Channel Coding/Distance Characteristics and Error Probability Bounds - Revision history

Maintenance script: Join split $$ formula lines for correct rendering

2026-03-16T13:46:42Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:46, 16 March 2026
Line 275:		Line 275:
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Please note:}$  In real transmission systems,  due to thermal noise the following always applies:		$\text{Please note:}$  In real transmission systems,  due to thermal noise the following always applies:
	:$${\rm Pr(bit\:error)} > 0\hspace{0.05cm},\hspace{0.5cm}{\rm Pr(block\:error)} > {\rm Pr(bit\:error)}		:$${\rm Pr(bit\:error)} > 0\hspace{0.05cm},\hspace{0.5cm}{\rm Pr(block\:error)} > {\rm Pr(bit\:error)}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	For this a simple explanation attempt:   If the decoder decides in each block of length  $k$  exactly one bit wrong,		For this a simple explanation attempt:   If the decoder decides in each block of length  $k$  exactly one bit wrong,

Maintenance script: Add German interlanguage link

2026-03-16T12:27:35Z

Add German interlanguage link

← Older revision		Revision as of 14:27, 16 March 2026
Line 484:		Line 484:

	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Distanzeigenschaften_und_Fehlerwahrscheinlichkeitsschranken]]

Guenter at 17:24, 20 December 2022

2022-12-20T17:24:57Z

← Older revision		Revision as of 19:24, 20 December 2022
Line 450:		Line 450:
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input\|$\text{$\rm AWGN$  channel}$]].		$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input\|$\text{$\rm AWGN$  channel}$]].
	[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes ~~'''Korrektur'''~~]]		[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes]]

	*Red circles denote the  $\rm BER$  for our  "rate  $1/2$  standard encoder"  with memory  $m = 2$.<br>		*Red circles denote the  $\rm BER$  for our  "rate  $1/2$  standard encoder"  with memory  $m = 2$.<br>
Line 458:		Line 458:


	* Codes with large memory  $m$  lead to large gains over uncoded transmission~~  $($dashed curve$)$~~.		* Codes with large memory  $m$  lead to large gains over uncoded transmission.

Guenter at 16:12, 7 December 2022

2022-12-07T16:12:32Z

← Older revision		Revision as of 18:12, 7 December 2022
Line 180:		Line 180:
	First,  the state transition diagram must be redrawn for this purpose.  The graphic shows this on the left in the previous form as diagram  $\rm (A)$,  while the new diagram  $\rm (B)$  is given on the right.  It can be seen:		First,  the state transition diagram must be redrawn for this purpose.  The graphic shows this on the left in the previous form as diagram  $\rm (A)$,  while the new diagram  $\rm (B)$  is given on the right.  It can be seen:

	[[File:~~KC T 3 5 Diagram~~.png\|right\|frame\|State transition diagram in two different variants ~~'''KORREKTUR: Diagram'''~~\|class=fit]]		[[File:KC_T_3_5_Diagram_v3.png\|right\|frame\|State transition diagram in two different variants\|class=fit]]

	#   The state  $S_0$  is split into the start state  $S_0$  and the end state  $S_0\hspace{0.01cm}'$.  Thus,  all paths of the trellis that start in state  $S_0$  and return to it at some point can also be traced in the right graph  $\rm (B)$.  Excluded are direct transitions from  $S_0$  to  $S_0\hspace{0.01cm}'$  and thus also the zero path.<br>		#   The state  $S_0$  is split into the start state  $S_0$  and the end state  $S_0\hspace{0.01cm}'$.  Thus,  all paths of the trellis that start in state  $S_0$  and return to it at some point can also be traced in the right graph  $\rm (B)$.  Excluded are direct transitions from  $S_0$  to  $S_0\hspace{0.01cm}'$  and thus also the zero path.<br>

Guenter at 18:00, 2 December 2022

2022-12-02T18:00:44Z

← Older revision		Revision as of 20:00, 2 December 2022
Line 156:		Line 156:

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 2:}$  Thus, the enhanced path weighting enumerator function of our standard ~~coder '''KORREKTUR:~~ encoder~~'''~~ is:		$\text{Example 2:}$  Thus, the enhanced path weighting enumerator function of our standard encoder is:

	::<math>T_{\rm enh}(X,\ U) = U \cdot X^5 + 2 \cdot U^2 \cdot X^6 + 4 \cdot U^3 \cdot X^7+ \text{ ...} \hspace{0.1cm}		::<math>T_{\rm enh}(X,\ U) = U \cdot X^5 + 2 \cdot U^2 \cdot X^6 + 4 \cdot U^3 \cdot X^7+ \text{ ...} \hspace{0.1cm}

Hwang at 11:33, 1 December 2022

2022-12-01T11:33:19Z

← Older revision		Revision as of 13:33, 1 December 2022
Line 472:		Line 472:
	== Exercises for the chapter ==		== Exercises for the chapter ==
	<br>		<br>
	[[Aufgaben:Exercise_3.12:_Path_Weighting_Function\|"Exercise 3.12: Path Weighting Function"]]		[[Aufgaben:Exercise_3.12:_Path_Weighting_Function\|Exercise 3.12: Path Weighting Function]]

	[[Aufgaben:Exercise_3.12Z:_Ring_and_Feedback\|"Exercise 3.12Z: Ring and Feedback"]]		[[Aufgaben:Exercise_3.12Z:_Ring_and_Feedback\|Exercise 3.12Z: Ring and Feedback]]

	[[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again\|"Exercise 3.13: Path Weighting Function again"]]		[[Aufgaben:Exercise_3.13:_Path_Weighting_Function_again\|Exercise 3.13: Path Weighting Function again]]

	[[Aufgaben:Exercise_3.14:_Error_Probability_Bounds\|"Exercise 3.14: Error Probability Bounds"]]		[[Aufgaben:Exercise_3.14:_Error_Probability_Bounds\|Exercise 3.14: Error Probability Bounds]]

	==References==		==References==

Hwang at 11:30, 1 December 2022

2022-12-01T11:30:39Z

← Older revision		Revision as of 13:30, 1 December 2022
Line 449:		Line 449:

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#~~AWGN_channel_at_Binary_Input~~\|$\text{$\rm AWGN$  channel}$]].		$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input\|$\text{$\rm AWGN$  channel}$]].
	[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes '''Korrektur''']]		[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes '''Korrektur''']]

Hwang at 11:28, 1 December 2022

2022-12-01T11:28:58Z

← Older revision		Revision as of 13:28, 1 December 2022
Line 386:		Line 386:
	Finally,  an upper bound is given for the bit error probability.  According to the graph,  we proceed as in  [Liv10]<ref name='Liv10'>Liva, G.:  Channel Coding.  Lecture manuscript, Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010.</ref>  assume the following conditions:<br>		Finally,  an upper bound is given for the bit error probability.  According to the graph,  we proceed as in  [Liv10]<ref name='Liv10'>Liva, G.:  Channel Coding.  Lecture manuscript, Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010.</ref>  assume the following conditions:<br>

	*~~Sent the~~ zero sequence  $\underline{x} = \underline{0}$   ⇒   Path  $\varphi_0$.<br>		*The zero sequence  $\underline{x} = \underline{0}$ was sent   ⇒   Path  $\varphi_0$.<br>

	*The duration of a path deviation  $($"error burst duration"$)$  is denoted by  $L$ .<br>		*The duration of a path deviation  $($"error burst duration"$)$  is denoted by  $L$ .<br>

Hwang at 11:26, 1 December 2022

2022-12-01T11:26:50Z

← Older revision		Revision as of 13:26, 1 December 2022
Line 349:		Line 349:

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	By  »'''summing over all paths'''«  and comparing with the (simple)  [[Channel_Coding/~~Distance_Characteristics_and_Error_Probability_Barriers~~#Path_weighting_enumerator_function\|$\text{path weighting function}$]]  $T(X)$  we get the result:		By  »'''summing over all paths'''«  and comparing with the (simple)  [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds#Path_weighting_enumerator_function\|$\text{path weighting function}$]]  $T(X)$  we get the result:

	::<math>{\rm Pr(burst\:error)} \le T(X = \beta),\hspace{0.5cm}{\rm with}\hspace{0.5cm}		::<math>{\rm Pr(burst\:error)} \le T(X = \beta),\hspace{0.5cm}{\rm with}\hspace{0.5cm}

Hwang at 11:23, 1 December 2022

2022-12-01T11:23:41Z

← Older revision		Revision as of 13:23, 1 December 2022
Line 325:		Line 325:
	:$$\Rightarrow \hspace{0.3cm}{\rm Pr(burst\:error)}= {\rm Pr} \left ( \cup_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}} \hspace{0.15cm} \big[\varphi_{\hspace{0.02cm}0} \mapsto \varphi_{\hspace{0.02cm}i}\big] \right )\hspace{0.05cm}.$$		:$$\Rightarrow \hspace{0.3cm}{\rm Pr(burst\:error)}= {\rm Pr} \left ( \cup_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}} \hspace{0.15cm} \big[\varphi_{\hspace{0.02cm}0} \mapsto \varphi_{\hspace{0.02cm}i}\big] \right )\hspace{0.05cm}.$$

	*An upper bound for this is provided by the so-called  [[Channel_Coding/Bounds_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability\|"$\text{Union Bound}$"]]:		*An upper bound for this is provided by the so-called  [[Channel_Coding/Bounds_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability\|$\text{Union Bound}$]]:

	::<math>{\rm Pr(Burst\:error)} \le \sum_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}}\hspace{0.15cm}		::<math>{\rm Pr(Burst\:error)} \le \sum_{\varphi_{\hspace{0.02cm}i} \in {\it \Phi}}\hspace{0.15cm}

← Older revision		Revision as of 13:30, 1 December 2022
Line 449:		Line 449:

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#~~AWGN_channel_at_Binary_Input~~\|$\text{$\rm AWGN$  channel}$]].		$\text{Example 5:}$  The graph illustrates the good correction capability   ⇒   small bit error rate  $\rm (BER)$  of convolutional codes at the  [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input\|$\text{$\rm AWGN$  channel}$]].
	[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes '''Korrektur''']]		[[File:EN_KC_T_3_5_S6.png\|right\|frame\|AWGN bit error probability of convolutional codes '''Korrektur''']]