Aufgaben:Exercise 3.12Z: Ring and Feedback - Revision history

Guenter at 16:27, 22 November 2022

2022-11-22T16:27:34Z

Guenter at 16:13, 22 November 2022

2022-11-22T16:13:55Z

Guenter at 16:09, 22 November 2022

2022-11-22T16:09:24Z

Guenter at 15:20, 21 November 2022

2022-11-21T15:20:10Z

Noah at 20:23, 20 October 2022

2022-10-20T20:23:28Z

Guenter at 08:48, 13 October 2022

2022-10-13T08:48:43Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.12Z: Ring und Rückkopplung to Aufgaben:Exercise 3.12Z: Ring and Feedback

2022-10-13T08:44:13Z

Guenter moved page Aufgabe 3.12Z: Ring und Rückkopplung to Exercise 3.12Z: Ring and Feedback

Javier: Text replacement - "„" to """

2021-05-28T14:37:08Z

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Javier: Text replacement - "”" to """

2021-05-28T14:29:29Z

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Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:53:34Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

@@ Line 21: / Line 21: @@
-Hints:
+<u>Hints:</u>
 * This exercise belongs to the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds| "Distance Characteristics and Error Probability Bounds"]].
@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>solutions 1 and 2</u>:
+'''(1)'''&nbsp; Correct are the&nbsp; <u>solutions 1 and 2</u>:
-*In general terms, one first goes from $S_1$ to $S_2$, remains $j$&ndash;times in the state $S_2 \ (j = 0, \ 1, \, 2, \ \text{ ...})$, and finally continues from $S_2$ to $S_3$.
+*In general terms,&nbsp; one first goes from&nbsp; $S_1$&nbsp; to&nbsp; $S_2$,&nbsp; remains&nbsp; $j$&ndash;times in the state&nbsp; $S_2 \ (j = 0, \ 1, \, 2, \ \text{ ...})$,&nbsp; and finally continues from&nbsp; $S_2$&nbsp; to&nbsp; $S_3$.
-'''(2)'''&nbsp; Correct is the <u>solution suggestion 2</u>:
+'''(2)'''&nbsp; Correct is the&nbsp; <u>solution suggestion 2</u>:
-*In accordance with the explanations for the subtask '''(1)''', one obtains for the substitution of the ring
+*In accordance with the explanations for subtask&nbsp; '''(1)''',&nbsp; one obtains for the substitution of the ring:
 :$$E \hspace{-0.15cm} \ = \ \hspace{-0.15cm} A \cdot B + A  \cdot C \cdot B + A  \cdot C^2 \cdot B + A  \cdot C^3 \cdot B + \text{ ...} \hspace{0.1cm}=A \cdot B \cdot [1 + C + C^2+ C^3 +\text{ ...}\hspace{0.1cm}]
 \hspace{0.05cm}.$$
-*The parenthesis expression gives $1/(1 \, &ndash;C)$.
+*The parenthesis expression gives&nbsp; $1/(1 \, &ndash;C)$.
 :$$E(X, U) =  \frac{A(X, U) \cdot B(X, U)}{1- C(X, U)}
 \hspace{0.05cm}.$$
@@ Line 76: / Line 76: @@
 '''(3)'''&nbsp; Correct are the <u>solutions 1, 3 and 4</u>:
-* one goes first from $S_1$ to $S_2 \ \Rightarrow \ A(X, \, U)$,
+* One goes first from&nbsp; $S_1$&nbsp; to&nbsp; $S_2 \ \Rightarrow \ A(X, \, U)$,
-'''(4)'''&nbsp; Thus, the correct solution is <u>suggested solution 1</u>:
+* finally from&nbsp; $S_3$&nbsp; to&nbsp; $S_4 \ \Rightarrow \ B(X, \, U)$,
 *According to the sample solution to subtask '''(3)''' applies:
 :$$F(X, U) = A(X, U) \cdot C(X, U) \cdot E(X, U) \cdot B(X, U)\hspace{0.05cm}$$
-*Here $E(X, \, U)$ describes the path "$j$&ndash;times" back to $S_2$ and again to $S_3 \ (j =0, \ 1, \ 2, \ \text{ ...})$:
+*Here&nbsp; $E(X, \, U)$&nbsp; describes the path&nbsp; "$j$&ndash;times"&nbsp; back to&nbsp; $S_2$&nbsp; and again to&nbsp; $S_3 \ (j =0, \ 1, \ 2, \ \text{ ...})$:
 :$$E(X, U) =  1 + D \cdot C + (1 + D)^2 + (1 + D)^3 + \text{ ...} \hspace{0.1cm}= \frac{1}{1-C \hspace{0.05cm} D}
 \hspace{0.3cm}

@@ Line 2: / Line 2: @@
 [[File:EN_KC_Z_3_12.png|right|frame|Ring and feedback in the state transition diagram]]
-In order to determine the path weighting enumerator function&nbsp; $T(X)$&nbsp; of a convolutional code from the state transition diagram, it is necessary to reduce the diagram until it can be represented by a single connection from the initial state to the final state.
+In order to determine the path weighting enumerator function &nbsp; $T(X)$ &nbsp; of a convolutional code from the state transition diagram,&nbsp; it is necessary to reduce the diagram until it can be represented by a single connection from the initial state to the final state.
 In the course of this diagram reduction can occur:
 * serial and parallel transitions,
 * a ring according to the sketch above,
 * a feedback according to the sketch below.
-For these two graphs, find the correspondences&nbsp; $E(X, \, U)$&nbsp; and&nbsp; $F(X, \, U)$&nbsp; depending on the given functions&nbsp; $A(X, \, U), \ B(X, \ U), \ C(X, \, U), \ D(X, \, U)$&nbsp;.
+For these two graphs,&nbsp; find the correspondences &nbsp; $E(X, \, U)$ &nbsp; and &nbsp; $F(X, \, U)$ &nbsp; depending on the given functions &nbsp; $A(X, \, U), \ B(X, \ U), \ C(X, \, U), \ D(X, \, U)$&nbsp;.
@@ Line 22: / Line 24: @@
 * This exercise belongs to the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds| "Distance Characteristics and Error Probability Bounds"]].
-* This exercise is intended to prove some of the statements on the&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Rules_for_manipulating_the_state_transition_diagram|"Rules for manipulating the state transition diagram"]]&nbsp; page.
+* This exercise is intended to prove some of the statements on the&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Rules_for_manipulating_the_state_transition_diagram|"Rules for manipulating the state transition diagram"]]&nbsp; section.
-* Applied these rules in the&nbsp; [[Aufgaben:Exercise_3.12:_Path_Weighting_Function|"Exercise 3.12"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again|"Exercise 3.13"]].

@@ Line 20: / Line 20: @@
 Hints:
-* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers| "Distance characteristics and error probability barriers"]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds| "Distance Characteristics and Error Probability Bounds"]].
 * This exercise is intended to prove some of the statements on the&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Rules_for_manipulating_the_state_transition_diagram|"Rules for manipulating the state transition diagram"]]&nbsp; page.
 * Applied these rules in the&nbsp; [[Aufgaben:Exercise_3.12:_Path_Weighting_Function|"Exercise 3.12"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again|"Exercise 3.13"]].

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers}}
-[[File:P_ID2710__KC_Z_3_12.png|right|frame|Ring and feedback in the state transition diagram]]
+[[File:EN_KC_Z_3_12.png|right|frame|Ring and feedback in the state transition diagram]]
 In order to determine the path weighting enumerator function&nbsp; $T(X)$&nbsp; of a convolutional code from the state transition diagram, it is necessary to reduce the diagram until it can be represented by a single connection from the initial state to the final state.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Distanzeigenschaften und Fehlerwahrscheinlichkeitsschranken}}
+{{quiz-Header|Buchseite=Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers}}
-[[File:P_ID2710__KC_Z_3_12.png|right|frame|Ring und Rückkopplung im Zustandsübergangsdiagramm]]
+[[File:P_ID2710__KC_Z_3_12.png|right|frame|Ring and feedback in the state transition diagram]]
-Um die Pfadgewichtsfunktion&nbsp; $T(X)$&nbsp; eines Faltungscodes aus dem Zustandsübergangsdiagramm bestimmen zu können, ist es erforderlich, das Diagramm so zu reduzieren, bis es durch eine einzige Verbindung vom Startzustand zum Endzustand dargestellt werden kann.
+In order to determine the path weighting enumerator function&nbsp; $T(X)$&nbsp; of a convolutional code from the state transition diagram, it is necessary to reduce the diagram until it can be represented by a single connection from the initial state to the final state.
-Im Zuge dieser Diagrammreduktion können auftreten:
+In the course of this diagram reduction can occur:
-* serielle und parallele Übergänge,
+* serial and parallel transitions,
-* ein Ring entsprechend der obigen Skizze,
+* a ring according to the sketch above,
-* eine Rückkopplung entsprechend der unteren Skizze.
+* a feedback according to the sketch below.
-Für diese beiden Graphen sind die Entsprechungen&nbsp; $E(X, \, U)$&nbsp; und&nbsp; $F(X, \, U)$&nbsp; in Abhängigkeit der angegebenen Funktionen&nbsp; $A(X, \, U), \ B(X, \ U), \ C(X, \, U), \ D(X, \, U)$&nbsp; zu ermitteln.
+For these two graphs, find the correspondences&nbsp; $E(X, \, U)$&nbsp; and&nbsp; $F(X, \, U)$&nbsp; depending on the given functions&nbsp; $A(X, \, U), \ B(X, \ U), \ C(X, \, U), \ D(X, \, U)$&nbsp;.
@@ Line 19: / Line 19: @@
-''Hinweise:''
+Hints:
-* Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Distanzeigenschaften_und_Fehlerwahrscheinlichkeitsschranken| Distanzeigenschaften und Fehlerwahrscheinlichkeitsschranken]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers| "Distance characteristics and error probability barriers"]].
-* Mit dieser Aufgabe sollen einige der Angaben auf der Seite&nbsp; [[Channel_Coding/Distanzeigenschaften_und_Fehlerwahrscheinlichkeitsschranken#Regeln_zur_Manipulation_des_Zustands.C3.BCbergangsdiagramms|Regeln zur Manipulation des Zustandsübergangsdiagramms]]&nbsp; bewiesen werden.
+* This exercise is intended to prove some of the statements on the&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Rules_for_manipulating_the_state_transition_diagram|"Rules for manipulating the state transition diagram"]]&nbsp; page.
-* Angewendet werden diese Regeln in der&nbsp; [[Aufgaben:Aufgabe_3.12:_Pfadgewichtsfunktion|Aufgabe 3.12]]&nbsp; und der&nbsp; [[Aufgaben:Aufgabe_3.13:_Nochmals_zu_den_Pfadgewichtsfunktionen|Aufgabe 3.13]].
+* Applied these rules in the&nbsp; [[Aufgaben:Exercise_3.12:_Path_Weighting_Function|"Exercise 3.12"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again|"Exercise 3.13"]].
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welche der aufgeführten Übergänge sind beim Ring möglich?
+{Which of the listed transitions are possible with the ring?
 |type="[]"}
 + $S_1 &#8594; S_2 &#8594; S_3$,
@@ Line 35: / Line 35: @@
 - $S_1 &#8594; S_2 &#8594; S_1 &#8594; S_2 &#8594; S_3$.
-{Wie lautet die Ersetzung&nbsp; $E(X, \, U)$&nbsp; eines Ringes?
+{What is the substitution&nbsp; $E(X, \, U)$&nbsp; of a ring?
 |type="()"}
 - $E(X, \, U) = [A(X, \, U) + B(X, \, U)] \ / \ [1 \, -C(X, \, U)]$,
@@ Line 41: / Line 41: @@
 - $E(X, \, U) = A(X, \, U) \cdot C(X, \, U) \ / \ [1 \, -B(X, \, U)]$.
-{Welche der aufgeführten Übergänge sind bei Rückkopplung möglich?
+{Which of the listed transitions are possible with feedback?
 |type="[]"}
 + $S_1 &#8594; S_2 &#8594; S_3 &#8594; S_4$,
@@ Line 48: / Line 48: @@
 + $S_1 &#8594; S_2 &#8594; S_3 &#8594; S_2 &#8594; S_3 &#8594; S_2 &#8594; S_3 &#8594; S_4$.
-{Wie lautet die Ersetzung&nbsp; $F(X, \, U)$&nbsp; einer Rückkopplung?
+{What is the substitution&nbsp; $F(X, \, U)$&nbsp; of a feedback?
 |type="()"}
 + $F(X, \, U) = A(X, \, U) \cdot B(X, \, U) \cdot C(X, \, U) \ / \ [1 \, -C(X, \, U) \cdot D(X, \, U)]$
@@ Line 54: / Line 54: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1 und 2</u>:
+'''(1)'''&nbsp; Correct are <u>solutions 1 and 2</u>:
-*Allgemein ausgedrückt: Man geht zunächst von $S_1$ nach $S_2$, verbleibt $j$&ndash;mal im Zustand $S_2 \ (j = 0, \ 1, \, 2, \ \text{ ...})$ und geht abschließend von $S_2$ nach $S_3$ weiter.
+*In general terms, one first goes from $S_1$ to $S_2$, remains $j$&ndash;times in the state $S_2 \ (j = 0, \ 1, \, 2, \ \text{ ...})$, and finally continues from $S_2$ to $S_3$.
-'''(2)'''&nbsp; Richtig ist der <u>Lösungsvorschlag 2</u>:
+'''(2)'''&nbsp; Correct is the <u>solution suggestion 2</u>:
-*Entsprechend den Ausführungen zur Teilaufgabe '''(1)''' erhält man für die Ersetzung des Ringes
+*In accordance with the explanations for the subtask '''(1)''', one obtains for the substitution of the ring
 :$$E \hspace{-0.15cm} \ = \ \hspace{-0.15cm} A \cdot B + A  \cdot C \cdot B + A  \cdot C^2 \cdot B + A  \cdot C^3 \cdot B + \text{ ...} \hspace{0.1cm}=A \cdot B \cdot [1 + C + C^2+ C^3 +\text{ ...}\hspace{0.1cm}]
 \hspace{0.05cm}.$$
-*Der Klammerausdruck ergibt $1/(1 \, &ndash;C)$.
+*The parenthesis expression gives $1/(1 \, &ndash;C)$.
 :$$E(X, U) =  \frac{A(X, U) \cdot B(X, U)}{1- C(X, U)}
 \hspace{0.05cm}.$$
-'''(3)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1, 3 und 4</u>:
+'''(3)'''&nbsp; Correct are the <u>solutions 1, 3 and 4</u>:
-* Man geht zunächst von $S_1$ nach $S_2 \ \Rightarrow \ A(X, \, U)$,
+* one goes first from $S_1$ to $S_2 \ \Rightarrow \ A(X, \, U)$,
-* dann von $S_2$ nach $S_3 \ \Rightarrow \ C(X, \, U)$,
+* then from $S_2$ to $S_3 \ \Rightarrow \ C(X, \, U)$,
-* anschließend $j$&ndash;mal zurück nach $S_2$ und wieder nach $S_3 \ (j = 0, \ 1, \ 2, \ \text{ ...} \ ) \ \Rightarrow \ E(X, \, U)$,
+* then $j$&ndash;times back to $S_2$ and again to $S_3 \ (j = 0, \ 1, \ 2, \ \text{ ...} \ ) \ \Rightarrow \ E(X, \, U)$,
-* abschließend von $S_3$ nach $S_4 \ \Rightarrow \ B(X, \, U)$,
+* finally from $S_3$ to $S_4 \ \Rightarrow \ B(X, \, U)$,
-'''(4)'''&nbsp; Richtig ist also der <u>Lösungsvorschlag 1</u>:
+'''(4)'''&nbsp; Thus, the correct solution is <u>suggested solution 1</u>:
-*Entsprechend der Musterlösung zur Teilaufgabe '''(3)''' gilt:
+*According to the sample solution to subtask '''(3)''' applies:
 :$$F(X, U) = A(X, U) \cdot C(X, U) \cdot E(X, U) \cdot B(X, U)\hspace{0.05cm}$$
-*Hierbei beschreibt $E(X, \, U)$ den Weg "$j$&ndash;mal" zurück nach $S_2$ und wieder nach $S_3 \ (j =0, \ 1, \ 2, \ \text{ ...})$:
+*Here $E(X, \, U)$ describes the path "$j$&ndash;times" back to $S_2$ and again to $S_3 \ (j =0, \ 1, \ 2, \ \text{ ...})$:
 :$$E(X, U) =  1 + D \cdot C + (1 + D)^2 + (1 + D)^3 + \text{ ...} \hspace{0.1cm}= \frac{1}{1-C \hspace{0.05cm} D}
 \hspace{0.3cm}

← Older revision		Revision as of 08:48, 13 October 2022
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−	[[Category:Channel Coding: Exercises\|^3.5 ~~Distanzeigenschaften~~^]]	+	[[Category:Channel Coding: Exercises\|^3.5 Distance Properties^]]

← Older revision		Revision as of 12:53, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^3.5 Distanzeigenschaften^]]	+	[[Category:Channel Coding: Exercises\|^3.5 Distanzeigenschaften^]]