Aufgaben:Exercise 5.3Z: Analysis of the BSC Model - Revision history

Guenter at 13:52, 5 September 2022

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Guenter at 13:42, 5 September 2022

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Hwang at 13:45, 26 August 2022

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Guenter at 17:23, 23 August 2022

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Guenter: Guenter moved page Aufgaben:Aufgabe 5.3Z: Analyse des BSC-Modells to Aufgaben:Exercise 5.3Z: Analysis of the BSC Model

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Guenter moved page Aufgabe 5.3Z: Analyse des BSC-Modells to Exercise 5.3Z: Analysis of the BSC Model

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

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

Guenter at 14:23, 25 March 2019

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Guenter: Guenter verschob die Seite Aufgaben:5.3Z Analyse des BSC-Modells nach Aufgaben:Aufgabe 5.3Z: Analyse des BSC-Modells

2018-01-04T06:40:22Z

Guenter verschob die Seite 5.3Z Analyse des BSC-Modells nach Aufgabe 5.3Z: Analyse des BSC-Modells

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; In the BSC model, the mean error probability $p_{\rm M}$ is always equal to the characteristic probability $p$.
+'''(1)'''&nbsp; In the BSC model,&nbsp;  the mean error probability $p_{\rm M}$&nbsp; is always equal to the characteristic probability&nbsp; $p$.
 *For the error correlation function and the error distance distribution are valid
 :$$\varphi_{e}(k) =
@@ Line 73: / Line 74: @@
   p^2 \end{array} \right.\quad
 \begin{array}{*{1}c} f{\rm or }\hspace{0.15cm}k = 0  \hspace{0.05cm},
-\\  f{\rm or }\hspace{0.15cm} k \ne 0 \hspace{0.05cm},\\ \end{array}
+\\  f{\rm or }\hspace{0.15cm} k \ne 0 \hspace{0.05cm},\\ \end{array}$$
-\hspace{0.4cm}V_a(k) = (1-p)^{k-1}\hspace{0.05cm}.$$
+:$$V_a(k) = (1-p)^{k-1}\hspace{0.05cm}.$$
-*$p$&nbsp; can be determined from all the given characteristics, except $V_a(k = 1)$. This EDD value is independent of&nbsp; $p$&nbsp; equal to &nbsp;$(1&ndash;p)^0 = 1$.
+*$p$&nbsp; can be determined from all the given characteristics,&nbsp; except $V_a(k = 1)$.&nbsp; This EDD value is independent of&nbsp; $p$ &nbsp; equal to &nbsp; $(1&ndash;p)^0 = 1$.
 *Therefore, the <u>solutions 1, 2, 4 and 5</u> are correct.
-'''(2)'''&nbsp; The relative error frequency of the given sequence is equal to $h_{\rm F} = 22/1000 \approx 0.022$.
+'''(2)'''&nbsp; The relative error frequency of the given sequence is equal to&nbsp; $h_{\rm F} = 22/1000 \approx 0.022$.
-*It is quite obvious that the error sequence was generated by the model $M_2$ &nbsp;&#8658;&nbsp; $p_{\rm M} = 0.02$.
+*It is quite obvious that the error sequence was generated by the model&nbsp; $M_2$&nbsp; &nbsp; &#8658; &nbsp; $p_{\rm M} = 0.02$.
 *Because of the short sequence, $h_{\rm F}$ does not match $p_{\rm M}$ exactly, but at least approximates &nbsp;&#8658;&nbsp; <u>solution 2</u>.
-'''(3)'''&nbsp; The mean error distance &ndash; that is, the expected value of the random variable&nbsp; $a$ &ndash; is equal to the inverse of the mean error probability &#8658; ${\rm E}\big[a\big] = 1/0.1 \ \underline {= 10}$.
+'''(3)'''&nbsp; The mean error distance &ndash; that is,&nbsp; the expected value of the random variable&nbsp; $a$&nbsp; &ndash; is equal to the inverse of the mean error probability &#8658;
-'''(4)'''&nbsp; According to the equation&nbsp; ${\rm Pr}(a = k) = (1&ndash;p)^{k&ndash;1} \cdot p$&nbsp; we obtain:
+'''(4)'''&nbsp; According to the equation &nbsp; ${\rm Pr}(a = k) = (1&ndash;p)^{k&ndash;1} \cdot p$ &nbsp; we obtain:
 :$${\rm Pr}(a = 1) \hspace{0.15cm}\underline {=
 .1}\hspace{0.05cm},$$
@@ Line 109: / Line 113: @@
 \hspace{0.15cm}\underline {=0.3874}\hspace{0.05cm},\hspace{0.2cm}V_a(k = 11)= 0.9^{10}
 \hspace{0.15cm}\underline {=0.3487}.$$
-To check in comparison with subtask (4):
+*To check in comparison with subtask&nbsp; '''(4)''':
 :$${\rm Pr}(a = 10) = V_a(k = 10) - V_a(k =
 ) = 0.3874 - 0.3487  {= 0.0387}\hspace{0.05cm}.$$

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)}}
-[[File:P_ID1832__Dig_Z_5_3.png|right|frame|Given error sequence]]
+[[File:P_ID1832__Dig_Z_5_3.png|right|frame|The given error sequence]]
 We consider two different BSC models with the following parameters:
 * Model $M_1 \text{:} \hspace{0.4cm} p = 0.01$,
@@ Line 8: / Line 8: @@
-The graph shows an error sequence of length&nbsp; $N = 1000$, but it is not known from which of the two models this sequence originates.
+The graph shows an error sequence of length&nbsp; $N = 1000$,&nbsp; but it is not known from which of the two models this sequence originates.
 The two models are to be analyzed on the basis of
-* the error distance probabilities
+* the&nbsp; '''error distance probabilities'''
 :$${\rm Pr}(a = k) = (1-p)^{k-1}\cdot p \hspace{0.05cm},$$
-* the error distance distribution
+* the&nbsp; '''error distance distribution'''&nbsp; $\rm (EDD)$
 :$$V_a(k) =  {\rm Pr}(a \ge k) =  (1-p)^{k-1}\hspace{0.05cm},$$
-* the error correlation function
+* the&nbsp; '''error correlation function'''&nbsp; $\rm (ECF)$
 :$$\varphi_{e}(k) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  {\rm
 E}\big[e_{\nu} \cdot e_{\nu + k}\big] \ \ = \ \
@@ Line 25: / Line 25: @@
+*By counting,&nbsp; we would see that the error sequence of length&nbsp; $N = 1000$&nbsp; contains exactly&nbsp; $22$&nbsp; "ones".
@@ Line 49: / Line 46: @@
 |type="()"}
 - Model $M_1$,
-+ Model $M_2$.
++ model $M_2$.
 {What is the mean error distance of model&nbsp; $M_1$?

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Digitalsignalübertragung/Binary Symmetric Channel (BSC)}}
+{{quiz-Header|Buchseite=Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)}}
-[[File:P_ID1832__Dig_Z_5_3.png|right|frame|Gegebene Fehlerfolge]]
+[[File:P_ID1832__Dig_Z_5_3.png|right|frame|Given error sequence]]
-Wir betrachten zwei unterschiedliche BSC&ndash;Modelle mit den folgenden Parametern:
+We consider two different BSC models with the following parameters:
-* Modell $M_1 \text{:} \hspace{0.4cm} p = 0.01$,
+* Model $M_1 \text{:} \hspace{0.4cm} p = 0.01$,
-* Modell $M_2 \text{:} \hspace{0.4cm} p = 0.02$.
+* Model $M_2 \text{:} \hspace{0.4cm} p = 0.02$.
-Die Grafik zeigt eine Fehlerfolge der Länge&nbsp; $N = 1000$, wobei allerdings nicht bekannt ist, von welchem der beiden Modelle diese Folge stammt.
+The graph shows an error sequence of length&nbsp; $N = 1000$, but it is not known from which of the two models this sequence originates.
-Die beiden Modelle sollen analysiert werden anhand
+The two models are to be analyzed on the basis of
-* der Fehlerabstandswahrscheinlichkeiten
+* the error distance probabilities
 :$${\rm Pr}(a = k) = (1-p)^{k-1}\cdot p \hspace{0.05cm},$$
-* der Fehlerabstandsverteilung
+* the error distance distribution
 :$$V_a(k) =  {\rm Pr}(a \ge k) =  (1-p)^{k-1}\hspace{0.05cm},$$
-* der Fehlerkorrelationsfunktion
+* the error correlation function
 :$$\varphi_{e}(k) \hspace{-0.1cm} \ = \ \hspace{-0.1cm}  {\rm
 E}\big[e_{\nu} \cdot e_{\nu + k}\big] \ \ = \ \
   \left\{ \begin{array}{c} p \\
   p^2 \end{array} \right.\quad
-\begin{array}{*{1}c} f{\rm \ddot{u}r }\hspace{0.15cm}k = 0  \hspace{0.05cm},
+\begin{array}{*{1}c} f{\rm or }\hspace{0.15cm}k = 0  \hspace{0.05cm},
-\\  f{\rm \ddot{u}r }\hspace{0.15cm} k \ne 0 \hspace{0.05cm}.\\ \end{array}$$
+\\  f{\rm or }\hspace{0.15cm} k \ne 0 \hspace{0.05cm}.\\ \end{array}$$
@@ Line 29: / Line 29: @@
-''Hinweise:''
+''Notes:''
-* Die Aufgabe gehört zum Kapitel&nbsp; [[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)| Binary Symmetric Channel (BSC)]].
+* The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)| "Binary Symmetric Channel (BSC)"]].
-*Durch Abzählen würde man erkennen, dass die Fehlerfolge der Länge&nbsp; $N = 1000$&nbsp; genau&nbsp; $22$&nbsp; Einsen enthält.
+*By counting, we would see that the error sequence of length&nbsp; $N = 1000$&nbsp; contains exactly&nbsp; $22$&nbsp; ones.
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Aus welchen Kenngrößen kann auf die mittlere Fehlerwahrscheinlichkeit&nbsp; $p_{\rm M}$&nbsp; des BSC&ndash;Modells zurückgeschlossen werden?
+{Which parameters can be used to infer the mean error probability&nbsp; $p_{\rm M}$&nbsp; of the BSC model?
 |type="[]"}
-+ FKF&ndash;Wert&nbsp; $\varphi_e(k = 0)$,
++ ECF value&nbsp; $\varphi_e(k = 0)$,
-+ FKF&ndash;Wert&nbsp; $\varphi_e(k = 10)$,
++ ECF value&nbsp; $\varphi_e(k = 10)$,
-- FAV&ndash;Wert&nbsp; $V_a(k = 1)$,
+- EDD value&nbsp; $V_a(k = 1)$,
-+ FAV&ndash;Wert&nbsp; $V_a(k = 2)$,
++ EDD value&nbsp; $V_a(k = 2)$,
-+ FAV&ndash;Wert&nbsp; $V_a(k = 10)$.
++ EDD value&nbsp; $V_a(k = 10)$.
-{Von welchem Modell stammt die angegebene Fehlerfolge?
+{From which model does the given error sequence originate?
 |type="()"}
-- Modell $M_1$,
+- Model $M_1$,
-+ Modell $M_2$.
++ Model $M_2$.
-{Wie groß ist der mittlere Fehlerabstand von Modell&nbsp; $M_1$?
+{What is the mean error distance of model&nbsp; $M_1$?
 |type="{}"}
 ${\rm E}\big[a\big] \ = \ ${ 10 3% }
-{Wie groß sind für das Modell&nbsp; $M_1$&nbsp; die folgenden Wahrscheinlichkeiten?
+{What are the following probabilities for model&nbsp; $M_1$?&nbsp;
 |type="{}"}
 ${\rm Pr}(a = 1) \ = \ ${ 0.1 3% }
@@ Line 61: / Line 61: @@
 ${\rm Pr}(a = {\rm E}\big[a\big]) \ = \ ${ 0.0387 3% }
-{Berechnen Sie für das Modell&nbsp; $M_1$&nbsp; folgende Werte der Fehlerabstandsverteilung:
+{Calculate the following values of the error distance distribution for model&nbsp; $M_1$:&nbsp;
 |type="{}"}
 $V_a(k = 2) \ = \ ${ 0.9 3% }
@@ Line 68: / Line 68: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Beim BSC&ndash;Modell ist die mittlere Fehlerwahrscheinlichkeit $p_{\rm M}$ stets gleich der charakteristischen Wahrscheinlichkeit $p$.
+'''(1)'''&nbsp; In the BSC model, the mean error probability $p_{\rm M}$ is always equal to the characteristic probability $p$.
-*Für die Fehlerkorrelationsfunktion und die Fehlerabstandsverteilung gelten
+*For the error correlation function and the error distance distribution are valid
 :$$\varphi_{e}(k) =
   \left\{ \begin{array}{c} p \\
   p^2 \end{array} \right.\quad
-\begin{array}{*{1}c} f{\rm \ddot{u}r }\hspace{0.15cm}k = 0  \hspace{0.05cm},
+\begin{array}{*{1}c} f{\rm or }\hspace{0.15cm}k = 0  \hspace{0.05cm},
-\\  f{\rm \ddot{u}r }\hspace{0.15cm} k \ne 0 \hspace{0.05cm},\\ \end{array}
+\\  f{\rm or }\hspace{0.15cm} k \ne 0 \hspace{0.05cm},\\ \end{array}
 \hspace{0.4cm}V_a(k) = (1-p)^{k-1}\hspace{0.05cm}.$$
-*$p$&nbsp; lässt sich aus allen angegebenen Kenngrößen ermitteln, nur nicht aus $V_a(k = 1)$. Dieser FAV&ndash;Wert ist unabhängig von&nbsp; $p$&nbsp; gleich &nbsp;$(1&ndash;p)^0 = 1$.
+*$p$&nbsp; can be determined from all the given characteristics, except $V_a(k = 1)$. This EDD value is independent of&nbsp; $p$&nbsp; equal to &nbsp;$(1&ndash;p)^0 = 1$.
-*Zutreffend sind somit die <u>Lösungsvorschläge 1, 2, 4 und 5</u>.
+*Therefore, the <u>solutions 1, 2, 4 and 5</u> are correct.
-'''(2)'''&nbsp; Die relative Fehlerhäufigkeit der angegebenen Folge ist gleich $h_{\rm F} = 22/1000 \approx 0.022$.
+'''(2)'''&nbsp; The relative error frequency of the given sequence is equal to $h_{\rm F} = 22/1000 \approx 0.022$.
-*Es ist ganz offensichtlich, dass die Fehlerfolge vom Modell $M_2$ &nbsp;&#8658;&nbsp; $p_{\rm M} = 0.02$ generiert wurde.
+*It is quite obvious that the error sequence was generated by the model $M_2$ &nbsp;&#8658;&nbsp; $p_{\rm M} = 0.02$.
-*Aufgrund der kurzen Folge stimmt $h_{\rm F}$ mit $p_{\rm M}$ zwar nicht exakt überein, aber zumindest näherungsweise &nbsp;&#8658;&nbsp; <u>Vorschlag 2</u>.
+*Because of the short sequence, $h_{\rm F}$ does not match $p_{\rm M}$ exactly, but at least approximates &nbsp;&#8658;&nbsp; <u>solution 2</u>.
-'''(3)'''&nbsp; Der mittlere Fehlerabstand &ndash; also der Erwartungswert der Zufallsgröße&nbsp; $a$ &ndash; ist gleich dem Kehrwert der mittleren Fehlerwahrscheinlichkeit &#8658; ${\rm E}\big[a\big] = 1/0.1 \ \underline {= 10}$.
+'''(3)'''&nbsp; The mean error distance &ndash; that is, the expected value of the random variable&nbsp; $a$ &ndash; is equal to the inverse of the mean error probability &#8658; ${\rm E}\big[a\big] = 1/0.1 \ \underline {= 10}$.
-'''(4)'''&nbsp; Entsprechend der Gleichung&nbsp; ${\rm Pr}(a = k) = (1&ndash;p)^{k&ndash;1} \cdot p$&nbsp; erhält man:
+'''(4)'''&nbsp; According to the equation&nbsp; ${\rm Pr}(a = k) = (1&ndash;p)^{k&ndash;1} \cdot p$&nbsp; we obtain:
 :$${\rm Pr}(a = 1) \hspace{0.15cm}\underline {=
 .1}\hspace{0.05cm},$$
@@ Line 104: / Line 104: @@
-'''(5)'''&nbsp; Aus der Beziehung&nbsp; $V_a(k) = (1&ndash;p)^{k&ndash;1}$&nbsp; erhält man
+'''(5)'''&nbsp; From the relation&nbsp; $V_a(k) = (1&ndash;p)^{k&ndash;1}$&nbsp; we obtain
 :$$V_a(k = 2) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0.9^1 \hspace{0.15cm}\underline {= 0.9 } \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm}{\rm Pr}(a = 1) = V_a(k =
@@ Line 112: / Line 112: @@
 \hspace{0.15cm}\underline {=0.3874}\hspace{0.05cm},\hspace{0.2cm}V_a(k = 11)= 0.9^{10}
 \hspace{0.15cm}\underline {=0.3487}.$$
-Zur Kontrolle im Vergleich zur Teilaufgabe (4):
+To check in comparison with subtask (4):
 :$${\rm Pr}(a = 10) = V_a(k = 10) - V_a(k =
 ) = 0.3874 - 0.3487  {= 0.0387}\hspace{0.05cm}.$$

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← Older revision		Revision as of 12:52, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^5.2 Binary Symmetric Channel (BSC)^]]	+	[[Category:Digital Signal Transmission: Exercises\|^5.2 Binary Symmetric Channel (BSC)^]]

← Older revision	Revision as of 16:34, 23 August 2022
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@@ Line 28: / Line 28: @@
 * Die Aufgabe gehört zum Kapitel [[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)| Binary Symmetric Channel (BSC)]].
 *Durch Abzählen erkennt man, dass die Fehlerfolge der Länge $N = 1000$ genau $22$ Einsen enthält.
-* Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.