Aufgaben:Exercise 3.13: Path Weighting Function again - Revision history

Guenter at 16:50, 22 November 2022

2022-11-22T16:50:48Z

Guenter at 16:40, 22 November 2022

2022-11-22T16:40:02Z

Guenter at 16:37, 22 November 2022

2022-11-22T16:37:45Z

Guenter at 16:36, 22 November 2022

2022-11-22T16:36:42Z

Guenter at 16:49, 21 November 2022

2022-11-21T16:49:44Z

Guenter at 15:22, 21 November 2022

2022-11-21T15:22:35Z

Hwang at 12:43, 17 November 2022

2022-11-17T12:43:03Z

Noah at 20:32, 20 October 2022

2022-10-20T20:32:43Z

Noah at 20:32, 20 October 2022

2022-10-20T20:32:26Z

Show changes

Guenter at 08:49, 13 October 2022

2022-10-13T08:49:13Z

@@ Line 56: / Line 56: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>All proposed solutions</u> are correct. In the adapted diagram &nbsp;$\rm (B)$&nbsp; all transitions are drawn:
+'''(1)'''&nbsp; <u>All proposed solutions</u>&nbsp; are correct.&nbsp; In the adapted diagram &nbsp;$\rm (B)$&nbsp; all transitions are drawn:
 [[File:EN_KC_A_3_13a.png|right|frame|Reduction of the state transition diagram]]
-*The transition from $S_0$ to $S_1$ is denoted by "$1 \ | \ 11$".
+*The transition from&nbsp; $S_0$&nbsp; to $S_1$&nbsp; is denoted by&nbsp; "$1 \ | \ 11$".
 *The output sequence $\underline{x}_i = (11)$ is expressed by $X^2$, the input bit $u_i = 1$ by $U$.
-*The same result is obtained for $G(X, \, U)$:
+*The output sequence&nbsp; $\underline{x}_i = (11)$&nbsp; is expressed by&nbsp; $X^2$,&nbsp; the input bit&nbsp; $u_i = 1$&nbsp; by&nbsp; $U$.
 :$$A(X, U) = G(X, U)= UX^2
 \hspace{0.05cm}.$$
-*The output sequences $\underline{x}_i = (01)$ as well as $\underline{x}_i = (10)$ are both marked with $X$.
+*The output sequences&nbsp; $\underline{x}_i = (01)$&nbsp; as well as&nbsp; $\underline{x}_i = (10)$&nbsp; are both marked with&nbsp; $X$.
 *Taking into account the input bits, we thus obtain:
 :$$u_i = 1\hspace{-0.1cm}:\hspace{0.15cm} B(X, U) = D(X, U)= UX\hspace{0.05cm},$$
 :$$u_i = 0\hspace{-0.1cm}:\hspace{0.15cm} C(X, U) = E(X, U)= X
 \hspace{0.05cm}.$$
-*The transition "$0 \ | \ 00$" from $S_2$ to $S_1$ is expressed by $F(X, \, U) = 1$.
+*The transition&nbsp; "$0 \ | \ 00$"&nbsp; from&nbsp; $S_2$&nbsp; to&nbsp; $S_1$&nbsp; is expressed by&nbsp; $F(X, \, U) = 1$.
 * One obtains for the red background in the diagram &nbsp;$\rm (B)$:
 :$$T_1(X, U) =  \frac{A(X, U) \cdot B(X, U)}{1- C(X, U)} = \frac{X \cdot X}{1- U \cdot X}
 \hspace{0.05cm}.$$
-* The two <i>parallel transitions</i> corresponding to the blue background in the diagram &nbsp;$\rm (C)$&nbsp; can be combined as follows:
+* The two&nbsp; "parallel transitions"&nbsp; corresponding to the blue background in the diagram &nbsp;$\rm (C)$&nbsp; can be combined as follows:
 :$$T_2(X, U) \hspace{-0.15cm} \ = \ \hspace{-0.15cm}  T_1(X, U) + B(X, U) =\frac{X^2}{1- U  X}+ U  X =
   \frac{X^2 + U- U^2X^2}{1- U  X}
 \hspace{0.05cm}.$$
-* The extended path weighting enumerator function results according to diagram &nbsp;$\rm (D)$&nbsp; as <i>feedback</i>:
+* The extended path weighting enumerator function results according to diagram &nbsp;$\rm (D)$&nbsp; as&nbsp; "feedback<":
 :$$T_{\rm enh}(X, U) =  \frac{A(X, U) \cdot G(X, U)\cdot T_2(X, U)}{1- F(X, U) \cdot T_2(X, U)}
 =  \frac{UX^2 \cdot UX^2\cdot \frac{X^2 + UX- U^2X^2}{1- U  X}}{1- 1 \cdot \frac{X^2 + UX- U^2X^2}{1- U  X}}\hspace{0.05cm}.$$
-Even after several attempts, the author has not succeeded in further simplifying this expression in a target-oriented way. He tends to the <u>proposed solution 3</u> with the addition "without guarantee".
+Even after several attempts,&nbsp; the author has not succeeded in further simplifying this expression in a target-oriented way.&nbsp; He tends to the&nbsp; <u>proposed solution 3</u>&nbsp; with the addition&nbsp; "without guarantee".
 *However, this result would imply that the extended path weighting enumerator function of the equivalent systematic code is different from that of the nonsystematic code.
-*We will still clarify this issue with an expert.
+*However,&nbsp; this result would imply that the extended path weighting enumerator function of the equivalent systematic code is different from that of the nonsystematic code.&nbsp; We will still clarify this issue with an expert.
-'''(3)'''&nbsp; Correct are the <u>proposed solutions 1 and 2</u>:
+'''(3)'''&nbsp; Correct are the&nbsp; <u>proposed solutions 1 and 2</u>:
-*If one sets the formal parameter $U = 1$ in the extended function $T_{\rm enh}(X, \, U)$, one obtains the solution proposition 1:
+*If one sets the formal parameter&nbsp; $U = 1$&nbsp; in the extended function&nbsp; $T_{\rm enh}(X, \, U)$,&nbsp; one obtains the solution proposition 1:
 :$$T(X) =   \frac{X^4 \cdot  \frac{X^2 + X- X^2}{1- X}}{1- \frac{X^2 + X- X^2}{1- X}}=
 \frac{X^5 }{1- X - X} =
 \frac{X^5 }{1- 2X} \hspace{0.05cm}.$$
-*Using the series expansion $1/(1 \, &ndash;x) = 1 + x + x^2 + \ \text{...}\ $, we arrive at the proposed solution 2.
+*Using the series expansion&nbsp; $1/(1 \, &ndash;x) = 1 + x + x^2 + \ \text{...}\ $,&nbsp; we arrive at the proposed solution 2.
 *That is, the simple path weighting enumerator function $T(X)$ matches for both codes.
 {{ML-Fuß}}

@@ Line 29: / Line 29: @@
-===Solution===
+===Questions===
 <quiz display=simple>
 {What expressions do the following abbreviations stand for?

← Older revision		Revision as of 16:37, 22 November 2022
Line 25:		Line 25:
	* This exercise belongs to the chapter  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds\| "Distance Characteristics and Error Probability Bounds"]].		* This exercise belongs to the chapter  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds\| "Distance Characteristics and Error Probability Bounds"]].

−	* For the solution of subtasks  '''(2)'''  and  '''(3)'''  we refer here again to the section  [[Channel_Coding/~~Distance_Characteristics_and_Error_Probability_Barriers~~#Rules_for_manipulating_the_state_transition_diagram\|"Rules for manipulating the state transition diagram"]] in the theory part.	+	* For the solution of subtasks  '''(2)'''  and  '''(3)'''  we refer here again to the section  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Rules_for_manipulating_the_state_transition_diagram\|"Rules for manipulating the state transition diagram"]] in the theory part.

@@ Line 2: / Line 2: @@
 [[File:EN_KC_A_3_13.png|right|frame|For the reduction of the state transition diagram]]
-In 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; section, for the example of our rate 1/2 standard code with memory&nbsp; $m = 2$&nbsp; and the transfer function matrix
+In 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; section,&nbsp; for the example of our rate $1/2$ standard encoder
-:$${\boldsymbol{\rm G}}(D) = \big ( 1 + D + D^2\hspace{0.05cm},\hspace{0.1cm} 1  + D^2 \hspace{0.05cm}\big )$$
+*with memory&nbsp; $m = 2$&nbsp; and
-the calculation of path weighting enumerator functions was described in great detail. The results given were:
+*the transfer function matrix
 :$$T_{\rm enh}(X, U) \hspace{-0.2cm} \ = \ \hspace{-0.2cm}   \frac{U\hspace{-0.05cm} X^5}{1- 2U\hspace{-0.05cm}X}  =U\hspace{-0.05cm}X^5 \cdot \big [ 1 + (2U\hspace{-0.08cm}X) + (2U\hspace{-0.08cm}X)^2 + \text{...}  \hspace{0.05cm} \big ] \hspace{0.01cm},$$
 :$$T(X) \hspace{-0.2cm} \ = \ \hspace{-0.2cm}   \frac{X^5}{1- 2X}  =  X^5 \cdot \big [ 1 + (2X) + (2X)^2 + \text{...} \hspace{0.05cm} \big ] \hspace{0.05cm}.$$
-Now the same calculations shall be done for the&nbsp; [[Channel_Coding/Algebraic_and_Polynomial_Description#Systematic_convolutional_codes|"equivalent systematic code"]]&nbsp; with the transfer function matrix
+Now the same calculations shall be done for the&nbsp; [[Channel_Coding/Algebraic_and_Polynomial_Description#Systematic_convolutional_codes|"equivalent systematic encoder"]]&nbsp; with the transfer function matrix
-:$${\boldsymbol{\rm G}}(D) = \big ( 1 \hspace{0.05cm},\hspace{0.1cm} (1 + D^2)/(1 + D + D^2) \hspace{0.05cm}\big )$$
+:$${\boldsymbol{\rm G}}(D) = \big ( 1 \hspace{0.05cm},\hspace{0.1cm} (1 + D^2)/(1 + D + D^2) \hspace{0.05cm}\big ).$$
@@ Line 26: / Line 23: @@
 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"]].
-* For the solution of the subtasks '''(2)'' and '''(3)''' we refer here again to the section&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers#Rules_for_manipulating_the_state_transition_diagram|"Rules for manipulating the state transition diagram"]] in the theory section.
@@ Line 49: / Line 47: @@
 + None of the proposals are correct.
-{What expressions apply to the "simple" path weighting enumerator function?
+{What expressions apply to the&nbsp; "simple path weighting enumerator function"?
 |type="[]"}
 + $T(X) = X^5 \ / \ (1 \, &ndash;2X)$.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Distance_Characteristics_and_Error_Probability_Barriers}}
-[[File:P_ID2711__KC_A_3_13.png|right|frame|For the reduction of the state transition diagram]]
+[[File:EN_KC_A_3_13.png|right|frame|For the reduction of the state transition diagram]]
 In 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; section, for the example of our rate 1/2 standard code with memory&nbsp; $m = 2$&nbsp; and the transfer function matrix
 :$${\boldsymbol{\rm G}}(D) = \big ( 1 + D + D^2\hspace{0.05cm},\hspace{0.1cm} 1  + D^2 \hspace{0.05cm}\big )$$

← Older revision		Revision as of 08:49, 13 October 2022
Line 109:		Line 109:


−	[[Category:Channel Coding: Exercises\|^3.5 ~~Distanzeigenschaften~~^]]	+	[[Category:Channel Coding: Exercises\|^3.5 Distance Properties^]]