Aufgaben:Exercise 3.12: Strictly Symmetrical Channels - Revision history

Guenter at 11:57, 24 September 2021

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Noah at 09:10, 24 September 2021

2021-09-24T09:10:56Z

Guenter at 14:13, 22 September 2021

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Guenter at 13:07, 22 September 2021

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Guenter at 12:57, 22 September 2021

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Noah at 12:14, 16 September 2021

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Guenter at 15:38, 30 August 2021

2021-08-30T15:38:25Z

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 Exercise 3.12: Strongly Symmetrical Channels to Exercise 3.12: Strictly Symmetrical Channels

Guenter: Guenter moved page Aufgaben:Aufgabe 3.12: Streng symmetrische Kanäle to Aufgaben:Exercise 3.12: Strongly Symmetrical Channels

2021-07-22T14:56:06Z

Guenter moved page Aufgabe 3.12: Streng symmetrische Kanäle to Exercise 3.12: Strongly Symmetrical Channels

Javier: Text replacement - "Category:Aufgaben zu Informationstheorie" to "Category:Information Theory: Exercises"

2021-03-23T12:58:03Z

Text replacement - "Category:Aufgaben zu Informationstheorie" to "Category:Information Theory: Exercises"

← Older revision		Revision as of 11:57, 24 September 2021
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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^]]

← Older revision		Revision as of 09:10, 24 September 2021
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−	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Anwendung auf die Digitalsignalübertragung~~	+	{{quiz-Header\|Buchseite=Information_Theory/Application_to_Digital_Signal_Transmission
	}}		}}

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

@@ Line 87: / Line 87: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Subchannel&nbsp; $\rm A$&nbsp; is a &nbsp; (''Binary Symmetric Channel'')&nbsp; with crossover probability&nbsp; $q$   &nbsp; &rArr; &nbsp; <u>Proposed solution 1</u>.
+'''(1)'''&nbsp;  Subchannel&nbsp; $\rm A$&nbsp; is a &nbsp; ("Binary Symmetric Channel")&nbsp; with crossover probability&nbsp; $q$   &nbsp; &rArr; &nbsp; <u>Proposed solution 1</u>.
+'''(2)'''&nbsp;  Subchannel&nbsp; $\rm B$&nbsp; is an&nbsp; "erasure channel".&nbsp; Both the sink entropy and irrelevance of this subchannel are zero  &nbsp; &rArr; &nbsp; <u>Proposed solution 3</u>.
@@ Line 99: / Line 97: @@
 :$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$
 Thus, the <u>proposed solution 2</u> is correct here.
@@ Line 109: / Line 106: @@
-'''(5)'''&nbsp;  In the BSEC (''Binary Symmetric Error & Erasure Channel'') model, for example:
+'''(5)'''&nbsp;  In the BSEC&nbsp; ("Binary Symmetric Error & Erasure Channel")&nbsp; model, for example:
 :$$ {\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =\varepsilon \hspace{0.05cm}.$$
 *In contrast, for our auxiliary model according to the graph below, the result is:
 :$${\rm Pr}(Y \hspace{-0.05cm} = 1\hspace{-0.05cm}\mid \hspace{-0.05cm} X \hspace{-0.05cm}= 0) =(1- \lambda) \cdot q \hspace{0.05cm}.$$
 *This gives $q = ε/(1 – λ)$  &nbsp; &rArr; &nbsp; <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&nbsp; (solid/dotted)&nbsp; to illustrate the relationship between the models.
-'''(6)'''&nbsp;  Using the results of subtasks&nbsp; '''(3)''',&nbsp; '''(4)'''&nbsp; and&nbsp; '''(5)'''&nbsp; we obtain in general for the&nbsp; ''Binary Symmetric Error & Erasure Channel'':
+'''(6)'''&nbsp;  Using the results of subtasks&nbsp; '''(3)''',&nbsp; '''(4)'''&nbsp; and&nbsp; '''(5)'''&nbsp; we obtain in general for the&nbsp; "Binary Symmetric Error & Erasure Channel":
 :$$C_{\rm BSEC} = (1- \lambda) \cdot \left [ 1 - H_{\rm bin}(\frac{\varepsilon}{1- \lambda}) \right ]\hspace{0.05cm},$$
 :or the numerical values for&nbsp; $ε = 0.08$&nbsp; and&nbsp; $λ = 0.2$:
 :$$C_{\rm BSEC} = 0.8 \cdot \big [ 1 - H_{\rm bin}(0.1) \big ] = 0.8 \cdot \left [ 1 - 0.469 \right ] \hspace{0.15cm} \underline {=0.425\,{\rm bit}}\hspace{0.05cm}.$$
+'''(7)'''&nbsp;  The "Binary Symmetric Channel"&nbsp; (BSC) is a special case of the BSEC with&nbsp; $λ = 0$:
 :$$ C_{\rm BSC} = 1 - H_{\rm bin}(\varepsilon) = 1 - H_{\rm bin}(0.08) = 1 - 0.402 \hspace{0.15cm} \underline {=0.598\,{\rm bit}}\hspace{0.05cm}.$$
+'''(8)''' The&nbsp; "Binary Erasure Channel"&nbsp; (BEC) is a special case of the BSEC with&nbsp; $ε = 0$:
 :$$C_{\rm BEC} = (1- \lambda) \cdot \big [ 1 - H_{\rm bin}(0) \big ] = 1- \lambda\hspace{0.05cm}.$$
 *With&nbsp; $λ = 0.2$&nbsp;, this results in&nbsp; $C_{\rm BEC} = 0.8 \ \rm bit.$

@@ Line 3: / Line 3: @@
 }}
-[[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&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$.&nbsp; A ''strongly symmetric channel'' is one that is ''uniformly dispersive'' &nbsp; &rArr; &nbsp; each input symbol&nbsp; $u$&nbsp; has the same set of transition probabilities:
+The upper diagram shows two strictly symmetric subchannels&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$.&nbsp;
 :$$\left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.02cm}U}(y\hspace{0.03cm} |\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}u \in U \right \} \hspace{0.05cm},$$
-* moreover, ''uniformly focusing''  &nbsp; &rArr; &nbsp; each output symbol&nbsp; $y$&nbsp; has the same set of transition probabilities:
+* moreover, '''uniformly focusing'''  &nbsp; &rArr; &nbsp; each output symbol&nbsp; $y$&nbsp; has the same set of transition probabilities:
 :$$ \left \{ P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.02cm}U}(y\hspace{0.03cm} |\hspace{0.03cm} u) \hspace{-0.05cm}: \hspace{0.25cm}y \in Y \right \} \hspace{0.05cm}.$$
-The random quantity&nbsp; $U = \{0,\ 1\}$&nbsp; occurs directly at the inputs of the subchannels&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$&nbsp;.
+The random quantity&nbsp; $U = \{0,\ 1\}$&nbsp; occurs directly at the inputs of the sub-channels&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$.
 The channel capacity of a strictly symmetrical channel can be calculated much more easily than in the asymmetrical case.&nbsp; However, this will not be discussed in detail in this exercise.
@@ Line 14: / Line 15: @@
 For the capacity of the total channel applies:
 :$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B}\hspace{0.05cm}$$
-Here&nbsp; $p_{\rm A}$&nbsp; denotes the probability that the subchannel&nbsp; $\rm A$&nbsp; is selected and&nbsp; $C_{\rm A}$&nbsp; indicates its capacity.&nbsp; The same applies to subchannel&nbsp; $\rm B$.
+Here&nbsp; $p_{\rm A}$&nbsp; denotes the probability that the sub-channel&nbsp; $\rm A$&nbsp; is selected and&nbsp; $C_{\rm A}$&nbsp; indicates its capacity.&nbsp; The same applies to sub-channel&nbsp; $\rm B$.
-Subsequently, the channel capacity of the&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|Binary Symmetric Error & Erasure Channel]]&nbsp; $\rm (BSEC)$&nbsp; is also to be determined according to the sketch below (grey background) by deriving the relationship between
+Subsequently, the channel capacity of the&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|Binary Symmetric Error & Erasure Channel]]&nbsp; $\rm (BSEC)$&nbsp; is also to be determined according to the sketch below&nbsp; (grey background)&nbsp; by deriving the relationship between
-*the parameters&nbsp; $p_{\rm A}$,&nbsp; $p_{\rm B}$&nbsp; and the crossover probability&nbsp; $q$&nbsp; of the partial channel model shown above, and
+*the parameters&nbsp; $p_{\rm A}$,&nbsp; $p_{\rm B}$&nbsp; and the crossover probability&nbsp; $q$&nbsp; of the sub-channel model shown above, and
 * the parameters&nbsp; $λ$&nbsp; and&nbsp; $\varepsilon$&nbsp; of the BSEC model.
@@ Line 37: / Line 38: @@
 <quiz display=simple>
-{What is the capacity&nbsp; $C_{\rm A}$&nbsp; of subchannel&nbsp; $\rm A$?
+{What is the capacity&nbsp; $C_{\rm A}$&nbsp; of sub-channel&nbsp; $\rm A$?
 |type="()"}
-+ $C_{\rm A} = 1 – H_{\rm bin}(q),$
++ $C_{\rm A} = 1 - H_{\rm bin}(q),$
--  $C_{\rm A} = p_{\rm A} · \big[1 – H_{\rm bin}(q)\big],$
+-  $C_{\rm A} = p_{\rm A} · \big[1 - H_{\rm bin}(q)\big],$
 - $C_{\rm A} = 0.$
-{What is the capacity&nbsp; $C_{\rm B}$&nbsp; of subchannel&nbsp; $\rm B$?
+{What is the capacity&nbsp; $C_{\rm B}$&nbsp; of sub-channel&nbsp; $\rm B$?
 |type="()"}
-- $C_{\rm B} = 1 – H_{\rm bin}(q),$
+- $C_{\rm B} = 1 - H_{\rm bin}(q),$
-- $C_{\rm B} = p_{\rm B} · \big[1 – H_{\rm bin}(q)\big],$
+- $C_{\rm B} = p_{\rm B} · \big[1 - H_{\rm bin}(q)\big],$
 + $C_{\rm B} = 0.$
-{What is the capacity&nbsp; $C$&nbsp; does the total channel have?
+{What is the capacity&nbsp; $C$&nbsp; of the total channel?
 |type="[]"}
-- $C = 1 – H_{\rm bin}(q),$
+- $C = 1 - H_{\rm bin}(q),$
-+ $C = p_{\rm A} · \big[1 – H_{\rm bin}(q)\big],$
++ $C = p_{\rm A} · \big[1 - H_{\rm bin}(q)\big],$
 - $C = 0.$
-{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?&nbsp; With
 |type="[]"}
 - $p_{\rm A} = λ,$
-+ $p_{\rm A} = 1 – λ,$
++ $p_{\rm A} = 1 - λ,$
 - $p_{\rm A} = ε$,
-- $p_{\rm A} = ε/(1 – λ)?$
+- $p_{\rm A} = ε/(1 - λ)?$
-{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?&nbsp; With
 |type="[]"}
 - $q = λ,$
-- $q = 1 – λ,$
+- $q = 1 - λ,$
 - $q = ε,$
-+ $q = ε/(1 – λ)?$
++ $q = ε/(1 - λ)?$
-{Calculate the BSEC&nbsp; (''Binary Symmetric Error & Erasure Channel'')&nbsp; channel capacity for &nbsp;$ε = 0.08$&nbsp; and &nbsp;$λ = 0.2.$
+{What is the channel capacity of the BSEC&nbsp; ("Binary Symmetric Error & Erasure Channel")&nbsp; for &nbsp;$ε = 0.08$&nbsp; and &nbsp;$λ = 0.2.$
 |type="{}"}
 $C_{\rm BSEC} \ = \ $ { 0.425 3% } $\ \rm bit$
-{What is the channel capacity of the BSC channel  (''Binary Symmetric Channel'') for &nbsp;$ε = 0.08$?
+{What is the channel capacity of the BSC&nbsp;  ("Binary Symmetric Channel")&nbsp; for &nbsp;$ε = 0.08$?
 |type="{}"}
 $C_{\rm BSC}\ = \ $ { 0.598 3% } $\ \rm bit$
-{What is the channel capacity of the BEC channel&nbsp; (''Binary Erasure Channel'')&nbsp; for &nbsp;$λ = 0.2$?
+{What is the channel capacity of the BEC&nbsp; ("Binary Erasure Channel")&nbsp; for &nbsp;$λ = 0.2$?
 |type="{}"}
 $C_{\rm BEC}\ = \ $ { 0.8 3% }  $\ \rm bit$

@@ Line 3: / Line 3: @@
 }}
-[[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&nbsp; $\rm A$&nbsp; und&nbsp; $\rm B$.&nbsp; Ein&nbsp; ''streng  symmetrischer Kanal''&nbsp; (englisch:&nbsp; ''Strongly Symmetric Channel'')&nbsp; ist dabei
 * gleichmäßig&nbsp;  '''dispersiv'''&nbsp;  (''uniformly dispersive'')  &nbsp; &rArr; &nbsp; jedes Eingangssymbol&nbsp; $u$&nbsp; hat die gleiche Menge an Übergangswahrscheinlichkeiten:
@@ Line 97: / Line 97: @@
-[[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)'''&nbsp;  Die Kapazität $C$ des Gesamtkanals kann mit der angegebenen Gleichung berechnet werden:
 :$$ C = p_{\rm A} \cdot C_{\rm A} + p_{\rm B} \cdot C_{\rm B} = p_{\rm A} \cdot \big[1 - H_{\rm bin}(q)\big]\hspace{0.05cm}.$$

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← Older revision		Revision as of 12:58, 23 March 2021
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−	[[Category:~~Aufgaben zu Informationstheorie~~\|^3.3 Anwendung auf DSÜ-Kanäle^]]	+	[[Category:Information Theory: Exercises\|^3.3 Anwendung auf DSÜ-Kanäle^]]