Aufgaben:Exercise 3.12: Path Weighting Function - Revision history

Guenter at 16:08, 22 November 2022

2022-11-22T16:08:09Z

Guenter at 15:56, 22 November 2022

2022-11-22T15:56:22Z

Guenter at 15:55, 22 November 2022

2022-11-22T15:55:12Z

Guenter at 15:18, 21 November 2022

2022-11-21T15:18:32Z

Noah at 20:06, 20 October 2022

2022-10-20T20:06:20Z

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Guenter at 08:48, 13 October 2022

2022-10-13T08:48:12Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.12: Pfadgewichtsfunktion to Aufgaben:Exercise 3.12: Path Weighting Function

2022-10-13T08:42:57Z

Guenter moved page Aufgabe 3.12: Pfadgewichtsfunktion to Exercise 3.12: Path Weighting Function

Javier: Text replacement - "”" to """

2021-05-28T14:28:25Z

Text replacement - "”" to """

Javier: Text replacement - "„" to """

2021-05-28T14:27:00Z

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

2021-03-23T12:48:54Z

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

@@ Line 62: / Line 62: @@
 ===Solution===
 {{ML-Kopf}}
-[[File:EN_KC_A_3_12a.png|right|frame|State transition diagram after modifications]]
+[[File:EN_KC_A_3_12a.png|right|frame|State transition diagram <br>after modifications]]
-'''(1)'''&nbsp; From the adjacent graph, one can see that <u>proposed solutions 1, 3, 4, and 5</u> are correct:
+'''(1)'''&nbsp; From the adjacent graph,&nbsp; one can see that&nbsp; <u>proposed solutions 1, 3, 4, and 5</u>&nbsp; are correct:
-*The state $S_0$ must be split into a start state $S_0$ and a final state ${S_0}'$.
+*The state&nbsp; $S_0$&nbsp; must be split into a start state&nbsp; $S_0$&nbsp; and a final state&nbsp; ${S_0}'$.
-*The reason for this is that for the following calculation of the path weighting enumerator function $T(X, \, U)$ all transitions from $S_0$ to $S_0$ must be excluded.
-*Each code symbol $x &#8712; \{0, \, 1\}$ is represented by $X^x$, where $X$ is a dummy variable with respect to the output sequence: $x = 0 \ \Rightarrow \ X^0 = 1, \ x = 1 \ \Rightarrow \ X^1 = X. $ It further follows $(00) \ \Rightarrow \ 1, \ (01) \ \Rightarrow \ X, \ (10) \ \Rightarrow \ X, \ (11) \ \Rightarrow \ X^2$.
+*The reason for this is that for the following calculation of the path weighting enumerator function&nbsp; $T(X, \, U)$&nbsp; all transitions from&nbsp; $S_0$&nbsp; to&nbsp; $S_0$&nbsp; must be excluded.
-'''(2)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 2 und 3</u>:
 *The reduced diagram is a "ring" according to the listing in the [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Rules_for_manipulating_the_state_transition_diagram|"Theory section"]]. It follows:
 :$$T_{\rm enh}(X, U) =  \frac{A(X, U) \cdot B(X, U)}{1- C(X, U)}
 \hspace{0.05cm}.$$
-*With $A(X, \, U) = UX^2, \ B(X, \, U) = X, \ C(X, \, U) = UX$ one obtains with the given series expansion:
+*With&nbsp; $A(X, \, U) = UX^2, \ B(X, \, U) = X, \ C(X, \, U) = UX$,&nbsp; one obtains with the given series expansion:
 :$$T_{\rm enh}(X, U) =   \frac{U \hspace{0.05cm} X^3}{1- U  \hspace{0.05cm}  X}  = U \hspace{0.05cm} X^3 \cdot \left [ 1 + (U  \hspace{0.05cm}  X) + (U  \hspace{0.05cm}  X)^2 +\text{...} \hspace{0.10cm} \right ]
 \hspace{0.05cm}.$$
@@ Line 81: / Line 85: @@
-'''(3)'''&nbsp; One gets from the extended path weighting enumerator function to $T(X)$ by setting the formal parameter $U = 1$. So <u>both proposed solutions</u> are correct.
+'''(3)'''&nbsp; One gets from the extended path weighting enumerator function to&nbsp; $T(X)$&nbsp; by setting the formal parameter&nbsp; $U = 1$.
-'''(4)'''&nbsp; The free distance $d_{\rm F}$ can be read from the path weighting enumerator function $T(X)$ as the lowest exponent of the dummy variable $X$ &nbsp; &rArr; &nbsp; $d_{\rm F} \ \underline{= 3}$.
+'''(4)'''&nbsp; The free distance&nbsp; $d_{\rm F}$&nbsp; can be read from the path weighting enumerator function&nbsp; $T(X)$&nbsp; as the lowest exponent of the dummy variable&nbsp; $X$ &nbsp; &rArr; &nbsp; $d_{\rm F} \ \underline{= 3}$.
 {{ML-Fuß}}

← Older revision		Revision as of 15:56, 22 November 2022
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−	can be determined, where  $X$  and  $U$  are dummy variables.  The method is explained in detail in the  [[Channel_Coding/~~Distance_Characteristics_and_Error_Probability_Barriers~~#Path_weighting_enumerator_function_from_state_transition_diagram\|"Theory part"]].	+	can be determined, where  $X$  and  $U$  are dummy variables.  The method is explained in detail in the  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Path_weighting_enumerator_function_from_state_transition_diagram\|"Theory part"]].

−	Finally, from  $T(X)$  the  [[Channel_Coding/~~Distance_Characteristics_and_Error_Probability_Barriers~~#Free_distance_vs._minimum_distance\|"free distance"]]  $d_{\rm F}$  has to be determined.	+	Finally, from  $T(X)$  the  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Free_distance_vs._minimum_distance\|"free distance"]]  $d_{\rm F}$  has to be determined.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers}}
-[[File:EN_KC_A_3_12.png|right|frame|Convolutional encoder with&nbsp; $m = 1$&nbsp; and state transition diagram]]
+[[File:EN_KC_A_3_12.png|right|frame|$m = 1$&nbsp; convolutional encoder and state transition diagram]]
-In&nbsp; [[Aufgaben:Exercise_3.6:_State_Transition_Diagram|"Exercise 3.6"]]&nbsp; the state transition diagram for the drawn convolutional encoder was constructed with properties
+In&nbsp; [[Aufgaben:Exercise_3.6:_State_Transition_Diagram|$\text{Exercise 3.6}$]]&nbsp; the state transition diagram for the drawn convolutional encoder with properties
 * Rate&nbsp; $R = 1/2$,
 * memory&nbsp; $m = 1$,
 * transfer function matrix&nbsp; $\mathbf{G}(D) = (1, \, D)$
-which is shown on the right.
+was constructed  which is shown on the right.
 * the extended path weighting enumerator function&nbsp; $T_{\rm enh}(X, \, U)$
-be determined, where&nbsp; $X$&nbsp; and&nbsp; $U$&nbsp; are dummy variables.
+can be determined, where&nbsp; $X$&nbsp; and&nbsp; $U$&nbsp; are dummy variables.&nbsp; The method is explained in detail in the&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Path_weighting_enumerator_function_from_state_transition_diagram|"Theory part"]].
 * Consider the series expansion in the solution
 :$$\frac{1}{1-x} = 1 + x + x^2 +  x^3 + \hspace{0.05cm}\text{...}\hspace{0.1cm}.$$
@@ Line 39: / Line 39: @@
 |type="[]"}
 + The state&nbsp; $S_0$&nbsp; must be split into&nbsp; $S_0$&nbsp; and&nbsp; $S_0\hspace{0.01cm}'$.
-- The state &nbsp; $S_1$&nbsp; must be split into &nbsp; $S_0$&nbsp; and $S_0\hspace{0.01cm}'$.
+- The state &nbsp; $S_1$&nbsp; must be split into &nbsp; $S_0$&nbsp; and&nbsp; $S_0\hspace{0.01cm}'$.
-+ The transition from&nbsp; $S_0$&nbsp; to&nbsp; $S_1$&nbsp; must be labeled $U\hspace{-0.05cm}X^2$.
++ The transition from&nbsp; $S_0$&nbsp; to&nbsp; $S_1$&nbsp; must be labeled&nbsp; $U\hspace{-0.05cm}X^2$.
-+ The transition from&nbsp; $S_1$&nbsp; to&nbsp; $S_1$&nbsp; is to be labeled $U\hspace{-0.05cm}X$.
++ The transition from&nbsp; $S_1$&nbsp; to&nbsp; $S_1$&nbsp; is to be labeled&nbsp; $U\hspace{-0.05cm}X$.
-+ The transition from&nbsp; $S_1$&nbsp; to&nbsp; $S_0\hspace{0.01cm}'$ shall be labeled $X$.
++ The transition from&nbsp; $S_1$&nbsp; to&nbsp; $S_0\hspace{0.01cm}'$ shall be labeled&nbsp; $X$.
 {What equations apply to the extended path weighting enumerator function&nbsp; $T_{\rm enh}(X, \, U)$?
@@ Line 50: / Line 50: @@
 + $T_{\rm enh}(X, \, U) = UX^3 + U^2X^4 + U^3X^5 + \hspace{0.05cm}\text{...}\hspace{0.1cm}$
-{What equations apply to the "simple" path weighting enumerator function&nbsp; $T(X)$?
+{What equations apply to the&nbsp; "simple path weighting enumerator function"&nbsp; $T(X)$?
 |type="[]"}
 + $T(X) = X^3/(1 \, &ndash;X)$,
 + $T(X) = X^3 + X^4 + X^5 +\hspace{0.05cm}\text{...}\hspace{0.1cm}$
-{What is the free distance of the code under consideration?
+{What is the free distance of the considered code?
 |type="{}"}
 $d_{\rm F} \ = \ ${ 3 }

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers}}
-[[File:P_ID2698__KC_A_3_12.png|right|frame|Convolutional encoder with&nbsp; $m = 1$&nbsp; and state transition diagram]]
+[[File:EN_KC_A_3_12.png|right|frame|Convolutional encoder with&nbsp; $m = 1$&nbsp; and state transition diagram]]
 In&nbsp; [[Aufgaben:Exercise_3.6:_State_Transition_Diagram|"Exercise 3.6"]]&nbsp; the state transition diagram for the drawn convolutional encoder was constructed with properties
 * Rate&nbsp; $R = 1/2$,
@@ Line 62: / Line 62: @@
 ===Solution===
 {{ML-Kopf}}
-[[File:P_ID2703__KC_A_3_12a.png|right|frame|State transition diagram after modifications]]
+[[File:EN_KC_A_3_12a.png|right|frame|State transition diagram after modifications]]
 '''(1)'''&nbsp; From the adjacent graph, one can see that <u>proposed solutions 1, 3, 4, and 5</u> are correct:
 *The state $S_0$ must be split into a start state $S_0$ and a final state ${S_0}'$.

← 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 12:48, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^3.5 Distanzeigenschaften^]]	+	[[Category:Channel Coding: Exercises\|^3.5 Distanzeigenschaften^]]