Aufgaben:Exercise 3.11: Erasure Channel - Revision history

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2026-03-16T15:57:01Z

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	We are looking for:		We are looking for:
	* the capacity  $C_{M\rm –EC}$  of this M–ary Erasure Channel,		* the capacity  $C_{M\rm –EC}$  of this M–ary Erasure Channel,
	* the capacity  $C_{\rm BEC}$  of the  [[Channel_Coding/~~Kanalmodelle_und_Entscheiderstrukturen~~#~~Binary_Erasure_Channel_.E2.80.93_BEC~~\|Binary Erasure Channels]]  as a special case of the above model.		* the capacity  $C_{\rm BEC}$  of the  [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_–_BEC\|Binary Erasure Channels]]  as a special case of the above 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~~#Information-theoretical_model_of_digital_signal_transmission\|Information-theoretical model of digital signal transmission]].		*Reference is made in particular to the page  [[Information_Theory/Application_to_Digital_Signal_Transmission#Information-theoretical_model_of_digital_signal_transmission\|Information-theoretical model of digital signal transmission]].
	*In the above diagram,  "erasures"  $($with probability  $λ)$  are drawn in blue.		*In the above diagram,  "erasures"  $($with probability  $λ)$  are drawn in blue.
	* "Correct transmission paths"  $($i.e., from  $X = μ$  to  $Y = μ)$  are shown in red  ($1 ≤ μ ≤ M$).		* "Correct transmission paths"  $($i.e., from  $X = μ$  to  $Y = μ)$  are shown in red  ($1 ≤ μ ≤ M$).
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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.~~10_Auslöschungskanal~~]]		[[de:Aufgaben:Aufgabe 3.11: Auslöschungskanal]]

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	'''(5)'''  The channel capacity  $C$  is equal to the maximum mutual information  $I(X; Y)$,  where the maximization with respect to  $P_X(X)$  has already been considered by the symmetric approach:		'''(5)'''  The channel capacity  $C$  is equal to the maximum mutual information  $I(X; Y)$,  where the maximization with respect to  $P_X(X)$  has already been considered by the symmetric approach:
	:$$ C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) = H(Y) - H(Y \hspace{-0.1cm}\mid \hspace{-0.1cm} X) = ( 1-\lambda) \cdot {\rm log}_2 \hspace{0.1cm} M + H_{\rm bin} (\lambda) - H_{\rm bin} (\lambda) = ( 1-\lambda) \cdot {\rm log}_2 \hspace{0.1cm} M \hspace{0.05cm}$$		:$$ C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) = H(Y) - H(Y \hspace{-0.1cm}\mid \hspace{-0.1cm} X) = ( 1-\lambda) \cdot {\rm log}_2 \hspace{0.1cm} M + H_{\rm bin} (\lambda) - H_{\rm bin} (\lambda) = ( 1-\lambda) \cdot {\rm log}_2 \hspace{0.1cm} M \hspace{0.05cm}$$
	:$$\Rightarrow \hspace{0.3cm} M = 4\text{:} \hspace{0.3cm} \underline {C=1.6\,\,{\rm bit}} \hspace{0.05cm}, \hspace{0.8cm}		:$$\Rightarrow \hspace{0.3cm} M = 4\text{:} \hspace{0.3cm} \underline {C=1.6\,\,{\rm bit}} \hspace{0.05cm}, \hspace{0.8cm}M = 2\text{:} \hspace{0.3cm} \underline {C=0.8\,\,{\rm bit}} \hspace{0.05cm}.$$
	M = 2\text{:} \hspace{0.3cm} \underline {C=0.8\,\,{\rm bit}} \hspace{0.05cm}.$$

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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:3.10_Auslöschungskanal]]

Guenter at 11:56, 24 September 2021

2021-09-24T11:56:06Z

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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^]]

Guenter at 09:53, 24 September 2021

2021-09-24T09:53:58Z

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	'''(2)'''  Correct is the <u>proposed solution 3</u>, i.e. exactly  $2M$  connections.  ~~There~~:		'''(2)'''  Correct is the <u>proposed solution 3</u>, i.e. exactly  $2M$  connections.  Because:
	*From each source symbol  $X = μ$  one gets to the sink symbol  $Y = μ$  as well as to the erasure  $Y = \text{E}$.		*From each source symbol  $X = μ$  one gets to the sink symbol  $Y = μ$  as well as to the erasure  $Y = \text{E}$.

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	'''(6)'''  The  ''Binary Erasure Channel''  $\rm (BEC)$  is a special case of the general model considered here with  $M = 2$:		'''(6)'''  The  "Binary Erasure Channel"  $\rm (BEC)$  is a special case of the general model considered here with  $M = 2$:
	:$$C_{\rm BEC} = 1-\lambda \hspace{0.05cm}.$$		:$$C_{\rm BEC} = 1-\lambda \hspace{0.05cm}.$$
	*Thus, the correct <u>solution proposal 1</u>.		*Thus, the correct <u>solution proposal 1</u>.
	*The second proposed solution, on the other hand, applies to the  ''Binary Symmetric Channel''  $\rm (BSC)$  with distortion probability  $λ$.		*The second proposed solution, on the other hand, applies to the  "Binary Symmetric Channel"  $\rm (BSC)$  with distortion probability  $λ$.

	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 09:43, 24 September 2021

2021-09-24T09:43:57Z

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	*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#Information-theoretical_model_of_digital_signal_transmission\|Information-theoretical model of digital signal transmission]].		*Reference is made in particular to the page  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Information-theoretical_model_of_digital_signal_transmission\|Information-theoretical model of digital signal transmission]].
	*In the above diagram, ~~cancellations~~  $($with probability  $λ)$  are drawn in blue.		*In the above diagram,  "erasures"  $($with probability  $λ)$  are drawn in blue.
	* "Correct transmission paths"  $($i.e., from  $X = μ$  to  $Y = μ)$  are shown in red  ($1 ≤ μ ≤ M$).		* "Correct transmission paths"  $($i.e., from  $X = μ$  to  $Y = μ)$  are shown in red  ($1 ≤ μ ≤ M$).

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	<quiz display=simple>		<quiz display=simple>
	{ What  $P_X(X)$  should be generally applied for channel capacity calculation?		{ What  $P_X(X)$  should be generally applied for the channel capacity calculation?
	\|type="[]"}		\|type="[]"}
	- $P_X(X) = (0.5, \ 0.5),$		- $P_X(X) = (0.5, \ 0.5),$

Noah at 08:24, 24 September 2021

2021-09-24T08:24:22Z

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

Noah at 12:10, 23 September 2021

2021-09-23T12:10:38Z

Show changes

Guenter at 11:35, 22 September 2021

2021-09-22T11:35:42Z

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	}}		}}

	[[File:P_ID2791__Inf_A_3_10.png\|right\|frame\|Erasure channel with four inputs and five outputs]]		[[File:P_ID2791__Inf_A_3_10.png\|right\|frame\|Erasure channel with <br>four inputs and five outputs]]
	~~A cancellation~~ channel is considered with		An erasure channel is considered with
	* the  $M$  inputs  $x ∈ X = \{1,\ 2, \ \text{...} \ ,\ M\}$,  and		* the  $M$  inputs  $x ∈ X = \{1,\ 2, \ \text{...} \ ,\ M\}$,  and
	* the  $M + 1$  outputs  $y ∈ Y = \{1,\ 2,\ \ \text{...} \ ,\ M,\ \text{E}\}.$		* the  $M + 1$  outputs  $y ∈ Y = \{1,\ 2,\ \ \text{...} \ ,\ M,\ \text{E}\}.$
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	:$${\rm Pr}(Y \hspace{-0.05cm} = {\rm E}\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = \lambda \hspace{0.05cm}.$$		:$${\rm Pr}(Y \hspace{-0.05cm} = {\rm E}\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = \lambda \hspace{0.05cm}.$$
	We are looking for:		We are looking for:
	* the capacity  $C_{M\rm –EC}$  of this ''M–ary Erasure ~~Channels''~~,		* the capacity  $C_{M\rm –EC}$  of this M–ary Erasure Channel,
	* the capacity  $C_{\rm BEC}$  of the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC\|Binary Erasure Channels]]  as a special case of the above model.		* the capacity  $C_{\rm BEC}$  of the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC\|Binary Erasure Channels]]  as a special case of the above model.

Noah at 11:30, 14 September 2021

2021-09-14T11:30:35Z

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	}}		}}

	[[File:P_ID2791__Inf_A_3_10.png\|right\|frame\|~~Auslöschungskanal mit vier Eingängen und fünf Ausgängen~~]]		[[File:P_ID2791__Inf_A_3_10.png\|right\|frame\|Erasure channel with four inputs and five outputs]]
	~~Betrachtet wird ein Auslöschungskanal mit~~		A cancellation channel is considered with
	* ~~den~~  $M$  ~~Eingängen~~  $x ∈ X = \{1,\ 2, \ \text{...} \ ,\ M\}$,  ~~und~~		* the  $M$  inputs  $x ∈ X = \{1,\ 2, \ \text{...} \ ,\ M\}$,  and
	* ~~den~~  $M + 1$  ~~Ausgängen~~  $y ∈ Y = \{1,\ 2,\ \ \text{...} \ ,\ M,\ \text{E}\}.$		* the  $M + 1$  outputs  $y ∈ Y = \{1,\ 2,\ \ \text{...} \ ,\ M,\ \text{E}\}.$


	~~Die Grafik zeigt das Modell für den Sonderfall~~  $M = 4$.  ~~Das Sinkensymbol~~  $y = \text{E}$  ~~berücksichtigt eine  ''Auslöschung''  (englisch:  ''Erasure'' )  für den Fall, dass der Empfänger keine hinreichend gesicherte Entscheidung treffen kann~~.		The graph shows the model for the special case  $M = 4$.  The sink symbol  $y = \text{E}$  takes into account an erasure for the case that the receiver cannot make a sufficiently certain decision.

	~~Die Übergangswahrscheinlichkeiten sind für~~  $1 ≤ μ ≤ M$  ~~wie folgt gegeben~~:		The transition probabilities are given for  $1 ≤ μ ≤ M$  as follows:
	:$${\rm Pr}(Y \hspace{-0.05cm} = \mu\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = 1-\lambda \hspace{0.05cm},$$		:$${\rm Pr}(Y \hspace{-0.05cm} = \mu\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = 1-\lambda \hspace{0.05cm},$$
	:$${\rm Pr}(Y \hspace{-0.05cm} = {\rm E}\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = \lambda \hspace{0.05cm}.$$		:$${\rm Pr}(Y \hspace{-0.05cm} = {\rm E}\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= \mu) = \lambda \hspace{0.05cm}.$$
	~~Gesucht werden~~:		We are looking for:
	* ~~die Kapazität~~  $C_{M\rm –EC}$  ~~dieses~~ ''M–ary Erasure Channels'',		* the capacity  $C_{M\rm –EC}$  of this ''M–ary Erasure Channels'',
	* ~~die Kapazität~~  $C_{\rm BEC}$  ~~des~~  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC\|Binary Erasure Channels]]  ~~als Sonderfall des obigen Modells~~.		* the capacity  $C_{\rm BEC}$  of the  [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC\|Binary Erasure Channels]]  as a special case of the above model.