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Information Theory/Discrete Sources with Memory - Revision history
2024-03-29T13:49:50Z
Revision history for this page on the wiki
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https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52706&oldid=prev
Guenter at 15:19, 14 February 2023
2023-02-14T15:19:50Z
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<a href="//en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52706&oldid=52663">Show changes</a>
Guenter
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52663&oldid=prev
Hwang at 11:25, 13 February 2023
2023-02-13T11:25:02Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:25, 13 February 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*For the ternary markov source&nbsp; $\rm MQ3$&nbsp; the entropy approximations of&nbsp; $H_1 = 1.500$&nbsp; above&nbsp; $H_2 = 1.375$&nbsp; up to the limit&nbsp; $H = 1.250$&nbsp; continuously decreasing.&nbsp; Because of&nbsp; $M = 3$&nbsp; &rArr; &nbsp; $H_0 = 1.585$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*For the ternary markov source&nbsp; $\rm MQ3$&nbsp; the entropy approximations of&nbsp; $H_1 = 1.500$&nbsp; above&nbsp; $H_2 = 1.375$&nbsp; up to the limit&nbsp; $H = 1.250$&nbsp; continuously decreasing.&nbsp; Because of&nbsp; $M = 3$&nbsp; &rArr; &nbsp; $H_0 = 1.585$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*For the quaternary Markov source&nbsp; $\rm MQ4$&nbsp; one receives&nbsp; $H_0 = H_1 = 2.000$&nbsp; (since four equally probable states) and&nbsp; $H_2 = 1.5$. &nbsp; From the&nbsp; $H_1$-&nbsp; and&nbsp; $H_2$-values all entropy approximations&nbsp; $H_k$&nbsp; and the final value&nbsp; $H = 1.000$&nbsp; can be calculated.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*For the quaternary Markov source&nbsp; $\rm MQ4$&nbsp; one receives&nbsp; $H_0 = H_1 = 2.000$&nbsp; (since four equally probable states) and&nbsp; $H_2 = 1.5$. &nbsp; From the&nbsp; $H_1$-&nbsp; and&nbsp; $H_2$-values all entropy approximations&nbsp; $H_k$&nbsp; and the final value&nbsp; $H = 1.000$&nbsp; can be calculated.</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>*The two models&nbsp; $\rm MQ3$&nbsp; and&nbsp; $\rm MQ4$&nbsp; were created during the attempt to calculate the&nbsp; [[Information_Theory/<del class="diffchange diffchange-inline">Sources_with_Memory</del>#<del class="diffchange diffchange-inline">The_Entropy_of_AMI.E2.80. 93Codes</del>|$\text{AMI code}$]]&nbsp; to be described information&ndash;theoretically by Markov sources.&nbsp; The symbols&nbsp; $\rm M$,&nbsp; $\rm N$&nbsp; and&nbsp; $\rm P$&nbsp; stand for&nbsp; "minus",&nbsp; "zero"&nbsp; and&nbsp; "plus".</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>*The two models&nbsp; $\rm MQ3$&nbsp; and&nbsp; $\rm MQ4$&nbsp; were created during the attempt to calculate the&nbsp; [[Information_Theory/<ins class="diffchange diffchange-inline">Discrete_Sources_with_Memory</ins>#<ins class="diffchange diffchange-inline">The_entropy_of_the_AMI_code</ins>|$\text{AMI code}$]]&nbsp; to be described information&ndash;theoretically by Markov sources.&nbsp; The symbols&nbsp; $\rm M$,&nbsp; $\rm N$&nbsp; and&nbsp; $\rm P$&nbsp; stand for&nbsp; "minus",&nbsp; "zero"&nbsp; and&nbsp; "plus".</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The entropy approximations&nbsp; $H_1$,&nbsp; $H_2$&nbsp; and&nbsp; $H_3$&nbsp; of the AMI code (green markers) were calculated in the&nbsp; [[Aufgaben:Aufgabe 1.4: Entropienäherungen für den AMI-Code|"Exercise 1.4"]].&nbsp; On the calculation of&nbsp; $H_4$,&nbsp; $H_5$, ... had to be omitted for reasons of effort.&nbsp; But the final value of&nbsp; $H_k$&nbsp; for&nbsp; $k \to \infty$ &nbsp; ⇒ &nbsp; $H = 1.000$&nbsp; is known.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The entropy approximations&nbsp; $H_1$,&nbsp; $H_2$&nbsp; and&nbsp; $H_3$&nbsp; of the AMI code (green markers) were calculated in the&nbsp; [[Aufgaben:Aufgabe 1.4: Entropienäherungen für den AMI-Code|"Exercise 1.4"]].&nbsp; On the calculation of&nbsp; $H_4$,&nbsp; $H_5$, ... had to be omitted for reasons of effort.&nbsp; But the final value of&nbsp; $H_k$&nbsp; for&nbsp; $k \to \infty$ &nbsp; ⇒ &nbsp; $H = 1.000$&nbsp; is known.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*You can see that the Markov model&nbsp; $\rm MQ3$&nbsp; for&nbsp; $H_0 = 1.585$,&nbsp; $H_1 = 1.500$&nbsp; and&nbsp; $H_2 = 1.375$&nbsp; yields exactly the same numerical values as the AMI code. &nbsp; On the other hand&nbsp; $H_3$&nbsp; &rArr; &nbsp; $(1.333$&nbsp; instead of&nbsp; $1.292)$&nbsp; and especially the final value&nbsp; $H$&nbsp; &rArr; &nbsp; $(1.250$&nbsp; compared to&nbsp; $1.000)$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*You can see that the Markov model&nbsp; $\rm MQ3$&nbsp; for&nbsp; $H_0 = 1.585$,&nbsp; $H_1 = 1.500$&nbsp; and&nbsp; $H_2 = 1.375$&nbsp; yields exactly the same numerical values as the AMI code. &nbsp; On the other hand&nbsp; $H_3$&nbsp; &rArr; &nbsp; $(1.333$&nbsp; instead of&nbsp; $1.292)$&nbsp; and especially the final value&nbsp; $H$&nbsp; &rArr; &nbsp; $(1.250$&nbsp; compared to&nbsp; $1.000)$.</div></td></tr>
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Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52662&oldid=prev
Hwang at 11:23, 13 February 2023
2023-02-13T11:23:28Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:23, 13 February 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Notes'': </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Notes'': </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>*In the&nbsp; [[Aufgaben:Exercise_1.<del class="diffchange diffchange-inline">6</del>:<del class="diffchange diffchange-inline">_Non-Binary_Markov_Sources</del>|"Exercise 1.5"]]&nbsp; the above equations are applied to the more general case of an asymmetric binary source.</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>*In the&nbsp; [[Aufgaben:Exercise_1.<ins class="diffchange diffchange-inline">5</ins>:<ins class="diffchange diffchange-inline">_Binary_Markov_Source</ins>|"Exercise 1.5"]]&nbsp; the above equations are applied to the more general case of an asymmetric binary source.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*All equations in this section also apply to non-binary Markov sources&nbsp; $(M > 2)$ as shown in the next section.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*All equations in this section also apply to non-binary Markov sources&nbsp; $(M > 2)$ as shown in the next section.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*a quaternary Markov source&nbsp; $\rm MQ4$&nbsp; $(M = 4$,&nbsp; red color$)$. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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 quaternary Markov source&nbsp; $\rm MQ4$&nbsp; $(M = 4$,&nbsp; red color$)$. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>In&nbsp; [[Aufgaben:<del class="diffchange diffchange-inline">1</del>.<del class="diffchange diffchange-inline">6_Nichtbinäre_Markovquellen</del>|"Exercise 1.6"]]&nbsp; the entropy approximations&nbsp; $H_k$&nbsp; and the total entropy&nbsp; $H$&nbsp; are calculated as the limit of&nbsp; $H_k$&nbsp; for&nbsp; $k \to \infty$&nbsp;. &nbsp; The results are shown in the following figure.&nbsp; All entropies specified there have the unit "bit/symbol".</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>In&nbsp; [[Aufgaben:<ins class="diffchange diffchange-inline">Exercise_1</ins>.<ins class="diffchange diffchange-inline">6:_Non-Binary_Markov_Sources</ins>|"Exercise 1.6"]]&nbsp; the entropy approximations&nbsp; $H_k$&nbsp; and the total entropy&nbsp; $H$&nbsp; are calculated as the limit of&nbsp; $H_k$&nbsp; for&nbsp; $k \to \infty$&nbsp;. &nbsp; The results are shown in the following figure.&nbsp; All entropies specified there have the unit "bit/symbol".</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>[[File:EN_Inf_T_1_2_S6b.png|right|frame|Entropies for&nbsp; $\rm MQ3$,&nbsp; $\rm MQ4$&nbsp; and the&nbsp; $\rm AMI$&nbsp; code]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_Inf_T_1_2_S6b.png|right|frame|Entropies for&nbsp; $\rm MQ3$,&nbsp; $\rm MQ4$&nbsp; and the&nbsp; $\rm AMI$&nbsp; code]]</div></td></tr>
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Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52661&oldid=prev
Hwang at 11:19, 13 February 2023
2023-02-13T11:19:26Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:19, 13 February 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l248" >Line 248:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These equations allow first information&ndash;theoretical statements about the Markov processes:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>These equations allow first information&ndash;theoretical statements about the Markov processes:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For&nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}} = p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$&nbsp; the symbols are equally likely &nbsp; ⇒ &nbsp; $p_{\text{A}} = p_{\text{B}}= 0.5$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = H_0 = 1 \hspace{0.05cm} \rm bit/symbol$, independent of the actual values of the (conditional) transition probabilities&nbsp; $p_{\text{A|B}}$&nbsp; and&nbsp; $p_{\text{B|A}}$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* For&nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}} = p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$&nbsp; the symbols are equally likely &nbsp; ⇒ &nbsp; $p_{\text{A}} = p_{\text{B}}= 0.5$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = H_0 = 1 \hspace{0.05cm} \rm bit/symbol$, independent of the actual values of the (conditional) transition probabilities&nbsp; $p_{\text{A|B}}$&nbsp; and&nbsp; $p_{\text{B|A}}$.</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>*But the source entropy&nbsp; $H$&nbsp; as the limit value&nbsp; $($for&nbsp; $k \to \infty)$&nbsp; of the&nbsp; [[Information_Theory/<del class="diffchange diffchange-inline">Discrete Memoryless Sources</del>#Generalization_to_.7F.27 .22.60UNIQ-<del class="diffchange diffchange-inline">MathJax109</del>-QINU.60.22.27.7F<del class="diffchange diffchange-inline">.E2.80.93Tuple_and_boundary.C3.BCtransition</del>|$\text{Entropy approximation of order&nbsp; $k$}$]] &nbsp; &rArr; &nbsp; $H_k$&nbsp; depends very much on the actual values of&nbsp; $p_{\text{A|B}}$ &nbsp;and&nbsp; $p_{\text{B|A}}$&nbsp; and not only on their quotients.&nbsp; This is shown by the following example.</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>*But the source entropy&nbsp; $H$&nbsp; as the limit value&nbsp; $($for&nbsp; $k \to \infty)$&nbsp; of the&nbsp; [[Information_Theory/<ins class="diffchange diffchange-inline">Discrete_Sources_with_Memory</ins>#Generalization_to_.7F.27.22.60UNIQ-<ins class="diffchange diffchange-inline">MathJax111</ins>-QINU.60.22.27.7F<ins class="diffchange diffchange-inline">-tuple_and_boundary_crossing</ins>|$\text{Entropy approximation of order&nbsp; $k$}$]] &nbsp; &rArr; &nbsp; $H_k$&nbsp; depends very much on the actual values of&nbsp; $p_{\text{A|B}}$ &nbsp;and&nbsp; $p_{\text{B|A}}$&nbsp; and not only on their quotients.&nbsp; This is shown by the following example.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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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Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52660&oldid=prev
Hwang at 11:17, 13 February 2023
2023-02-13T11:17:22Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:17, 13 February 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l207" >Line 207:</td>
<td colspan="2" class="diff-lineno">Line 207:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, we do not consider the spectral properties of the AMI code here, but interpret this code information-theoretically:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, we do not consider the spectral properties of the AMI code here, but interpret this code information-theoretically:</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>*Based on the symbol set size&nbsp; $M = 3$&nbsp; the decision content of the (ternary) encoded sequence is equal to&nbsp; $H_0 = \log_2 \ 3 ≈ 1.585 \hspace{0.05cm} \rm bit/symbol$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = 1.5 \hspace{0.05cm} \rm bit/<del class="diffchange diffchange-inline">Symbol</del>$, as shown in the following calculation:</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>*Based on the symbol set size&nbsp; $M = 3$&nbsp; the decision content of the (ternary) encoded sequence is equal to&nbsp; $H_0 = \log_2 \ 3 ≈ 1.585 \hspace{0.05cm} \rm bit/symbol$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = 1.5 \hspace{0.05cm} \rm bit/<ins class="diffchange diffchange-inline">symbol</ins>$, as shown in the following calculation:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$p_{\rm H} = p_{\rm L} = 1/2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$p_{\rm H} = p_{\rm L} = 1/2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52659&oldid=prev
Hwang at 11:13, 13 February 2023
2023-02-13T11:13:30Z
<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="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:13, 13 February 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l193" >Line 193:</td>
<td colspan="2" class="diff-lineno">Line 193:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==The entropy of the AMI code == </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==The entropy of the AMI code == </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></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>In chapter&nbsp; [[Digital_Signal_Transmission/<del class="diffchange diffchange-inline">Symbol</del>-<del class="diffchange diffchange-inline">Wise Coding with Pseudo Ternary Codes</del>#<del class="diffchange diffchange-inline">Properties_of_AMI Code</del>|"Symbol-wise Coding with Pseudo-Ternary Codes"]]&nbsp; of the book&nbsp; "Digital Signal Transmission", among other things, the AMI pseudo-ternary code is discussed.</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>In chapter&nbsp; [[Digital_Signal_Transmission/<ins class="diffchange diffchange-inline">Symbolwise_Coding_with_Pseudo</ins>-<ins class="diffchange diffchange-inline">Ternary_Codes</ins>#<ins class="diffchange diffchange-inline">Properties_of_the_AMI_code</ins>|"Symbol-wise Coding with Pseudo-Ternary Codes"]]&nbsp; of the book&nbsp; "Digital Signal Transmission", among other things, the AMI pseudo-ternary code is discussed.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_Inf_T_1_2_S4.png|right|frame|Signals and symbol sequences for AMI code]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_Inf_T_1_2_S4.png|right|frame|Signals and symbol sequences for AMI code]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l223" >Line 223:</td>
<td colspan="2" class="diff-lineno">Line 223:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>The&nbsp; [[Aufgaben:<del class="diffchange diffchange-inline">1</del>.<del class="diffchange diffchange-inline">4_Entropienäherungen_für_den_AMI-Code</del>|"Exercise 1.4"]]&nbsp; shows the considerable effort required to calculate the entropy approximation&nbsp; $H_3$. &nbsp; Moreover,&nbsp; $H_3$&nbsp; still deviates significantly from the final value&nbsp; $H = 1 \,{\rm bit/symbol} $.&nbsp; A faster result is achieved if the AMI code is described by a Markov chain as explained in the next section.</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>The&nbsp; [[Aufgaben:<ins class="diffchange diffchange-inline">Exercise_1</ins>.<ins class="diffchange diffchange-inline">4:_Entropy_Approximations_for_the_AMI_Code</ins>|"Exercise 1.4"]]&nbsp; shows the considerable effort required to calculate the entropy approximation&nbsp; $H_3$. &nbsp; Moreover,&nbsp; $H_3$&nbsp; still deviates significantly from the final value&nbsp; $H = 1 \,{\rm bit/symbol} $.&nbsp; A faster result is achieved if the AMI code is described by a Markov chain as explained in the next section.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52658&oldid=prev
Hwang at 11:01, 13 February 2023
2023-02-13T11:01:58Z
<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="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:01, 13 February 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l40" >Line 40:</td>
<td colspan="2" class="diff-lineno">Line 40:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We continue to look at the source symbol sequence&nbsp; $〈 q_1, \hspace{0.05cm} q_2,\hspace{0.05cm}\text{ ...} \hspace{0.05cm}, q_{ν-1}, \hspace{0.05cm}q_ν, \hspace{0.05cm}\hspace{0.05cm}q_{ν+1} .\hspace{0.05cm}\text{...} \hspace{0.05cm}〉$&nbsp; and now consider the entropy of two successive source symbols. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We continue to look at the source symbol sequence&nbsp; $〈 q_1, \hspace{0.05cm} q_2,\hspace{0.05cm}\text{ ...} \hspace{0.05cm}, q_{ν-1}, \hspace{0.05cm}q_ν, \hspace{0.05cm}\hspace{0.05cm}q_{ν+1} .\hspace{0.05cm}\text{...} \hspace{0.05cm}〉$&nbsp; and now consider the entropy of two successive source symbols. </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>*All source symbols&nbsp; $q_ν$&nbsp; are taken from an alphabet with the symbol set size&nbsp; $M$, so that for the combination&nbsp; $(q_ν, \hspace{0.05cm}q_{ν+1})$&nbsp; there are exactly&nbsp; $M^2$&nbsp; possible symbol pairs with the following [[Theory_of_Stochastic_Signals/<del class="diffchange diffchange-inline">Set Theory Basics</del>#<del class="diffchange diffchange-inline">Intersection</del>|$\text{combined probabilities}$]]:</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>*All source symbols&nbsp; $q_ν$&nbsp; are taken from an alphabet with the symbol set size&nbsp; $M$, so that for the combination&nbsp; $(q_ν, \hspace{0.05cm}q_{ν+1})$&nbsp; there are exactly&nbsp; $M^2$&nbsp; possible symbol pairs with the following [[Theory_of_Stochastic_Signals/<ins class="diffchange diffchange-inline">Set_Theory_Basics</ins>#<ins class="diffchange diffchange-inline">Intersection_set</ins>|$\text{combined probabilities}$]]:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Pr}(q_{\nu}\cap q_{\nu+1})\le {\rm Pr}(q_{\nu}) \cdot {\rm Pr}( q_{\nu+1})</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$${\rm Pr}(q_{\nu}\cap q_{\nu+1})\le {\rm Pr}(q_{\nu}) \cdot {\rm Pr}( q_{\nu+1})</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l57" >Line 57:</td>
<td colspan="2" class="diff-lineno">Line 57:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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="background-color: #f8f9fa; color: #222; font-size: 88%; 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>*In order to achieve a consistent nomenclature, we now label the entropy defined in chapter&nbsp; [[Information_Theory/<del class="diffchange diffchange-inline">Discrete Memoryless Sources</del>#<del class="diffchange diffchange-inline">Model_and_Prerequisites</del>|"Memoryless Message Sources"]]&nbsp; with&nbsp; $H_1$:</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>*In order to achieve a consistent nomenclature, we now label the entropy defined in chapter&nbsp; [[Information_Theory/<ins class="diffchange diffchange-inline">Discrete_Memoryless_Sources</ins>#<ins class="diffchange diffchange-inline">Model_and_requirements</ins>|"Memoryless Message Sources"]]&nbsp; with&nbsp; $H_1$:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>:$$H_1 = \sum_{q_{\nu}\hspace{0.05cm} \in \hspace{0.05cm}\{ \hspace{0.05cm}q_{\mu}\hspace{0.01cm} \}} {\rm Pr}(q_{\nu}) \cdot {\rm log_2}\hspace{0.1cm}\frac {1}{{\rm Pr}(q_{\nu})} \hspace{0.5cm}({\rm unit\hspace{-0.1cm}: \hspace{0.1cm}bit/symbol})</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$H_1 = \sum_{q_{\nu}\hspace{0.05cm} \in \hspace{0.05cm}\{ \hspace{0.05cm}q_{\mu}\hspace{0.01cm} \}} {\rm Pr}(q_{\nu}) \cdot {\rm log_2}\hspace{0.1cm}\frac {1}{{\rm Pr}(q_{\nu})} \hspace{0.5cm}({\rm unit\hspace{-0.1cm}: \hspace{0.1cm}bit/symbol})</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l70" >Line 70:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>The previous equations each indicate an&nbsp; "ensemble mean value". &nbsp; The probabilities required for the calculation of&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; can, however, also be calculated as time averages from a very long sequence or, more precisely, approximated by the corresponding&nbsp; [[Theory_of_Stochastic_Signals/<del class="diffchange diffchange-inline">From Random Experiment to Random Variable</del>#Bernoulli<del class="diffchange diffchange-inline">'s_Law_of_Large_Numbers</del>|$\text{relative frequencies}$]].</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>The previous equations each indicate an&nbsp; "ensemble mean value". &nbsp; The probabilities required for the calculation of&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; can, however, also be calculated as time averages from a very long sequence or, more precisely, approximated by the corresponding&nbsp; [[Theory_of_Stochastic_Signals/<ins class="diffchange diffchange-inline">From_Random_Experiment_to_Random_Variable</ins>#Bernoulli<ins class="diffchange diffchange-inline">.27s_law_of_large_numbers</ins>|$\text{relative frequencies}$]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>Let us now illustrate the calculation of entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; with three examples.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Let us now illustrate the calculation of entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; with three examples.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l116" >Line 116:</td>
<td colspan="2" class="diff-lineno">Line 116:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The repeated symbols are marked by corresponding lower case letters.&nbsp; But it still applies&nbsp; $M=2$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The repeated symbols are marked by corresponding lower case letters.&nbsp; But it still applies&nbsp; $M=2$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Because of the equally probable binary symbols, this also results in&nbsp; $H_1 = H_0 = 1 \hspace{0.05cm} \rm bit/symbol$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Because of the equally probable binary symbols, this also results in&nbsp; $H_1 = H_0 = 1 \hspace{0.05cm} \rm bit/symbol$.</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>*As shown in&nbsp; [[Aufgaben:<del class="diffchange diffchange-inline">1</del>.<del class="diffchange diffchange-inline">3_Entropienäherungen</del>|"Exercise 1.3"]]&nbsp; for the compound probabilities we obtain&nbsp; $p_{\rm AA}=p_{\rm BB} = 3/8$&nbsp; and&nbsp; $p_{\rm AB}=p_{\rm BA} = 1/8$.&nbsp; Hence</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>*As shown in&nbsp; [[Aufgaben:<ins class="diffchange diffchange-inline">Exercise_1</ins>.<ins class="diffchange diffchange-inline">3:_Entropy_Approximations</ins>|"Exercise 1.3"]]&nbsp; for the compound probabilities we obtain&nbsp; $p_{\rm AA}=p_{\rm BB} = 3/8$&nbsp; and&nbsp; $p_{\rm AB}=p_{\rm BA} = 1/8$.&nbsp; Hence</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\begin{align*}H_2 ={1}/{2} \cdot \big [ 2 \cdot {3}/{8} \cdot {\rm log}_2\hspace{0.1cm} {8}/{3} + </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\begin{align*}H_2 ={1}/{2} \cdot \big [ 2 \cdot {3}/{8} \cdot {\rm log}_2\hspace{0.1cm} {8}/{3} + </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 \cdot {1}/{8} \cdot {\rm log}_2\hspace{0.1cm}8\big ] = {3}/{8} \cdot {\rm log}_2\hspace{0.1cm}8 - {3}/{8} \cdot{\rm log}_2\hspace{0.1cm}3 + {1}/{8} \cdot {\rm log}_2\hspace{0.1cm}8 \approx 0.906 \,{\rm bit/symbol} < H_1 = H_0</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 \cdot {1}/{8} \cdot {\rm log}_2\hspace{0.1cm}8\big ] = {3}/{8} \cdot {\rm log}_2\hspace{0.1cm}8 - {3}/{8} \cdot{\rm log}_2\hspace{0.1cm}3 + {1}/{8} \cdot {\rm log}_2\hspace{0.1cm}8 \approx 0.906 \,{\rm bit/symbol} < H_1 = H_0</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52657&oldid=prev
Hwang at 10:57, 13 February 2023
2023-02-13T10:57:03Z
<p></p>
<a href="//en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52657&oldid=52033">Show changes</a>
Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=52033&oldid=prev
Hwang at 12:45, 23 January 2023
2023-01-23T12:45:26Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 12:45, 23 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l204" >Line 204:</td>
<td colspan="2" class="diff-lineno">Line 204:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Each binary symbol&nbsp; $q_ν =\rm L$&nbsp; is represented by the code symbol&nbsp; $c_ν =\rm N$&nbsp;.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Each binary symbol&nbsp; $q_ν =\rm L$&nbsp; is represented by the code symbol&nbsp; $c_ν =\rm N$&nbsp;.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In contrast,&nbsp; $q_ν =\rm H$&nbsp; alternates with&nbsp; $c_ν =\rm P$&nbsp; and&nbsp; $c_ν =\rm M$&nbsp; coded &nbsp; ⇒ &nbsp; name "Alternate Mark Inversion".</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In contrast,&nbsp; $q_ν =\rm H$&nbsp; alternates with&nbsp; $c_ν =\rm P$&nbsp; and&nbsp; $c_ν =\rm M$&nbsp; coded &nbsp; ⇒ &nbsp; name "Alternate Mark Inversion".</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>*This special encoding adds redundancy with the sole purpose of ensuring that the <del class="diffchange diffchange-inline">code </del>sequence does not contain a DC component. </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>*This special encoding adds redundancy with the sole purpose of ensuring that the <ins class="diffchange diffchange-inline">encoded </ins>sequence does not contain a DC component. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, we do not consider the spectral properties of the AMI code here, but interpret this code information-theoretically:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, we do not consider the spectral properties of the AMI code here, but interpret this code information-theoretically:</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>*Based on the symbol set size&nbsp; $M = 3$&nbsp; the decision content of the (ternary) <del class="diffchange diffchange-inline">code </del>sequence is equal to&nbsp; $H_0 = \log_2 \ 3 ≈ 1.585 \hspace{0.05cm} \rm bit/symbol$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = 1.5 \hspace{0.05cm} \rm bit/Symbol$, as shown in the following calculation:</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>*Based on the symbol set size&nbsp; $M = 3$&nbsp; the decision content of the (ternary) <ins class="diffchange diffchange-inline">encoded </ins>sequence is equal to&nbsp; $H_0 = \log_2 \ 3 ≈ 1.585 \hspace{0.05cm} \rm bit/symbol$.&nbsp; The first entropy approximation returns&nbsp; $H_1 = 1.5 \hspace{0.05cm} \rm bit/Symbol$, as shown in the following calculation:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$p_{\rm H} = p_{\rm L} = 1/2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$p_{\rm H} = p_{\rm L} = 1/2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Information_Theory/Discrete_Sources_with_Memory&diff=50056&oldid=prev
Hwang at 23:31, 12 November 2022
2022-11-12T23:31:20Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 23:31, 12 November 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l175" >Line 175:</td>
<td colspan="2" class="diff-lineno">Line 175:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </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>$\text{Summary of the results of the last <del class="diffchange diffchange-inline">pages</del>:}$&nbsp;</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Summary of the results of the last <ins class="diffchange diffchange-inline">sections</ins>:}$&nbsp;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Generally it applies to the&nbsp; '''entropy of a message source''':</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Generally it applies to the&nbsp; '''entropy of a message source''':</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l268" >Line 268:</td>
<td colspan="2" class="diff-lineno">Line 268:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This example is worth noting:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This example is worth noting:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*If you had not used the Markov properties of the red and green sequences, you would have reached the respective result&nbsp; $H ≈ 0.72 \hspace{0.1cm} \rm bit/symbol$&nbsp; only after lengthy calculations.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*If you had not used the Markov properties of the red and green sequences, you would have reached the respective result&nbsp; $H ≈ 0.72 \hspace{0.1cm} \rm bit/symbol$&nbsp; only after lengthy calculations.</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>*The following <del class="diffchange diffchange-inline">pages </del>show that for a source with Markov properties the final value&nbsp; $H$&nbsp; can be determined from the entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; alone. &nbsp; Likewise, all entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; can also be calculated from&nbsp; $H_k$&nbsp; for&nbsp; $k$&ndash;tuples in a simple manner &nbsp; ⇒ &nbsp; $H_3$,&nbsp; $H_4$,&nbsp; $H_5$, ... &nbsp; $H_{100}$, ...</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>*The following <ins class="diffchange diffchange-inline">sections </ins>show that for a source with Markov properties the final value&nbsp; $H$&nbsp; can be determined from the entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; alone. &nbsp; Likewise, all entropy approximations&nbsp; $H_1$&nbsp; and&nbsp; $H_2$&nbsp; can also be calculated from&nbsp; $H_k$&nbsp; for&nbsp; $k$&ndash;tuples in a simple manner &nbsp; ⇒ &nbsp; $H_3$,&nbsp; $H_4$,&nbsp; $H_5$, ... &nbsp; $H_{100}$, ...</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l274" >Line 274:</td>
<td colspan="2" class="diff-lineno">Line 274:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:Inf_T_1_2_S5_vers2.png|right|frame|Markov processes with&nbsp; $M = 2$&nbsp; states]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:Inf_T_1_2_S5_vers2.png|right|frame|Markov processes with&nbsp; $M = 2$&nbsp; states]]</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>We continue to assume the first-order symmetric binary Markov source.&nbsp; As <del class="diffchange diffchange-inline">on </del>the previous <del class="diffchange diffchange-inline">page</del>, we use the following nomenclature for</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>We continue to assume the first-order symmetric binary Markov source.&nbsp; As <ins class="diffchange diffchange-inline">in </ins>the previous <ins class="diffchange diffchange-inline">section</ins>, we use the following nomenclature for</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*the transition probabilities &nbsp; &rArr; &nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}}$, &nbsp; $p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$,&nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}A}}= 1- p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$, &nbsp; $p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}B}} = 1 - p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}}$, &nbsp; </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*the transition probabilities &nbsp; &rArr; &nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}}$, &nbsp; $p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$,&nbsp; $p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}A}}= 1- p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}A}}$, &nbsp; $p_{\rm {B\hspace{0.01cm}|\hspace{0.01cm}B}} = 1 - p_{\rm {A\hspace{0.01cm}|\hspace{0.01cm}B}}$, &nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*the ergodic probabilities &nbsp; &rArr; &nbsp; $p_{\text{A}}$&nbsp; and&nbsp; $p_{\text{B}}$,</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>*the ergodic probabilities &nbsp; &rArr; &nbsp; $p_{\text{A}}$&nbsp; and&nbsp; $p_{\text{B}}$,</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l331" >Line 331:</td>
<td colspan="2" class="diff-lineno">Line 331:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Notes'': </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Notes'': </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In the&nbsp; [[Aufgaben:Exercise_1.6:_Non-Binary_Markov_Sources|Exercise 1.5]]&nbsp; the above equations are applied to the more general case of an asymmetric binary source.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*In the&nbsp; [[Aufgaben:Exercise_1.6:_Non-Binary_Markov_Sources|Exercise 1.5]]&nbsp; the above equations are applied to the more general case of an asymmetric binary source.</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>*All equations <del class="diffchange diffchange-inline">on </del>this <del class="diffchange diffchange-inline">page </del>also apply to non-binary Markov sources&nbsp; $(M > 2)$ as shown <del class="diffchange diffchange-inline">on </del>the next <del class="diffchange diffchange-inline">page</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>*All equations <ins class="diffchange diffchange-inline">in </ins>this <ins class="diffchange diffchange-inline">section </ins>also apply to non-binary Markov sources&nbsp; $(M > 2)$ as shown <ins class="diffchange diffchange-inline">in </ins>the next <ins class="diffchange diffchange-inline">section</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l366" >Line 366:</td>
<td colspan="2" class="diff-lineno">Line 366:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; 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>In the model&nbsp; $\rm MQ4$&nbsp; the state&nbsp; "Zero"&nbsp; was split into two states&nbsp; $\rm N$&nbsp; and&nbsp; $\rm O$&nbsp; (see upper right figure <del class="diffchange diffchange-inline">on </del>this <del class="diffchange diffchange-inline">page</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>In the model&nbsp; $\rm MQ4$&nbsp; the state&nbsp; "Zero"&nbsp; was split into two states&nbsp; $\rm N$&nbsp; and&nbsp; $\rm O$&nbsp; (see upper right figure <ins class="diffchange diffchange-inline">in </ins>this <ins class="diffchange diffchange-inline">section</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>*Here applies to the state&nbsp; $\rm N$: &nbsp; The current binary symbol&nbsp; $\rm L$&nbsp; is encoded with the amplitude value&nbsp; $0$&nbsp; (zero), as per the AMI rule.&nbsp; The next occurring&nbsp; $\rm H$ symbol, on the other hand, is displayed as&nbsp; $-1$&nbsp; (minus), because the last symol&nbsp; $\rm H$&nbsp; was encoded as&nbsp; $+1$&nbsp; (plus).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Here applies to the state&nbsp; $\rm N$: &nbsp; The current binary symbol&nbsp; $\rm L$&nbsp; is encoded with the amplitude value&nbsp; $0$&nbsp; (zero), as per the AMI rule.&nbsp; The next occurring&nbsp; $\rm H$ symbol, on the other hand, is displayed as&nbsp; $-1$&nbsp; (minus), because the last symol&nbsp; $\rm H$&nbsp; was encoded as&nbsp; $+1$&nbsp; (plus).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Also with the state&nbsp; $\rm O$&nbsp; the current binary symbol&nbsp; $\rm L$&nbsp; is represented with the ternary value&nbsp; $0$.&nbsp; In contrast to the state&nbsp; $\rm N$&nbsp; however, the next occurring&nbsp; $\rm H$ symbol is now encoded as&nbsp; $+1$&nbsp; (plus) since the last&nbsp; $\rm H$ symbol was encoded as&nbsp; $-1$&nbsp; (minus).</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*Also with the state&nbsp; $\rm O$&nbsp; the current binary symbol&nbsp; $\rm L$&nbsp; is represented with the ternary value&nbsp; $0$.&nbsp; In contrast to the state&nbsp; $\rm N$&nbsp; however, the next occurring&nbsp; $\rm H$ symbol is now encoded as&nbsp; $+1$&nbsp; (plus) since the last&nbsp; $\rm H$ symbol was encoded as&nbsp; $-1$&nbsp; (minus).</div></td></tr>
</table>
Hwang