>><i><ins class="diffchange diffchange-inline">R</ins></i> = <ins class="diffchange diffchange-inline">1</ins></u> <ins class="diffchange diffchange-inline">belegt</ins>, <ins class="diffchange diffchange-inline">unabhängig von den Auftrittswahrscheinlichkeiten. </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">Schritt 2:&nbsp;&nbsp;&nbsp; <i>V</i> = <i>U</i> + <i>p</i><sub>B</sub> = 0.07 + 0.08 = <u>0.15</u>,</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">Für die weiteren Werte gilt</ins>:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:Schritt <del class="diffchange diffchange-inline">3</del>:&nbsp;&nbsp;&nbsp; <del class="diffchange diffchange-inline"><i>W</i> </del>= <del class="diffchange diffchange-inline"><i>p</i><sub>F</sub> </del>+ <del class="diffchange diffchange-inline"><i>p</i><sub>G</sub> </del>= 0.<del class="diffchange diffchange-inline">12 </del>+ 0.<del class="diffchange diffchange-inline">10 </del>= <del class="diffchange diffchange-inline"><u></del>0.<del class="diffchange diffchange-inline">22</u></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>:Schritt <ins class="diffchange diffchange-inline">1</ins>:&nbsp;&nbsp;&nbsp; <ins class="diffchange diffchange-inline">$p_{\rm U} </ins>= <ins class="diffchange diffchange-inline">p_{\rm A} </ins>+ <ins class="diffchange diffchange-inline">p_{\rm H} </ins>= 0.<ins class="diffchange diffchange-inline">04 </ins>+ 0.<ins class="diffchange diffchange-inline">03 \hspace{0.15cm}\underline{ </ins>=0.<ins class="diffchange diffchange-inline">07}$</ins>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:Schritt <del class="diffchange diffchange-inline">4</del>:&nbsp;&nbsp;&nbsp; <del class="diffchange diffchange-inline"><i>X</i> </del>= <del class="diffchange diffchange-inline"><i>V</i> </del>+ <del class="diffchange diffchange-inline"><i>p</i><sub>C</sub> </del>= 0.<del class="diffchange diffchange-inline">15 </del>+ 0.<del class="diffchange diffchange-inline">14 </del>= 0.<del class="diffchange diffchange-inline">29</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>:Schritt <ins class="diffchange diffchange-inline">2</ins>:&nbsp;&nbsp;&nbsp; <ins class="diffchange diffchange-inline">$p_{\rm V} </ins>= <ins class="diffchange diffchange-inline">p_{\rm U} </ins>+ <ins class="diffchange diffchange-inline">p_{\rm B} </ins>= 0.<ins class="diffchange diffchange-inline">07 </ins>+ 0.<ins class="diffchange diffchange-inline">08 \hspace{0.15cm}\underline{ </ins>=0.<ins class="diffchange diffchange-inline">15}$</ins>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:Schritt <del class="diffchange diffchange-inline">5</del>:&nbsp;&nbsp;&nbsp; <del class="diffchange diffchange-inline"><i>Y</i> </del>= <del class="diffchange diffchange-inline"><i>W</i> </del>+ <del class="diffchange diffchange-inline"><i>p</i><sub>E</sub> </del>= 0.<del class="diffchange diffchange-inline">22 </del>+ 0.<del class="diffchange diffchange-inline">24 </del>= 0.<del class="diffchange diffchange-inline">46</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>:Schritt <ins class="diffchange diffchange-inline">3</ins>:&nbsp;&nbsp;&nbsp; <ins class="diffchange diffchange-inline">$p_{\rm W} </ins>= <ins class="diffchange diffchange-inline">p_{\rm F} </ins>+ <ins class="diffchange diffchange-inline">p_{\rm G} </ins>= 0.<ins class="diffchange diffchange-inline">12 </ins>+ 0.<ins class="diffchange diffchange-inline">10 \hspace{0.15cm}\underline{ </ins>=0.<ins class="diffchange diffchange-inline">22}$</ins>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:Schritt <del class="diffchange diffchange-inline">6</del>:&nbsp;&nbsp;&nbsp; <del class="diffchange diffchange-inline"><i>Z</i> </del>= <del class="diffchange diffchange-inline"><i>X</i> </del>+ <del class="diffchange diffchange-inline"><i>p</i><sub>D</sub> </del>= 0.<del class="diffchange diffchange-inline">29 </del>+ 0.<del class="diffchange diffchange-inline">25 </del>= <del class="diffchange diffchange-inline"><u></del>0.<del class="diffchange diffchange-inline">54</u>.</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>:Schritt <ins class="diffchange diffchange-inline">4</ins>:&nbsp;&nbsp;&nbsp; <ins class="diffchange diffchange-inline">$p_{\rm X} </ins>= <ins class="diffchange diffchange-inline">p_{\rm V} </ins>+ <ins class="diffchange diffchange-inline">p_{\rm C} </ins>= 0.<ins class="diffchange diffchange-inline">15 </ins>+ 0.<ins class="diffchange diffchange-inline">14 \hspace{0.15cm}\underline{ </ins>=0.<ins class="diffchange diffchange-inline">29}$,</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Damit ergibt sich das hier dargestelle Baumdiagramm</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">:Schritt 5:&nbsp;&nbsp;&nbsp; $p_{\rm Y} = p_{\rm W} + p_{\rm E} = 0</ins>.<ins class="diffchange diffchange-inline">22 + 0.24 \hspace{0.15cm}\underline{ =0.46}$,</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 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;">:Schritt 6:&nbsp;&nbsp;&nbsp; $p_{\rm Z} = p_{\rm X} + p_{\rm D} = 0.29 + 0.25 \hspace{0.15cm}\underline{ =0.54}$.</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;"><br clear=all></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''(3)'''&nbsp; Den Huffman&ndash;Code für das Symbol $\rm A$ erhält man, wenn man den Weg von der $\rm Root$ (gelber Punkt) zum Symbol $\rm A$ zurückverfolgt und jeder roten Verbindungslinie eine <b>1</b> zuordnet, jeder blauen eine <b>0</b>.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">'''(3)'''</del>&nbsp; <del class="diffchange diffchange-inline">Den Huffman</del>&ndash;<del class="diffchange diffchange-inline">Code für das Symbol </del><b><del class="diffchange diffchange-inline">A</del><<del class="diffchange diffchange-inline">/b> erhält man, wenn man den Weg von der <i</del>><del class="diffchange diffchange-inline">Root</del></<del class="diffchange diffchange-inline">i</del>> <del class="diffchange diffchange-inline">(gelber Punkt) zum Symbol <b>A</b> zurückverfolgt und jeder roten Verbindungslinie eine &bdquo;<b>1</del></b><del class="diffchange diffchange-inline">&rdquo; zuordnet</del>, <del class="diffchange diffchange-inline">jeder blauen eine &bdquo;<b>0</b>&rdquo;.</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">* Symbol $\rm A$:&nbsp;&nbsp;</ins>&nbsp; <ins class="diffchange diffchange-inline">rot&ndash;rot</ins>&ndash;<ins class="diffchange diffchange-inline">rot&ndash;blau&ndash;rot &#8594; </ins><b><<ins class="diffchange diffchange-inline">u</ins>><ins class="diffchange diffchange-inline">11101</ins></<ins class="diffchange diffchange-inline">u</ins>></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>A</b></del>:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;rot<del class="diffchange diffchange-inline">&ndash;blau</del>&ndash;rot &#8594; <b><u><del class="diffchange diffchange-inline">11101</del></u></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm B$</ins>:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;rot&ndash;rot &#8594; <b><u><ins class="diffchange diffchange-inline">1111</ins></u></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>B</b></del>:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;<del class="diffchange diffchange-inline">rot&ndash;rot </del>&#8594; <b><u><del class="diffchange diffchange-inline">1111</del></u></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm C$</ins>:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;<ins class="diffchange diffchange-inline">blau </ins>&#8594; <b><u><ins class="diffchange diffchange-inline">110</ins></u></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>C</b></del>:&nbsp;&nbsp;&nbsp; rot&ndash;<del class="diffchange diffchange-inline">rot</del>&ndash;<del class="diffchange diffchange-inline">blau </del>&#8594; <b><u><del class="diffchange diffchange-inline">110</del></u></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm D$</ins>:&nbsp;&nbsp;&nbsp; rot&ndash;<ins class="diffchange diffchange-inline">blau</ins>&ndash; &#8594; <b><u><ins class="diffchange diffchange-inline">10</ins></u></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>D</b></del>:&nbsp;&nbsp;&nbsp<del class="diffchange diffchange-inline">; rot&ndash</del>;blau&ndash; &#8594; <b><del class="diffchange diffchange-inline"><u>10</u></del></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm E$</ins>:&nbsp;&nbsp;&nbsp; blau&ndash;<ins class="diffchange diffchange-inline">rot </ins>&#8594; <b><ins class="diffchange diffchange-inline">01</ins></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>E</b></del>:&nbsp;&nbsp;&nbsp; blau&ndash;rot &#8594; <b><del class="diffchange diffchange-inline">01</del></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm F$</ins>:&nbsp;&nbsp;&nbsp<ins class="diffchange diffchange-inline">; blau&ndash</ins>;blau&ndash;rot &#8594; <b><ins class="diffchange diffchange-inline">001</ins></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>F</b></del>:&nbsp;&nbsp;&nbsp; blau&ndash;blau&ndash;<del class="diffchange diffchange-inline">rot </del>&#8594; <b><del class="diffchange diffchange-inline">001</del></b>,</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>* Symbol <ins class="diffchange diffchange-inline">$\rm G$</ins>:&nbsp;&nbsp;&nbsp; blau&ndash;blau&ndash;<ins class="diffchange diffchange-inline">blau </ins>&#8594; <b><ins class="diffchange diffchange-inline">000</ins></b>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>* Symbol <del class="diffchange diffchange-inline"><b>G</b></del>:&nbsp;&nbsp;&nbsp; <del class="diffchange diffchange-inline">blau</del>&ndash;blau&ndash;blau &#8594; <b><del class="diffchange diffchange-inline">000</del></b><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>* Symbol <ins class="diffchange diffchange-inline">$\rm H$</ins>:&nbsp;&nbsp;&nbsp; <ins class="diffchange diffchange-inline">rot&ndash;rot&ndash;rot</ins>&ndash;blau&ndash;blau &#8594; <b><ins class="diffchange diffchange-inline">11100</ins></b><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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Symbol <b>H</b>:&nbsp;&nbsp;&nbsp; rot&ndash;rot&ndash;rot&ndash;blau&ndash;blau &#8594; <b>11100</b>.</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 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;">'''(4)'''&nbsp; Die Codierung unter Punkt '''(3)''' hat ergeben, dass</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">'''(4)'''&nbsp; Die Codierung unter Punkt (3) hat ergeben</del>, <del class="diffchange diffchange-inline">dass</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">* die Symbole $\rm D$ und $\rm E$ mit zwei Bit</ins>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* die Symbole <del class="diffchange diffchange-inline"><b>D</b> </del>und <del class="diffchange diffchange-inline"><b>E</b> </del>mit <del class="diffchange diffchange-inline">zwei </del>Bit,</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 Symbole <ins class="diffchange diffchange-inline">$\rm C$, $\rm F$ </ins>und <ins class="diffchange diffchange-inline">$\rm G$ </ins>mit <ins class="diffchange diffchange-inline">drei </ins>Bit,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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">die Symbole <b>C</b>, <b>F</b> und <b>G</b> </del>mit <del class="diffchange diffchange-inline">drei </del>Bit,</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">das Symbol $\rm B$ </ins>mit <ins class="diffchange diffchange-inline">vier </ins>Bit, <ins class="diffchange diffchange-inline">und</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <del class="diffchange diffchange-inline">das Symbol <b>B</b> </del>mit <del class="diffchange diffchange-inline">vier </del>Bit<del class="diffchange diffchange-inline">, und</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">die Symbole $\rm A$ und $\rm H$ jeweils </ins>mit <ins class="diffchange diffchange-inline">fünf </ins>Bit</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* die Symbole <b>A</b> und <b>H</b> jeweils mit fünf Bit</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>dargestellt werden. Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&dquo;):</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>dargestellt werden. Damit erhält man für die mittlere Codewortlänge (in &bdquo;bit/Quellensymbol&dquo;):</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l163" >Zeile 163:</td>
<td colspan="2" class="diff-lineno">Zeile 163:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>'''(5)'''&nbsp; Richtig ist allein <u>Antwort 1</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>'''(5)'''&nbsp; Richtig ist allein <ins class="diffchange diffchange-inline">die </ins><u>Antwort 1</u>:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Die mittlere Codewortlänge <del class="diffchange diffchange-inline"><i>L</i><sub></del>M<del class="diffchange diffchange-inline"></sub> </del>kann nicht kleiner sein als die Quellenentropie <del class="diffchange diffchange-inline"><i></del>H<del class="diffchange diffchange-inline"></i></del>. Damit scheiden die Antworten 2 und 3 aus.</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 mittlere Codewortlänge <ins class="diffchange diffchange-inline">$L_{\rm </ins>M<ins class="diffchange diffchange-inline">}$ </ins>kann nicht kleiner sein als die Quellenentropie <ins class="diffchange diffchange-inline">$</ins>H<ins class="diffchange diffchange-inline">$</ins>. Damit scheiden die Antworten 2 und 3 aus.</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>*Aus den vorgegebenen Auftrittswahrscheinlichkeiten kann die Quellenentropie zu <del class="diffchange diffchange-inline"><i></del>H<del class="diffchange diffchange-inline"></i> </del>= 2.71 bit/Quellensymbol berechnet werden. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Aus den vorgegebenen Auftrittswahrscheinlichkeiten kann die Quellenentropie zu <ins class="diffchange diffchange-inline">$</ins>H = 2.71<ins class="diffchange diffchange-inline">$ </ins>bit/Quellensymbol berechnet werden. </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Man erkennt, dass <del class="diffchange diffchange-inline">die vorliegende </del>Huffman&ndash;Codierung die <del class="diffchange diffchange-inline">durch das Quellencodierungstheorem </del>vorgegebene Grenze <del class="diffchange diffchange-inline"><i>L</i><sub></del>M, min<del class="diffchange diffchange-inline"></sub> </del>= <del class="diffchange diffchange-inline"><i></del>H<del class="diffchange diffchange-inline"></i> </del>= 2.71 bit/Quellensymbol nahezu erreicht.</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>*Man erkennt, dass <ins class="diffchange diffchange-inline">diese </ins>Huffman&ndash;Codierung die vorgegebene Grenze <ins class="diffchange diffchange-inline">(Quellencodierungstheorem) $L_{\rm </ins>M, <ins class="diffchange diffchange-inline">\ </ins>min <ins class="diffchange diffchange-inline">} </ins>= H = 2.71<ins class="diffchange diffchange-inline">$ </ins>bit/Quellensymbol nahezu erreicht.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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_2.6:_Zur_Huffman-Codierung&diff=26350&oldid=prev
Guenter am 28. September 2018 um 07:29 Uhr
2018-09-28T07:29:26Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 28. September 2018, 07:29 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l72" >Zeile 72:</td>
<td colspan="2" class="diff-lineno">Zeile 72:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wahrscheinlichkeit $ \ = \ $ { 0.15 1% } &nbsp; bei Knoten $\rm V$</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>Wahrscheinlichkeit $ \ = \ $ { 0.15 1% } &nbsp; bei Knoten $\rm V$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wahrscheinlichkeit $ \ = \ $ { 0.22 1% } &nbsp; bei Knoten $\rm W$</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>Wahrscheinlichkeit $ \ = \ $ { 0.22 1% } &nbsp; bei Knoten $\rm W$</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;">Wahrscheinlichkeit $ \ = \ $ { 0.29 1% } &nbsp; bei Knoten $\rm X$</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;">Wahrscheinlichkeit $ \ = \ $ { 0.46 1% } &nbsp; bei Knoten $\rm Y$</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>Wahrscheinlichkeit $ \ = \ $ { 0.54 1% } &nbsp; bei Knoten $\rm Z$</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>Wahrscheinlichkeit $ \ = \ $ { 0.54 1% } &nbsp; bei Knoten $\rm Z$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wahrscheinlichkeit $ \ = \ $ { 1 1% } &nbsp; bei $\rm Root$</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>Wahrscheinlichkeit $ \ = \ $ { 1 1% } &nbsp; bei $\rm Root$</div></td></tr>
</table>
Guenter
https://www.lntwww.de/index.php?title=Aufgaben:Aufgabe_2.6:_Zur_Huffman-Codierung&diff=26349&oldid=prev
Guenter am 28. September 2018 um 07:00 Uhr
2018-09-28T07:00:03Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="de">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Version vom 28. September 2018, 07:00 Uhr</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l67" >Zeile 67:</td>
<td colspan="2" class="diff-lineno">Zeile 67:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>{Welche Zahlenwerte (Wahrscheinlichkeiten sollten bei den Knoten im Baumdiagramm stehen?</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>{Welche Zahlenwerte (Wahrscheinlichkeiten<ins class="diffchange diffchange-inline">) </ins>sollten bei den Knoten im Baumdiagramm stehen?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Wahrscheinlichkeit $ \ = \ $ { 0.07 1% } &nbsp; bei Knoten $\rm U$</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>Wahrscheinlichkeit $ \ = \ $ { 0.07 1% } &nbsp; bei Knoten $\rm U$</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l90" >Zeile 90:</td>
<td colspan="2" class="diff-lineno">Zeile 90:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>{Wie groß ist die Quellenentropie $H$? <i>Hinweis:</i> Es gibt genau eine Lösung.</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 die Quellenentropie $H$? <ins class="diffchange diffchange-inline">&nbsp; </ins><i>Hinweis:</i> Es gibt genau eine Lösung.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>+ $H = 2.71\ \rm bit/Quellensymbol.$</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>+ $H = 2.71\ \rm bit/Quellensymbol.$</div></td></tr>
</table>
Guenter