Aufgaben:Exercise 3.12: Strictly Symmetrical Channels - Revision history

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	Here  $p_{\rm A}$  denotes the probability that the sub-channel  $\rm A$  is selected and  $C_{\rm A}$  indicates its capacity.  The same applies to sub-channel  $\rm B$.		Here  $p_{\rm A}$  denotes the probability that the sub-channel  $\rm A$  is selected and  $C_{\rm A}$  indicates its capacity.  The same applies to sub-channel  $\rm B$.

	Subsequently, the channel capacity of the  [[Channel_Coding/~~Kanalmodelle_und_Entscheiderstrukturen~~#Binary_Symmetric_Error_~~.26_Erasure_Channel_.E2.80.93_BSEC~~\|Binary Symmetric Error & Erasure Channel]]  $\rm (BSEC)$  is also to be determined according to the sketch below  (grey background)  by deriving the relationship between		Subsequently, the channel capacity of the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Error_&_Erasure_Channel_–_BSEC\|Binary Symmetric Error & Erasure Channel]]  $\rm (BSEC)$  is also to be determined according to the sketch below  (grey background)  by deriving the relationship between
	*the parameters  $p_{\rm A}$,  $p_{\rm B}$  and the crossover probability  $q$  of the sub-channel model shown above, and		*the parameters  $p_{\rm A}$,  $p_{\rm B}$  and the crossover probability  $q$  of the sub-channel model shown above, and
	* the parameters  $λ$  and  $\varepsilon$  of the BSEC model.		* the parameters  $λ$  and  $\varepsilon$  of the BSEC model.
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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to Digital Signal Transmission]].		*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to Digital Signal Transmission]].
	*Reference is made in particular to the page  [[Information_Theory/~~Anwendung_auf_die_Digitalsignalübertragung~~#Properties_of_symmetrical_channels\|Properties of symmetrical channels]].		*Reference is made in particular to the page  [[Information_Theory/Application_to_Digital_Signal_Transmission#Properties_of_symmetrical_channels\|Properties of symmetrical channels]].

	*According to  [[Aufgaben:Aufgabe_3.10Z:_BSC–Kanalkapazität\|Exercise 3.10Z]] , the following applies to the channel capacity of the BSC model with the crossover probability  $\varepsilon$:		*According to  [[Aufgaben:Aufgabe_3.10Z:_BSC–Kanalkapazität\|Exercise 3.10Z]] , the following applies to the channel capacity of the BSC model with the crossover probability  $\varepsilon$:
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	[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]
	[[de:Aufgaben:3.~~11_Streng_symmetrische_Kanäle~~]]		[[de:Aufgaben:Aufgabe 3.12: Streng symmetrische Kanäle]]

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	[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]
			[[de:Aufgaben:3.11_Streng_symmetrische_Kanäle]]

Guenter at 11:57, 24 September 2021

2021-09-24T11:57:04Z

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	[[Category:Information Theory: Exercises\|^3.3 ~~Anwendung auf DSÜ-Kanäle~~^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]

Noah at 09:10, 24 September 2021

2021-09-24T09:10:56Z

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	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Anwendung auf die Digitalsignalübertragung~~		{{quiz-Header\|Buchseite=Information_Theory/Application_to_Digital_Signal_Transmission
	}}		}}

Guenter at 14:13, 22 September 2021

2021-09-22T14:13:43Z

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	'''(8)''' The  "Binary Erasure Channel"  (BEC) is a special case of the BSEC with  $ε = 0$:		'''(8)''' The  "Binary Erasure Channel"  (BEC) is a special case of the BSEC with  $ε = 0$:
	:$$C_{\rm BEC} = (1- \lambda) \cdot \big [ 1 - H_{\rm bin}(0) \big ] = 1- \lambda\hspace{0.05cm}.$$		:$$C_{\rm BEC} = (1- \lambda) \cdot \big [ 1 - H_{\rm bin}(0) \big ] = 1- \lambda\hspace{0.05cm}.$$
	*With  $λ = 0.2$ , this results in  $C_{\rm BEC} = 0.8 \ \rm bit.$		*With  $λ = 0.2$ , this results in  $C_{\rm BEC}\hspace{0.15cm} \underline { = 0.8 \ \rm bit}.$

	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:07, 22 September 2021

2021-09-22T13:07:34Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Subchannel  $\rm A$  is a   (''Binary Symmetric Channel'')  with crossover probability  $q$   ⇒   <u>Proposed solution 1</u>.		'''(1)'''  Subchannel  $\rm A$  is a   ("Binary Symmetric Channel")  with crossover probability  $q$   ⇒   <u>Proposed solution 1</u>.


			'''(2)'''  Subchannel  $\rm B$  is an  "erasure channel".  Both the sink entropy and irrelevance of this subchannel are zero   ⇒   <u>Proposed solution 3</u>.
	'''(2)'''  Subchannel  $\rm B$  is an  ''erasure channel''.  Both the sink entropy and irrelevance of this subchannel are zero   ⇒   <u>Proposed solution 3</u>.



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	:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$		:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$
	Thus, the <u>proposed solution 2</u> is correct here.		Thus, the <u>proposed solution 2</u> is correct here.



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	'''(5)'''  In the BSEC (''Binary Symmetric Error & Erasure Channel'') model, for example:		'''(5)'''  In the BSEC  ("Binary Symmetric Error & Erasure Channel")  model, for example:
	:$$ {\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =\varepsilon \hspace{0.05cm}.$$		:$$ {\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =\varepsilon \hspace{0.05cm}.$$
	*In contrast, for our auxiliary model according to the graph below, the result is:		*In contrast, for our auxiliary model according to the graph below, the result is:
	:$${\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =(1- \lambda) \cdot q \hspace{0.05cm}.$$		:$${\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =(1- \lambda) \cdot q \hspace{0.05cm}.$$
	*This gives $q = ε/(1 – λ)$   ⇒   <u>Proposed solution 4</u>.		*This gives $q = ε/(1 – λ)$   ⇒   <u>Proposed solution 4</u>.
	*The graph uses colours and line type (solid/dotted) to illustrate the relationship between the models.		*The graph uses colours and line type  (solid/dotted)  to illustrate the relationship between the models.



	'''(6)'''  Using the results of subtasks  '''(3)''',  '''(4)'''  and  '''(5)'''  we obtain in general for the  ''Binary Symmetric Error & Erasure Channel'':		'''(6)'''  Using the results of subtasks  '''(3)''',  '''(4)'''  and  '''(5)'''  we obtain in general for the  "Binary Symmetric Error & Erasure Channel":
	:$$C_{\rm BSEC} = (1- \lambda) \cdot \left [ 1 - H_{\rm bin}(\frac{\varepsilon}{1- \lambda}) \right ]\hspace{0.05cm},$$		:$$C_{\rm BSEC} = (1- \lambda) \cdot \left [ 1 - H_{\rm bin}(\frac{\varepsilon}{1- \lambda}) \right ]\hspace{0.05cm},$$
	:or the numerical values for  $ε = 0.08$  and  $λ = 0.2$:		:or the numerical values for  $ε = 0.08$  and  $λ = 0.2$:
	:$$C_{\rm BSEC} = 0.8 \cdot \big [ 1 - H_{\rm bin}(0.1) \big ] = 0.8 \cdot \left [ 1 - 0.469 \right ] \hspace{0.15cm} \underline {=0.425\,{\rm bit}}\hspace{0.05cm}.$$		:$$C_{\rm BSEC} = 0.8 \cdot \big [ 1 - H_{\rm bin}(0.1) \big ] = 0.8 \cdot \left [ 1 - 0.469 \right ] \hspace{0.15cm} \underline {=0.425\,{\rm bit}}\hspace{0.05cm}.$$

			'''(7)'''  The "Binary Symmetric Channel"  (BSC) is a special case of the BSEC with  $λ = 0$:

	'''(7)'''  The ''Binary Symmetric Channel''  (BSC) is a special case of the BSEC with  $λ = 0$:
	:$$ C_{\rm BSC} = 1 - H_{\rm bin}(\varepsilon) = 1 - H_{\rm bin}(0.08) = 1 - 0.402 \hspace{0.15cm} \underline {=0.598\,{\rm bit}}\hspace{0.05cm}.$$		:$$ C_{\rm BSC} = 1 - H_{\rm bin}(\varepsilon) = 1 - H_{\rm bin}(0.08) = 1 - 0.402 \hspace{0.15cm} \underline {=0.598\,{\rm bit}}\hspace{0.05cm}.$$

			'''(8)''' The  "Binary Erasure Channel"  (BEC) is a special case of the BSEC with  $ε = 0$:

	'''(8)''' The  ''Binary Erasure Channel''  (BEC) is a special case of the BSEC with  $ε = 0$:
	:$$C_{\rm BEC} = (1- \lambda) \cdot \big [ 1 - H_{\rm bin}(0) \big ] = 1- \lambda\hspace{0.05cm}.$$		:$$C_{\rm BEC} = (1- \lambda) \cdot \big [ 1 - H_{\rm bin}(0) \big ] = 1- \lambda\hspace{0.05cm}.$$
	*With  $λ = 0.2$ , this results in  $C_{\rm BEC} = 0.8 \ \rm bit.$		*With  $λ = 0.2$ , this results in  $C_{\rm BEC} = 0.8 \ \rm bit.$

Guenter at 12:57, 22 September 2021

2021-09-22T12:57:21Z

← Older revision		Revision as of 14:57, 22 September 2021
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	}}		}}

	[[File:EN_Inf_A_3_11.png\|right\|frame\|Predefined ~~partial~~ channel model (top) <br>and BSEC model (bottom)]]		[[File:EN_Inf_A_3_11.png\|right\|frame\|Predefined sub-channel model (top) <br>and BSEC model (bottom)]]
	The upper diagram shows two strictly symmetric subchannels  $\rm A$  and  $\rm B$.  A ''strongly symmetric channel'' is one that is ''uniformly dispersive''   ⇒   each input symbol  $u$  has the same set of transition probabilities:		The upper diagram shows two strictly symmetric subchannels  $\rm A$  and  $\rm B$.
			*A  '''strongly symmetric channel'''  is one that is  "uniformly dispersive"   ⇒   each input symbol  $u$  has the same set of transition probabilities:
	:$$\left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}\|\hspace{0.02cm}U}(y\hspace{0.03cm} \|\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}u \in U \right \} \hspace{0.05cm},$$		:$$\left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}\|\hspace{0.02cm}U}(y\hspace{0.03cm} \|\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}u \in U \right \} \hspace{0.05cm},$$
	* moreover, ''uniformly focusing''   ⇒   each output symbol  $y$  has the same set of transition probabilities:		* moreover, '''uniformly focusing'''   ⇒   each output symbol  $y$  has the same set of transition probabilities:
	:$$ \left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}\|\hspace{0.02cm}U}(y\hspace{0.03cm} \|\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}y \in Y \right \} \hspace{0.05cm}.$$		:$$ \left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}\|\hspace{0.02cm}U}(y\hspace{0.03cm} \|\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}y \in Y \right \} \hspace{0.05cm}.$$
	The random quantity  $U = \{0,\ 1\}$  occurs directly at the inputs of the ~~subchannels~~  $\rm A$  and  $\rm B$~~ ~~.		The random quantity  $U = \{0,\ 1\}$  occurs directly at the inputs of the sub-channels  $\rm A$  and  $\rm B$.

	The channel capacity of a strictly symmetrical channel can be calculated much more easily than in the asymmetrical case.  However, this will not be discussed in detail in this exercise.		The channel capacity of a strictly symmetrical channel can be calculated much more easily than in the asymmetrical case.  However, this will not be discussed in detail in this exercise.
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	For the capacity of the total channel applies:		For the capacity of the total channel applies:
	:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B}\hspace{0.05cm}$$		:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B}\hspace{0.05cm}$$
	Here  $p_{\rm A}$  denotes the probability that the ~~subchannel~~  $\rm A$  is selected and  $C_{\rm A}$  indicates its capacity.  The same applies to ~~subchannel~~  $\rm B$.		Here  $p_{\rm A}$  denotes the probability that the sub-channel  $\rm A$  is selected and  $C_{\rm A}$  indicates its capacity.  The same applies to sub-channel  $\rm B$.

	Subsequently, the channel capacity of the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC\|Binary Symmetric Error & Erasure Channel]]  $\rm (BSEC)$  is also to be determined according to the sketch below (grey background) by deriving the relationship between		Subsequently, the channel capacity of the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC\|Binary Symmetric Error & Erasure Channel]]  $\rm (BSEC)$  is also to be determined according to the sketch below  (grey background)  by deriving the relationship between
	*the parameters  $p_{\rm A}$,  $p_{\rm B}$  and the crossover probability  $q$  of the ~~partial~~ channel model shown above, and		*the parameters  $p_{\rm A}$,  $p_{\rm B}$  and the crossover probability  $q$  of the sub-channel model shown above, and
	* the parameters  $λ$  and  $\varepsilon$  of the BSEC model.		* the parameters  $λ$  and  $\varepsilon$  of the BSEC model.

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	<quiz display=simple>		<quiz display=simple>
	{What is the capacity  $C_{\rm A}$  of ~~subchannel~~  $\rm A$?		{What is the capacity  $C_{\rm A}$  of sub-channel  $\rm A$?
	\|type="()"}		\|type="()"}
	+ $C_{\rm A} = 1 – H_{\rm bin}(q),$		+ $C_{\rm A} = 1 - H_{\rm bin}(q),$
	- $C_{\rm A} = p_{\rm A} · \big[1 – H_{\rm bin}(q)\big],$		- $C_{\rm A} = p_{\rm A} · \big[1 - H_{\rm bin}(q)\big],$
	- $C_{\rm A} = 0.$		- $C_{\rm A} = 0.$

	{What is the capacity  $C_{\rm B}$  of ~~subchannel~~  $\rm B$?		{What is the capacity  $C_{\rm B}$  of sub-channel  $\rm B$?
	\|type="()"}		\|type="()"}
	- $C_{\rm B} = 1 – H_{\rm bin}(q),$		- $C_{\rm B} = 1 - H_{\rm bin}(q),$
	- $C_{\rm B} = p_{\rm B} · \big[1 – H_{\rm bin}(q)\big],$		- $C_{\rm B} = p_{\rm B} · \big[1 - H_{\rm bin}(q)\big],$
	+ $C_{\rm B} = 0.$		+ $C_{\rm B} = 0.$

	{What is the capacity  $C$  ~~does~~ the total channel ~~have~~?		{What is the capacity  $C$  of the total channel?
	\|type="[]"}		\|type="[]"}
	- $C = 1 – H_{\rm bin}(q),$		- $C = 1 - H_{\rm bin}(q),$
	+ $C = p_{\rm A} · \big[1 – H_{\rm bin}(q)\big],$		+ $C = p_{\rm A} · \big[1 - H_{\rm bin}(q)\big],$
	- $C = 0.$		- $C = 0.$

	{How do you get from the considered ~~partial~~ channel model to the BSEC model? With		{How do you get from the considered sub-channel model to the BSEC model?  With
	\|type="[]"}		\|type="[]"}
	- $p_{\rm A} = λ,$		- $p_{\rm A} = λ,$
	+ $p_{\rm A} = 1 – λ,$		+ $p_{\rm A} = 1 - λ,$
	- $p_{\rm A} = ε$,		- $p_{\rm A} = ε$,
	- $p_{\rm A} = ε/(1 – λ)?$		- $p_{\rm A} = ε/(1 - λ)?$

	{How do you get from the considered ~~partial~~ channel model to the BSEC model? With		{How do you get from the considered sub-channel model to the BSEC model?  With
	\|type="[]"}		\|type="[]"}
	- $q = λ,$		- $q = λ,$
	- $q = 1 – λ,$		- $q = 1 - λ,$
	- $q = ε,$		- $q = ε,$
	+ $q = ε/(1 – λ)?$		+ $q = ε/(1 - λ)?$


	{~~Calculate~~ the BSEC  (''Binary Symmetric Error & Erasure Channel'')  ~~channel capacity~~ for  $ε = 0.08$  and  $λ = 0.2.$		{What is the channel capacity of the BSEC  ("Binary Symmetric Error & Erasure Channel")  for  $ε = 0.08$  and  $λ = 0.2.$
	\|type="{}"}		\|type="{}"}
	$C_{\rm BSEC} \ = \ $ { 0.425 3% } $\ \rm bit$		$C_{\rm BSEC} \ = \ $ { 0.425 3% } $\ \rm bit$

	{What is the channel capacity of the BSC ~~channel~~ (''Binary Symmetric Channel'') for  $ε = 0.08$?		{What is the channel capacity of the BSC  ("Binary Symmetric Channel")  for  $ε = 0.08$?
	\|type="{}"}		\|type="{}"}
	$C_{\rm BSC}\ = \ $ { 0.598 3% } $\ \rm bit$		$C_{\rm BSC}\ = \ $ { 0.598 3% } $\ \rm bit$

	{What is the channel capacity of the BEC ~~channel~~  (''Binary Erasure Channel'')  for  $λ = 0.2$?		{What is the channel capacity of the BEC  ("Binary Erasure Channel")  for  $λ = 0.2$?
	\|type="{}"}		\|type="{}"}
	$C_{\rm BEC}\ = \ $ { 0.8 3% } $\ \rm bit$		$C_{\rm BEC}\ = \ $ { 0.8 3% } $\ \rm bit$

Noah at 12:14, 16 September 2021

2021-09-16T12:14:04Z

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Guenter at 15:38, 30 August 2021

2021-08-30T15:38:25Z

← Older revision		Revision as of 17:38, 30 August 2021
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	[[File:~~P_ID2801__Inf_A_3_11_neu~~.png\|right\|frame\|Vorgegebenes Teilkanalmodell (oben) <br>und BSEC–Modell (unten)]]		[[File:EN_Inf_A_3_11.png\|right\|frame\|Vorgegebenes Teilkanalmodell (oben) <br>und BSEC–Modell (unten)]]
	Die obere Grafik zeigt zwei streng symmetrische Teilkanäle  $\rm A$  und  $\rm B$.  Ein  ''streng symmetrischer Kanal''  (englisch:  ''Strongly Symmetric Channel'')  ist dabei		Die obere Grafik zeigt zwei streng symmetrische Teilkanäle  $\rm A$  und  $\rm B$.  Ein  ''streng symmetrischer Kanal''  (englisch:  ''Strongly Symmetric Channel'')  ist dabei
	* gleichmäßig  '''dispersiv'''  (''uniformly dispersive'')   ⇒   jedes Eingangssymbol  $u$  hat die gleiche Menge an Übergangswahrscheinlichkeiten:		* gleichmäßig  '''dispersiv'''  (''uniformly dispersive'')   ⇒   jedes Eingangssymbol  $u$  hat die gleiche Menge an Übergangswahrscheinlichkeiten:
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	[[File:~~P_ID2804__Inf_A_3_11e_neu~~.png\|right\|frame\|BSEC–Modell (oben) und <br>vorgegebenes Teilkanalmodell (unten)]]		[[File:EN_Inf_A_3_11e.png\|right\|frame\|BSEC–Modell (oben) und <br>vorgegebenes Teilkanalmodell (unten)]]
	'''(3)'''  Die Kapazität $C$ des Gesamtkanals kann mit der angegebenen Gleichung berechnet werden:		'''(3)'''  Die Kapazität $C$ des Gesamtkanals kann mit der angegebenen Gleichung berechnet werden:
	:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$		:$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$

Guenter: Guenter moved page Aufgaben:Exercise 3.12: Strongly Symmetrical Channels to Aufgaben:Exercise 3.12: Strictly Symmetrical Channels

2021-07-22T15:15:53Z

Guenter moved page Aufgaben:Exercise 3.12: Strongly Symmetrical Channels to Aufgaben:Exercise 3.12: Strictly Symmetrical Channels

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