Aufgaben:Exercise 5.1: Error Distance Distribution - Revision history

Guenter at 13:13, 1 October 2022

2022-10-01T13:13:58Z

Hwang at 02:35, 18 September 2022

2022-09-18T02:35:47Z

Guenter at 15:21, 3 September 2022

2022-09-03T15:21:40Z

Guenter at 14:36, 3 September 2022

2022-09-03T14:36:25Z

Guenter at 15:24, 1 September 2022

2022-09-01T15:24:06Z

Hwang at 07:53, 19 August 2022

2022-08-19T07:53:18Z

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Guenter at 15:32, 15 August 2022

2022-08-15T15:32:17Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.1: Fehlerabstandsverteilung to Aufgaben:Exercise 5.1: Error Distance Distribution

2022-08-15T15:30:09Z

Guenter moved page Aufgabe 5.1: Fehlerabstandsverteilung to Exercise 5.1: Error Distance Distribution

Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:50:20Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Guenter at 16:47, 22 March 2019

2019-03-22T16:47:53Z

@@ Line 9: / Line 9: @@
-As an example, we consider the sequences:
+As an example,&nbsp; we consider the sequences:
 :$$<\hspace{-0.1cm}e_{\nu} \hspace{-0.1cm}>  \ = \ <
 \hspace{-0.1cm}0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, \text{...}

@@ Line 119: / Line 119: @@
 :$${\rm E}[a] =  2 \cdot 0.1 +3 \cdot 0.2 +4 \cdot 0.2 +5 \cdot 0.2 +6 \cdot 0.3=
 .4.$$
-*Without this specification,&nbsp; only the statement&nbsp; ${\rm E}[a] &#8805; 4.4$&nbsp; is possible. But this means that the condition&nbsp; $p_{\rm M} < 1/4.4 < 0.227$&nbsp; is valid for the mean error probability.&nbsp; The statement 3 is therefore also not true with certainty.
+*Without this specification,&nbsp; only the statement&nbsp; ${\rm E}[a] &#8805; 4.4$&nbsp; is possible. But this means that the condition&nbsp; $p_{\rm M} < 1/4.4 < 0.227$&nbsp; is valid for the mean error probability.&nbsp; Statement 3 is therefore also not true with certainty.
 {{ML-Fuß}}
 [[Category:Digital Signal Transmission: Exercises|^5.1 Digital Channel Models^]]

@@ Line 78: / Line 78: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Evaluation of the error distance sequence indicates errors at $\nu = 2, 5, 6, 10, 12, 17, 18, 19, 22, 26, 27$ and $29$.
+'''(1)'''&nbsp; Evaluation of the error distance sequence indicates errors at&nbsp; $\nu = 2, 5, 6, 10, 12, 17, 18, 19, 22, 26, 27$&nbsp; and&nbsp; $29$.
 *It follows: &nbsp; $e_{\rm 16} \ \underline {= 0}$, &nbsp; &nbsp; $e_{\rm 17} \ \underline {= 1}$, &nbsp; &nbsp; $e_{\rm 18} \ \underline {= 1}$.
@@ Line 86: / Line 87: @@
-'''(3)'''&nbsp; ${\rm Pr}(a = k) = V_a(k) \, &ndash;V_a(k+1)$ holds. From this we obtain for the individual probabilities:
+'''(3)'''&nbsp; ${\rm Pr}(a = k) = V_a(k) \, &ndash;V_a(k+1)$&nbsp; holds.&nbsp; From this we obtain for the individual probabilities:
 :$${\rm Pr}(a = 1)\hspace{-0.1cm} \ = \ \hspace{-0.1cm}V_a(1) - V_a(2) = 1 - 0.7\hspace{0.15cm}\underline {= 0.3}\hspace{0.05cm},$$
 :$${\rm Pr}(a = 2)\hspace{-0.1cm} \ = \ \hspace{-0.1cm}V_a(2) - V_a(3) = 0.7 - 0.45 \hspace{0.15cm}\underline {= 0.25}\hspace{0.05cm},$$
@@ Line 95: / Line 96: @@
-'''(4)'''&nbsp; From $V_a(k=6) = {\rm Pr}(a &#8805; 6) = 0$, it follows directly for the maximum error distance $k_{\rm max} \ \underline {= 5}$.
+'''(4)'''&nbsp; From&nbsp; $V_a(k=6) = {\rm Pr}(a &#8805; 6) = 0$,&nbsp; it follows directly for the maximum error distance&nbsp;
-'''(5)'''&nbsp; Using the probabilities calculated in '''(3)''', the expected value we are looking for is:
+'''(5)'''&nbsp; Using the probabilities calculated in subtask&nbsp; '''(3)''',&nbsp; the expected value we are looking for is:
 :$${\rm E}\big[a \big] = \sum_{k = 1}^{5} k \cdot {\rm Pr}(a = k) =  1 \cdot 0.3 +2 \cdot 0.25 +3 \cdot 0.2 +4 \cdot 0.15 +5 \cdot 0.1\hspace{0.15cm}\underline { = 2.5}
   \hspace{0.05cm}.$$
-'''(6)'''&nbsp; The mean error probability is the inverse of the mean error distance: &nbsp; $p_{\rm M} \ \underline {= 0.4}$.
+'''(6)'''&nbsp; The mean error probability is the inverse of the average error distance: &nbsp;
-'''(7)'''&nbsp; With certainty, only <u>statement 1</u> is true:
+'''(7)'''&nbsp; With certainty,&nbsp; only <u>statement 1</u>&nbsp; is true:
-*The first statement is true because ${\rm Pr}(a = 1) = V_a(1) \, &ndash; V_a(2) = 0$.
+*The first statement is true because ${\rm Pr}(a = 1) = V_a(1) - V_a(2) = 0$.
 * The second statement is not certain because $V_a(6)$ gives only the sum of the probabilities ${\rm Pr}(a &#8805; 6)$, but not ${\rm Pr}(a = 6)$ alone.
-*Only with the additional specification $V_a(7) = 0$ would statement 2 be true.
+* The second statement is not certain because&nbsp; $V_a(6)$&nbsp; gives only the sum of the probabilities ${\rm Pr}(a &#8805; 6)$,&nbsp; but not ${\rm Pr}(a = 6)$ alone.&nbsp;
 * Likewise, for the expected value ${\rm E}[a]$, no definite statement is possible due to missing information. With $V_a(7) = 0$ the result would be:
 :$${\rm E}[a] =  2 \cdot 0.1 +3 \cdot 0.2 +4 \cdot 0.2 +5 \cdot 0.2 +6 \cdot 0.3=
-.4$$
+.4.$$
-*Without this specification, only the statement ${\rm E}[a] &#8805; 4.4$ is possible. But this means that the condition $p_{\rm M} < 1/4.4 < 0.227$ is valid for the mean error probability.
+*Without this specification,&nbsp; only the statement&nbsp; ${\rm E}[a] &#8805; 4.4$&nbsp; is possible. But this means that the condition&nbsp; $p_{\rm M} < 1/4.4 < 0.227$&nbsp; is valid for the mean error probability.&nbsp; The statement 3 is therefore also not true with certainty.
 {{ML-Fuß}}
 [[Category:Digital Signal Transmission: Exercises|^5.1 Digital Channel Models^]]

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models}}
-[[File:EN_Dig_A_5_1.png|right|frame|Error distance distributions]]
+[[File:EN_Dig_A_5_1.png|right|frame|Error distance distribution]]
 Any digital channel model can be described in the same way by
 * the error sequence &nbsp;$&#9001;e_{\rm \nu}&#9002;$, and
 * the error distance sequence &nbsp;$&#9001;a_{\rm \nu \hspace{0.05cm}'}&#9002;$.
@@ Line 15: / Line 16: @@
 \hspace{-0.1cm}> \hspace{0.05cm}.$$
-One can see from this, for example:
+One can see from this,&nbsp; for example:
 * The error distance&nbsp; $a_2 = 3$&nbsp; means that there are two error-free symbols between the first and the second error.
 * In contrast,&nbsp; $a_3 = 1$&nbsp; indicates that the second error is immediately followed by a third.
-The different indices &nbsp;$(\nu$&nbsp; and&nbsp; $\nu\hspace{0.05cm} '$, each starting with &nbsp;$1$)&nbsp; are necessary because there is no synchrony between the error distance sequence and the error sequence.
+The different indices &nbsp;$(\nu$&nbsp; and&nbsp; $\nu\hspace{0.05cm} '$,&nbsp; each starting with &nbsp;$1$)&nbsp; are necessary because there is no synchrony between the error distance sequence and the error sequence.
-In the graph, for two different models &nbsp;$M_1$&nbsp; and &nbsp;$M_2$,&nbsp; the error distance distribution (EDD) is given as
+In the graph,&nbsp; for two different models &nbsp;$M_1$&nbsp; and &nbsp;$M_2$,&nbsp; the&nbsp; "error distance distribution"&nbsp; $\rm (EDD)$&nbsp; is given as
 :$$V_a(k) =  {\rm Pr}(a \ge k) = 1 - \sum_{\kappa = 1}^{k}  {\rm Pr}(a = \kappa)\hspace{0.05cm}$$
@@ Line 30: / Line 32: @@
+Note:&nbsp;  The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models|"Parameters of Digital Channel Models"]].
@@ Line 50: / Line 49: @@
 $V_a(k = 1) \ = \ $ { 1 }
-{For model&nbsp; $M_1$,&nbsp; determine the probabilities of the error distances.
+{For model&nbsp; $M_1$,&nbsp;&nbsp; determine the probabilities of the error distances.
 |type="{}"}
 ${\rm Pr}(a = 1) \ = \ $ { 0.3 3% }
@@ Line 62: / Line 61: @@
 $k_{\rm max} \ = \ ${ 5 }
-{Calculate the mean error distance for model&nbsp; $M_1$.&nbsp;
+{Calculate the average error distance for model&nbsp; $M_1$.&nbsp;
 |type="{}"}
 ${\rm E}\big[a \big] \ = \ ${ 2.5 3% }
@@ Line 74: / Line 73: @@
 + Two errors cannot directly follow each other.
 - The most frequent error distance is&nbsp; $a = 6$.
-- The average error probability is&nbsp; $p_{\rm M} = 0.25$.
+- The mean error probability is&nbsp; $p_{\rm M} = 0.25$.
 </quiz>

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models}}
-[[File:P_ID1827__Dig_A_5_1.png|right|frame|Error distance distributions]]
+[[File:EN_Dig_A_5_1.png|right|frame|Error distance distributions]]
 Any digital channel model can be described in the same way by
 * the error sequence &nbsp;$&#9001;e_{\rm \nu}&#9002;$, and

← Older revision		Revision as of 15:32, 15 August 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^5.1 ~~Zu den Digitalen Kanalmodellen~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^5.1 Digital Channel Models^]]

← Older revision		Revision as of 12:50, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^5.1 Zu den Digitalen Kanalmodellen^]]	+	[[Category:Digital Signal Transmission: Exercises\|^5.1 Zu den Digitalen Kanalmodellen^]]

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digitalsignalübertragung/Beschreibungsgrößen digitaler Kanalmodelle}}
-[[File:P_ID1827__Dig_A_5_1.png|right|frame|Zwei Fehlerabstandsverteilungen]]
+[[File:P_ID1827__Dig_A_5_1.png|right|frame|Fehlerabstandsverteilungen]]
 Ein jedes digitales Kanalmodell kann in gleicher Weise beschrieben werden durch
 * die Fehlerfolge &nbsp;$&#9001;e_{\rm \nu}&#9002;$, und