Aufgaben:Exercise 1.4: Maximum Likelihood Decision - Revision history

Hwang at 16:54, 1 November 2022

2022-11-01T16:54:51Z

Guenter at 15:15, 28 June 2022

2022-06-28T15:15:57Z

Guenter at 14:37, 14 June 2022

2022-06-14T14:37:15Z

Guenter at 14:15, 12 June 2022

2022-06-12T14:15:21Z

Guenter at 14:04, 12 June 2022

2022-06-12T14:04:13Z

Noah at 15:16, 29 May 2022

2022-05-29T15:16:09Z

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Guenter at 09:23, 11 May 2022

2022-05-11T09:23:12Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.4: Maximum–Likelihood–Entscheidung to Aufgaben:Exercise 1.4: Maximum Likelihood Decision

2022-05-11T09:14:40Z

Guenter moved page Aufgabe 1.4: Maximum–Likelihood–Entscheidung to Exercise 1.4: Maximum Likelihood Decision

Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:48:33Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

Javier: Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

2020-07-09T12:50:20Z

Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

@@ Line 60: / Line 60: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; Correct&nbsp; <u>answer 3</u> &nbsp; &rArr; &nbsp;  $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$:
-*The Hamming distances between the specific received word $\underline{y} = (1, 0, 0, 0, 1)$ and the four possible codewords $\underline{x}_{i}$ are as follows:
+*The Hamming distances between the specific received word $\underline{y} = (1, 0, 0, 0, 1)$ and the four possible code words $\underline{x}_{i}$ are as follows:
 :$$d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_0) = 2\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_1) = 4\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 1\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_3) = 3\hspace{0.05cm}.$$
 *A decision is made for the sequence with the smallest Hamming distance $d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 1$.

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_KC_A_1_3.png|right|frame|Maximum likelihood decoding model]]
+[[File:EN_KC_A_1_4_neu.png|right|frame|Maximum likelihood decoding model]]
 We consider the digital transmission system according to the graph.&nbsp; Considered are:

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2384__KC_A_1_4.png|right|frame|Maximum likelihood decoding model '''Korrektur''']]
+[[File:EN_KC_A_1_3.png|right|frame|Maximum likelihood decoding model]]
 We consider the digital transmission system according to the graph.&nbsp; Considered are:

@@ Line 59: / Line 59: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct <u>answer 3</u>:
+'''(1)'''&nbsp; Correct&nbsp; <u>answer 3</u> &nbsp; &rArr; &nbsp;  $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$:
-*The Hamming distances between the specific received word $\underline{y} = (1, 0, 0, 1)$ and the four possible codewords $\underline{x}_{i}$ are as follows:
+*The Hamming distances between the specific received word $\underline{y} = (1, 0, 0, 0, 1)$ and the four possible codewords $\underline{x}_{i}$ are as follows:
 :$$d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_0) = 2\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_1) = 4\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 1\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_3) = 3\hspace{0.05cm}.$$
 *A decision is made for the sequence with the smallest Hamming distance $d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 1$.
@@ Line 66: / Line 66: @@
-'''(2)'''&nbsp; For $\underline{y} = (0, 0, 0, 1, 0)$ the <u>answers 1 and 2</u> are correct, as the following calculation shows:
+'''(2)'''&nbsp; For&nbsp; $\underline{y} = (0, 0, 0, 1, 0)$&nbsp; the&nbsp; <u>answers 1 and 2</u>&nbsp; are correct,&nbsp; as the following calculation shows:
 :$$d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_0) = 1\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_1) = 1\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 4\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_3) = 4\hspace{0.05cm}.$$
-'''(3)'''&nbsp; Correct <u>answer 3</u>:
+'''(3)'''&nbsp; Correct <u>answer 3</u> &nbsp; &rArr; &nbsp;  $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$:
-*According to the Hamming distance, a decision in favor of $x_{2}$ would be just as possible as for $x_{3}$ if the vector $\underline{y} = (1, 0, 1, 1, 1)$ is received:
+*According to the Hamming distance,&nbsp;  a decision in favor of&nbsp; $x_{2}$&nbsp; would be just as possible as for&nbsp; $x_{3}$&nbsp; if the vector&nbsp; $\underline{y} = (1, 0, 1, 1, 1)$&nbsp; is received:
 :$$d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_0) = 4\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_1) = 4\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_2) = 1\hspace{0.05cm}, \hspace{0.3cm} d_{\rm H}(\underline{y}, \hspace{0.05cm}\underline{x}_3) = 1\hspace{0.05cm}.$$
-*But the received vector $\underline{y}$ is different from $x_{2}$ with respect to the fourth bit and from $x_{3}$ in the second bit.
+*But the received vector&nbsp; $\underline{y}$&nbsp; is different from&nbsp; $x_{2}$&nbsp; with respect to the fourth bit and from&nbsp; $x_{3}$&nbsp; in the second bit.
-* Since the fourth bit is more uncertain than the second, it will choose $x_{2}$ .
+* Since the fourth bit is more uncertain than the second,&nbsp; it will choose&nbsp; $x_{2}$ .
-'''(4)'''&nbsp; Since this is a systematic code, the decision for $\underline{z} = (1, 0, 1, 0, 1)$ is equivalent to the decision
+'''(4)'''&nbsp; Since this is a systematic code,&nbsp; the decision for&nbsp; $\underline{z} = (1, 0, 1, 0, 1)$&nbsp; is equivalent to the decision
 :$$v_{1}  \ \underline{ = 1}, \ v_{2} \ \underline{= 0}.$$
-*It is not certain that $\underline{u} = (1, 0)$ was actually sent.
+*It is not certain that&nbsp; $\underline{u} = (1, 0)$&nbsp; was actually sent.
-*But the probability is highest for this given the received vector $\underline{y} = (1, 0, 1, 1)$.
+*But the probability is highest for this given the received vector $\underline{y} = (1, 0, 1, 1, 1)$.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2384__KC_A_1_4.png|right|frame|Maximum Likelihood Decoding Model]]
+[[File:P_ID2384__KC_A_1_4.png|right|frame|Maximum likelihood decoding model '''Korrektur''']]
-We consider the digital transmission system according to the graph. Considered are:
+We consider the digital transmission system according to the graph.&nbsp; Considered are:
-*a systematic&nbsp; $(5, 2)$ block code&nbsp; $\mathcal{C}$&nbsp; with the code words
+*a systematic&nbsp; $(5, 2)$&nbsp; block code&nbsp; $\mathcal{C}$&nbsp; with the code words
 :$$\underline{x}_{0} \hspace{-0.1cm} \ =  \  \hspace{-0.1cm} (0, 0, 0, 0, 0) \hspace{0.05cm},$$ $$\underline{x}_{1} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (0, 1, 0, 1, 0) \hspace{0.05cm},$$ $$\underline{x}_{2} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (1, 0, 1, 0, 1) \hspace{0.05cm},$$ $$\underline{x}_{3} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (1, 1, 1, 1, 1) \hspace{0.05cm};$$
-*a digital (binary) channel model that crosses the vector&nbsp; $\underline{x} \in {\rm GF} (2^{5})$&nbsp; over into the vector&nbsp; $\underline{y} \in {\rm GF} (2^{5})$&nbsp; (Noah)
+*a digital (binary) channel model that changes the vector&nbsp; $\underline{x} \in {\rm GF} (2^{5})$&nbsp; over into the vector&nbsp; $\underline{y} \in {\rm GF} (2^{5})$,
-*a&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum_likelihood_decision_at_the_BSC_channel|Maximum Likelihood Decoder]]&nbsp; (short: &nbsp; ML–Decoder) &nbsp;with the decision rule
+*a&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum_likelihood_decision_at_the_BSC_channel|Maximum likelihood decoder]]&nbsp; (short: &nbsp; ML decoder) &nbsp;with the decision rule
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}|\hspace{0.05cm} \underline{y} ) = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} d_{\rm H}(\underline{y} \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i}).$$
-Here,&nbsp; $d_{\rm H} (\underline{y}, \ \underline{x_{i}})$&nbsp; the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|Hamming distance]]&nbsp; between the received word&nbsp; $\underline{y}$&nbsp; and the (possibly) sent codeword&nbsp; $\underline{x_{i}}$.
+Here,&nbsp; $d_{\rm H} (\underline{y}, \ \underline{x_{i}})$&nbsp; is the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|Hamming distance]]&nbsp; between the received word&nbsp; $\underline{y}$&nbsp; and the&nbsp; (possibly)&nbsp; sent code word&nbsp; $\underline{x_{i}}$.
+Note:&nbsp; This exercise belongs to the chapter&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures|"Channel Models and Decision Structures"]].
@@ Line 29: / Line 25: @@
 <quiz display=simple>
-{Let&nbsp; $\underline{y} = (1, 0, 0, 1)$. Which decisions satisfy the maximum likelihood criterion?
+{Let&nbsp; $\underline{y} = (1, 0, 0, 0, 1)$.&nbsp; Which decisions satisfy the maximum likelihood criterion?
 |type="[]"}
-- $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0)$,
+- $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0, 0)$,
 - $\underline{z} = \underline{x}_{1} = (0, 1, 0, 1, 0)$,
 + $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$,
-- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1)$.
+- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1, 1)$.
-{Let&nbsp; $\underline{y} = (0, 0, 0, 1, 0)$. Which decisions satisfy the maximum likelihood criterion?
+{Let&nbsp; $\underline{y} = (0, 0, 0, 1, 0)$.&nbsp; Which decisions satisfy the maximum likelihood criterion?
 |type="[]"}
-+ $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0)$,
++ $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0, 0)$,
 + $\underline{z} = \underline{x}_{1} = (0, 1, 0, 1, 0)$,
 - $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$,
-- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1)$.
+- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1, 1)$.
-{What decision does the ML decoder make for&nbsp; $\underline{y} = (1, 0, 1, 1, 1)$ when it is told that the last two symbols are uncertain?
+{What decision does the maximum likelihood decoder make for&nbsp; $\underline{y} = (1, 0, 1, 1, 1)$ when it is told that the last two symbols are uncertain?
 |type="[]"}
-- $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0)$,
+- $\underline{z} = \underline{x}_{0} = (0, 0, 0, 0, 0)$,
 - $\underline{z} = \underline{x}_{1} = (0, 1, 0, 1, 0)$,
 + $\underline{z} = \underline{x}_{2} = (1, 0, 1, 0, 1)$,
-- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1)$.
+- $\underline{z} = \underline{x}_{3} = (1, 1, 1, 1, 1)$.

← Older revision		Revision as of 09:23, 11 May 2022
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−	[[Category:Channel Coding: Exercises\|^1.2 ~~Kanal und Entscheiderstrukturen~~	+	[[Category:Channel Coding: Exercises\|^1.2 Channel and Decision Device^]]
−	^]]

← Older revision	Revision as of 09:14, 11 May 2022
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← Older revision		Revision as of 12:48, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^1.2 Kanal und Entscheiderstrukturen	+	[[Category:Channel Coding: Exercises\|^1.2 Kanal und Entscheiderstrukturen
	^]]		^]]

@@ Line 9: / Line 9: @@
 :$$\underline{x}_{0} \hspace{-0.1cm} \ =  \  \hspace{-0.1cm} (0, 0, 0, 0, 0) \hspace{0.05cm},$$ $$\underline{x}_{1} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (0, 1, 0, 1, 0) \hspace{0.05cm},$$ $$\underline{x}_{2} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (1, 0, 1, 0, 1) \hspace{0.05cm},$$ $$\underline{x}_{3} \hspace{-0.1cm} \  =  \  \hspace{-0.1cm} (1, 1, 1, 1, 1) \hspace{0.05cm};$$
 *ein digitales (binäres) Kanalmodell, das den Vektor&nbsp; $\underline{x} \in {\rm GF} (2^{5})$&nbsp; in den Vektor&nbsp; $\underline{y} \in {\rm GF} (2^{5})$&nbsp; verfälscht;
-*ein&nbsp; [[Kanalcodierung/Kanalmodelle_und_Entscheiderstrukturen#Maximum-Likelihood.E2.80.93Entscheidung_beim_BSC.E2.80.93Kanal|Maximum–Likelihood–Decoder]]&nbsp; (kurz: &nbsp; ML–Decoder) &nbsp;mit der Entscheidungsregel
+*ein&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Maximum-Likelihood.E2.80.93Entscheidung_beim_BSC.E2.80.93Kanal|Maximum–Likelihood–Decoder]]&nbsp; (kurz: &nbsp; ML–Decoder) &nbsp;mit der Entscheidungsregel
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}|\hspace{0.05cm} \underline{y} ) = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} d_{\rm H}(\underline{y} \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i}).$$
-Hier bezeichnet&nbsp; $d_{\rm H} (\underline{y}, \ \underline{x_{i}})$&nbsp; die&nbsp; [[Kanalcodierung/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming–Distanz]]&nbsp; zwischen dem Empfangswort&nbsp; $\underline{y}$&nbsp; und dem (möglicherweise) gesendeten Codewort&nbsp; $\underline{x_{i}}$.
+Hier bezeichnet&nbsp; $d_{\rm H} (\underline{y}, \ \underline{x_{i}})$&nbsp; die&nbsp; [[Channel_Coding/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming–Distanz]]&nbsp; zwischen dem Empfangswort&nbsp; $\underline{y}$&nbsp; und dem (möglicherweise) gesendeten Codewort&nbsp; $\underline{x_{i}}$.
@@ Line 21: / Line 21: @@
 ''Hinweis'':
-*Die Aufgabe gehört zum Kapitel&nbsp; [[Kanalcodierung/Kanalmodelle_und_Entscheiderstrukturen|Kanalmodelle und  Entscheiderstrukturen]].
+*Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen|Kanalmodelle und  Entscheiderstrukturen]].